Analysis of Option Pricing Models Using Time Series and Numerical Methods

Abstract

Option Pricing is a popular topic in practical finance and in academic. To research about option pricing, the random activity of underlying assets is studied in the first place and an ARIMA model (Autoregressive Integrated Moving Average Model), which could be used to forecast the future trend of an underlying asset, is researched and implemented. There are many ways to determine the price of an option, among which the Black-Scholes formula is the most influential one. However, there are many circumstances that the Black-Scholes formula does not have an analytic solution. Regarding this situation, three numerical methods, including Binary Tree, Monte Carlo simulation and Finite Difference are implemented and compared. The strengths and limitations of all three methods are discussed. Furthermore, the classical Black-Scholes model has extensions to incorporate the changes of implied volatility with respect to maturity and strike price. Further research in volatility theory was carried out in the Final Year Project. Particularly, Bruno Dupire's volatility model is studied and the volatility is treated as a function of the current underlying asset price and time. The pricing model was implemented under C++ with low degree of error.