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Mathematical Finance

Impact factor: 1.0 5-Year impact factor: 1.463 Print ISSN: 0960-1627 Online ISSN: 1467-9965 Publisher: Wiley Blackwell (Blackwell Publishing)

Subjects: Business, Finance, Economics, Mathematical Methods Social Sciences

Most recent papers:

  • Semi‐efficient valuations and put‐call parity.
    Martin Herdegen, Martin Schweizer.
    Mathematical Finance. September 06, 2017
    We propose an approach to the valuation of payoffs in general semimartingale models of financial markets where prices are nonnegative. Each asset price can hit 0; we only exclude that this ever happens simultaneously for all assets. We start from two simple, economically motivated axioms, namely, absence of arbitrage (in the sense of NUPBR) and absence of relative arbitrage among all buy‐and‐hold strategies (called static efficiency). A valuation process for a payoff is then called semi‐efficient consistent if the financial market enlarged by that process still satisfies this combination of properties. It turns out that this approach lies in the middle between the extremes of valuing by risk‐neutral expectation and valuing by absence of arbitrage alone. We show that this always yields put‐call parity, although put and call values themselves can be nonunique, even for complete markets. We provide general formulas for put and call values in complete markets and show that these are symmetric and that both contain three terms in general. We also show that our approach recovers all the put‐call parity respecting valuation formulas in the classic theory as special cases, and we explain when and how the different terms in the put and call valuation formulas disappear or simplify. Along the way, we also define and characterize completeness for general semimartingale financial markets and connect this to the classic theory.
    September 06, 2017   doi: 10.1111/mafi.12162   open full text
  • Risk management with weighted VaR.
    Pengyu Wei.
    Mathematical Finance. August 29, 2017
    This article studies the optimal portfolio selection of expected utility‐maximizing investors who must also manage their market‐risk exposures. The risk is measured by a so‐called weighted value‐at‐risk (WVaR) risk measure, which is a generalization of both value‐at‐risk (VaR) and expected shortfall (ES). The feasibility, well‐posedness, and existence of the optimal solution are examined. We obtain the optimal solution (when it exists) and show how risk measures change asset allocation patterns. In particular, we characterize three classes of risk measures: the first class will lead to models that do not admit an optimal solution, the second class can give rise to endogenous portfolio insurance, and the third class, which includes VaR and ES, two popular regulatory risk measures, will allow economic agents to engage in “regulatory capital arbitrage,” incurring larger losses when losses occur.
    August 29, 2017   doi: 10.1111/mafi.12160   open full text
  • Option pricing in the moderate deviations regime.
    Peter Friz, Stefan Gerhold, Arpad Pinter.
    Mathematical Finance. August 25, 2017
    We consider call option prices close to expiry in diffusion models, in an asymptotic regime (“moderately out of the money”) that interpolates between the well‐studied cases of at‐the‐money and out‐of‐the‐money regimes. First and higher order small‐time moderate deviation estimates of call prices and implied volatilities are obtained. The expansions involve only simple expressions of the model parameters, and we show how to calculate them for generic local and stochastic volatility models. Some numerical computations for the Heston model illustrate the accuracy of our results.
    August 25, 2017   doi: 10.1111/mafi.12156   open full text
  • Error analysis of finite difference and Markov chain approximations for option pricing.
    Lingfei Li, Gongqiu Zhang.
    Mathematical Finance. August 25, 2017
    Mijatović and Pistorius proposed an efficient Markov chain approximation method for pricing European and barrier options in general one‐dimensional Markovian models. However, sharp convergence rates of this method for realistic financial payoffs, which are nonsmooth, are rarely available. In this paper, we solve this problem for general one‐dimensional diffusion models, which play a fundamental role in financial applications. For such models, the Markov chain approximation method is equivalent to the method of lines using the central difference. Our analysis is based on the spectral representation of the exact solution and the approximate solution. By establishing the convergence rate for the eigenvalues and the eigenfunctions, we obtain sharp convergence rates for the transition density and the price of options with nonsmooth payoffs. In particular, we show that for call‐/put‐type payoffs, convergence is second order, while for digital‐type payoffs, convergence is generally only first order. Furthermore, we provide theoretical justification for two well‐known smoothing techniques that can restore second‐order convergence for digital‐type payoffs and explain oscillations observed in the convergence for options with nonsmooth payoffs. As an extension, we also establish sharp convergence rates for European options for a rich class of Markovian jump models constructed from diffusions via subordination. The theoretical estimates are confirmed using numerical examples.
    August 25, 2017   doi: 10.1111/mafi.12161   open full text
  • The valuation of American options in a multidimensional exponential Lévy model.
    Tomasz Klimsiak, Andrzej Rozkosz.
    Mathematical Finance. August 25, 2017
    We consider the problem of valuation of American options written on dividend‐paying assets whose price dynamics follow a multidimensional exponential Lévy model. We carefully examine the relation between the option prices, related partial integro‐differential variational inequalities, and reflected backward stochastic differential equations. In particular, we prove regularity results for the value function and obtain the early exercise premium formula for a broad class of payoff functions.
    August 25, 2017   doi: 10.1111/mafi.12163   open full text
  • Analytical approximations of local‐Heston volatility model and error analysis.
    R. Bompis, E. Gobet.
    Mathematical Finance. August 18, 2017
    This paper studies the expansion of an option price (with bounded Lipschitz payoff) in a stochastic volatility model including a local volatility component. The stochastic volatility is a square root process, which is widely used for modeling the behavior of the variance process (Heston model). The local volatility part is of general form, requiring only appropriate growth and boundedness assumptions. We rigorously establish tight error estimates of our expansions, using Malliavin calculus. The error analysis, which requires a careful treatment because of the lack of weak differentiability of the model, is interesting on its own. Moreover, in the particular case of call–put options, we also provide expansions of the Black–Scholes implied volatility that allow to obtain very simple formulas that are fast to compute compared to the Monte Carlo approach and maintain a very competitive accuracy.
    August 18, 2017   doi: 10.1111/mafi.12154   open full text
  • Liquidity effects of trading frequency.
    Roman Gayduk, Sergey Nadtochiy.
    Mathematical Finance. August 18, 2017
    In this paper, we present a discrete‐time modeling framework, in which the shape and dynamics of a Limit Order Book (LOB) arise endogenously from an equilibrium between multiple market participants (agents). We use the proposed modeling framework to analyze the effects of trading frequency on market liquidity in a very general setting. In particular, we demonstrate the dual effect of high trading frequency. On the one hand, the higher frequency increases market efficiency, if the agents choose to provide liquidity in equilibrium. On the other hand, it also makes markets more fragile, in the sense that the agents choose to provide liquidity in equilibrium only if they are market neutral (i.e., their beliefs satisfy certain martingale property). Even a very small deviation from market neutrality may cause the agents to stop providing liquidity, if the trading frequency is sufficiently high, which represents an endogenous liquidity crisis (also known as flash crash) in the market. This framework enables us to provide more insight into how such a liquidity crisis unfolds, connecting it to the so‐called adverse selection effect.
    August 18, 2017   doi: 10.1111/mafi.12157   open full text
  • Consistent recalibration of yield curve models.
    Philipp Harms, David Stefanovits, Josef Teichmann, Mario V. Wüthrich.
    Mathematical Finance. August 18, 2017
    The analytical tractability of affine (short rate) models, such as the Vasiček and the Cox–Ingersoll–Ross (CIR) models, has made them a popular choice for modeling the dynamics of interest rates. However, in order to properly account for the dynamics of real data, these models must exhibit time‐dependent or even stochastic parameters. This breaks their tractability, and modeling and simulating become an arduous task. We introduce a new class of Heath–Jarrow–Morton (HJM) models that both fit the dynamics of real market data and remain tractable. We call these models consistent recalibration (CRC) models. CRC models appear as limits of concatenations of forward rate increments, each belonging to a Hull–White extended affine factor model with possibly different parameters. That is, we construct HJM models from “tangent” affine models. We develop a theory for continuous path versions of such models and discuss their numerical implementations within the Vasiček and CIR frameworks.
    August 18, 2017   doi: 10.1111/mafi.12159   open full text
  • The optimal method for pricing Bermudan options by simulation.
    Alfredo Ibáñez, Carlos Velasco.
    Mathematical Finance. August 18, 2017
    Least‐squares methods enable us to price Bermudan‐style options by Monte Carlo simulation. They are based on estimating the option continuation value by least‐squares. We show that the Bermudan price is maximized when this continuation value is estimated near the exercise boundary, which is equivalent to implicitly estimating the optimal exercise boundary by using the value‐matching condition. Localization is the key difference with respect to global regression methods, but is fundamental for optimal exercise decisions and requires estimation of the continuation value by iterating local least‐squares (because we estimate and localize the exercise boundary at the same time). In the numerical example, in agreement with this optimality, the new prices or lower bounds (i) improve upon the prices reported by other methods and (ii) are very close to the associated dual upper bounds. We also study the method's convergence.
    August 18, 2017   doi: 10.1111/mafi.12158   open full text
  • On the market viability under proportional transaction costs.
    Erhan Bayraktar, Xiang Yu.
    Mathematical Finance. August 10, 2017
    This paper studies the market viability with proportional transaction costs. Instead of requiring the existence of strictly consistent price systems as in the literature, we show that strictly consistent local martingale systems (SCLMS) can successfully serve as the dual elements such that the market viability can be verified. We introduce two weaker notions of no arbitrage conditions on market models named no unbounded profit with bounded risk (NUPBR) and no local arbitrage with bounded portfolios (NLABPs). In particular, we show that the NUPBR and NLABP conditions in the robust sense are equivalent to the existence of SCLMS for general market models. We also discuss the implications for the utility maximization problem.
    August 10, 2017   doi: 10.1111/mafi.12155   open full text
  • Small‐cost asymptotics for long‐term growth rates in incomplete markets.
    Yaroslav Melnyk, Frank Thomas Seifried.
    Mathematical Finance. May 04, 2017
    This paper provides a rigorous asymptotic analysis of long‐term growth rates under both proportional and Morton–Pliska transaction costs. We consider a general incomplete financial market with an unspanned Markov factor process that includes the Heston stochastic volatility model and the Kim–Omberg stochastic excess return model as special cases. Using a dynamic programming approach, we determine the leading‐order expansions of long‐term growth rates and explicitly construct strategies that are optimal at the leading order. We further analyze the asymptotic performance of Morton–Pliska strategies in settings with proportional transaction costs. We find that the performance of the optimal Morton–Pliska strategy is the same as that of the optimal one with costs increased by a factor of 2. Finally, we demonstrate that our strategies are in fact pathwise optimal, in the sense that they maximize the long‐run growth rate path by path.
    May 04, 2017   doi: 10.1111/mafi.12152   open full text
  • On American VIX options under the generalized 3/2 and 1/2 models.
    Jérôme Detemple, Yerkin Kitapbayev.
    Mathematical Finance. May 04, 2017
    In this paper, we extend the 3/2 model for VIX studied by Goard and Mazur and introduce the generalized 3/2 and 1/2 classes of volatility processes. Under these models, we study the pricing of European and American VIX options, and for the latter, we obtain an early exercise premium representation using a free‐boundary approach and local time‐space calculus. The optimal exercise boundary for the volatility is obtained as the unique solution to an integral equation of Volterra type. We also consider a model mixing these two classes and formulate the corresponding optimal stopping problem in terms of the observed factor process. The price of an American VIX call is then represented by an early exercise premium formula. We show the existence of a pair of optimal exercise boundaries for the factor process and characterize them as the unique solution to a system of integral equations.
    May 04, 2017   doi: 10.1111/mafi.12153   open full text
  • On the C‐property and w∗‐representations of risk measures.
    Niushan Gao, Foivos Xanthos.
    Mathematical Finance. May 02, 2017
    We identify a large class of Orlicz spaces LΦ(μ) for which the topology σ(LΦ(μ),LΦ(μ)n∼) fails the C‐property introduced by Biagini and Frittelli. We also establish a variant of the C‐property and use it to prove a w∗‐representation theorem for proper convex increasing functionals, satisfying a suitable version of Delbaen's Fatou property, on Orlicz spaces LΦ(μ) with limt→∞Φ(t)t=∞. Our results apply, in particular, to risk measures on all Orlicz spaces LΦ(P) other than L1(P).
    May 02, 2017   doi: 10.1111/mafi.12150   open full text
  • On peacocks and lyrebirds: Australian options, Brownian bridges, and the average of submartingales.
    Christian‐Oliver Ewald, Marc Yor.
    Mathematical Finance. April 18, 2017
    We introduce a class of stochastic processes, which we refer to as lyrebirds. These extend a class of stochastic processes, which have recently been coined peacocks, but are more commonly known as processes that are increasing in the convex order. We show how these processes arise naturally in the context of Asian and Australian options and consider further applications, such as the arithmetic average of a Brownian bridge and the average of submartingales, including the case of Asian and Australian options where the underlying features constant elasticity of variance or is of Merton jump diffusion type.
    April 18, 2017   doi: 10.1111/mafi.12144   open full text
  • Fair bilateral pricing under funding costs and exogenous collateralization.
    Tianyang Nie, Marek Rutkowski.
    Mathematical Finance. April 18, 2017
    Bielecki and Rutkowski introduced and studied a generic nonlinear market model, which includes several risky assets, multiple funding accounts, and margin accounts. In this paper, we examine the pricing and hedging of contract from the perspective of both the hedger and the counterparty with arbitrary initial endowments. We derive inequalities for unilateral prices and we study the range of fair bilateral prices. We also examine the positive homogeneity and monotonicity of unilateral prices with respect to the initial endowments. Our study hinges on results from Nie and Rutkowski for backward stochastic differential equations (BSDEs) driven by continuous martingales, but we also derive the pricing partial differential equations (PDEs) for path‐independent contingent claims of a European style in a Markovian framework.
    April 18, 2017   doi: 10.1111/mafi.12145   open full text
  • Arbitrage‐free XVA.
    Maxim Bichuch, Agostino Capponi, Stephan Sturm.
    Mathematical Finance. April 18, 2017
    We develop a framework for computing the total valuation adjustment (XVA) of a European claim accounting for funding costs, counterparty credit risk, and collateralization. Based on no‐arbitrage arguments, we derive backward stochastic differential equations associated with the replicating portfolios of long and short positions in the claim. This leads to the definition of buyer's and seller's XVA, which in turn identify a no‐arbitrage interval. In the case that borrowing and lending rates coincide, we provide a fully explicit expression for the unique XVA, expressed as a percentage of the price of the traded claim, and for the corresponding replication strategies. In the general case of asymmetric funding, repo, and collateral rates, we study the semilinear partial differential equations characterizing buyer's and seller's XVA and show the existence of a unique classical solution to it. To illustrate our results, we conduct a numerical study demonstrating how funding costs, repo rates, and counterparty risk contribute to determine the total valuation adjustment.
    April 18, 2017   doi: 10.1111/mafi.12146   open full text
  • Conic martingales from stochastic integrals.
    Monique Jeanblanc, Frédéric Vrins.
    Mathematical Finance. April 18, 2017
    In this paper, we introduce the concept of conic martingales. This class refers to stochastic processes that have the martingale property but that evolve within given (possibly time‐dependent) boundaries. We first review some results about the martingale property of solution to driftless stochastic differential equations. We then provide a simple way to construct and handle such processes. Specific attention is paid to martingales in [0, 1]. One of these martingales proves to be analytically tractable. It is shown that up to shifting and rescaling constants, it is the only martingale (with the trivial constant, Brownian motion, and geometric Brownian motion) having a separable diffusion coefficient σ(t,y)=g(t)h(y) and that can be obtained via a time‐homogeneous mapping of Gaussian diffusions. The approach is exemplified by modeling stochastic conditional survival probabilities in the univariate and bivariate cases.
    April 18, 2017   doi: 10.1111/mafi.12147   open full text
  • A note on the long rate in factor models of the term structure.
    Jan Kort.
    Mathematical Finance. April 18, 2017
    In this paper, we consider factor models of the term structure based on a Brownian filtration. We show that the existence of a nondeterministic long rate in a factor model of the term structure implies, as a consequence of the Dybvig–Ingersoll–Ross theorem, that the model has an equivalent representation in which one of the state variables is nondecreasing. For two‐dimensional factor models, we prove moreover that if the long rate is nondeterministic, the yield curve flattens out, and the factor process is asymptotically nondeterministic, then the term structure is unbounded. Finally, we provide an explicit example of a three‐dimensional affine factor model with a nondeterministic yet finite long rate in which the volatility of the factor process does not vanish over time.
    April 18, 2017   doi: 10.1111/mafi.12151   open full text
  • Optimal cash holdings under heterogeneous beliefs.
    Robert Jarrow, Andrey Krishenik, Andreea Minca.
    Mathematical Finance. April 18, 2017
    This paper explores a one‐period model for a firm that finances its operations through debt provided by heterogeneous creditors. Creditors differ in their beliefs about the firm's investment outcomes. We show the existence of Stackelberg equilibria in which the firm holds cash reserves in order to provide incentives for pessimistic creditors to invest in the firm. We find interest rates and cash holdings to be complementary tools for increasing debt capacity. In markets with a high concentration of capital across a small interval of pessimistic creditors or by a few large creditors, cash holdings is the preferred tool to increase the debt capacity of the firm.
    April 18, 2017   doi: 10.1111/mafi.12148   open full text
  • Super‐replication in fully incomplete markets.
    Yan Dolinsky, Ariel Neufeld.
    Mathematical Finance. March 24, 2017
    In this work, we introduce the notion of fully incomplete markets. We prove that for these markets, the super‐replication price coincides with the model‐free super‐replication price. Namely, the knowledge of the model does not reduce the super‐replication price. We provide two families of fully incomplete models: stochastic volatility models and rough volatility models. Moreover, we give several computational examples. Our approach is purely probabilistic.
    March 24, 2017   doi: 10.1111/mafi.12149   open full text
  • Investing With Liquid And Illiquid Assets.
    Maxim Bichuch, Paolo Guasoni.
    Mathematical Finance. December 16, 2016
    We find optimal trading policies for long‐term investors with constant relative risk aversion and constant investment opportunities, which include one safe asset, liquid risky assets, and an illiquid risky asset trading with proportional costs. Access to liquid assets creates a diversification motive, which reduces illiquid trading, and a hedging motive, which both reduces illiquid trading and increases liquid trading. A further tempering effect depresses the liquid asset's weight when the illiquid asset's weight is close to ideal, to keep it near that level by reducing its volatility. Multiple liquid assets lead to portfolio separation in four funds: the safe asset, the myopic portfolio, the illiquid asset, and its hedging portfolio.
    December 16, 2016   doi: 10.1111/mafi.12135   open full text
  • Indifference Pricing For Contingent Claims: Large Deviations Effects.
    Scott Robertson, Konstantinos Spiliopoulos.
