Option pricing under stochastic interest rate

Abstract

This study proposes a newly generated model to price options under a stochastic interest rate environment. The developed model introduces a numerical procedure with the arbitrage-free condition by working on a binomial tree. The model handles price changes in stocks and interest rates. Principally, it is assumed that the movements in the stock prices are defined by the Cox, Ross, Rubinstein (CRR) model. The CRR model proposes a numerical method to price options by assuming the interest rate is constant throughout the option's life. Moreover, the thesis claims that interest rates vary based on the Black, Derman, Toy (BDT) model. The BDT model defines the evolution of interest rates in the future. It presents a numerical procedure by using the binomial tree. Crucially, the interest rate is log-normally distributed in the BDT model; hence, the short rate cannot take a negative value. Also, the BDT model assumes that it has a mean-reverting property which means that the interest rate shows a tendency to converge to the average of interest rates in the long term. Additionally, this study utilizes the CRR and BDT model in order to derive a new option valuation framework. Also, the proposed model gives a numerical solution rather than an analytical formula due to the BDT model's structure. This thesis focuses on pricing the European options that can expire only on the maturity date. Furthermore, a group of options with different strike prices and different maturities is valued according to the developed model and the CRR model to observe interest rate impacts under two parameters: strike price and time-to-maturity. Finally, the estimated prices by both models are compared with actual-market prices to determine the accuracies of the models. Then, it is detected that the effect of the stochastic interest rate behavior on which maturities is significant.