The present study examined delay discounting of hypothetical monetary rewards over a wide range of amounts (from $20 to $10 million) in order to determine how reward amount affects the parameters of the hyperboloid discounting function and to compare fits of the hyperboloid model with fits of two discounting models used in neuroeconomics: the quasi‐hyperbolic and the double‐exponential. Of the three models assessed, the hyperboloid provided the best fit to the delay discounting data. The present delay discounting results may be compared to those of a previous study on probability discounting (Myerson, Green, & Morris, 2011) that used the same extended range of amounts. The hyperboloid function accurately described both types of discounting, but reward amount had opposite effects on the degree of discounting. Importantly, the amount of delayed reward affected the rate parameter of the hyperboloid discounting function but not its exponent, whereas the opposite was true for the amount of probabilistic reward. The finding that the exponent of the hyperboloid discounting function remains relatively constant across a wide range of delayed amounts provides strong support for a psychophysical scaling interpretation, and stands in stark contrast to the finding that the exponent of the hyperboloid function increases with the amount of probabilistic reward. Taken together, these findings argue that delay and probability discounting involve fundamentally different decision‐making mechanisms.