Abstract:

The price fluctuation of liquid financial assets is usually modeled as a stochastic process which satisfies some form of the "efficient markets hypothesis". Such assumptions can be made precise in terms of martingale measures. We discuss the role of these martingale measures in analyzing financial derivatives such as options, viewed as non-linear functionals of the underlying stochastic process.

Uniqueness of the martingale measure provides the mathematical key to a perfect "hedge" of a financial derivative by means of a dynamic trading strategy in the underlying assets, and in particular to pricing formulas of Black-Scholes type. But for realistic models the martingale measure is no longer unique, and intrinsic risks appear on the level of derivatives. We discuss various mathematical approaches to the problem of pricing and hedging in such a setting.