Option Pricing Model Under Stochastic Interest Rate

Option pricing theory is one of core problems in financial mathematics. In1973, Blackand Scholes put forward the famous Black-Scholes option pricing model and also obtained itspricing formula based on the assumption that the stock price followed the geometricBrownian motion. However, many scholars agreed that the fractional Brownian motion ismore adapted to the financial market since the fractional Brownian motion has the long-termdependence and fat tail.And the interest rate is not constant, because interest is random.In this dissertation, we assume that the stock price obeys the stochastic differentialequation driven by fractional Brownian motion. The financial market model, with stochasticinterest rate, is built in fractional Brownian motion environment, we discuss the pricingproblem of Binary option and Gap option. The thesis includes five chapters.In chapter one, we introduce the history and current research of option pricing, the basisof selected topic and the main content in this dissertation.In chapter two, we introduce the definition and characteristic of fractional Brownianmotion, we also introduce the insurance actuary method for European option pricing.In chapter three, we assume that the stock price obeys the stochastic differential equationdriven by fractional Brownian motion, and interest rate satisfies the Vasicek model driven byfractional Brownian motion, The finacial market mathmatical model is built. By means of theinsurance actuarial method and the stochastic analysis theory of fractional Brownian motion,we obtain the pricing formula for the Gap option.In chapter four, in the finacial market mathmatical model with random interest rate, inthe fractional Brownian motion environment, by means of the insurance actuarial method andthe stochastic analysis theory of fractional Brownian motion, we discuss the Binary optionpricing problem and obtain the pricing formula for the Binary option.In chapter five, we summarize the main results in this dissertation and point out someissues which need further improvement.