Stochastic Models for Asset Pricing

Abstract

Stochastic calculus has been applied to the problems of pricing financial derivatives since 1973 when Black and Scholes published their famous paper ”The pricing of options and corporate liabilities” in the journal of political economy. In this work, we introduce basic concepts of probability theory which gives a better understanding in the study of stochastic processes, such as Markov process, Martingale and Brownian motion. We then construct the Itˆo’s integral under stochastic calculus and it was used to study stochastic differential equations. The lognormal model was used to model asset prices showing its usefulness in financial mathematics. Finally, we show how the famous Black-Scholes model for option pricing was obtained from the lognormal asset model.