Stochastic methods for finance and stock markets

Since about 1900, stochastic methods have been used for modeling economic processes and for analyzing random phenomena in finance and stock markets. Recently, main emphasis is laid on price building processes, the calculation of premiums in high risk markets or insurance companies, the evaluation of capital assets and investments, and in option pricing. These latter issues have been largely fostered by the theory of Merton, Black, and Scholes which has finally been rewarded by a Nobel price. A large number of mathematically oriented monographs or practical introductions to the subject have been published between 1996 and 2000.

The current course presents, after a brief introduction into the terminology and basic facts of finance markets, the mathematical theory of martingales, a special type of stochastic processes which is the main tool for option pricing, both in the discrete and continuous time case. Basic theorems for arbitrage-free markets are derived which yield, e.g., the classical pricing formula of Black-Scholes. In continuous time, the Brown-Wiener processes are used for modeling the logarithmic evolution of stock prices, the corresponding analysis deals with stochastic integral and differential equations.

The following list of topics shows how these concepts and theorems are introduced and derived, respectively:

1. Stock markets, futures, derivatives, options etc.
2. Conditional distributions and conditional expectation of random variables
3. Time-discrete martingales
4. Models for finance markets with discrete time, investment strategies, arbitrage
5. Complete markets and option pricing
6. The market model of Cox, Ross und Rubinstein
7. Time-continuous stochastic processes, Ito integral, Ito formula
8. Black-Scholes model and corresponding option pricing formula.

Literature: See website http://www.stochastik.rwth-aachen.de/bock/english/courses/finance_lit.html

Excerpt:

• BAXTER, M., RENNIE, A.: Financial calculus. Cambridge Univ. Press, 1997.
• BIAIS, B. et al. (eds.): Financial mathematics. Springer, Berlin, 1997.
• DEMPSTER, M.A.H., PLISKA, S.R.: Mathematics of derivative securities. Cambridge Univ. Press, 1997.
• IRLE, A.: Finanzmathematik - Die Bewertung von Derivaten. Teubner, Stuttgart, 1998.
• LAMBERTON, D., LAPEYRE, B.: Introduction to stochastic calculus applied to finance. Chapman and Hall, London, 1996.
• PLISKA, S.R.: Introduction to mathematical finance. Discrete time models. Blackwell, Malden (Mass.), 1998.
• SANDMANN, K.: Einfuehrung in die Stochastik der Finanzmaerkte. Springer, Berlin, 1999.
• SHAW, W.: Modelling mathematical derivatives with MATHEMATICA. Cambridge Univ. Press, 1997.

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