    Mathematical Finance. November 24, 2016
    We study utility indifference prices and optimal purchasing quantities for a nontraded contingent claim in an incomplete semimartingale market with vanishing hedging errors. We make connections with the theory of large deviations. We concentrate on sequences of semicomplete markets where in the nth market, the claim Bn admits the decomposition Bn=Dn+Yn. Here, Dn is replicable by trading in the underlying assets Sn, but Yn is independent of Sn. Under broad conditions, we may assume that Yn vanishes in accordance with a large deviations principle (LDP) as n grows. In this setting, for an exponential investor, we identify the limit of the average indifference price pn(qn), for qn units of Bn, as n→∞. We show that if |qn|→∞, the limiting price typically differs from the price obtained by assuming bounded positions supn|qn|<∞, and the difference is explicitly identifiable using large deviations theory. Furthermore, we show that optimal purchase quantities occur at the large deviations scaling, and hence large positions arise endogenously in this setting.
    November 24, 2016   doi: 10.1111/mafi.12137   open full text
  • Shareholder Risk Measures.
    Delia Coculescu, Jean‐Charles Rochet.
    Mathematical Finance. November 24, 2016
    The aim of this paper is to put forward a new family of risk measures that could guide investment decisions of private companies. But at the difference of the classical approach of Artzner, Delbaen, Eber, and Heath and the subsequent extensions of this model, our risk measures are built to reflect the risk perception of shareholders rather than regulators. Instead of an axiomatic approach, we derive risk measures from the optimal policies of a shareholder value‐maximizing company. We study these optimal policies and the related risk measures that we call shareholder risk measures. We emphasize the fact that due to the specific corporate environment, in particular the limited shareholders' liability and the possibility to pay out dividends from cash reserves, these risk measures are not convex. Also, they depend on the specific economic situation of the firm, in particular its current cash level, and thus they are not translation invariant. This paper bridges the gap between two important branches of mathematical finance: risk measures and optimal dividends.
    November 24, 2016   doi: 10.1111/mafi.12142   open full text
  • Profit Sharing In Hedge Funds.
    Xue Dong He, Steven Kou.
    Mathematical Finance. November 21, 2016
    In a new scheme for hedge fund managerial compensation known as the first‐loss scheme, a fund manager uses her investment in the fund to cover any fund losses first; by contrast, in the traditional scheme currently used in most US funds, the manager does not cover investors' losses in the fund. We propose a framework based on cumulative prospect theory to compute and compare the trading strategies, fund risk, and managers' and investors' utilities in these two schemes analytically. The model is calibrated to the historical attrition rates of US hedge funds. We find that with reasonable parameter values, both fund managers' and investors' utilities can be improved and fund risk can be reduced simultaneously by replacing the traditional scheme (with 10% internal capital and 20% performance fee) with a first‐loss scheme (with 10% first‐loss capital and 30% performance fee). When the performance fee in the first‐loss scheme is 40% (a current market practice), however, such substitution renders investors worse off.
    November 21, 2016   doi: 10.1111/mafi.12143   open full text
  • International Reserve Management: A Drift‐Switching Reflected Jump‐Diffusion Model.
    Ning Cai, Xuewei Yang.
    Mathematical Finance. November 18, 2016
    We study the cost of shocks, that is, jump risk, with respect to reserve management when the reserve process is formulated as a drift‐switching jump diffusion with a reflecting barrier at 0. Inspired by the Brownian drift switching model, our model results in a more realistic dynamic behavior of international reserves than the buffer stock model. The new model can capture both the jump behavior in reserve dynamics and the leptokurtic feature of the increment distribution which has a higher peak and two asymmetric heavier tails than the normal distribution. Through the selection of an initial distribution that reflects certain steady state behaviors, the reserve process becomes a regenerative process. This selection enables us to derive a closed‐form expression for the total expected discounted cost of managing reserves, thus helping us to numerically find management strategies that minimize costs. The numerical results show that shocks at the reserve level have a significant effect on reserve management strategies and that model misspecification can result in nonnegligible additional costs.
    November 18, 2016   doi: 10.1111/mafi.12134   open full text
  • Asymptotic Equivalence Of Risk Measures Under Dependence Uncertainty.
    Jun Cai, Haiyan Liu, Ruodu Wang.
    Mathematical Finance. November 17, 2016
    In this paper, we study the aggregate risk of inhomogeneous risks with dependence uncertainty, evaluated by a generic risk measure. We say that a pair of risk measures is asymptotically equivalent if the ratio of the worst‐case values of the two risk measures is almost one for the sum of a large number of risks with unknown dependence structure. The study of asymptotic equivalence is particularly important for a pair of a noncoherent risk measure and a coherent risk measure, as the worst‐case value of a noncoherent risk measure under dependence uncertainty is typically difficult to obtain. The main contribution of this paper is to establish general asymptotic equivalence results for the classes of distortion risk measures and convex risk measures under different mild conditions. The results implicitly suggest that it is only reasonable to implement a coherent risk measure for the aggregation of a large number of risks with uncertainty in the dependence structure, a relevant situation for risk management practice.
    November 17, 2016   doi: 10.1111/mafi.12140   open full text
  • Bounding Wrong‐Way Risk In Cva Calculation.
    Paul Glasserman, Linan Yang.
    Mathematical Finance. November 17, 2016
    A credit valuation adjustment (CVA) is an adjustment applied to the value of a derivative contract or a portfolio of derivatives to account for counterparty credit risk. Measuring CVA requires combining models of market and credit risk to estimate a counterparty's risk of default together with the market value of exposure to the counterparty at default. Wrong‐way risk refers to the possibility that a counterparty's likelihood of default increases with the market value of the exposure. We develop a method for bounding wrong‐way risk, holding fixed marginal models for market and credit risk and varying the dependence between them. Given simulated paths of the two models, a linear program computes the worst‐case CVA. We analyze properties of the solution and prove convergence of the estimated bound as the number of paths increases. The worst case can be overly pessimistic, so we extend the procedure by constraining the deviation of the joint model from a baseline reference model. Measuring the deviation through relative entropy leads to a tractable convex optimization problem that can be solved through the iterative proportional fitting procedure. Here, too, we prove convergence of the resulting estimate of the penalized worst‐case CVA and the joint distribution that attains it. We consider extensions with additional constraints and illustrate the method with examples.
    November 17, 2016   doi: 10.1111/mafi.12141   open full text
  • Modeling Sovereign Risks: From A Hybrid Model To The Generalized Density Approach.
    Ying Jiao, Shanqiu Li.
    Mathematical Finance. November 17, 2016
    Motivated by the European sovereign debt crisis, we propose a hybrid sovereign default model that combines an accessible part taking into account the evolution of the sovereign solvency and the impact of critical political events, and a totally inaccessible part for the idiosyncratic credit risk. We obtain closed‐form formulas for the probability that the default occurs at critical political dates in a Markovian setting. Moreover, we introduce a generalized density framework for the hybrid default time and deduce the compensator process of default. Finally, we apply the hybrid model and the generalized density to the valuation of sovereign bonds and explain the significant jumps in long‐term government bond yields during the sovereign crisis.
    November 17, 2016   doi: 10.1111/mafi.12136   open full text
  • Robust Utility Maximization With Lévy Processes.
    Ariel Neufeld, Marcel Nutz.
    Mathematical Finance. November 11, 2016
    We study a robust portfolio optimization problem under model uncertainty for an investor with logarithmic or power utility. The uncertainty is specified by a set of possible Lévy triplets, that is, possible instantaneous drift, volatility, and jump characteristics of the price process. We show that an optimal investment strategy exists and compute it in semi‐closed form. Moreover, we provide a saddle point analysis describing a worst‐case model.
    November 11, 2016   doi: 10.1111/mafi.12139   open full text
  • Dynamic Defaultable Term Structure Modeling Beyond The Intensity Paradigm.
    Frank Gehmlich, Thorsten Schmidt.
    Mathematical Finance. November 10, 2016
    The two main approaches in credit risk are the structural approach pioneered by Merton and the reduced‐form framework proposed by Jarrow and Turnbull and by Artzner and Delbaen. The goal of this paper is to provide a unified view on both approaches. This is achieved by studying reduced‐form approaches under weak assumptions. In particular, we do not assume the global existence of a default intensity and allow default at fixed or predictable times, such as coupon payment dates, with positive probability. In this generalized framework, we study dynamic term structures prone to default risk following the forward‐rate approach proposed by Heath, Jarrow, and Morton. It turns out that previously considered models lead to arbitrage possibilities when default can happen at a predictable time. A suitable generalization of the forward‐rate approach contains an additional stochastic integral with atoms at predictable times and necessary and sufficient conditions for an appropriate no‐arbitrage condition are given. For efficient implementations, we develop a new class of affine models that do not satisfy the standard assumption of stochastic continuity. The chosen approach is intimately related to the theory of enlargement of filtrations, for which we provide an example by means of filtering theory where the Azéma supermartingale contains upward and downward jumps, both at predictable and totally inaccessible stopping times.
    November 10, 2016   doi: 10.1111/mafi.12138   open full text
  • Indifference Prices And Implied Volatilities.
    Matthew Lorig.
    Mathematical Finance. May 26, 2016
    We consider a general local‐stochastic volatility model and an investor with exponential utility. For a European‐style contingent claim, whose payoff may depend on either a traded or nontraded asset, we derive an explicit approximation for both the buyer's and seller's indifference prices. For European calls on a traded asset, we translate indifference prices into an explicit approximation of the buyer's and seller's implied volatility surfaces. For European claims on a nontraded asset, we establish rigorous error bounds for the indifference price approximation. Finally, we implement our indifference price and implied volatility approximations in two examples.
    May 26, 2016   doi: 10.1111/mafi.12129   open full text
  • Optimal Liquidation And Adverse Selection In Dark Pools.
    Peter Kratz, Torsten Schöneborn.
    Mathematical Finance. May 25, 2016
    We consider an investor who has access both to a traditional venue and a dark pool for liquidating a position in a single asset. While trade execution is certain on the traditional exchange, she faces linear price impact costs. On the other hand, dark pool orders suffer from adverse selection and trade execution is uncertain. Adverse selection decreases order sizes in the dark pool while it speeds up trading at the exchange. For small orders, it is optimal to avoid the dark pool completely. Adverse selection can prevent profitable round‐trip trading strategies that otherwise would arise if permanent price impact were included in the model.
    May 25, 2016   doi: 10.1111/mafi.12126   open full text
  • Social Discounting And The Long Rate Of Interest.
    Dorje C. Brody, Lane P. Hughston.
    Mathematical Finance. May 24, 2016
    The well‐known theorem of Dybvig, Ingersoll, and Ross shows that the long zero‐coupon rate can never fall. This result, which, although undoubtedly correct, has been regarded by many as surprising, stems from the implicit assumption that the long‐term discount function has an exponential tail. We revisit the problem in the setting of modern interest rate theory, and show that if the long “simple” interest rate (or Libor rate) is finite, then this rate (unlike the zero‐coupon rate) acts viably as a state variable, the value of which can fluctuate randomly in line with other economic indicators. New interest rate models are constructed, under this hypothesis and certain generalizations thereof, that illustrate explicitly the good asymptotic behavior of the resulting discount bond systems. The conditions necessary for the existence of such “hyperbolic” and “generalized hyperbolic” long rates are those of so‐called social discounting, which allow for long‐term cash flows to be treated as broadly “just as important” as those of the short or medium term. As a consequence, we are able to provide a consistent arbitrage‐free valuation framework for the cost‐benefit analysis and risk management of long‐term social projects, such as those associated with sustainable energy, resource conservation, and climate change.
    May 24, 2016   doi: 10.1111/mafi.12122   open full text
  • Efficient Pricing Of Barrier Options And Credit Default Swaps In Lévy Models With Stochastic Interest Rate.
    Svetlana Boyarchenko, Sergei Levendorskiĭ.
    Mathematical Finance. May 23, 2016
    Recently, advantages of conformal deformations of the contours of integration in pricing formulas for European options have been demonstrated in the context of wide classes of Lévy models, the Heston model, and other affine models. Similar deformations were used in one‐factor Lévy models to price options with barrier and lookback features and credit default swaps (CDSs). In the present paper, we generalize this approach to models, where the dynamics of the assets is modeled as eXt, where X is a Lévy process, and the interest rate rt is stochastic. Assuming that X and r are independent, and Lr, the infinitesimal generator of the pricing semigroup in the model for the short rate, satisfies weak regularity conditions, which hold for popular models of the short rate, we develop a variation of the pricing procedure for Lévy models which is almost as fast as in the case of the constant interest rate. Numerical examples show that about 0.15 second suffices to calculate prices of 8 options of same maturity in a two‐factor model with the error tolerance 5·10−5 and less; in a three‐factor model, accuracy of order 0.001–0.005 is achieved in about 0.2 second. Similar results are obtained for quanto CDS, where an additional stochastic factor is the exchange rate. We suggest a class of Lévy models with the stochastic interest rate driven by 1–3 factors, which allows for fast calculations. This class can satisfy the current regulatory requirements for banks mandating sufficiently sophisticated credit risk models.
    May 23, 2016   doi: 10.1111/mafi.12121   open full text
  • Liquidation Of An Indivisible Asset With Independent Investment.
    Emilie Fabre, Guillaume Royer, Nizar Touzi.
    Mathematical Finance. May 23, 2016
    We provide an extension of the explicit solution of a mixed optimal stopping–optimal stochastic control problem introduced by Henderson and Hobson. The problem examines whether the optimal investment problem on a local martingale financial market is affected by the optimal liquidation of an independent indivisible asset. The indivisible asset process is defined by a homogeneous scalar stochastic differential equation, and the investor's preferences are defined by a general expected utility function. The value function is obtained in explicit form, and we prove the existence of an optimal stopping–investment strategy characterized as the limit of an explicit maximizing strategy. Our approach is based on the standard dynamic programming approach.
    May 23, 2016   doi: 10.1111/mafi.12127   open full text
  • Convergence Of A Least‐Squares Monte Carlo Algorithm For American Option Pricing With Dependent Sample Data.
    Daniel Z. Zanger.
    Mathematical Finance. May 23, 2016
    We analyze the convergence of the Longstaff–Schwartz algorithm relying on only a single set of independent Monte Carlo sample paths that is repeatedly reused for all exercise time‐steps. We prove new estimates on the stochastic component of the error of this algorithm whenever the approximation architecture is any uniformly bounded set of L2 functions of finite Vapnik–Chervonenkis dimension (VC‐dimension), but in particular need not necessarily be either convex or closed. We also establish new overall error estimates, incorporating bounds on the approximation error as well, for certain nonlinear, nonconvex sets of neural networks.
    May 23, 2016   doi: 10.1111/mafi.12125   open full text
  • The 4/2 Stochastic Volatility Model: A Unified Approach For The Heston And The 3/2 Model.
    Martino Grasselli.
    Mathematical Finance. May 19, 2016
    We introduce a new stochastic volatility model that includes, as special instances, the Heston (1993) and the 3/2 model of Heston (1997) and Platen (1997). Our model exhibits important features: first, instantaneous volatility can be uniformly bounded away from zero, and second, our model is mathematically and computationally tractable, thereby enabling an efficient pricing procedure. This called for using the Lie symmetries theory for partial differential equations; doing so allowed us to extend known results on Bessel processes. Finally, we provide an exact simulation scheme for the model, which is useful for numerical applications.
    May 19, 2016   doi: 10.1111/mafi.12124   open full text
  • Leveraged Etf Implied Volatilities From Etf Dynamics.
    Tim Leung, Matthew Lorig, Andrea Pascucci.
    Mathematical Finance. May 19, 2016
    The growth of the exchange‐traded fund (ETF) industry has given rise to the trading of options written on ETFs and their leveraged counterparts (LETFs). We study the relationship between the ETF and LETF implied volatility surfaces when the underlying ETF is modeled by a general class of local‐stochastic volatility models. A closed‐form approximation for prices is derived for European‐style options whose payoffs depend on the terminal value of the ETF and/or LETF. Rigorous error bounds for this pricing approximation are established. A closed‐form approximation for implied volatilities is also derived. We also discuss a scaling procedure for comparing implied volatilities across leverage ratios. The implied volatility expansions and scalings are tested in three settings: Heston, limited constant elasticity of variance (CEV), and limited SABR; the last two are regularized versions of the well‐known CEV and SABR models.
    May 19, 2016   doi: 10.1111/mafi.12128   open full text
  • Utility Maximization In A Large Market.
    Oleksii Mostovyi.
    Mathematical Finance. May 19, 2016
    We study the problem of expected utility maximization in a large market, i.e., a market with countably many traded assets. Assuming that agents have von Neumann–Morgenstern preferences with stochastic utility function and that consumption occurs according to a stochastic clock, we obtain the “usual” conclusions of the utility maximization theory. We also give a characterization of the value function in a large market in terms of a sequence of value functions in finite‐dimensional models.
    May 19, 2016   doi: 10.1111/mafi.12123   open full text
  • Density Of Skew Brownian Motion And Its Functionals With Application In Finance.
    Alexander Gairat, Vadim Shcherbakov.
    Mathematical Finance. May 19, 2016
    We derive the joint density of a Skew Brownian motion, its last visit to the origin, its local and occupation times. The result enables us to obtain explicit analytical formulas for pricing European options under both a two‐valued local volatility model and a displaced diffusion model with constrained volatility.
    May 19, 2016   doi: 10.1111/mafi.12120   open full text
  • Portfolio Optimization And Stochastic Volatility Asymptotics.
    Jean‐Pierre Fouque, Ronnie Sircar, Thaleia Zariphopoulou.
    Mathematical Finance. September 30, 2015
    We study the Merton portfolio optimization problem in the presence of stochastic volatility using asymptotic approximations when the volatility process is characterized by its timescales of fluctuation. This approach is tractable because it treats the incomplete markets problem as a perturbation around the complete market constant volatility problem for the value function, which is well understood. When volatility is fast mean‐reverting, this is a singular perturbation problem for a nonlinear Hamilton–Jacobi–Bellman partial differential equation, while when volatility is slowly varying, it is a regular perturbation. These analyses can be combined for multifactor multiscale stochastic volatility models. The asymptotics shares remarkable similarities with the linear option pricing problem, which follows from some new properties of the Merton risk tolerance function. We give examples in the family of mixture of power utilities and also use our asymptotic analysis to suggest a “practical” strategy that does not require tracking the fast‐moving volatility. In this paper, we present formal derivations of asymptotic approximations, and we provide a convergence proof in the case of power utility and single‐factor stochastic volatility. We assess our approximation in a particular case where there is an explicit solution.
    September 30, 2015   doi: 10.1111/mafi.12109   open full text
  • Robust Fundamental Theorem For Continuous Processes.
    Sara Biagini, Bruno Bouchard, Constantinos Kardaras, Marcel Nutz.
