Brownian Motion and Stochastic Calculus

This chapter is about stochastic calculus, i.e., calculus that involves random variables and Brownian motions in particular. The original Brownian motion refers to the trajectory of pollen moving around in a dish of water. The trajectory of such a particle is very random in the sense that its future position is not deterministic, moreover the particle makes countless zig-zags even over an infinitesimally small time interval. This description turns out to fit stock prices very well, as stock price movements also tend to be rather unpredictable over a short horizon. Brownian motion plays a crucial role in the derivation of option prices. This appendix provides the reader with some elementary tools in probability crucial to understanding option pricing studies. Here, we aim to provide an intuitive discussion leaving it to the more mathematically inclined readers to refer to more advanced texts in mathematical finance.1 There are also probability related books that the interested reader may wish to consult such as Williams (1991). For an alternative soft introduction to the probabilistic tools presented here, we may consultNeftci (1996).2