Modern financial mathematics has its origins in the seminal papers by Black and Scholes (1973) and by Merton (1973), where the Itô's formula has been used for deriving the Black-Scholes equation. Harrison and Kreps (1979) and Harrison and Pliska (1981) have further showed that a natural mathematical framework for the analyse of financial markets is martingale theory and stochastic analysis. Since then this framework has played a dominating role in financial mathematics. In this note, I will give a brief review on the martingale approach to option pricing and illustrate this approach through several examples: pricing foreign currency option, exchange option, lookback options and Asian options.
1 Work supported in part by City University of Hong Kong, contract 9000906, and by National Science Foundation of China, grant 79790130.