    Mathematical Finance. September 30, 2015
    We study a continuous‐time financial market with continuous price processes under model uncertainty, modeled via a family P of possible physical measures. A robust notion NA 1(P) of no‐arbitrage of the first kind is introduced; it postulates that a nonnegative, nonvanishing claim cannot be superhedged for free by using simple trading strategies. Our first main result is a version of the fundamental theorem of asset pricing: NA 1(P) holds if and only if every P∈P admits a martingale measure that is equivalent up to a certain lifetime. The second main result provides the existence of optimal superhedging strategies for general contingent claims and a representation of the superhedging price in terms of martingale measures.
    September 30, 2015   doi: 10.1111/mafi.12110   open full text
  • A State‐Constrained Differential Game Arising In Optimal Portfolio Liquidation.
    Alexander Schied, Tao Zhang.
    Mathematical Finance. September 29, 2015
    We consider n risk‐averse agents who compete for liquidity in an Almgren–Chriss market impact model. Mathematically, this situation can be described by a Nash equilibrium for a certain linear quadratic differential game with state constraints. The state constraints enter the problem as terminal boundary conditions for finite and infinite time horizons. We prove existence and uniqueness of Nash equilibria and give closed‐form solutions in some special cases. We also analyze qualitative properties of the equilibrium strategies and provide corresponding financial interpretations.
    September 29, 2015   doi: 10.1111/mafi.12108   open full text
  • Pricing For Large Positions In Contingent Claims.
    Scott Robertson.
    Mathematical Finance. September 29, 2015
    Approximations to utility indifference prices are provided for a contingent claim in the large position size limit. Results are valid for general utility functions on the real line and semi‐martingale models. It is shown that as the position size approaches infinity, the utility function's decay rate for large negative wealths is the primary driver of prices. For utilities with exponential decay, one may price like an exponential investor. For utilities with a power decay, one may price like a power investor after a suitable adjustment to the rate at which the position size becomes large. In a sizable class of diffusion models, limiting indifference prices are explicitly computed for an exponential investor. Furthermore, the large claim limit arises endogenously as the hedging error for the claim vanishes.
    September 29, 2015   doi: 10.1111/mafi.12107   open full text
  • The General Structure Of Optimal Investment And Consumption With Small Transaction Costs.
    Jan Kallsen, Johannes Muhle‐Karbe.
    Mathematical Finance. September 29, 2015
    We investigate the general structure of optimal investment and consumption with small proportional transaction costs. For a safe asset and a risky asset with general continuous dynamics, traded with random and time‐varying but small transaction costs, we derive simple formal asymptotics for the optimal policy and welfare. These reveal the roles of the investors' preferences as well as the market and cost dynamics, and also lead to a fully dynamic model for the implied trading volume. In frictionless models that can be solved in closed form, explicit formulas for the leading‐order corrections due to small transaction costs are obtained.
    September 29, 2015   doi: 10.1111/mafi.12106   open full text
  • Explicit Implied Volatilities For Multifactor Local‐Stochastic Volatility Models.
    Matthew Lorig, Stefano Pagliarani, Andrea Pascucci.
    Mathematical Finance. September 29, 2015
    We consider an asset whose risk‐neutral dynamics are described by a general class of local‐stochastic volatility models and derive a family of asymptotic expansions for European‐style option prices and implied volatilities. We also establish rigorous error estimates for these quantities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under four different model dynamics: constant elasticity of variance local volatility, Heston stochastic volatility, three‐halves stochastic volatility, and SABR local‐stochastic volatility.
    September 29, 2015   doi: 10.1111/mafi.12105   open full text
  • On Arbitrage And Duality Under Model Uncertainty And Portfolio Constraints.
    Erhan Bayraktar, Zhou Zhou.
    Mathematical Finance. September 29, 2015
    We consider the fundamental theorem of asset pricing (FTAP) and the hedging prices of options under nondominated model uncertainty and portfolio constraints in discrete time. We first show that no arbitrage holds if and only if there exists some family of probability measures such that any admissible portfolio value process is a local super‐martingale under these measures. We also get the nondominated optional decomposition with constraints. From this decomposition, we obtain the duality of the super‐hedging prices of European options, as well as the sub‐ and super‐hedging prices of American options. Finally, we get the FTAP and the duality of super‐hedging prices in a market where stocks are traded dynamically and options are traded statically.
    September 29, 2015   doi: 10.1111/mafi.12104   open full text
  • Option Pricing And Hedging With Execution Costs And Market Impact.
    Olivier Guéant, Jiang Pu.
    Mathematical Finance. September 25, 2015
    This paper considers the pricing and hedging of a call option when liquidity matters, that is, either for a large nominal or for an illiquid underlying asset. In practice, as opposed to the classical assumptions of a price‐taking agent in a frictionless market, traders cannot be perfectly hedged because of execution costs and market impact. They indeed face a trade‐off between hedging errors and costs that can be solved by using stochastic optimal control. Our modeling framework, which is inspired by the recent literature on optimal execution, makes it possible to account for both execution costs and the lasting market impact of trades. Prices are obtained through the indifference pricing approach. Numerical examples are provided, along with comparisons to standard methods.
    September 25, 2015   doi: 10.1111/mafi.12102   open full text
  • Shadow Prices For Continuous Processes.
    Christoph Czichowsky, Walter Schachermayer, Junjian Yang.
    Mathematical Finance. September 18, 2015
    In a financial market with a continuous price process and proportional transaction costs, we investigate the problem of utility maximization of terminal wealth. We give sufficient conditions for the existence of a shadow price process, i.e., a least favorable frictionless market leading to the same optimal strategy and utility as in the original market under transaction costs. The crucial ingredients are the continuity of the price process and the hypothesis of “no unbounded profit with bounded risk.” A counterexample reveals that these hypotheses cannot be relaxed.
    September 18, 2015   doi: 10.1111/mafi.12103   open full text
  • Impact Of Time Illiquidity In A Mixed Market Without Full Observation.
    Salvatore Federico, Paul Gassiat, Fausto Gozzi.
    Mathematical Finance. September 18, 2015
    We study a problem of optimal investment/consumption over an infinite horizon in a market with two possibly correlated assets: one liquid and one illiquid. The liquid asset is observed and can be traded continuously, while the illiquid one can be traded only at discrete random times, corresponding to the jumps of a Poisson process with intensity λ, is observed at the trading dates, and is partially observed between two different trading dates. The problem is a nonstandard mixed discrete/continuous optimal control problem, which we solve by a dynamic programming approach. When the utility has a general form, we prove that the value function is the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation and characterize the optimal allocation in the illiquid asset. In the case of power utility, we establish the regularity of the value function needed to prove the verification theorem, providing the complete theoretical solution of the problem. This enables us to perform numerical simulations, so as to analyze the impact of time illiquidity and how this impact is affected by the degree of observation.
    September 18, 2015   doi: 10.1111/mafi.12101   open full text
  • Approximate Hedging Problem With Transaction Costs In Stochastic Volatility Markets.
    Thai Huu Nguyen, Serguei Pergamenshchikov.
    Mathematical Finance. June 30, 2015
    This paper studies the problem of option replication in general stochastic volatility markets with transaction costs, using a new specification for the volatility adjustment in Leland's algorithm. We prove several limit theorems for the normalized replication error of Leland's strategy, as well as that of the strategy suggested by Lépinette. The asymptotic results obtained not only generalize the existing results, but also enable us to fix the underhedging property pointed out by Kabanov and Safarian. We also discuss possible methods to improve the convergence rate and to reduce the option price inclusive of transaction costs.
    June 30, 2015   doi: 10.1111/mafi.12094   open full text
  • A Primal–Dual Algorithm For Bsdes.
    Christian Bender, Nikolaus Schweizer, Jia Zhuo.
    Mathematical Finance. June 26, 2015
    We generalize the primal–dual methodology, which is popular in the pricing of early‐exercise options, to a backward dynamic programming equation associated with time discretization schemes of (reflected) backward stochastic differential equations (BSDEs). Taking as an input some approximate solution of the backward dynamic program, which was precomputed, e.g., by least‐squares Monte Carlo, this methodology enables us to construct a confidence interval for the unknown true solution of the time‐discretized (reflected) BSDE at time 0. We numerically demonstrate the practical applicability of our method in two 5‐dimensional nonlinear pricing problems where tight price bounds were previously unavailable.
    June 26, 2015   doi: 10.1111/mafi.12100   open full text
  • Trading With Small Price Impact.
    Ludovic Moreau, Johannes Muhle‐Karbe, H. Mete Soner.
    Mathematical Finance. June 23, 2015
    An investor trades a safe and several risky assets with linear price impact to maximize expected utility from terminal wealth. In the limit for small impact costs, we explicitly determine the optimal policy and welfare, in a general Markovian setting allowing for stochastic market, cost, and preference parameters. These results shed light on the general structure of the problem at hand, and also unveil close connections to optimal execution problems and to other market frictions such as proportional and fixed transaction costs.
    June 23, 2015   doi: 10.1111/mafi.12098   open full text
  • A First‐Order Bspde For Swing Option Pricing: Classical Solutions.
    Christian Bender, Nikolai Dokuchaev.
    Mathematical Finance. June 21, 2015
    In a companion paper, we studied a control problem related to swing option pricing in a general non‐Markovian setting. The main result there shows that the value process of this control problem can uniquely be characterized in terms of a first‐order backward stochastic partial differential equation (BSPDE) and a pathwise differential inclusion. In this paper, we additionally assume that the cash flow process of the swing option is left‐continuous in expectation. Under this assumption, we show that the value process is continuously differentiable in the space variable that represents the volume in which the holder of the option can still exercise until maturity. This gives rise to an existence and uniqueness result for the corresponding BSPDE in a classical sense. We also explicitly represent the space derivative of the value process in terms of a nonstandard optimal stopping problem over a subset of predictable stopping times. This representation can be applied to derive a dual minimization problem in terms of martingales.
    June 21, 2015   doi: 10.1111/mafi.12096   open full text
  • Mean‐Variance Policy For Discrete‐Time Cone‐Constrained Markets: Time Consistency In Efficiency And The Minimum‐Variance Signed Supermartingale Measure.
    Xiangyu Cui, Duan Li, Xun Li.
    Mathematical Finance. June 19, 2015
    The discrete‐time mean‐variance portfolio selection formulation, which is a representative of general dynamic mean‐risk portfolio selection problems, typically does not satisfy time consistency in efficiency (TCIE), i.e., a truncated precommitted efficient policy may become inefficient for the corresponding truncated problem. In this paper, we analytically investigate the effect of portfolio constraints on the TCIE of convex cone‐constrained markets. More specifically, we derive semi‐analytical expressions for the precommitted efficient mean‐variance policy and the minimum‐variance signed supermartingale measure (VSSM) and examine their relationship. Our analysis shows that the precommitted discrete‐time efficient mean‐variance policy satisfies TCIE if and only if the conditional expectation of the density of the VSSM (with respect to the original probability measure) is nonnegative, or once the conditional expectation becomes negative, it remains at the same negative value until the terminal time. Our finding indicates that the TCIE property depends only on the basic market setting, including portfolio constraints. This motivates us to establish a general procedure for constructing TCIE dynamic portfolio selection problems by introducing suitable portfolio constraints.
    June 19, 2015   doi: 10.1111/mafi.12093   open full text
  • Risk‐Minimization For Life Insurance Liabilities With Dependent Mortality Risk.
    Francesca Biagini, Camila Botero, Irene Schreiber.
    Mathematical Finance. June 19, 2015
    In this paper, we study the pricing and hedging of typical life insurance liabilities for an insurance portfolio with dependent mortality risk by means of the well‐known risk‐minimization approach. As the insurance portfolio consists of individuals of different age cohorts in order to capture the cross‐generational dependency structure of the portfolio, we introduce affine models for the mortality intensities based on Gaussian random fields that deliver analytically tractable results. We also provide specific examples consistent with historical mortality data and correlation structures. Main novelties of this work are the explicit computations of risk‐minimizing strategies for life insurance liabilities written on an insurance portfolio composed of primary financial assets (a risky asset and a money market account) and a family of longevity bonds, and the simultaneous consideration of different age cohorts.
    June 19, 2015   doi: 10.1111/mafi.12095   open full text
  • Model Uncertainty And Scenario Aggregation.
    Mathieu Cambou, Damir Filipović.
    Mathematical Finance. June 19, 2015
    This paper provides a coherent method for scenario aggregation addressing model uncertainty. It is based on divergence minimization from a reference probability measure subject to scenario constraints. An example from regulatory practice motivates the definition of five fundamental criteria that serve as a basis for our method. Standard risk measures, such as value‐at‐risk and expected shortfall, are shown to be robust with respect to minimum divergence scenario aggregation. Various examples illustrate the tractability of our method.
    June 19, 2015   doi: 10.1111/mafi.12097   open full text
  • Dynamic Trading Volume.
    Paolo Guasoni, Marko Weber.
    Mathematical Finance. June 19, 2015
    We derive the process followed by trading volume, in a market with finite depth and constant investment opportunities, where a large investor, with a long horizon and constant relative risk aversion, trades a safe and a risky asset. Trading volume approximately follows a Gaussian, mean‐reverting diffusion, and increases with depth, volatility, and risk aversion. Unlike the frictionless theory, finite depth excludes leverage and short sales because such positions may not be solvent even with continuous trading.
    June 19, 2015   doi: 10.1111/mafi.12099   open full text
  • An Analytical Solution For The Two‐Sided Parisian Stopping Time, Its Asymptotics, And The Pricing Of Parisian Options.
    Angelos Dassios, Jia Wei Lim.
    Mathematical Finance. June 16, 2015
    In this paper, we obtain a recursive formula for the density of the two‐sided Parisian stopping time. This formula does not require any numerical inversion of Laplace transforms, and is similar to the formula obtained for the one‐sided Parisian stopping time derived in Dassios and Lim. However, when we study the tails of the two distributions, we find that the two‐sided stopping time has an exponential tail, while the one‐sided stopping time has a heavier tail. We derive an asymptotic result for the tail of the two‐sided stopping time distribution and propose an alternative method of approximating the price of the two‐sided Parisian option.
    June 16, 2015   doi: 10.1111/mafi.12091   open full text
  • Optimal Investment For All Time Horizons And Martin Boundary Of Space‐Time Diffusions.
    Sergey Nadtochiy, Michael Tehranchi.
    Mathematical Finance. June 16, 2015
    This paper is concerned with the axiomatic foundation and explicit construction of a general class of optimality criteria that can be used for investment problems with multiple time horizons, or when the time horizon is not known in advance. Both the investment criterion and the optimal strategy are characterized by the Hamilton–Jacobi–Bellman equation on a semi‐infinite time interval. In the case where this equation can be linearized, the problem reduces to a time‐reversed parabolic equation, which cannot be analyzed via the standard methods of partial differential equations. Under the additional uniform ellipticity condition, we make use of the available description of all minimal solutions to such equations, along with some basic facts from potential theory and convex analysis, to obtain an explicit integral representation of all positive solutions. These results allow us to construct a large family of the aforementioned optimality criteria, including some closed‐form examples in relevant financial models.
    June 16, 2015   doi: 10.1111/mafi.12092   open full text
  • Real Options With Competition And Regime Switching.
    Alain Bensoussan, SingRu Hoe, ZhongFeng Yan, George Yin.
    Mathematical Finance. December 18, 2014
    In this paper, we examine irreversible investment decisions in duopoly games with a variable economic climate. Integrating timing flexibility, competition, and changes in the economic environment in the form of a cash flow process with regime switching, the problem is formulated as a stopping‐time game under Stackelberg leader‐follower competition, in which both players determine their respective optimal market entry time. By extending the variational inequality approach, we solve for the free boundaries and obtain optimal investment strategies for each player. Despite the lack of regularity in the leader's obstacle and the cash flow regime uncertainty, the regime‐dependent optimal policies for both the leader and the follower are obtained. In addition, we perform comprehensive numerical experiments to demonstrate the properties of solutions and to gain insights into the implications of regime switching.
    December 18, 2014   doi: 10.1111/mafi.12085   open full text
  • No‐Arbitrage In A Numéraire‐Independent Modeling Framework.
    Martin Herdegen.
    Mathematical Finance. December 18, 2014
    The classic approach to modeling financial markets consists of four steps. First, one fixes a currency unit. Second, one describes in that unit the evolution of financial assets by a stochastic process. Third, one chooses in that unit a numéraire, usually the price process of a positive asset. Fourth, one divides the original price process by the numéraire and considers the class of admissible strategies for trading. This approach has one fundamental drawback: Almost all concepts, definitions, and results, including no‐arbitrage conditions like NA, NFLVR, and NUPBR depend by their very definition, at least formally, on initial choices of a currency unit and a numéraire. In this paper, we develop a new framework for modeling financial markets, which is not based on ex‐ante choices of a currency unit and a numéraire. In particular, we introduce a “numéraire‐independent” notion of no‐arbitrage and derive its dual characterization. This yields a numéraire‐independent version of the fundamental theorem of asset pricing (FTAP). We also explain how the classic approach and other recent approaches to modeling financial markets and studying no‐arbitrage can be embedded in our framework.
    December 18, 2014   doi: 10.1111/mafi.12088   open full text
  • Tug‐Of‐War, Market Manipulation, And Option Pricing.
    Kaj Nyström, Mikko Parviainen.
    Mathematical Finance. December 18, 2014
    We develop an option pricing model based on a tug‐of‐war game. This two‐player zero‐sum stochastic differential game is formulated in the context of a multidimensional financial market. The issuer and the holder try to manipulate asset price processes in order to minimize and maximize the expected discounted reward. We prove that the game has a value and that the value function is the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the nonlinear and completely degenerate infinity Laplace operator.
    December 18, 2014   doi: 10.1111/mafi.12090   open full text
  • Expectations Of Functions Of Stochastic Time With Application To Credit Risk Modeling.
    Ovidiu Costin, Michael B. Gordy, Min Huang, Pawel J. Szerszen.
    Mathematical Finance. December 15, 2014
    We develop two novel approaches to solving for the Laplace transform of a time‐changed stochastic process. We discard the standard assumption that the background process (Xt) is Lévy. Maintaining the assumption that the business clock (Tt) and the background process are independent, we develop two different series solutions for the Laplace transform of the time‐changed process X̃t=X(Tt). In fact, our methods apply not only to Laplace transforms, but more generically to expectations of smooth functions of random time. We apply the methods to introduce stochastic time change to the standard class of default intensity models of credit risk, and show that stochastic time‐change has a very large effect on the pricing of deep out‐of‐the‐money options on credit default swaps.
    December 15, 2014   doi: 10.1111/mafi.12082   open full text
  • Model‐Independent Lower Bound On Variance Swaps.
    Nabil Kahalé.
    Mathematical Finance. December 15, 2014
    It is well known that, under a continuity assumption on the price of a stock S, the realized variance of S for maturity T can be replicated by a portfolio of calls and puts maturing at T. This paper assumes that call prices on S maturing at T are known for all strikes but makes no continuity assumptions on S. We derive semiexplicit expressions for the supremum lower bound Vinf on the hedged payoff, at maturity T, of a long position in the realized variance of S. Equivalently, Vinf is the supremum strike K such that an investor with a long position in a variance swap with strike K can ensure a nonnegative payoff at T. We study examples with constant implied volatilities and with a volatility skew. In our examples, Vinf is close to the fair variance strike obtained under the continuity assumption.
