Research On Reset Option Pricing Under Markov Modelat Ed Fractional Brownian Motion Model

Reset option is a kind of weak path dependent exotic option which is first proposed by Gray and Whaley in 1997.Its strike price can be reset or adjusted in the process of using.This feature provides more profit opportunities for investors.Therefore,it is of great practical significance to study reset option.Previous studies have found that stock price has a certain dependence on historical price,fractional Brownian motion has the characteristics of self similarity and longterm dependence,which can describe the change of stock price;double fractional Brownian motion can describe the special situation without independent increment and stationary increment.Therefore,the risk asset model driven by fractional Brownian motion and double fractional Brownian motion is closer to the real financial market.In recent years,the researches on option pricing under fractional Brownian motion model and double fractional Brownian motion model have become one of the hot issues in option pricing.Considering the influence of business cycle on stock price,we adopt continuous time Markov chain to describe different states in different business cycles,and extend the existing fractional(double fractional)Brownian motion model to Markov modulated fractional(double fractional)Brownian motion model,which makes the models more practical.In this paper,we only study the reset option with a single reset time.The main results are obtained as follows:(1)The reset option pricing is studied reset option in Markov modulated fractional Brownian motion environment.It is assumed that the expected rate of return,risk-free interest rate and volatility are affected by the market economy states of,and the economic states in the business cycle is described by continuous time Markov chain.By the actuarial method,the call and put pricing formula of the reset option in Markov modulated fractional Brownian motion environment is obtained.Then numerical simulations are carried out by matlab software.It highlights that the reset option has more investment opportunities and more profit opportunities than other options,and analyzes the impact of continuous Markov chain on the option price;(2)The pricing problem of reset option under Markov modulated double fractional Brownian motion model is studied.It is assumed that the expected rate of return,riskfree interest rate and volatility are affected by the states of market economy,and the economic states in the business cycle is described by continuous time Markov chain.The corresponding mathematical model of financial market is established.By actuarial pricing method,the call and put pricing formulas of reset option in the Markov modulated double fractional Brownian motion we are numerically simulations are carried out by matlab software.The necessity of introducing continuous time Markov chain is verified,and the effects of reset option pricing model in Markov modulated fractional Brownian motion environment and Markov modulated double fractional Brownian motion environment on option price are compared and analyzed.