Stability of Stochastic Pricing Models under Volatility Fluctuations

The standard theory of the stochastic models used to value financial derivatives contracts involves models whose input parameters are deterministic functions and often constants. Because of the random nature of the changes in the market prices of the financial instruments, the coefficients of these models are inevitably susceptible to random perturbation from their initial estimates. In this paper we will investigate the behavior of some of the most widely used models when small changes are applied to their volatility component. Starting with the Black-Scholes model for the price of a European call option, we will continue our analysis of the traditional models for pricing American options, Asian options, Barrier options, as well as some of the models for the short term rate of interest. In addition to obtaining convergence results for all of the models, we will examine a method of controlling the deviations in the volatility parameter of the Black-Scholes model and the resulting estimate can be used for further extensions on the topic. Moreover, we will present an example of how to calculate probabilities of unlikely events, using a technique called importance sampling. We will concentrate only on the case of discrete random variables but the same algorithm can be applied to estimate the probabilities of the deviations of the pricing functions of the models under question. The latter is left for future research.