    December 15, 2014   doi: 10.1111/mafi.12083   open full text
  • On The Martingale Property In Stochastic Volatility Models Based On Time‐Homogeneous Diffusions.
    Carole Bernard, Zhenyu Cui, Don McLeish.
    Mathematical Finance. December 15, 2014
    Lions and Musiela give sufficient conditions to verify when a stochastic exponential of a continuous local martingale is a martingale or a uniformly integrable martingale. Blei and Engelbert and Mijatović and Urusov give necessary and sufficient conditions in the case of perfect correlation (ρ=1). For financial applications, such as checking the martingale property of the stock price process in correlated stochastic volatility models, we extend their work to the arbitrary correlation case (−1≤ρ≤1). We give a complete classification of the convergence properties of both perpetual and capped integral functionals of time‐homogeneous diffusions and generalize results in Mijatović and Urusov with direct proofs avoiding the use of separating times (concept introduced by Cherny and Urusov and extensively used in the proofs of Mijatović and Urusov).
    December 15, 2014   doi: 10.1111/mafi.12084   open full text
  • Local Variance Gamma And Explicit Calibration To Option Prices.
    Peter Carr, Sergey Nadtochiy.
    Mathematical Finance. December 15, 2014
    In some options markets (e.g., commodities), options are listed with only a single maturity for each underlying. In others (e.g., equities, currencies), options are listed with multiple maturities. In this paper, we analyze a special class of pure jump Markov martingale models and provide an algorithm for calibrating such models to match the market prices of European options with multiple strikes and maturities. This algorithm matches option prices exactly and only requires solving several one‐dimensional root‐search problems and applying elementary functions. We show how to construct a time‐homogeneous process which meets a single smile, and a piecewise time‐homogeneous process which can meet multiple smiles.
    December 15, 2014   doi: 10.1111/mafi.12086   open full text
  • Robust Portfolios And Weak Incentives In Long‐Run Investments.
    Paolo Guasoni, Johannes Muhle‐Karbe, Hao Xing.
    Mathematical Finance. December 15, 2014
    When the planning horizon is long, and the safe asset grows indefinitely, isoelastic portfolios are nearly optimal for investors who are close to isoelastic for high wealth, and not too risk averse for low wealth. We prove this result in a general arbitrage‐free, frictionless, semimartingale model. As a consequence, optimal portfolios are robust to the perturbations in preferences induced by common option compensation schemes, and such incentives are weaker when their horizon is longer. Robust option incentives are possible, but require several, arbitrarily large exercise prices, and are not always convex.
    December 15, 2014   doi: 10.1111/mafi.12087   open full text
  • The Numéraire Property And Long‐Term Growth Optimality For Drawdown‐Constrained Investments.
    Constantinos Kardaras, Jan Obłój, Eckhard Platen.
    Mathematical Finance. December 15, 2014
    We consider the portfolio choice problem for a long‐run investor in a general continuous semimartingale model. We combine the decision criterion of pathwise growth optimality with a flexible specification of attitude toward risk, encoded by a linear drawdown constraint imposed on admissible wealth processes. We define the constrained numéraire property through the notion of expected relative return and prove that drawdown‐constrained numéraire portfolio exists and is unique, but may depend on the investment horizon. However, when sampled at the times of its maximum and asymptotically as the time‐horizon becomes distant, the drawdown‐constrained numéraire portfolio is given explicitly through a model‐independent transformation of the unconstrained numéraire portfolio. The asymptotically growth‐optimal strategy is obtained as limit of numéraire strategies on finite horizons.
    December 15, 2014   doi: 10.1111/mafi.12081   open full text
  • Optimal Investment With Intermediate Consumption And Random Endowment.
    Oleksii Mostovyi.
    Mathematical Finance. December 15, 2014
    We consider an optimal investment problem with intermediate consumption and random endowment, in an incomplete semimartingale model of the financial market. We establish the key assertions of the utility maximization theory, assuming that both primal and dual value functions are finite in the interiors of their domains and that the random endowment at maturity can be dominated by the terminal value of a self‐financing wealth process. In order to facilitate the verification of these conditions, we present alternative, but equivalent conditions, under which the conclusions of the theory hold.
    December 15, 2014   doi: 10.1111/mafi.12089   open full text
  • Multivariate Risk Measures: A Constructive Approach Based On Selections.
    Ilya Molchanov, Ignacio Cascos.
    Mathematical Finance. September 11, 2014
    Since risky positions in multivariate portfolios can be offset by various choices of capital requirements that depend on the exchange rules and related transaction costs, it is natural to assume that the risk measures of random vectors are set‐valued. Furthermore, it is reasonable to include the exchange rules in the argument of the risk measure and so consider risk measures of set‐valued portfolios. This situation includes the classical Kabanov's transaction costs model, where the set‐valued portfolio is given by the sum of a random vector and an exchange cone, but also a number of further cases of additional liquidity constraints. We suggest a definition of the risk measure based on calling a set‐valued portfolio acceptable if it possesses a selection with all individually acceptable marginals. The obtained selection risk measure is coherent (or convex), law invariant, and has values being upper convex closed sets. We describe the dual representation of the selection risk measure and suggest efficient ways of approximating it from below and from above. In the case of Kabanov's exchange cone model, it is shown how the selection risk measure relates to the set‐valued risk measures considered by Kulikov (2008, Theory Probab. Appl. 52, 614–635), Hamel and Heyde (2010, SIAM J. Financ. Math. 1, 66–95), and Hamel, Heyde, and Rudloff (2013, Math. Financ. Econ. 5, 1–28).
    September 11, 2014   doi: 10.1111/mafi.12078   open full text
  • Price Setting Of Market Makers: A Filtering Problem With Endogenous Filtration.
    Christoph Kühn, Matthias Riedel.
    Mathematical Finance. September 05, 2014
    This paper studies the price‐setting problem of market makers under risk neutrality and perfect competition in continuous time. The classic approach of Glosten–Milgrom is followed. Bid and ask prices are defined as conditional expectations of a true value of the asset given the market makers' partial information that includes the customers' trading decisions. The true value is modeled as a Markov process that can be observed by the customers with some noise at Poisson times. A mathematically rigorous analysis of the price‐setting problem is carried out, solving a filtering problem with endogenous filtration that depends on the bid and ask price processes quoted by the market maker. The existence and uniqueness of the bid and ask price processes is shown under some conditions.
    September 05, 2014   doi: 10.1111/mafi.12079   open full text
  • Sensitivity Analysis Of Nonlinear Behavior With Distorted Probability.
    Xi‐Ren Cao, Xiangwei Wan.
    Mathematical Finance. September 04, 2014
    In this paper, we propose a sensitivity‐based analysis to study the nonlinear behavior under nonexpected utility with probability distortions (or “distorted utility” for short). We first discover the “monolinearity” of distorted utility, which means that after properly changing the underlying probability measure, distorted utility becomes locally linear in probabilities, and the derivative of distorted utility is simply an expectation of the sample path derivative under the new measure. From the monolinearity property, simulation algorithms for estimating the derivative of distorted utility can be developed, leading to gradient‐based search algorithms for the optimum of distorted utility. We then apply the sensitivity‐based approach to the portfolio selection problem under distorted utility with complete and incomplete markets. For the complete markets case, the first‐order condition is derived and optimal wealth deduced. For the incomplete markets case, a dual characterization of optimal policies is provided; a solvable incomplete market example with unhedgeable interest rate risk is also presented. We expect this sensitivity‐based approach to be generally applicable to optimization problems involving probability distortions.
    September 04, 2014   doi: 10.1111/mafi.12076   open full text
  • Price‐Admissibility Conditions For Arbitrage‐Free Linear Price Function Models For The Term Structure Of Interest Rates.
    Andrew F. Siegel.
    Mathematical Finance. September 03, 2014
    To assure price admissibility—that all bond prices, yields, and forward rates remain positive—we show how to control the state variables within the class of arbitrage‐free linear price function models for the evolution of interest rate yield curves over time. Price admissibility is necessary to preclude cash‐and‐carry arbitrage, a market imperfection that can happen even with a risk‐neutral diffusion process and positive bond prices. We assure price admissibility by (i) defining the state variables to be scaled partial sums of weighted coefficients of the exponential terms in the bond pricing function, (ii) identifying a simplex within which these state variables remain price admissible, and (iii) choosing a general functional form for the diffusion that selectively diminishes near the simplex boundary. By assuring that prices, yields, and forward rates remain positive with tractable diffusions for the physical and risk‐neutral measures, an obstacle is removed from the wider acceptance of interest rate methods that are linear in prices.
    September 03, 2014   doi: 10.1111/mafi.12075   open full text
  • Fast Swaption Pricing In Gaussian Term Structure Models.
    Jaehyuk Choi, Sungchan Shin.
    Mathematical Finance. September 03, 2014
    We propose a fast and accurate numerical method for pricing European swaptions in multifactor Gaussian term structure models. Our method can be used to accelerate the calibration of such models to the volatility surface. The pricing of an interest rate option in such a model involves evaluating a multidimensional integral of the payoff of the claim on a domain where the payoff is positive. In our method, we approximate the exercise boundary of the state space by a hyperplane tangent to the maximum probability point on the boundary and simplify the multidimensional integration into an analytical form. The maximum probability point can be determined using the gradient descent method. We demonstrate that our method is superior to previous methods by comparing the results to the price obtained by numerical integration.
    September 03, 2014   doi: 10.1111/mafi.12077   open full text
  • Coherence And Elicitability.
    Johanna F. Ziegel.
    Mathematical Finance. September 03, 2014
    The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In statistical decision theory, risk measures for which such verification and comparison is possible, are called elicitable. It is known that quantile‐based risk measures such as value at risk are elicitable. In this paper, the existing result of the nonelicitability of expected shortfall is extended to all law‐invariant spectral risk measures unless they reduce to minus the expected value. Hence, it is unclear how to perform forecast verification or comparison. However, the class of elicitable law‐invariant coherent risk measures does not reduce to minus the expected value. We show that it consists of certain expectiles.
    September 03, 2014   doi: 10.1111/mafi.12080   open full text
  • Fire Sales Forensics: Measuring Endogenous Risk.
    Rama Cont, Lakshithe Wagalath.
    Mathematical Finance. September 01, 2014
    We propose a tractable framework for quantifying the impact of loss‐triggered fire sales on portfolio risk, in a multi‐asset setting. We derive analytical expressions for the impact of fire sales on the realized volatility and correlations of asset returns in a fire sales scenario and show that our results provide a quantitative explanation for the spikes in volatility and correlations observed during such deleveraging episodes. These results are then used to develop an econometric framework for the forensic analysis of fire sales episodes, using observations of market prices. We give conditions for the identifiability of model parameters from time series of asset prices, propose a statistical test for the presence of fire sales, and an estimator for the magnitude of fire sales in each asset class. Pathwise consistency and large sample properties of the estimator are studied in the high‐frequency asymptotic regime. We illustrate our methodology by applying it to the forensic analysis of two recent deleveraging episodes: the Quant Crash of August 2007 and the Great Deleveraging following the default of Lehman Brothers in Fall 2008.
    September 01, 2014   doi: 10.1111/mafi.12071   open full text
  • Optimal Investment In Credit Derivatives Portfolio Under Contagion Risk.
    Lijun Bo, Agostino Capponi.
    Mathematical Finance. September 01, 2014
    We consider the optimal portfolio problem of a power investor who wishes to allocate her wealth between several credit default swaps (CDSs) and a money market account. We model contagion risk among the reference entities in the portfolio using a reduced‐form Markovian model with interacting default intensities. Using the dynamic programming principle, we establish a lattice dependence structure between the Hamilton‐Jacobi‐Bellman equations associated with the default states of the portfolio. We show existence and uniqueness of a classical solution to each equation and characterize them in terms of solutions to inhomogeneous Bernoulli type ordinary differential equations. We provide a precise characterization for the directionality of the CDS investment strategy and perform a numerical analysis to assess the impact of default contagion. We find that the increased intensity triggered by default of a very risky entity strongly impacts size and directionality of the investor strategy. Such findings outline the key role played by default contagion when investing in portfolios subject to multiple sources of default risk.
    September 01, 2014   doi: 10.1111/mafi.12074   open full text
  • A Note On The Quantile Formulation.
    Zuo Quan Xu.
    Mathematical Finance. September 01, 2014
    Many investment models in discrete or continuous‐time settings boil down to maximizing an objective of the quantile function of the decision variable. This quantile optimization problem is known as the quantile formulation of the original investment problem. Under certain monotonicity assumptions, several schemes to solve such quantile optimization problems have been proposed in the literature. In this paper, we propose a change‐of‐variable and relaxation method to solve the quantile optimization problems without using the calculus of variations or making any monotonicity assumptions. The method is demonstrated through a portfolio choice problem under rank‐dependent utility theory (RDUT). We show that this problem is equivalent to a classical Merton's portfolio choice problem under expected utility theory with the same utility function but a different pricing kernel explicitly determined by the given pricing kernel and probability weighting function. With this result, the feasibility, well‐posedness, attainability, and uniqueness issues for the portfolio choice problem under RDUT are solved. It is also shown that solving functional optimization problems may reduce to solving probabilistic optimization problems. The method is applicable to general models with law‐invariant preference measures including portfolio choice models under cumulative prospect theory (CPT) or RDUT, Yaari's dual model, Lopes' SP/A model, and optimal stopping models under CPT or RDUT.
    September 01, 2014   doi: 10.1111/mafi.12072   open full text
  • Stability Of The Exponential Utility Maximization Problem With Respect To Preferences.
    Hao Xing.
    Mathematical Finance. August 28, 2014
    This paper studies stability of the exponential utility maximization when there are small variations on agent's utility function. Two settings are considered. First, in a general semimartingale model where random endowments are present, a sequence of utilities defined on R converges to the exponential utility. Under a uniform condition on their marginal utilities, convergence of value functions, optimal payoffs, and optimal investment strategies are obtained, their rate of convergence is also determined. Stability of utility‐based pricing is studied as an application. Second, a sequence of utilities defined on R+ converges to the exponential utility after shifting and scaling. Their associated optimal strategies, after appropriate scaling, converge to the optimal strategy for the exponential hedging problem. This complements Theorem 3.2 in [Nutz, M. (2012): Risk aversion asymptotics for power utility maximization. Probab. Theory & Relat. Fields 152, 703–749], which establishes the convergence for a sequence of power utilities.
    August 28, 2014   doi: 10.1111/mafi.12073   open full text
  • Multivariate Subordination Of Markov Processes With Financial Applications.
    Rafael Mendoza‐Arriaga, Vadim Linetsky.
    Mathematical Finance. June 19, 2014
    This paper develops the procedure of multivariate subordination for a collection of independent Markov processes with killing. Starting from d independent Markov processes Xi with killing and an independent d‐dimensional time change T, we construct a new process by time, changing each of the Markov processes Xi with a coordinate Ti. When T is a d‐dimensional Lévy subordinator, the time changed process (Yi:=Xi(Ti(t)) is a time‐homogeneous Markov process with state‐dependent jumps and killing in the product of the state spaces of Xi. The dependence among jumps of its components is governed by the d‐dimensional Lévy measure of the subordinator. When T is a d‐dimensional additive subordinator, Y is a time‐inhomogeneous Markov process. When Ti=∫0tVsids with Vi forming a multivariate Markov process, (Yi,Vi) is a Markov process, where each Vi plays a role of stochastic volatility of Yi. This construction provides a rich modeling architecture for building multivariate models in finance with time‐ and state‐dependent jumps, stochastic volatility, and killing (default). The semigroup theory provides powerful analytical and computational tools for securities pricing in this framework. To illustrate, the paper considers applications to multiname unified credit‐equity models and correlated commodity models.
    June 19, 2014   doi: 10.1111/mafi.12061   open full text
  • A First‐Order Bspde For Swing Option Pricing.
    Christian Bender, Nikolai Dokuchaev.
    Mathematical Finance. May 20, 2014
    We study an optimal control problem related to swing option pricing in a general non‐Markovian setting in continuous time. As a main result we uniquely characterize the value process in terms of a first‐order nonlinear backward stochastic partial differential equation and a differential inclusion. Based on this result we also determine the set of optimal controls and derive a dual minimization problem.
    May 20, 2014   doi: 10.1111/mafi.12067   open full text
  • Valuation Of Barrier Options Via A General Self‐Duality.
    Elisa Alòs, Zhanyu Chen, Thorsten Rheinländer.
    Mathematical Finance. May 20, 2014
    Classical put–call symmetry relates the price of puts and calls under a suitable dual market transform. One well‐known application is the semistatic hedging of path‐dependent barrier options with European options. This, however, in its classical form requires the price process to observe rather stringent and unrealistic symmetry properties. In this paper, we develop a general self‐duality theorem to develop valuation schemes for barrier options in stochastic volatility models with correlation.
    May 20, 2014   doi: 10.1111/mafi.12063   open full text
  • Utility Maximization Under Model Uncertainty In Discrete Time.
    Marcel Nutz.
    Mathematical Finance. May 20, 2014
    We give a general formulation of the utility maximization problem under nondominated model uncertainty in discrete time and show that an optimal portfolio exists for any utility function that is bounded from above. In the unbounded case, integrability conditions are needed as nonexistence may arise even if the value function is finite.
    May 20, 2014   doi: 10.1111/mafi.12068   open full text
  • High‐Order Short‐Time Expansions For Atm Option Prices Of Exponential Lévy Models.
    José E. Figueroa‐López, Ruoting Gong, Christian Houdré.
    Mathematical Finance. May 20, 2014
    The short‐time asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In this work, a novel second‐order approximation for at‐the‐money (ATM) option prices is derived for a large class of exponential Lévy models with or without Brownian component. The results hereafter shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of ATM option prices near expiration. In the presence of a Brownian component, the second‐order term, in time‐t, is of the form d2t(3−Y)/2, with d2 only depending on Y, the degree of jump activity, on σ, the volatility of the continuous component, and on an additional parameter controlling the intensity of the “small” jumps (regardless of their signs). This extends the well‐known result that the leading first‐order term is σt1/2/2π. In contrast, under a pure‐jump model, the dependence on Y and on the separate intensities of negative and positive small jumps are already reflected in the leading term, which is of the form d1t1/Y. The second‐order term is shown to be of the form d̃2t and, therefore, its order of decay turns out to be independent of Y. The asymptotic behavior of the corresponding Black–Scholes implied volatilities is also addressed. Our method of proof is based on an integral representation of the option price involving the tail probability of the log‐return process under the share measure and a suitable change of probability measure under which the pure‐jump component of the log‐return process becomes a Y‐stable process. Our approach is sufficiently general to cover a wide class of Lévy processes, which satisfy the latter property and whose Lévy density can be closely approximated by a stable density near the origin. Our numerical results show that the first‐order term typically exhibits rather poor performance and that the second‐order term can significantly improve the approximation's accuracy, particularly in the absence of a Brownian component.
