Bounds on option prices for semimartingale market models

Authors: Alexander A. Gushchin and Ernesto Mordecki.

Abstract: We propose a methodology for the determination of the range of option prices of a European option in a general semimartingale market model, with a convex payoff function. Prices are obtained as expectations along 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 the discounted stock prices (adapted from the theory of statistical experiments), which allows us to find extremal distributions and, correspondingly, upper and lower bounds for the range of option prices. 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 (the most natural for this problem) terms of predictable characteristics of the stochastic logarithm of the discounted stock price process. Particular attention is given to two examples: discrete time and diffusion with jumps market models.

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