On the stability of call/put option's prices in incomplete models under statistical estimations

Abstract

In exponential semi-martingale setting for risky asset we estimate the difference of prices of options when initial physical measure $P$ and corresponding martingale measure $Q$ change to $\tilde{P}$ and $\tilde{Q}$ respectively. Then, we estimate $L_1$-distance of option's prices for corresponding parametric models with known and estimated parameters. The results are applied to exponential Levy models with special choice of martingale measure as Esscher measure, minimal entropy measure and $f^q$-minimal martingale measure. We illustrate our results by considering GMY and CGMY models.