    May 20, 2014   doi: 10.1111/mafi.12064   open full text
  • Gambling In Contests With Regret.
    Han Feng, David Hobson.
    Mathematical Finance. May 20, 2014
    This paper discusses the gambling contest introduced in Seel and Strack (, Gambling in Contests, Journal of Economic Theory, 148(5), 2033–2048) and considers the impact of adding a penalty associated with failure to follow a winning strategy. The Seel and Strack model consists of n‐agents each of whom privately observes a transient diffusion process and chooses when to stop it. The player with the highest stopped value wins the contest, and each player's objective is to maximize her probability of winning the contest. We give a new derivation of the results of Seel and Strack based on a Lagrangian approach. Moreover, we consider an extension of the problem to a behavioral finance context in the sense of regret theory. In particular, an agent is penalized when her chosen strategy does not win the contest, but there existed an alternative strategy that would have resulted in victory.
    May 20, 2014   doi: 10.1111/mafi.12069   open full text
  • Arrow–Debreu Equilibria For Rank‐Dependent Utilities.
    Jianming Xia, Xun Yu Zhou.
    Mathematical Finance. May 20, 2014
    We provide conditions on a one‐period‐two‐date pure exchange economy with rank‐dependent utility agents under which Arrow–Debreu equilibria exist. When such an equilibrium exists, we show that the state‐price density is a weighted marginal rate of intertemporal substitution of a representative agent, where the weight depends on the differential of the probability weighting function. Based on the result, we find that asset prices depend upon agents' subjective beliefs regarding overall consumption growth, and we offer a direction for possible resolution of the equity premium puzzle.
    May 20, 2014   doi: 10.1111/mafi.12070   open full text
  • Benchmarked Risk Minimization.
    Ke Du, Eckhard Platen.
    Mathematical Finance. May 20, 2014
    This paper discusses the problem of hedging not perfectly replicable contingent claims using the numéraire portfolio. The proposed concept of benchmarked risk minimization leads beyond the classical no‐arbitrage paradigm. It provides in incomplete markets a generalization of the pricing under classical risk minimization, pioneered by Föllmer, Sondermann, and Schweizer. The latter relies on a quadratic criterion, requests square integrability of claims and gains processes, and relies on the existence of an equivalent risk‐neutral probability measure. Benchmarked risk minimization avoids these restrictive assumptions and provides symmetry with respect to all primary securities. It employs the real‐world probability measure and the numéraire portfolio to identify the minimal possible price for a contingent claim. Furthermore, the resulting benchmarked (i.e., numéraire portfolio denominated) profit and loss is only driven by uncertainty that is orthogonal to benchmarked‐traded uncertainty, and forms a local martingale that starts at zero. Consequently, sufficiently different benchmarked profits and losses, when pooled, become asymptotically negligible through diversification. This property makes benchmarked risk minimization the least expensive method for pricing and hedging diversified pools of not fully replicable benchmarked contingent claims. In addition, when hedging it incorporates evolving information about nonhedgeable uncertainty, which is ignored under classical risk minimization.
    May 20, 2014   doi: 10.1111/mafi.12065   open full text
    Yuhong Xu.
    Mathematical Finance. May 20, 2014
    This paper deals with multidimensional dynamic risk measures induced by conditional g‐expectations. A notion of multidimensional g‐expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem, and viability on a rectangle of solutions to multidimensional backward stochastic differential equations, some necessary and sufficient conditions are given for the constancy, monotonicity, positivity, and translatability properties of multidimensional conditional g‐expectations and multidimensional dynamic risk measures; we prove that a multidimensional dynamic g‐risk measure is nonincreasingly convex if and only if the generator g satisfies a quasi‐monotone increasingly convex condition. A general dual representation is given for the multidimensional dynamic convex g‐risk measure in which the penalty term is expressed more precisely. It is shown that model uncertainty leads to the convexity of risk measures. As to applications, we show how this multidimensional approach can be applied to measure the insolvency risk of a firm with interacting subsidiaries; optimal risk sharing for γ‐tolerant g‐risk measures, and risk contribution for coherent g‐risk measures are investigated. Insurance g‐risk measure and other ways to induce g‐risk measures are also studied at the end of the paper.
    May 20, 2014   doi: 10.1111/mafi.12062   open full text
  • Do Arbitrage‐Free Prices Come From Utility Maximization?
    Pietro Siorpaes.
    Mathematical Finance. May 20, 2014
    In this paper we ask whether, given a stock market and an illiquid derivative, there exists arbitrage‐free prices at which a utility‐maximizing agent would always want to buy the derivative, irrespectively of his own initial endowment of derivatives and cash. We prove that this is false for any given investor if one considers all initial endowments with finite utility, and that it can instead be true if one restricts to the endowments in the interior. We show, however, how the endowments on the boundary can give rise to very odd phenomena; for example, an investor with such an endowment would choose not to trade in the derivative even at prices arbitrarily close to some arbitrage price.
    May 20, 2014   doi: 10.1111/mafi.12066   open full text
  • A Model‐Free Version Of The Fundamental Theorem Of Asset Pricing And The Super‐Replication Theorem.
    B. Acciaio, M. Beiglböck, F. Penkner, W. Schachermayer.
    Mathematical Finance. December 06, 2013
    We propose a Fundamental Theorem of Asset Pricing and a Super‐Replication Theorem in a model‐independent framework. We prove these theorems in the setting of finite, discrete time and a market consisting of a risky asset S as well as options written on this risky asset. As a technical condition, we assume the existence of a traded option with a superlinearly growing payoff‐function, e.g., a power option. This condition is not needed when sufficiently many vanilla options maturing at the horizon T are traded in the market.
    December 06, 2013   doi: 10.1111/mafi.12060   open full text
  • On Valuing Stochastic Perpetuities Using New Long Horizon Stock Price Models Distinguishing Booms, Busts, And Balanced Markets.
    Dilip B. Madan, Marc Yor.
    Mathematical Finance. December 02, 2013
    For longer horizons, assuming no dividend distributions, models for discounted stock prices in balanced markets are formulated as conditional expectations of nontrivial terminal random variables defined at infinity. Observing that extant models fail to have this property, new models are proposed. The new concept of a balanced market proposed here permits a distinction between such markets and unduly optimistic or pessimistic ones. A tractable example is developed and termed the discounted variance gamma model. Calibrations to market data provide empirical support. Additionally, procedures are presented for the valuation of path dependent stochastic perpetuities. Evidence is provided for long dated equity linked claims paying coupon for time spent by equity above a lower barrier, being underpriced by extant models relative to the new discounted ones. Given the popularity of such claims, the resulting mispricing could possibly take some corrections. Furthermore for these new discounted models, implied volatility curves do not flatten out at the larger maturities.
    December 02, 2013   doi: 10.1111/mafi.12056   open full text
  • Model‐Independent No‐Arbitrage Conditions On American Put Options.
    Alexander M. G. Cox, Christoph Hoeggerl.
    Mathematical Finance. December 02, 2013
    We consider the pricing of American put options in a model‐independent setting: that is, we do not assume that asset prices behave according to a given model, but aim to draw conclusions that hold in any model. We incorporate market information by supposing that the prices of European options are known. In this setting, we are able to provide conditions on the American put prices which are necessary for the absence of arbitrage. Moreover, if we further assume that there are finitely many European and American options traded, then we are able to show that these conditions are also sufficient. To show sufficiency, we construct a model under which both American and European options are correctly priced at all strikes simultaneously. In particular, we need to carefully consider the optimal stopping strategy in the construction of our process.
    December 02, 2013   doi: 10.1111/mafi.12058   open full text
  • Stochastic Local Intensity Loss Models With Interacting Particle Systems.
    Aurélien Alfonsi, Céline Labart, Jérôme Lelong.
    Mathematical Finance. December 02, 2013
    It is well known from the work of Schönbucher that the marginal laws of a loss process can be matched by a unit increasing time inhomogeneous Markov process, whose deterministic jump intensity is called local intensity. The stochastic local intensity (SLI) models such as the one proposed by Arnsdorf and Halperin allow to get a stochastic jump intensity while keeping the same marginal laws. These models involve a nonlinear stochastic differential equation (SDE) with jumps. The first contribution of this paper is to prove the existence and uniqueness of such processes. This is made by means of an interacting particle system, whose convergence rate toward the nonlinear SDE is analyzed. Second, this approach provides a powerful way to compute pathwise expectations with the SLI model: we show that the computational cost is roughly the same as a crude Monte Carlo algorithm for standard SDEs.
    December 02, 2013   doi: 10.1111/mafi.12059   open full text
  • The Incentives Of Hedge Fund Fees And High‐Water Marks.
    Paolo Guasoni, Jan Obłój.
    Mathematical Finance. December 02, 2013
    Hedge fund managers receive a large fraction of their funds' profits, paid when funds exceed their high‐water marks. We study the incentives of such performance fees. A manager with long‐horizon, constant investment opportunities and relative risk aversion, chooses a constant Merton portfolio. However, the effective risk aversion shrinks toward one in proportion to performance fees. Risk shifting implications are ambiguous and depend on the manager's own risk aversion. Managers with equal investment opportunities but different performance fees and risk aversions may coexist in a competitive equilibrium. The resulting leverage increases with performance fees—a prediction that we confirm empirically.
    December 02, 2013   doi: 10.1111/mafi.12057   open full text
  • Behavioral Portfolio Selection: Asymptotics And Stability Along A Sequence Of Models.
    Christian Reichlin.
    Mathematical Finance. October 11, 2013
    We consider a sequence of financial markets that converges weakly in a suitable sense and maximize a behavioral preference functional in each market. For expected concave utilities, it is well known that the maximal expected utilities and the corresponding final positions converge to the corresponding quantities in the limit model. We prove similar results for nonconcave utilities and distorted expectations as employed in behavioral finance, and we illustrate by a counterexample that these results require a stronger notion of convergence of the underlying models compared to the concave utility maximization. We use the results to analyze the stability of behavioral portfolio selection problems and to provide numerically tractable methods to solve such problems in complete continuous‐time models.
    October 11, 2013   doi: 10.1111/mafi.12053   open full text
  • General Intensity Shapes In Optimal Liquidation.
    Olivier Guéant, Charles‐Albert Lehalle.
    Mathematical Finance. October 09, 2013
    The classical literature on optimal liquidation, rooted in Almgren–Chriss models, tackles the optimal liquidation problem using a trade‐off between market impact and price risk. It answers the general question of optimal scheduling but the very question of the actual way to proceed with liquidation is rarely dealt with. Our model, which incorporates both price risk and nonexecution risk, is an attempt to tackle this question using limit orders. The very general framework we propose to model liquidation with limit orders generalizes existing ones in two ways. We consider a risk‐averse agent, whereas the model of Bayraktar and Ludkovski only tackles the case of a risk‐neutral one. We consider very general functional forms for the execution process intensity, whereas Guéant, Lehalle and Fernandez‐Tapia are restricted to exponential intensity. Eventually, we link the execution cost function of Almgren–Chriss models to the intensity function in our model, providing then a way to see Almgren–Chriss models as a limit of ours.
    October 09, 2013   doi: 10.1111/mafi.12052   open full text
  • A New Look At Short‐Term Implied Volatility In Asset Price Models With Jumps.
    Aleksandar Mijatović, Peter Tankov.
    Mathematical Finance. October 09, 2013
    We analyze the behavior of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalization of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, which is a function of both the diffusion component of the process and the jump activity (Blumenthal–Getoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short‐end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the viewpoint of option prices: in the latter, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the former, the wings have a constant model‐independent slope. This result gives a theoretical justification for the preference of the infinite variation Lévy models over the finite variation ones in the calibration based on short‐maturity option prices.
    October 09, 2013   doi: 10.1111/mafi.12055   open full text
  • Resilience To Contagion In Financial Networks.
    Hamed Amini, Rama Cont, Andreea Minca.
    Mathematical Finance. October 09, 2013
    We derive rigorous asymptotic results for the magnitude of contagion in a large counterparty network and give an analytical expression for the asymptotic fraction of defaults, in terms of network characteristics. Our results extend previous studies on contagion in random graphs to inhomogeneous‐directed graphs with a given degree sequence and arbitrary distribution of weights. We introduce a criterion for the resilience of a large financial network to the insolvency of a small group of financial institutions and quantify how contagion amplifies small shocks to the network. Our results emphasize the role played by “contagious links” and show that institutions which contribute most to network instability have both large connectivity and a large fraction of contagious links. The asymptotic results show good agreement with simulations for networks with realistic sizes.
    October 09, 2013   doi: 10.1111/mafi.12051   open full text
  • Comparing Local Risks By Acceptance And Rejection.
    Amnon Schreiber.
    Mathematical Finance. October 09, 2013
    It is said that risky asset h acceptance dominates risky asset k if any decision maker who rejects the investment in h also rejects the investment in k. While in general acceptance dominance is a partial order, we show that it becomes a complete order if only infinitely short investment time horizons are considered. Two indices that induce different variants of this order are proposed, absolute acceptance dominance and relative acceptance dominance, and their properties are studied. We then show that many indices of riskiness that are compatible with the acceptance dominance order coincide with our indices in the continuous‐time setup.
    October 09, 2013   doi: 10.1111/mafi.12054   open full text
  • Measuring Distribution Model Risk.
    Thomas Breuer, Imre Csiszár.
    Mathematical Finance. October 09, 2013
    We propose to interpret distribution model risk as sensitivity of expected loss to changes in the risk factor distribution, and to measure the distribution model risk of a portfolio by the maximum expected loss over a set of plausible distributions defined in terms of some divergence from an estimated distribution. The divergence may be relative entropy or another f‐divergence or Bregman distance. We use the theory of minimizing convex integral functionals under moment constraints to give formulae for the calculation of distribution model risk and to explicitly determine the worst case distribution from the set of plausible distributions. We also evaluate related risk measures describing divergence preferences.
    October 09, 2013   doi: 10.1111/mafi.12050   open full text
    O. Bardou, N. Frikha, G. Pagès.
    Mathematical Finance. October 09, 2013
    In this paper, we investigate a method based on risk minimization to hedge observable but nontradable source of risk on financial or energy markets. The optimal portfolio strategy is obtained by minimizing dynamically the conditional value‐at‐risk (CVaR) using three main tools: a stochastic approximation algorithm, optimal quantization, and variance reduction techniques (importance sampling and linear control variable), as the quantities of interest are naturally related to rare events. As a first step, we investigate the problem of CVaR regression, which corresponds to a static portfolio strategy where the number of units of each tradable assets is fixed at time 0 and remains unchanged till maturity. We devise a stochastic approximation algorithm and study its a.s. convergence and weak convergence rate. Then, we extend our approach to the dynamic case under the assumption that the process modeling the nontradable source of risk and financial assets prices is Markovian. Finally, we illustrate our approach by considering several portfolios in connection with energy markets.
    October 09, 2013   doi: 10.1111/mafi.12049   open full text
  • Linked Recursive Preferences And Optimality.
    Shlomo Levental, Sumit Sinha, Mark Schroder.
    Mathematical Finance. October 09, 2013
    We study a class of optimization problems involving linked recursive preferences in a continuous‐time Brownian setting. Such links can arise when preferences depend directly on the level or volatility of wealth, in principal–agent (optimal compensation) problems with moral hazard, and when the impact of social influences on preferences is modeled via utility (and utility diffusion) externalities. We characterize the necessary first‐order conditions, which are also sufficient under additional conditions ensuring concavity. We also examine applications to optimal consumption and portfolio choice, and applications to Pareto optimal allocations.
    October 09, 2013   doi: 10.1111/mafi.12047   open full text
  • Robust Utility Maximization In Nondominated Models With 2 Bsde: The Uncertain Volatility Model.
    Anis Matoussi, Dylan Possamaï, Chao Zhou.
    Mathematical Finance. June 18, 2013
    The problem of robust utility maximization in an incomplete market with volatility uncertainty is considered, in the sense that the volatility of the market is only assumed to lie between two given bounds. The set of all possible models (probability measures) considered here is nondominated. We propose studying this problem in the framework of second‐order backward stochastic differential equations (2BSDEs for short) with quadratic growth generators. We show for exponential, power, and logarithmic utilities that the value function of the problem can be written as the initial value of a particular 2BSDE and prove existence of an optimal strategy. Finally, several examples which shed more light on the problem and its links with the classical utility maximization one are provided. In particular, we show that in some cases, the upper bound of the volatility interval plays a central role, exactly as in the option pricing problem with uncertain volatility models.
    June 18, 2013   doi: 10.1111/mafi.12031   open full text
  • Portfolio Liquidation In Dark Pools In Continuous Time.
    Peter Kratz, Torsten Schöneborn.
    Mathematical Finance. June 18, 2013
    We consider an illiquid financial market where a risk averse investor has to liquidate a portfolio within a finite time horizon [0, T] and can trade continuously at a traditional exchange (the “primary venue”) and in a dark pool. At the primary venue, trading yields a linear price impact. In the dark pool, no price impact costs arise but order execution is uncertain, modeled by a multidimensional Poisson process. We characterize the costs of trading by a linear‐quadratic functional which incorporates both the price impact costs of trading at the primary exchange and the market risk of the position. The solution of the cost minimization problem is characterized by a matrix differential equation with singular boundary condition; by means of stochastic control theory, we provide a verification argument. If a single‐asset position is to be liquidated, the investor slowly trades out of her position at the primary venue, with the remainder being placed in the dark pool at any point in time. For multi‐asset liquidations this is generally not the case; for example, it can be optimal to oversize orders in the dark pool in order to turn a poorly balanced portfolio into a portfolio bearing less risk.
    June 18, 2013   doi: 10.1111/mafi.12037   open full text
  • No‐Arbitrage Pricing For Dividend‐Paying Securities In Discrete‐Time Markets With Transaction Costs.
    Tomasz R. Bielecki, Igor Cialenco, Rodrigo Rodriguez.
    Mathematical Finance. June 18, 2013
    We prove a version of First Fundamental Theorem of Asset Pricing under transaction costs for discrete‐time markets with dividend‐paying securities. Specifically, we show that the no‐arbitrage condition under the efficient friction assumption is equivalent to the existence of a risk‐neutral measure. We derive dual representations for the superhedging ask and subhedging bid price processes of a contingent claim contract. Our results are illustrated with a vanilla credit default swap contract.
    June 18, 2013   doi: 10.1111/mafi.12038   open full text
  • Bessel Processes, Stochastic Volatility, And Timer Options.
    Chenxu Li.
