The Acturial Pricing Of Product Option

Option pricing theory is one of the main problems in the research of modern financial mathematics.It is also widely used in the risk control of modern financial securities market.With the rapid development of economy,the financial market is also developing.The traditional option pricing method has gradually failed to keep up with the pace of the times,and its disadvantages are becoming more and more obvious.The classical option pricing model assumes that the market is a complete and ideal market without arbitrage and equilibrium,but this situation does not exist in the real world.Some scholars try to apply the actuarial method to the study of option pricing and transform option pricing into a fair premium problem.The basic idea is that risk-free assets are discounted at the risk-free interest rate and risk assets are discounted at the expected rate of return.This method does not make any requirements for the financial market.It is applicable to a wide range of financial markets.In the 1970 s,Black and Scholes firstly proposed the B-S model of option pricing.However,some assumptions in the B-S model can not fully reflect some phenomena in the actual market.So many scholars extended the B-S model.For example,the drift rate and volatility are extended as functions of time,the constant interest rate is extended as random interest rate.These extensions make the option pricing model more suitable to the actual market.Product option is an option that takes the product of two asset prices or the product of stock index as the underlying assets.Product option is widely used in practical transactions,such as foreign equity currency option and corporate income option.Geometric Asian option takes the geometric average value of stock price within the term as the settlement price of the option,which reduces the impact of price fluctuation.It can avoid price manipulation near the end of the period.This paper studies the pricing problem of product option using actuarial method,establishes the pricing model of geometric Asian product option by combining geometric Asian option and product option,and further obtains the actuarial pricing formula of Asian product option.The main contents are as follows:Firstly,suppose that the underlying asset of the product option follows the exponential O-U process,the interest rate follows the Hull-White model,the drift rate ?(t)and volatility(t)are both deterministic functions about time t,and the strike price K is a constant.The acturial pricing of the product option is given,and the pricing formula of the product option is derived by the theory of stochastic analysis.Secondly,on the basis of the above model,assuming that the strike price is uncertain,the acturial pricing formula of product option is derived by the theory of stochastic analysis and the method of measure transformation,respectively.Finally,geometric Asian option is firstly applied to product option,and the pricing model of geometric Asian product option is established.Assuming that the asset price follows Geometric Fractional Brownian motion,the interest rate follows Hull-White interest rate model and the strike price is uncertain,the defintion of acturial pricing of geometric Asian product is given and its actuarial pricing formular is derived.