Valuation On The Compound Power Options And Their Applications In Double Heston Jump-diffusion Model

Option is a type of contract that gives the holder the right to buy or sell an asset at the agreed price within the agreed time or period.Options are widely used in the risk management of assets in financial field,and are favored by investors because of their rich variety and flexible function.The establishment of the Chicago Board of Options Exchange in 1973 marked the beginning of the truly organized trading era of options.It has been only a few decades,but there are more than a thousand options which include many exotic options that are used to meet some special needs of markets and investors.For example,compound option is an option on an option,and it can avoid the risk of market volatility.Power option has strong leverage because of its return depend on the power of the underlying stock price at maturity.And before applying options,how to measure risk and balance the relationship between investment and income cannot be ignored.So the option pricing problem arises.Complex option and market model make much trouble in option pricing;for instance,it's hard to obtain analytic solution of pricing formula.But it is also more meaningful to real financial market.In this paper,we propose a new type of option——compound power option,that is,the compound option with power option as the target option,and study whether it has both advantages of high leverage and risk management.For options pricing research,how to improve the fitting result between model and real financial market has been one of the topical and important subjects in finance.The classical Black-Scholes model that was first proposed has many limitations.For instance,it cannot depict the volatility clustering and time variation of financial time series data,and cannot capture market emergencies and so on.However,social emergencies could impact financial markets and market interest rates are not static in fact.In order to explain the market phenomena more accurately,scholars unremit-tingly refine the Black-Scholes model.Stochastic volatility model and jump-diffusion model are two main types of refined models.Stochastic volatility model considers the correlation between underlying assets and assets volatility,and describes volatility change process with stochastic pro-cess.But the model cannot explain the jump risk that caused by extreme market.Jump-diffusion model can explain the abnormal jump of asset price,but it neglects the price fluctuation and in-terest rate change.In addition,single stochastic volatility model generally has fitting defects that cannot fit the thick tail of asset income distribution and the persistence of volatility at the same time,while multi-factor stochastic volatility model can significantly improve the flexibility of as-set income edge distribution.Double Heston Jump-diffusion model gathers the advantages of all three which ensure the compound power option pricing formula fitter to real market.This paper study compound power option pricing under the double Heston Jump-diffusion model and derive the pricing formula by using Feynman-Kac theorem,Ito formula,characteristic function of multi-dimensional random variable and Fourier inverse transform method,etc.Then analyzing the price difference of compound power option under five different models and the influence of mean recovery rate,fluctuation variance and correlation coefficient on option price by numerical examples.The conclusion shows that compound power option has higher income than the standard compound option.It combines the advantages of both options and performs better risk control ability,leverage and investment demands.This result shows that different options with complementary advantages have potential for combination,which helps to enrich the options types.On the other hand,the data demonstrates that the double Heston Jump-diffusion model fits the market characteristics better and is superior in options pricing.Then,under the same model,the paper use multi-dimensional Fourier inverse transformation,mathematical induction method,distribution function and joint eigenfunction of high-dimensional random variable to obtain multi-stage compound power option pricing formula and derive ex-tendible power option and multi-period extendible power option pricing formulas by compound power option pricing thought.Numerical analysis of the pricing formulas shows that multi-stage compound power option has stronger risk control ability,and the extendible power option also shows good risk control ability and leverage.The paper's conclusion not only enriches derivatives types and the theoretical system of compound options pricing,but also provides ideas and methods to combine options.Compound power options have been proved to have certain market value,and it supports financial derivatives market for more fresh blood.