    Mathematical Finance. June 18, 2013
    Motivated by analytical valuation of timer options (an important innovation in realized variance‐based derivatives), we explore their novel mathematical connection with stochastic volatility and Bessel processes (with constant drift). Under the Heston (1993) stochastic volatility model, we formulate the problem through a first‐passage time problem on realized variance, and generalize the standard risk‐neutral valuation theory for fixed maturity options to a case involving random maturity. By time change and the general theory of Markov diffusions, we characterize the joint distribution of the first‐passage time of the realized variance and the corresponding variance using Bessel processes with drift. Thus, explicit formulas for a useful joint density related to Bessel processes are derived via Laplace transform inversion. Based on these theoretical findings, we obtain a Black–Scholes–Merton‐type formula for pricing timer options, and thus extend the analytical tractability of the Heston model. Several issues regarding the numerical implementation are briefly discussed.
    June 18, 2013   doi: 10.1111/mafi.12041   open full text
  • Optimal High‐Frequency Trading In A Pro Rata Microstructure With Predictive Information.
    Fabien Guilbaud, Huyên Pham.
    Mathematical Finance. June 18, 2013
    We propose a framework to study optimal trading policies in a one‐tick pro rata limit order book, as typically arises in short‐term interest rate futures contracts. The high‐frequency trader chooses to post either market orders or limit orders, which are represented, respectively, by impulse controls and regular controls. We discuss the consequences of the two main features of this microstructure: first, the limit orders are only partially executed, and therefore she has no control on the executed quantity. Second, the high‐frequency trader faces the overtrading risk, which is the risk of large variations in her inventory. The consequences of this risk are investigated in the context of optimal liquidation. The optimal trading problem is studied by stochastic control and dynamic programming methods, and we provide the associated numerical resolution procedure and prove its convergence. We propose dimension reduction techniques in several cases of practical interest. We also detail a high‐frequency trading strategy in the case where a (predictive) directional information on the price is available. Each of the resulting strategies is illustrated by numerical tests.
    June 18, 2013   doi: 10.1111/mafi.12042   open full text
  • Hope, Fear, And Aspirations.
    Xue Dong He, Xun Yu Zhou.
    Mathematical Finance. June 18, 2013
    We propose a rank‐dependent portfolio choice model in continuous time that captures the role in decision making of three emotions: hope, fear, and aspirations. Hope and fear are modeled through an inverse‐S shaped probability weighting function and aspirations through a probabilistic constraint. By employing the recently developed approach of quantile formulation, we solve the portfolio choice problem both thoroughly and analytically. These solutions motivate us to introduce a fear index, a hope index, and a lottery‐likeness index to quantify the impacts of three emotions, respectively, on investment behavior. We find that a sufficiently high level of fear endogenously necessitates portfolio insurance. On the other hand, hope is reflected in the agent's perspective on good states of the world: a higher level of hope causes the agent to include more scenarios under the notion of good states and leads to greater payoffs in sufficiently good states. Finally, an exceedingly high level of aspirations results in the construction of a lottery‐type payoff, indicating that the agent needs to enter into a pure gamble in order to achieve his goal. We also conduct numerical experiments to demonstrate our findings.
    June 18, 2013   doi: 10.1111/mafi.12044   open full text
  • Optimal Selling Rules For Monetary Invariant Criteria: Tracking The Maximum Of A Portfolio With Negative Drift.
    Romuald Elie, Gilles‐Edouard Espinosa.
    Mathematical Finance. June 08, 2013
    Considering a positive portfolio diffusion X with negative drift, we investigate optimal stopping problems of the form infθEfXθsups∈[0,τ]Xs,where f is a nonincreasing function, τ is the next random time where the portfolio X crosses zero and θ is any stopping time smaller than τ. Hereby, our motivation is the obtention of an optimal selling strategy minimizing the relative distance between the liquidation value of the portfolio and its highest possible value before it reaches zero. This paper unifies optimal selling rules observed for the quadratic absolute distance criteria in this stationary framework with bang–bang type ones observed for monetary invariant criteria but in finite horizon. More precisely, we provide a verification result for the general stopping problem of interest and derive the exact solution for two classical criteria f of the literature. For the power utility criterion f:y↦−yλ with λ>0, instantaneous selling is always optimal, which is consistent with previous observations for the Black‐Scholes model in finite observation. On the contrary, for a relative quadratic error criterion, f:y↦(1−y)2, selling is optimal as soon as the process X crosses a specified function φ of its running maximum X*. These results reinforce the idea that optimal stopping problems of similar type lead easily to selling rules of very different nature. Nevertheless, our numerical experiments suggest that the practical optimal selling rule for the relative quadratic error criterion is in fact very close to immediate selling.
    June 08, 2013   doi: 10.1111/mafi.12036   open full text
  • A General Equilibrium Model Of A Multifirm Moral‐Hazard Economy With Financial Markets.
    Jaeyoung Sung, Xuhu Wan.
    Mathematical Finance. June 06, 2013
    We present a general equilibrium model of a moral‐hazard economy with many firms and financial markets, where stocks and bonds are traded. Contrary to the principal‐agent literature, we argue that optimal contracting in an infinite economy is not about a tradeoff between risk sharing and incentives, but it is all about incentives. Even when the economy is finite, optimal contracts do not depend on principals’ risk aversion, but on market prices of risks. We also show that optimal contracting does not require relative performance evaluation, that the second best risk‐free interest rate is lower than that of the first best, and that the second‐best equity premium can be higher or lower than that of the first best. Moral hazard can contribute to the resolution of the risk‐free rate puzzle. Its potential to explain the equity premium puzzle is examined.
    June 06, 2013   doi: 10.1111/mafi.12032   open full text
  • Optimal Liquidation In A Limit Order Book For A Risk‐Averse Investor.
    Arne Løkka.
    Mathematical Finance. June 06, 2013
    In a limit order book model with exponential resilience, general shape function, and an unaffected stock price following the Bachelier model, we consider the problem of optimal liquidation for an investor with constant absolute risk aversion. We show that the problem can be reduced to a two‐dimensional deterministic problem which involves no buy orders. We derive an explicit expression for the value function and the optimal liquidation strategy. The analysis is complicated by the fact that the intervention boundary, which determines the optimal liquidation strategy, is discontinuous if there are levels in the limit order book with relatively little market depth. Despite this complication, the equation for the intervention boundary is fairly simple. We show that the optimal liquidation strategy possesses the natural properties one would expect, and provide an explicit example for the case where the limit order book has a constant shape function.
    June 06, 2013   doi: 10.1111/mafi.12033   open full text
  • Optimal Investment Under Relative Performance Concerns.
    Gilles‐Edouard Espinosa, Nizar Touzi.
    Mathematical Finance. June 06, 2013
    We consider the problem of optimal investment when agents take into account their relative performance by comparison to their peers. Given N interacting agents, we consider the following optimization problem for agent i, 1≤i≤N: supπi∈AiEUixxx(1−λi)XTπi+λixxxXTπi−X¯Ti,πxxxxxx,where Ui is the utility function of agent i, πi his portfolio, Xπi his wealth, X¯i,π the average wealth of his peers, and λi is the parameter of relative interest for agent i. Together with some mild technical conditions, we assume that the portfolio of each agent i is restricted in some subset Ai. We show existence and uniqueness of a Nash equilibrium in the following situations: ‐ unconstrained agents, ‐ constrained agents with exponential utilities and Black–Scholes financial market. We also investigate the limit when the number of agents N goes to infinity. Finally, when the constraints sets are vector spaces, we study the impact of the λis on the risk of the market.
    June 06, 2013   doi: 10.1111/mafi.12034   open full text
  • Option Pricing And Hedging With Small Transaction Costs.
    Jan Kallsen, Johannes Muhle‐Karbe.
    Mathematical Finance. June 06, 2013
    An investor with constant absolute risk aversion trades a risky asset with general Itô‐dynamics, in the presence of small proportional transaction costs. In this setting, we formally derive a leading‐order optimal trading policy and the associated welfare, expressed in terms of the local dynamics of the frictionless optimizer. By applying these results in the presence of a random endowment, we obtain asymptotic formulas for utility indifference prices and hedging strategies in the presence of small transaction costs.
    June 06, 2013   doi: 10.1111/mafi.12035   open full text
  • Markets For Inflation‐Indexed Bonds As Mechanisms For Efficient Monetary Policy.
    Christian‐Oliver Ewald, Johannes Geissler.
    Mathematical Finance. June 06, 2013
    We consider a continuous‐time framework featuring a central bank, private agents, and a financial market. The central bank's objective is to maximize a functional, which measures the classical trade‐off between output and inflation over time plus income from the sales of inflation‐indexed bonds minus payments for the liabilities that the inflation‐indexed bonds produce at maturity. Private agents are assumed to have adaptive expectations. The financial market is modeled in continuous‐time Black–Scholes–Merton style and financial agents are averse against inflation risk, attaching an inflation risk premium to nominal bonds. Following this route, we explain demand for inflation‐indexed securities on the financial market from a no‐arbitrage assumption and derive pricing formulas for inflation‐linked bonds and calls, which lead to a supply‐demand equilibrium. Furthermore, we study the consequences that the sales of inflation‐indexed securities have on the observed inflation rate and price level. Similar to the study of Walsh, we find that the inflationary bias is significantly reduced, and hence that markets for inflation‐indexed bonds provide a mechanism to reduce inflationary bias and increase central bank's credibility.
    June 06, 2013   doi: 10.1111/mafi.12039   open full text
  • Comment On “Skewness‐Aware Asset Allocation”.
    Kwangil Bae.
    Mathematical Finance. June 06, 2013
    This paper discusses risk measures proposed by Low et al. One of their new risk measures is skewness‐aware deviation, which is closely related to constant absolute risk aversion utility functions. This measure captures downside risk more effectively than traditional variance does. The authors also propose a second measure, skewness‐aware variance, which is derived from skewness‐aware deviation. This measure simplifies asset allocation problems and empirical results indicate that it captures risk better than traditional variance. However, this measure is also found to be inconsistent due to factor selection. Additionally, in the aspect of skewness‐aware deviation, optimal portfolios based upon skewness‐aware variance are sometimes less efficient than optimal portfolios that base themselves on traditional variance.
    June 06, 2013   doi: 10.1111/mafi.12040   open full text
  • Optimal Insurance Design Under Rank‐Dependent Expected Utility.
    Carole Bernard, Xuedong He, Jia‐An Yan, Xun Yu Zhou.
    Mathematical Finance. February 18, 2013
    We consider an optimal insurance design problem for an individual whose preferences are dictated by the rank‐dependent expected utility (RDEU) theory with a concave utility function and an inverse‐S shaped probability distortion function. This type of RDEU is known to describe human behavior better than the classical expected utility. By applying the technique of quantile formulation, we solve the problem explicitly. We show that the optimal contract not only insures large losses above a deductible but also insures small losses fully. This is consistent, for instance, with the demand for warranties. Finally, we compare our results, analytically and numerically, both to those in the expected utility framework and to cases in which the distortion function is convex or concave.
    February 18, 2013   doi: 10.1111/mafi.12027   open full text
  • Risk Measures On And Value At Risk With Probability/Loss Function.
    Marco Frittelli, Marco Maggis, Ilaria Peri.
    Mathematical Finance. February 18, 2013
    We propose a generalization of the classical notion of the V@Rλ that takes into account not only the probability of the losses, but the balance between such probability and the amount of the loss. This is obtained by defining a new class of law invariant risk measures based on an appropriate family of acceptance sets. The V@Rλ and other known law invariant risk measures turn out to be special cases of our proposal. We further prove the dual representation of Risk Measures on
    February 18, 2013   doi: 10.1111/mafi.12028   open full text
  • Correlation Under Stress In Normal Variance Mixture Models.
    Michael Kalkbrener, Natalie Packham.
    Mathematical Finance. February 18, 2013
    We investigate correlations of asset returns in stress scenarios where a common risk factor is truncated. Our analysis is performed in the class of normal variance mixture (NVM) models, which encompasses many distributions commonly used in financial modeling. For the special cases of jointly normally and t‐distributed asset returns we derive closed formulas for the correlation under stress. For the NVM distribution, we calculate the asymptotic limit of the correlation under stress, which depends on whether the variables are in the maximum domain of attraction of the Fréchet or Gumbel distribution. It turns out that correlations in heavy‐tailed NVM models are less sensitive to stress than in medium‐ or light‐tailed models. Our analysis sheds light on the suitability of this model class to serve as a quantitative framework for stress testing, and as such provides valuable information for risk and capital management in financial institutions, where NVM models are frequently used for assessing capital adequacy. We also demonstrate how our results can be applied for more prudent stress testing.
    February 18, 2013   doi: 10.1111/mafi.12029   open full text
  • Dual Representations For General Multiple Stopping Problems.
    Christian Bender, John Schoenmakers, Jianing Zhang.
    Mathematical Finance. February 18, 2013
    In this paper, we study the dual representation for generalized multiple stopping problems, hence the pricing problem of general multiple exercise options. We derive a dual representation which allows for cash flows which are subject to volume constraints modeled by integer‐valued adapted processes and refraction periods modeled by stopping times. As such, this extends the works by Schoenmakers (2012), Bender (2011a), Bender (2011b), Aleksandrov and Hambly (2010), and Meinshausen and Hambly (2004) on multiple exercise options, which either take into consideration a refraction period or volume constraints, but not both simultaneously. We also allow more flexible cash flow structures than the additive structure in the above references. For example, some exponential utility problems are covered by our setting. We supplement the theoretical results with an explicit Monte Carlo algorithm for constructing confidence intervals for prices of multiple exercise options and illustrate it with a numerical study on the pricing of a swing option in an electricity market.
    February 18, 2013   doi: 10.1111/mafi.12030   open full text
  • Convex Risk Measures For Good Deal Bounds.
    Takuji Arai, Masaaki Fukasawa.
    Mathematical Finance. February 11, 2013
    We study convex risk measures describing the upper and lower bounds of a good deal bound, which is a subinterval of a no‐arbitrage pricing bound. We call such a convex risk measure a good deal valuation and give a set of equivalent conditions for its existence in terms of market. A good deal valuation is characterized by several equivalent properties and in particular, we see that a convex risk measure is a good deal valuation only if it is given as a risk indifference price. An application to shortfall risk measure is given. In addition, we show that the no‐free‐lunch (NFL) condition is equivalent to the existence of a relevant convex risk measure, which is a good deal valuation. The relevance turns out to be a condition for a good deal valuation to be reasonable. Further, we investigate conditions under which any good deal valuation is relevant.
    February 11, 2013   doi: 10.1111/mafi.12020   open full text
  • Optimal Trade Execution And Price Manipulation In Order Books With Time‐Varying Liquidity.
    Antje Fruth, Torsten Schöneborn, Mikhail Urusov.
    Mathematical Finance. February 11, 2013
    In financial markets, liquidity is not constant over time but exhibits strong seasonal patterns. In this paper, we consider a limit order book model that allows for time‐dependent, deterministic depth and resilience of the book and determine optimal portfolio liquidation strategies. In a first model variant, we propose a trading‐dependent spread that increases when market orders are matched against the order book. In this model, no price manipulation occurs and the optimal strategy is of the wait region/buy region type often encountered in singular control problems. In a second model, we assume that there is no spread in the order book. Under this assumption, we find that price manipulation can occur, depending on the model parameters. Even in the absence of classical price manipulation, there may be transaction triggered price manipulation. In specific cases, we can state the optimal strategy in closed form.
    February 11, 2013   doi: 10.1111/mafi.12022   open full text
  • Multifractional Stochastic Volatility Models.
    Sylvain Corlay, Joachim Lebovits, Jacques Lévy Véhel.
    Mathematical Finance. February 11, 2013
    The aim of this work is to advocate the use of multifractional Brownian motion (mBm) as a relevant model in financial mathematics. mBm is an extension of fractional Brownian motion where the Hurst parameter is allowed to vary in time. This enables the possibility to accommodate for varying local regularity, and to decouple it from long‐range dependence properties. While we believe that mBm is potentially useful in a variety of applications in finance, we focus here on a multifractional stochastic volatility Hull & White model that is an extension of the model studied in Comte and Renault. Using the stochastic calculus with respect to mBm developed in Lebovits and Lévy Véhel, we solve the corresponding stochastic differential equations. Since the solutions are of course not explicit, we take advantage of recently developed numerical techniques, namely functional quantization‐based cubature methods, to get accurate approximations. This allows us to test the behavior of our model (as well as the one in Comte and Renault) with respect to its parameters, and in particular its ability to explain some features of the implied volatility surface. An advantage of our model is that it is able both to fit smiles at different maturities, and to take volatility persistence into account in a more precise way than Comte and Renault.
    February 11, 2013   doi: 10.1111/mafi.12024   open full text
  • On The Consistency Of Regression‐Based Monte Carlo Methods For Pricing Bermudan Options In Case Of Estimated Financial Models.
    Andreas Fromkorth, Michael Kohler.
    Mathematical Finance. February 11, 2013
    In many applications of regression‐based Monte Carlo methods for pricing, American options in discrete time parameters of the underlying financial model have to be estimated from observed data. In this paper suitably defined nonparametric regression‐based Monte Carlo methods are applied to paths of financial models where the parameters converge toward true values of the parameters. For various Black–Scholes, GARCH, and Levy models it is shown that in this case the price estimated from the approximate model converges to the true price.
    February 11, 2013   doi: 10.1111/mafi.12025   open full text
  • Pricing Swaptions Under Multifactor Gaussian Hjm Models.
    João Pedro Vidal Nunes, Pedro Miguel Silva Prazeres.
    Mathematical Finance. February 07, 2013
    Several approximations have been proposed in the literature for the pricing of European‐style swaptions under multifactor term structure models. However, none of them provides an estimate for the inherent approximation error. Until now, only the Edgeworth expansion technique of Collin‐Dufresne and Goldstein is able to characterize the order of the approximation error. Under a multifactor HJM Gaussian framework, this paper proposes a new approximation for European‐style swaptions, which is able to set bounds on the magnitude of the approximation error and is based on the conditioning approach initiated by Curran and Rogers and Shi. All the proposed pricing bounds will arise as a simple by‐product of the Nielsen and Sandmann setup, and will be shown to provide a better accuracy–efficiency trade‐off than all the approximations already proposed in the literature.
    February 07, 2013   doi: 10.1111/mafi.12019   open full text
  • On Optimal Investment For A Behavioral Investor In Multiperiod Incomplete Market Models.
    Laurence Carassus, Miklós Rásonyi.
    Mathematical Finance. February 07, 2013
    We study the optimal investment problem for a behavioral investor in an incomplete discrete‐time multiperiod financial market model. For the first time in the literature, we provide easily verifiable and interpretable conditions for well‐posedness. Under two different sets of assumptions, we also establish the existence of optimal strategies.
    February 07, 2013   doi: 10.1111/mafi.12018   open full text
  • Arbitrage Bounds For Prices Of Weighted Variance Swaps.
    Mark Davis, Jan Obłój, Vimal Raval.
