Stochastic Calculus and Its Impact On Analyzing Option Pricing

Abstract

This research study the behaviour of the correction of Black-Scholes portfolios based on historical stock price data. We build a model that simulates one sample path of the stock price stochastic process at discrete time steps and track the correction over a time interval as it relates to the change in stock price over time. We also study the effect that relaxing this assumption has on the self-financing property of the replicating portfolio. We show that as the frequency of the time steps increases, the correction is more likely to be close to 0. We also show that the majority of historical stock return series that studied have caused the replicated portfolio to have a positive correction. We conclude that the Black-Scholes model can be used to find the no-arbitrage rational price for an option, a financial instrument that derives its value from the value of an underlying asset.