Bounds on Option Prices for Semimartingale Market Models

We propose a methodology for determining the range of option prices of a European option with a convex payoff function in a general semimartingale market model. Prices are obtained as expectations with respect to the set of equivalent martingale measures. Since the set of prices is an interval on the real line, two main questions are considered: (i) how to find upper and lower estimates for the range of prices, and (ii) how to establish the attainability of these estimates. To solve the first question, we introduce a partial ordering in the set of distributions of discounted stock prices (adapted from the theory of statistical experiments), which allows us to find extremal distributions and, accordingly, the upper and lower bounds for the range of option prices. The weak convergence of probability measures is used to answer the second question, whether the bounds obtained at the first step are exact. Exploiting stochastic calculus, we give answers to both questions in terms (the most natural for this problem) of predictable characteristics of the stochastic logarithm of a discounted stock price process. Special attention is given to two examples: a discrete-time and a diffusion-with-jumps market models.