    Mathematical Finance. February 07, 2013
    We develop a theory of robust pricing and hedging of a weighted variance swap given market prices for a finite number of co‐maturing put options. We assume the put option prices do not admit arbitrage and deduce no‐arbitrage bounds on the weighted variance swap along with super‐ and sub‐replicating strategies that enforce them. We find that market quotes for variance swaps are surprisingly close to the model‐free lower bounds we determine. We solve the problem by transforming it into an analogous question for a European option with a convex payoff. The lower bound becomes a problem in semi‐infinite linear programming which we solve in detail. The upper bound is explicit. We work in a model‐independent and probability‐free setup. In particular, we use and extend Föllmer’s pathwise stochastic calculus. Appropriate notions of arbitrage and admissibility are introduced. This allows us to establish the usual hedging relation between the variance swap and the “log contract” and similar connections for weighted variance swaps. Our results take form of a FTAP: we show that the absence of (weak) arbitrage is equivalent to the existence of a classical model which reproduces the observed prices via risk‐neutral expectations of discounted payoffs.
    February 07, 2013   doi: 10.1111/mafi.12021   open full text
  • Time‐Consistent And Market‐Consistent Evaluations.
    Antoon Pelsser, Mitja Stadje.
    Mathematical Finance. February 07, 2013
    We consider evaluation methods for payoffs with an inherent financial risk as encountered for instance for portfolios held by pension funds and insurance companies. Pricing such payoffs in a way consistent to market prices typically involves combining actuarial techniques with methods from mathematical finance. We propose to extend standard actuarial principles by a new market‐consistent evaluation procedure which we call “two‐step market evaluation.” This procedure preserves the structure of standard evaluation techniques and has many other appealing properties. We give a complete axiomatic characterization for two‐step market evaluations. We show further that in a dynamic setting with continuous stock prices every evaluation which is time‐consistent and market‐consistent is a two‐step market evaluation. We also give characterization results and examples in terms of g‐expectations in a Brownian‐Poisson setting.
    February 07, 2013   doi: 10.1111/mafi.12026   open full text
  • Risk Metrics And Fine Tuning Of High‐Frequency Trading Strategies.
    ÁLvaro Cartea, Sebastian Jaimungal.
    Mathematical Finance. February 07, 2013
    We propose risk metrics to assess the performance of high‐frequency (HF) trading strategies that seek to maximize profits from making the realized spread where the holding period is extremely short (fractions of a second, seconds, or at most minutes). The HF trader maximizes expected terminal wealth and is constrained by both capital and the amount of inventory that she can hold at any time. The risk metrics enable the HF trader to fine tune her strategies by trading off different metrics of inventory risk, which also proxy for capital risk, against expected profits. The dynamics of the midprice of the asset are driven by information flows which are impounded in the midprice by market participants who update their quotes in the limit order book. Furthermore, the midprice also exhibits stochastic jumps as a consequence of the arrival of market orders that have an impact on prices which can give rise to market momentum (expected prices to trend up or down). The HF trader’s optimal strategy incorporates a buffer to cover adverse selection costs and manages inventories to maximize the expected gains from market momentum.
    February 07, 2013   doi: 10.1111/mafi.12023   open full text
  • Bilateral Counterparty Risk Under Funding Constraints—Part I: Pricing.
    Stéphane Crépey.
    Mathematical Finance. December 12, 2012
    This and the follow‐up paper deal with the valuation and hedging of bilateral counterparty risk on over‐the‐counter derivatives. Our study is done in a multiple‐curve setup reflecting the various funding constraints (or costs) involved, allowing one to investigate the question of interaction between bilateral counterparty risk and funding. The first task is to define a suitable notion of no arbitrage price in the presence of various funding costs. This is the object of this paper, where we develop an “additive, multiple curve” extension of the classical “multiplicative (discounted), one curve” risk‐neutral pricing approach. We derive the dynamic hedging interpretation of such an “additive risk‐neutral” price, starting by consistency with pricing by replication in the case of a complete market. This is illustrated by a completely solved example building over previous work by Burgard and Kjaer.
    December 12, 2012   doi: 10.1111/mafi.12004   open full text
  • Bilateral Counterparty Risk Under Funding Constraints—Part Ii: Cva.
    Stéphane Crépey.
    Mathematical Finance. December 12, 2012
    The correction in value of an over‐the‐counter derivative contract due to counterparty risk under funding constraints is represented as the value of a dividend‐paying option on the value of the contract clean of counterparty risk and excess funding costs. This representation allows one to analyze the structure of this correction, the so‐called Credit Valuation Adjustment (CVA for short), in terms of replacement cost/benefits, credit cost/benefits, and funding cost/benefits. We develop a reduced‐form backward stochastic differential equations (BSDE) approach to the problem of pricing and hedging the CVA. In the Markov setup, explicit CVA pricing and hedging schemes are formulated in terms of semilinear partial differential equations.
    December 12, 2012   doi: 10.1111/mafi.12005   open full text
  • Game Call Options Revisited.
    S. C. P. Yam, S. P. Yung, W. Zhou.
    Mathematical Finance. November 02, 2012
    In this paper, having been inspired by the work of Kunita and Seko, we study the pricing of δ‐penalty game call options on a stock with a dividend payment. For the perpetual case, our result reveals that the optimal stopping region for the option seller depends crucially on the dividend rate d. More precisely, we show that when the penalty δ is small, there are two critical dividends 0 < d1 < d2 < ∞ such that the optimal stopping region for the option seller takes one of the following forms: (1) an interval if d < d1; (2) a singleton if d∈ [d1, d2]; or (3) an empty set if d > d2. When d∈ [d1, d2], the value function is not continuously differentiable at the optimal stopping boundary for the option seller, therefore our result in the perpetual case cannot be established by the free boundary approach with smooth‐fit conditions imposed on both free boundaries. For the finite time horizon case, the dependence of the optimal stopping region for the option seller on the time to maturity is exhibited; more precisely, when both δ and d are small, we show that there are two critical times 0 < T1 < T2 < T, such that the optimal stopping region for the option seller takes one of the following forms: (1) an interval if t < T1; (2) a singleton if t∈ [T1, T2]; or (3) an empty set if t > T2. In summary, for both the perpetual and the finite horizon cases, we characterize in terms of model parameters how the optimal stopping region for the option seller shrinks when the dividend rate d increases and the time to maturity decreases; these results complete the original work of Emmerling for the perpetual case and Kunita and Seko for the finite maturity case. In addition, for the finite time horizon case, we also extend the probabilistic method for the establishment of existence and regularity results of the classical American option pricing problem to the game option setting. Finally, we characterize the pair of optimal stopping boundaries for both the seller and the buyer as the unique pair of solutions to a couple of integral equations and provide numerical illustrations.
    November 02, 2012   doi: 10.1111/mafi.12000   open full text
  • Admissibility Of Generic Market Models Of Forward Swap Rates.
    Libo Li, Marek Rutkowski.
    Mathematical Finance. November 02, 2012
    Our main goal is to re‐examine and extend certain results from the papers by Galluccio et al. and Pietersz and van Regenmortel. We establish several results providing alternate necessary and sufficient conditions for admissibility of a family of forward swaps, that is, the property that it is supported by a (positive) family of bonds associated with the underlying tenor structure. We also derive the generic expression for the joint dynamics of a family of forward swap rates under a single probability measure and we show that these dynamics are uniquely determined by a selection of volatility processes with respect to the set of driving martingales.
    November 02, 2012   doi: 10.1111/mafi.12001   open full text
  • Boundary Evolution Equations For American Options.
    Daniel Mitchell, Jonathan Goodman, Kumar Muthuraman.
    Mathematical Finance. November 02, 2012
    We consider the problem of finding optimal exercise policies for American options, both under constant and stochastic volatility settings. Rather than work with the usual equations that characterize the price exclusively, we derive and use boundary evolution equations that characterize the evolution of the optimal exercise boundary. Using these boundary evolution equations we show how one can construct very efficient computational methods for pricing American options that avoid common sources of error. First, we detail a methodology for standard static grids and then describe an improvement that defines a grid that evolves dynamically while solving the problem. When integral representations are available, as in the Black–Scholes setting, we also describe a modified integral method that leverages on the representation to solve the boundary evolution equations. Finally we compare runtime and accuracy to other popular numerical methods. The ideas and methodology presented herein can easily be extended to other optimal stopping problems.
    November 02, 2012   doi: 10.1111/mafi.12002   open full text
  • An Online Portfolio Selection Algorithm With Regret Logarithmic In Price Variation.
    Elad Hazan, Satyen Kale.
    Mathematical Finance. November 02, 2012
    We present a novel efficient algorithm for portfolio selection which theoretically attains two desirable properties: 1 Worst‐case guarantee: the algorithm is universal in the sense that it asymptotically performs almost as well as the best constant rebalanced portfolio determined in hindsight from the realized market prices. Furthermore, it attains the tightest known bounds on the regret, or the log‐wealth difference relative to the best constant rebalanced portfolio. We prove that the regret of algorithm is bounded by O(log Q), where Q is the quadratic variation of the stock prices. This is the first improvement upon Cover’s (1991) seminal work that attains a regret bound of O(log T), where T is the number of trading iterations. 2 Average‐case guarantee: in the Geometric Brownian Motion (GBM) model of stock prices, our algorithm attains tighter regret bounds, which are provably impossible in the worst‐case. Hence, when the GBM model is a good approximation of the behavior of market, the new algorithm has an advantage over previous ones, albeit retaining worst‐case guarantees. We derive this algorithm as a special case of a novel and more general method for online convex optimization with exp‐concave loss functions.1
    November 02, 2012   doi: 10.1111/mafi.12006   open full text
  • Time‐Changed Ornstein–Uhlenbeck Processes And Their Applications In Commodity Derivative Models.
    Lingfei Li, Vadim Linetsky.
    Mathematical Finance. November 02, 2012
    This paper studies subordinate Ornstein–Uhlenbeck (OU) processes, i.e., OU diffusions time changed by Lévy subordinators. We construct their sample path decomposition, show that they possess mean‐reverting jumps, study their equivalent measure transformations, and the spectral representation of their transition semigroups in terms of Hermite expansions. As an application, we propose a new class of commodity models with mean‐reverting jumps based on subordinate OU processes. Further time changing by the integral of a Cox–Ingersoll–Ross process plus a deterministic function of time, we induce stochastic volatility and time inhomogeneity, such as seasonality, in the models. We obtain analytical solutions for commodity futures options in terms of Hermite expansions. The models are consistent with the initial futures curve, exhibit Samuelson’s maturity effect, and are flexible enough to capture a variety of implied volatility smile patterns observed in commodities futures options.
    November 02, 2012   doi: 10.1111/mafi.12003   open full text
  • Pricing Derivatives On Multiscale Diffusions: An Eigenfunction Expansion Approach.
    Matthew Lorig.
    Mathematical Finance. November 02, 2012
    Using tools from spectral analysis, singular and regular perturbation theory, we develop a systematic method for analytically computing the approximate price of a large class derivative‐assets. The payoff of the derivative‐assets may be path‐dependent. In addition, the process underlying the derivatives may exhibit killing (i.e., jump to default) as well as combined local/nonlocal stochastic volatility. The nonlocal component of volatility may be multiscale, in the sense that it may be driven by one fast‐varying and one slow‐varying factor. The flexibility of our modeling framework is contrasted by the simplicity of our method. We reduce the derivative pricing problem to that of solving a single eigenvalue equation. Once the eigenvalue equation is solved, the approximate price of a derivative can be calculated formulaically. To illustrate our method, we calculate the approximate price of three derivative‐assets: a vanilla option on a defaultable stock, a path‐dependent option on a nondefaultable stock, and a bond in a short‐rate model.
    November 02, 2012   doi: 10.1111/mafi.12007   open full text
  • Portfolios Of American Options Under General Preferences: Results And Counterexamples.
    Vicky Henderson, Jia Sun, A. Elizabeth Whalley.
    Mathematical Finance. November 02, 2012
    We consider the optimal exercise of a portfolio of American call options in an incomplete market. Options are written on a single underlying asset but may have different characteristics of strikes, maturities, and vesting dates. Our motivation is to model the decision faced by an employee who is granted options periodically on the stock of her company, and who is not permitted to trade this stock. The first part of our study considers the optimal exercise of single options. We prove results under minimal assumptions and give several counterexamples where these assumptions fail—describing the shape and nesting properties of the exercise regions. The second part of the study considers portfolios of options with differing characteristics. The main result is that options with comonotonic strike, maturity, and vesting date should be exercised in order of increasing strike. It is true under weak assumptions on preferences and requires no assumptions on prices. Potentially the exercise ordering result can significantly reduce the complexity of computations in a particular example. This is illustrated by solving the resulting dynamic programming problem in a constant absolute risk aversion utility indifference model.
    November 02, 2012   doi: 10.1111/mafi.12008   open full text
  • Default And Systemic Risk In Equilibrium.
    Agostino Capponi, Martin Larsson.
    Mathematical Finance. November 02, 2012
    We develop a finite horizon continuous time market model, where risk‐averse investors maximize utility from terminal wealth by dynamically investing in a risk‐free money market account, a stock, and a defaultable bond, whose prices are determined via equilibrium. We analyze the endogenous interaction arising between the stock and the defaultable bond via the interplay between equilibrium behavior of investors, risk preferences and cyclicality properties of the default intensity. We find that the equilibrium price of the stock experiences a jump at default, despite that the default event has no causal impact on the underlying economic fundamentals. We characterize the direction of the jump in terms of a relation between investor preferences and the cyclicality properties of the default intensity. We conduct a similar analysis for the market price of risk and for the investor wealth process, and determine how heterogeneity of preferences affects the exposure to default carried by different investors.
    November 02, 2012   doi: 10.1111/mafi.12009   open full text
  • Black–Scholes Representation For Asian Options.
    Jan Vecer.
    Mathematical Finance. November 02, 2012
    Asian options are securities with a payoff that depends on the average of the underlying stock price over a certain time interval. We identify three natural assets that appear in pricing of the Asian options, namely a stock S, a zero coupon bond BT with maturity T, and an abstract asset A (an “average asset”) that pays off a weighted average of the stock price number of units of a dollar at time T. It turns out that each of these assets has its own martingale measure, allowing us to obtain Black–Scholes type formulas for the fixed strike and the floating strike Asian options. The model independent formulas are analogous to the Black–Scholes formula for the plain vanilla options; they are expressed in terms of probabilities under the corresponding martingale measures that the Asian option will end up in the money. Computation of these probabilities is relevant for hedging. In contrast to the plain vanilla options, the probabilities for the Asian options do not admit a simple closed form solution. However, we show that it is possible to obtain the numerical values in the geometric Brownian motion model efficiently, either by solving a partial differential equation numerically, or by computing the Laplace transform. Models with stochastic volatility or pure jump models can be also priced within the Black–Scholes framework for the Asian options.
    November 02, 2012   doi: 10.1111/mafi.12012   open full text
  • General Properties Of Isoelastic Utility Economies.
    Joel M. Vanden.
    Mathematical Finance. November 02, 2012
    This paper studies the class of single‐good Arrow–Debreu economies in which all agents have isoelastic utility functions and homogeneous beliefs, but have possibly different cautiousness parameters and endowments. For each economy in this class, the equilibrium stochastic discount factor is an exponential function of the inverse mapping of a completely monotone function, evaluated at the aggregate consumption. This fact allows for general properties of the class to be studied analytically in terms of known properties of completely monotone functions. For example, conditions are presented under which the agents’ cautiousness parameters and a distribution of initial wealth can be recovered from an equilibrium stochastic discount factor, even if nothing is known about the agents’ endowments. This is a multiagent inverse problem since information about economic primitives is extracted from equilibrium prices. Several example economies are used to illustrate the results.
    November 02, 2012   doi: 10.1111/mafi.12010   open full text
  • The Effect Of Trading Futures On Short Sale Constraints.
    Robert Jarrow, Philip Protter, Sergio Pulido.
    Mathematical Finance. November 02, 2012
    It is commonly believed that the trading of futures on a commodity enables the market to overcome short selling constraints on the spot commodity itself. This belief is embedded in the notion that trading strategies involving futures contracts enable traders to replicate the payoffs as if they were short the spot commodity. The purpose of this paper is to investigate this common belief in a general arbitrage‐free semimartingale financial model with trading in futures and a short selling prohibition on the spot commodity. We show via various examples that, in general, this common belief is incorrect. Furthermore, we provide a set of sufficient conditions, albeit very restrictive, under which the common belief is true.
    November 02, 2012   doi: 10.1111/mafi.12013   open full text
  • Swaption Pricing In Affine And Other Models.
    Don H. Kim.
    Mathematical Finance. November 02, 2012
    This paper shows that Singleton and Umantsev’s method for swaption pricing in affine models can be simplified and extended to other models. Two alternative methods for approximating the option exercise boundary are introduced: one based on the multivariate Taylor series expansion, and the other based on duration‐matched zero‐coupon bond approximation. Applied to affine models and quadratic‐Gaussian models, these methods are found to give accurate swaption prices.
    November 02, 2012   doi: 10.1111/mafi.12014   open full text
  • From Smile Asymptotics To Market Risk Measures.
    Ronnie Sircar, Stephan Sturm.
    Mathematical Finance. November 02, 2012
    The left tail of the implied volatility skew, coming from quotes on out‐of‐the‐money put options, can be thought to reflect the market’s assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In particular, we make use of indifference pricing by dynamic convex risk measures, which are given as solutions of backward stochastic differential equations, to establish a link between these two approaches to risk measurement. We derive a characterization of the implied volatility in terms of the solution of a nonlinear partial differential equation and provide a small time‐to‐maturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parameterized class, distorted entropic dynamic risk measures, which we introduce here, such that the asymptotic volatility skew under indifference pricing can be matched with the market skew. We demonstrate this in a calibration exercise to market implied volatility data.
    November 02, 2012   doi: 10.1111/mafi.12015   open full text
  • Closed Form Pricing Formulas For Discretely Sampled Generalized Variance Swaps.
    Wendong Zheng, Yue Kuen Kwok.
    Mathematical Finance. November 02, 2012
    Most of the existing pricing models of variance derivative products assume continuous sampling of the realized variance processes, though actual contractual specifications compute the realized variance based on sampling at discrete times. We present a general analytic approach for pricing discretely sampled generalized variance swaps under the stochastic volatility models with simultaneous jumps in the asset price and variance processes. The resulting pricing formula of the gamma swap is in closed form while those of the corridor variance swaps and conditional variance swaps take the form of one‐dimensional Fourier integrals. We also verify through analytic calculations the convergence of the asymptotic limit of the pricing formulas of the discretely sampled generalized variance swaps under vanishing sampling interval to the analytic pricing formulas of the continuously sampled counterparts. The proposed methodology can be applied to any affine model and other higher moments swaps as well. We examine the exposure to convexity (volatility of variance) and skew (correlation between the equity returns and variance process) of these discretely sampled generalized variance swaps. We explore the impact on the fair strike prices of these exotic variance swaps with respect to different sets of parameter values, like varying sampling frequencies, jump intensity, and width of the monitoring corridor.
