In 1900, a young French mathematician, named Louis Bachelier has had the idea of using the Brownian motion (a motion discovered by British biologist Robert Brown) to modelize the asset prices of French stock market in his thesis, titled "Theory of Speculation". His idea has been strongly criticized by eminent scientific people due to the fact that the Brownian motion may assume negative values though an asset price can not! 73 years later Fischer Black and Myron Scholes have used almost the same idea, where the sole difference was to take the exponential of the Brownian motion and they have received the Nobel Prize of Economy with Robert Merton. Since then their model has been extended and generalized in all the directions. Today all the delicate activities in the stock markets of the world are realized thanks to these models.

A. S. Üstünel

References

L. Bachelier: "Théorie de la Spéculation", Gauthier-Villars (1900).

F. Black and M. Scholes: The Pricing of Options and Corporate Liabilities". Journal of Political Economy 81(3) (1973), 637-654.

J. P. Fouque, G. Papanicolaou, K. R. Sircar: "Derivatives in Financial Markets with Stochastic Volatility". Cambridge University Press, 2000.

M. Harrison and D. Kreps: "Martingales and Arbitrage in Multiperiod Security Markets". Journal of Economic Theory, 20, (1979), 381408.

J. C. Hull: "Options, Futures and Other Derivative Securities", Seventh Edition, (2008), Prentice Hall.

K. Ito: "On a Stochastic Integral Equation". Proc. Imperial Academy Tokyo, 22, (1946), 3235.

P. Protter: "Stochastic Integration and Differential Equations: A New Approach", (1990), Springer-Verlag.

P. Samuelson: "Rational Theory of Warrant Pricing". Industrial Management Review, 6, 2, (1965), 1331.

P. J. Schonbucher: "Credit Derivative Pricing Models", (2003), Wiley.

S. E. Shreve: "Stochastic Calculus for Finance I: The Binomial Asset Pricing Model", (2004), Springer.

S. E. Shreve: "Stochastic Calculus for Finance II: Continuous-Time Models", (2004), Springer.

J. M. Steele: "Stochastic Calculus and Financial Applications", (2001), Springer.

A. S. Üstünel: "Introduction to Financial Mathematics". Bilkent Lecture Notes, 2015.̈

A. S. Üstünel: "Introduction to Analysis on Wiener Space". Lecture Notes in Math., Vol. 1516. Springer, 1995.