Research On The Minimum Or Maximum Option Pricing Under Several Fractional Brownian Motion Models With Dividends

The minimum or maximum option is also called extreme value option,which is determined by the changes of the price of multiple original assets.The holders of the minimum or maximum options have the right to obtain the best return among multiple original assets on the expiry date,which allows holders to obtain the greatest benefits among multiple assets,so it is of great significance to study the issue of the minimum or maximum option pricing.Existing works have found that the changing process of stock price always shows with a sharp peak and thick tail distribution.And the past stock prices will also have an effect on the future stock prices.Fractional Brownian motion has self-similarity and long-term relevance,which can better describe the changes of stock prices in the actual financial market.In recent years,increasing attention has been paid to option pricing under the fractional Brownian motion model.This article considers the continuous payment of dividends and studies the pricing of the minimum or maximum option under several types of fractional Brownian motion models.The main results are obtained as follows:(1)The minimum or maximum option pricing is discussed when the price of the underlying asset(stock)with dividends is subject to the fractional Brownian motion model under the conditions that all parameters are positive constants.Assuming that the stock price obeys the stochastic differential equation driven by the fractional Brownian motion,we use the risk-neutral pricing method to obtain the maximum and minimum under the fractional Brownian motion with dividents call and put option pricing,formulas and parity formulas.Matlab is used to analyze the fractional Brownian motion model with dividends,the standard Brownian motion model and the fractional Brownian motion model without dividends under the minimum or maximum option pricing.It follows that if the impact of stock dividends on option prices is ignored,then the minimum or maximum option pricing will be underestimated;(2)The minimum or maximum option pricing under the fractional Brownian motion model with dividends is studied when the parameters are all time deterministic functions.Assuming that the stock price obeys the stochastic differential equation driven by the variable coefficients fractional Brownian motion.We use the martingale method to obtain the minimum or maximum option pricing formula.Then we use Matlab to analyze the influence of each parameter under the fractional Brownian motion model with variable coefficients on the minimum or maximum option pricing,we obtain that the risk-free interest rate and bonus rate have a linear relationship with the minimum or maximum option pricing and the volatility has a nonlinear relationship with the minimum or maximum option pricing;(3)The minimum or maximum option pricing under the Markov modulated fractional Brownian motion model with dividends is studied.Considering that the economic states are not fixed,which changes over time.The Markov chain is used to describe the economic cycle of the market.We use the principle of risk-neutral pricing to obtain the call and put formulas of the minimum or maximum option under the Markov modulated fractional Brownian motion model.Matlab is used to analyze the impact of the standard Brownian motion model,the fractional Brownian motion model and the Markov modulated fractional Brownian motion model on the minimum or maximum option pricing.It follows that if we ignore the state changes in the actual economic market has changed,the option price will be overestimate seriously.