    November 02, 2012   doi: 10.1111/mafi.12016   open full text
  • Static Fund Separation Of Long‐Term Investments.
    Paolo Guasoni, Scott Robertson.
    Mathematical Finance. November 02, 2012
    This paper proves a class of static fund separation theorems, valid for investors with a long horizon and constant relative risk aversion, and with stochastic investment opportunities. An optimal portfolio decomposes as a constant mix of a few preference‐free funds, which are common to all investors. The weight in each fund is a constant that may depend on an investor’s risk aversion, but not on the state variable, which changes over time. Vice versa, the composition of each fund may depend on the state, but not on the risk aversion, since a fund appears in the portfolios of different investors. We prove these results for two classes of models with a single state variable, and several assets with constant correlations with the state. In the linear class, the state is an Ornstein–Uhlenbeck process, risk premia are affine in the state, while volatilities and the interest rate are constant. In the square root class, the state follows a square root diffusion, expected returns and the interest rate are affine in the state, while volatilities are linear in the square root of the state.
    November 02, 2012   doi: 10.1111/mafi.12017   open full text
  • Large Portfolio Asymptotics For Loss From Default.
    Kay Giesecke, Konstantinos Spiliopoulos, Richard B. Sowers, Justin A. Sirignano.
    Mathematical Finance. October 31, 2012
    We prove a law of large numbers for the loss from default and use it for approximating the distribution of the loss from default in large, potentially heterogeneous portfolios. The density of the limiting measure is shown to solve a nonlinear stochastic partial differential equation, and certain moments of the limiting measure are shown to satisfy an infinite system of stochastic differential equations. The solution to this system leads to the distribution of the limiting portfolio loss, which we propose as an approximation to the loss distribution for a large portfolio. Numerical tests illustrate the accuracy of the approximation, and highlight its computational advantages over a direct Monte Carlo simulation of the original stochastic system.
    October 31, 2012   doi: 10.1111/mafi.12011   open full text
  • Optimal Consumption And Investment For A Large Investor: An Intensity‐Based Control Framework.
    Michael Busch, Ralf Korn, Frank Thomas Seifried.
    Mathematical Finance. June 19, 2012
    We introduce a new stochastic control framework where in addition to controlling the local coefficients of a jump‐diffusion process, it is also possible to control the intensity of switching from one state of the environment to the other. Building upon this framework, we develop a large investor model for optimal consumption and investment that generalizes the regime‐switching approach of Bäuerle and Rieder (2004).
    June 19, 2012   doi: 10.1111/j.1467-9965.2012.00528.x   open full text
  • Arbitrage‐Free Multifactor Term Structure Models: A Theory Based On Stochastic Control.
    Andrea Gombani, Wolfgang J. Runggaldier.
    Mathematical Finance. June 19, 2012
    We present an alternative approach to the pricing of bonds and bond derivatives in a multivariate factor model for the term structure of interest rates that is based on the solution of an optimal stochastic control problem. It can also be seen as an alternative to the classical approach of computing forward prices by forward measures and as such can be extended to other situations where traditionally a change of measure is involved based on a change of numeraire. We finally provide explicit formulas for the computation of bond options in a bivariate linear‐quadratic factor model.
    June 19, 2012   doi: 10.1111/j.1467-9965.2012.00527.x   open full text
  • No‐Arbitrage Pricing Under Systemic Risk: Accounting For Cross‐Ownership.
    Tom Fischer.
    Mathematical Finance. June 19, 2012
    We generalize Merton’s asset valuation approach to systems of multiple financial firms where cross‐ownership of equities and liabilities is present. The liabilities, which may include debts and derivatives, can be of differing seniority. We derive equations for the prices of equities and recovery claims under no‐arbitrage. An existence result and a uniqueness result are proven. Examples and an algorithm for the simultaneous calculation of all no‐arbitrage prices are provided. A result on capital structure irrelevance for groups of firms regarding externally held claims is discussed, as well as financial leverage and systemic risk caused by cross‐ownership.
    June 19, 2012   doi: 10.1111/j.1467-9965.2012.00526.x   open full text
  • Dynamic Portfolio Optimization With A Defaultable Security And Regime‐Switching.
    Agostino Capponi, José E. Figueroa‐López.
    Mathematical Finance. June 19, 2012
    We consider a portfolio optimization problem in a defaultable market with finitely‐many economical regimes, where the investor can dynamically allocate her wealth among a defaultable bond, a stock, and a money market account. The market coefficients are assumed to depend on the market regime in place, which is modeled by a finite state continuous time Markov process. By separating the utility maximization problem into a predefault and postdefault component, we deduce two coupled Hamilton–Jacobi–Bellman equations for the post‐ and predefault optimal value functions, and show a novel verification theorem for their solutions. We obtain explicit constructions of value functions and investment strategies for investors with logarithmic and Constant Relative Risk Aversion utilities, and provide a precise characterization of the directionality of the bond investment strategies in terms of corporate returns, forward rates, and expected recovery at default. We illustrate the dependence of the optimal strategies on time, losses given default, and risk aversion level of the investor through a detailed economic and numerical analysis.
    June 19, 2012   doi: 10.1111/j.1467-9965.2012.00522.x   open full text
  • The Two Fundamental Theorems Of Asset Pricing For A Class Of Continuous‐Time Financial Markets.
    Andrew Lyasoff.
    Mathematical Finance. June 19, 2012
    The paper is concerned with the first and the second fundamental theorems of asset pricing in the case of nonexploding financial markets, in which the excess‐returns from risky securities represent continuous semimartingales with absolutely continuous predictable characteristics. For such markets, the notions of “arbitrage” and “completeness” are characterized as properties of the distribution law of the excess‐returns. It is shown that any form of arbitrage is tantamount to guaranteed arbitrage, which leads to a somewhat stronger version of the first fundamental theorem. New proofs of the first and the second fundamental theorems, which rely exclusively on methods from stochastic analysis, are established.
    June 19, 2012   doi: 10.1111/j.1467-9965.2012.00530.x   open full text
  • Rethinking Dynamic Capital Structure Models With Roll‐Over Debt.
    Jean‐Paul Décamps, Stéphane Villeneuve.
    Mathematical Finance. June 19, 2012
    Dynamic capital structure models with roll‐over debt rely on widely accepted arguments that have never been formalized. This paper clarifies the literature and provides a rigorous formulation of the equity holders’ decision problem within a game theory framework. We spell out the linkage between default policies in a rational expectations equilibrium and optimal stopping theory. We prove that there exists a unique equilibrium in constant barrier strategies, which coincides with that derived in the literature. Furthermore, that equilibrium is the unique equilibrium when the firm loses all its value at default time. Whether the result holds when there is a recovery at default remains a conjecture.
    June 19, 2012   doi: 10.1111/j.1467-9965.2012.00532.x   open full text
  • Pricing And Semimartingale Representations Of Vulnerable Contingent Claims In Regime‐Switching Markets.
    Agostino Capponi, José E. Figueroa‐López, Jeffrey Nisen.
    Mathematical Finance. June 19, 2012
    Using a suitable change of probability measure, we obtain a Poisson series representation for the arbitrage‐free price process of vulnerable contingent claims in a regime‐switching market driven by an underlying continuous‐time Markov process. As a result of this representation, along with a short‐time asymptotic expansion of the claim’s price process, we develop an efficient novel method for pricing claims whose payoffs may depend on the full path of the underlying Markov chain. The proposed approach is applied to price not only simple European claims such as defaultable bonds, but also a new type of path‐dependent claims that we term self‐decomposable, as well as the important class of vulnerable call and put options on a stock. We provide a detailed error analysis and illustrate the accuracy and computational complexity of our method on several market traded instruments, such as defaultable bond prices, barrier options, and vulnerable call options. Using again our Poisson series representation, we show differentiability in time of the predefault price function of European vulnerable claims, which enables us to rigorously deduce Feynman‐Kač representations for the predefault pricing function and new semimartingale representations for the price process of the vulnerable claim under both risk‐neutral and objective probability measures.
    June 19, 2012   doi: 10.1111/j.1467-9965.2012.00533.x   open full text
  • The Affine Libor Models.
    Martin Keller‐Ressel, Antonis Papapantoleon, Josef Teichmann.
    Mathematical Finance. June 14, 2012
    We provide a general and flexible approach to LIBOR modeling based on the class of affine factor processes. Our approach respects the basic economic requirement that LIBOR rates are nonnegative, and the basic requirement from mathematical finance that LIBOR rates are analytically tractable martingales with respect to their own forward measure. Additionally, and most importantly, our approach also leads to analytically tractable expressions of multi‐LIBOR payoffs. This approach unifies therefore the advantages of well‐known forward price models with those of classical LIBOR rate models. Several examples are added and prototypical volatility smiles are shown. We believe that the CIR process‐based LIBOR model might be of particular interest for applications, since closed form valuation formulas for caps and swaptions are derived.
    June 14, 2012   doi: 10.1111/j.1467-9965.2012.00531.x   open full text
  • On The Lower Arbitrage Bound Of American Contingent Claims.
    Beatrice Acciaio, Gregor Svindland.
    Mathematical Finance. June 14, 2012
    We prove that in a discrete‐time market model the lower arbitrage bound of an American contingent claim is itself an arbitrage‐free price if and only if it corresponds to the price of the claim optimally exercised under some equivalent martingale measure.
    June 14, 2012   doi: 10.1111/j.1467-9965.2012.00519.x   open full text
  • Arbitrage‐Free Bilateral Counterparty Risk Valuation Under Collateralization And Application To Credit Default Swaps.
    Damiano Brigo, Agostino Capponi, Andrea Pallavicini.
    Mathematical Finance. June 14, 2012
    We develop an arbitrage‐free valuation framework for bilateral counterparty risk, where collateral is included with possible rehypothecation. We show that the adjustment is given by the sum of two option payoff terms, where each term depends on the netted exposure, i.e., the difference between the on‐default exposure and the predefault collateral account. We then specialize our analysis to credit default swaps (CDS) as underlying portfolios, and construct a numerical scheme to evaluate the adjustment under a doubly stochastic default framework. In particular, we show that for CDS contracts a perfect collateralization cannot be achieved, even under continuous collateralization, if the reference entity’s and counterparty’s default times are dependent. The impact of rehypothecation, collateral margining frequency, and default correlation‐induced contagion is illustrated with numerical examples.
    June 14, 2012   doi: 10.1111/j.1467-9965.2012.00520.x   open full text
  • A Method For Pricing American Options Using Semi‐Infinite Linear Programming.
    Sören Christensen.
    Mathematical Finance. June 14, 2012
    We introduce a new approach for the numerical pricing of American options. The main idea is to choose a finite number of suitable excessive functions (randomly) and to find the smallest majorant of the gain function in the span of these functions. The resulting problem is a linear semi‐infinite programming problem, that can be solved using standard algorithms. This leads to good upper bounds for the original problem. For our algorithms no discretization of space and time and no simulation is necessary. Furthermore it is applicable even for high‐dimensional problems. The algorithm provides an approximation of the value not only for one starting point, but for the complete value function on the continuation set, so that the optimal exercise region and, for example, the Greeks can be calculated. We apply the algorithm to (one‐ and) multidimensional diffusions and show it to be fast and accurate.
    June 14, 2012   doi: 10.1111/j.1467-9965.2012.00523.x   open full text
  • Dynamic Coherent Acceptability Indices And Their Applications To Finance.
    Tomasz R. Bielecki, Igor Cialenco, Zhao Zhang.
    Mathematical Finance. June 14, 2012
    In this paper, we present a theoretical framework for studying coherent acceptability indices (CAIs) in a dynamic setup. We study dynamic CAIs (DCAIs) and dynamic coherent risk measures (DCRMs), and we establish a duality between them. We derive a representation theorem for DCRMs in terms of a so‐called dynamically consistent sequence of sets of probability measures. Based on these results, we give a specific construction of DCAIs. We also provide examples of DCAIs, both abstract and also some that generalize selected classical financial measures of portfolio performance.
    June 14, 2012   doi: 10.1111/j.1467-9965.2012.00524.x   open full text
  • Limit Theorems For Partial Hedging Under Transaction Costs.
    Yan Dolinsky.
    Mathematical Finance. June 14, 2012
    We study shortfall risk minimization for American options with path‐dependent payoffs under proportional transaction costs in the Black–Scholes (BS) model. We show that for this case the shortfall risk is a limit of similar terms in an appropriate sequence of binomial models. We also prove that in the continuous time BS model, for a given initial capital, there exists a portfolio strategy which minimizes the shortfall risk. In the absence of transactions costs (complete markets) similar limit theorems were obtained by Dolinsky and Kifer for game options. In the presence of transaction costs the markets are no longer complete and additional machinery is required. Shortfall risk minimization for American options under transaction costs was not studied before.
    June 14, 2012   doi: 10.1111/j.1467-9965.2012.00525.x   open full text
  • Liquidation In Limit Order Books With Controlled Intensity.
    Erhan Bayraktar, Michael Ludkovski.
    Mathematical Finance. June 14, 2012
    We consider a framework for solving optimal liquidation problems in limit order books. In particular, order arrivals are modeled as a point process whose intensity depends on the liquidation price. We set up a stochastic control problem in which the goal is to maximize the expected revenue from liquidating the entire position held. We solve this optimal liquidation problem for power‐law and exponential‐decay order book models explicitly and discuss several extensions. We also consider the continuous selling (or fluid) limit when the trading units are ever smaller and the intensity is ever larger. This limit provides an analytical approximation to the value function and the optimal solution. Using techniques from viscosity solutions we show that the discrete state problem and its optimal solution converge to the corresponding quantities in the continuous selling limit uniformly on compacts.
    June 14, 2012   doi: 10.1111/j.1467-9965.2012.00529.x   open full text
  • Transform Analysis For Point Processes And Applications In Credit Risk.
    Kay Giesecke, Shilin Zhu.
    Mathematical Finance. February 29, 2012
    This paper develops a formula for a transform of a vector point process with totally inaccessible arrivals. The transform is expressed in terms of a Laplace transform under an equivalent probability measure of the point process compensator. The Laplace transform of the compensator can be calculated explicitly for a wide range of model specifications, because it is analogous to the value of a simple security. The transform formula extends the computational tractability offered by extant security pricing models to a point process and its applications, which include valuation and risk management problems arising in single‐name and portfolio credit risk.
    February 29, 2012   doi: 10.1111/j.1467-9965.2011.00512.x   open full text
  • Pricing Chained Options With Curved Barriers.
    Doobae Jun, Hyejin Ku.
    Mathematical Finance. February 13, 2012
    This paper studies barrier options which are chained together, each with payoff contingent on curved barriers. When the underlying asset price hits a primary curved barrier, a secondary barrier option is given to a primary barrier option holder. Then if the asset price hits another curved barrier, a third barrier option is given, and so on. We provide explicit price formulas for these options when two or more barrier options with exponential barriers are chained together. We then extend the results to the options with general curved barriers.
    February 13, 2012   doi: 10.1111/j.1467-9965.2011.00513.x   open full text
  • Running For The Exit: Distressed Selling And Endogenous Correlation In Financial Markets.
    Rama Cont, Lakshithe Wagalath.
    Mathematical Finance. February 13, 2012
    We propose a simple multiperiod model of price impact from trading in a market with multiple assets, which illustrates how feedback effects due to distressed selling and short selling lead to endogenous correlations between asset classes. We show that distressed selling by investors exiting a fund and short selling of the fund’s positions by traders may have nonnegligible impact on the realized correlations between returns of assets held by the fund. These feedback effects may lead to positive realized correlations between fundamentally uncorrelated assets, as well as an increase in correlations across all asset classes and in the fund’s volatility which is exacerbated in scenarios in which the fund undergoes large losses. By studying the diffusion limit of our discrete time model, we obtain analytical expressions for the realized covariance and show that the realized covariance may be decomposed as the sum of a fundamental covariance and a liquidity‐dependent “excess” covariance. Finally, we examine the impact of these feedback effects on the volatility of other funds. Our results provide insight into the nature of spikes in correlation associated with the failure or liquidation of large funds.
    February 13, 2012   doi: 10.1111/j.1467-9965.2011.00510.x   open full text
  • Rating Based Lévy Libor Model.
    Ernst Eberlein, Zorana Grbac.
    Mathematical Finance. February 03, 2012
    In this paper, we consider modeling of credit risk within the Libor market models. We extend the classical definition of the default‐free forward Libor rate and develop the rating based Libor market model to cover defaultable bonds with credit ratings. As driving processes for the dynamics of the default‐free and the predefault term structure of Libor rates, time‐inhomogeneous Lévy processes are used. Credit migration is modeled by a conditional Markov chain, whose properties are preserved under different forward Libor measures. Conditions for absence of arbitrage in the model are derived and valuation formulae for some common credit derivatives in this setup are presented.
    February 03, 2012   doi: 10.1111/j.1467-9965.2011.00514.x   open full text
  • Mean–Variance Portfolio Optimization With State‐Dependent Risk Aversion.
    Tomas Björk, Agatha Murgoci, Xun Yu Zhou.
    Mathematical Finance. February 03, 2012
    The objective of this paper is to study the mean–variance portfolio optimization in continuous time. Since this problem is time inconsistent we attack it by placing the problem within a game theoretic framework and look for subgame perfect Nash equilibrium strategies. This particular problem has already been studied in Basak and Chabakauri where the authors assumed a constant risk aversion parameter. This assumption leads to an equilibrium control where the dollar amount invested in the risky asset is independent of current wealth, and we argue that this result is unrealistic from an economic point of view. In order to have a more realistic model we instead study the case when the risk aversion depends dynamically on current wealth. This is a substantially more complicated problem than the one with constant risk aversion but, using the general theory of time‐inconsistent control developed in Björk and Murgoci, we provide a fairly detailed analysis on the general case. In particular, when the risk aversion is inversely proportional to wealth, we provide an analytical solution where the equilibrium dollar amount invested in the risky asset is proportional to current wealth. The equilibrium for this model thus appears more reasonable than the one for the model with constant risk aversion.
    February 03, 2012   doi: 10.1111/j.1467-9965.2011.00515.x   open full text
  • Optimal Liquidation Of Derivative Portfolios.
    Vicky Henderson, David Hobson.
    Mathematical Finance. May 13, 2011
    We consider the problem facing a risk‐averse agent who seeks to liquidate or exercise a portfolio of (infinitely divisible) perpetual American‐style options on a single underlying asset. The optimal liquidation strategy is of threshold form and can be characterized explicitly as the solution of a calculus of variations problem. Apart from a possible initial exercise of a tranche of options, the optimal behavior involves liquidating the portfolio in infinitesimal amounts, but at times which are singular with respect to calendar time. We consider a number of illustrative examples involving CRRA and CARA utility, stocks, and portfolios of options with different strikes, and a model where the act of exercising has an impact on the underlying asset price.
    May 13, 2011   doi: 10.1111/j.1467-9965.2011.00477.x   open full text