Discrete-Time Approximation of Functionals in Models of Ornstein–Uhlenbeck Type, with Applications to Finance

Abstract

The paper provides benchmark approximation techniques for handling in models in continuous-time functional relationships in variables that are sampled discretely over time. The methods are demonstrated for value-functional-type of expectations in models of Ornstein–Uhlenbeck type. Based on Laguerre reduction series, a three-step program for these functionals is shown to result in both a continuous and a discrete time setting. The program is illustrated in the options case and in models based on GIG-distributions, yielding novel series representations for calibration when variance is discretely-sampled in particular. By numerical examples it is shown how the series enable computation accuracies of some 3 decimal places, for example, with just a single digit number of terms; for this the paper considers discretely-sampled situations with dimensions of up to 4 digits, and even in these dimensions significant discrepancies with the continuously-sampled values are found to persist.