Valuations On Two Types Of Reset Options In A Fractional Brownian Motion

Since 1973,American researchers Blake,F.and Shores,M.and S.Scholes established the mathematical model of option pricing,namely - option pricing formula,and the theory of option pricing has developed rapidly,therefore,option pricing has become one of the core problems in financial mathematics and financial engineering.After a lot of data analysis,the researchers concluded that the stock price is long-term dependent,which is different from Brownian motion,which shows that the stock price does not obey the Brownian motion completely.And the stock price appears a ”peak fat tail” distribution,and the stock price fluctuation is not random walk,but has a long-term dependence in different periods,therefore,brownian motion can not describe the law of stock price change very well.When 0.5 < < 1(H is the Exponent),the fractional Brownian motion has long dependence,this property makes the fractional Brownian motion better simulate the market in the study of option pricing,and makes the result more reasonable and more realistic.However,with the rapid development of the financial market,the general traditional options can not meet the needs of financial companies and investors for financial market returns,and financial institutions continue to introduce new financial derivative products,therefore the new type option also called the exotic or the special option appears subsequently.There are many types of exotic options,such as Asian,barrier,look-back,forward effect,reset option,and so on.Most exotic options are traded outside the financial market.Compared with the standard option,the singular option is very sensitive to some parameters,which makes its value judgment more complicated.Since the introduction of exotic options,many scholars have made remarkable achievements in the study of these options.In the complex and changeable financial market and the economic environment of fast and slow development,the short board of the single exotic option gradually appears,in order to better meet the needs of investors and enrich the financial market,financial institutions have introduced a kind of ”combined” exotic option,that is,two or more exotic options are combined together.Such as Asian reset options,so that the option has the characteristics of Asian options,reset options.In this paper,the fractional Brownian motion model is used to study the ”combination” option,I.E.Two kinds of Asian time-point reset options.The first is the geometric mean Asian time-point reset option pricing under the fractional Brownian motion model.Based on fractional risk pricing theory and actuarial method,the geometric mean Asian time-point reset call option pricing formula with fixed strike price under fractional Brownian motion is derived,and its parameter sensitivity is analyzed,through numerical calculation,the influence of model parameters on option price is analyzed.In the second class,we discuss the forward-effective asian-time reset option pricing of the geometric average of the floating strike price under the fractional Brownian motion model.For Two different time zones,the explicit solution of the geometric mean forward-effective Asian time-reset call option with floating strike price under fractional Brownian motion is derived by using the stochastic analysis theory and actuarial theory,the influence of model parameters on option price is obtained by numerical analysis.