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

TL;DR

This paper investigates how option prices in incomplete markets are affected by changes in statistical estimations of underlying models, providing bounds on price differences under various measure changes.

Contribution

It offers new estimates for option price differences in incomplete models when model parameters and measures are estimated from data, with applications to Levy models.

Findings

Derived bounds for option price differences under measure changes.

Applied results to GMY and CGMY Levy models.

Analyzed effects of parameter estimation on option prices.

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.