Option Price Decomposition Formula Under Stochastic Volatility

In the financial market,in order to avoid the risk of trading as much as possible,a kind of financial derivative product-option has been produced.With the wide use of options,option pricing has attracted more and more attention of scholars.As an important subject with a long history in the field of financial mathematics,there are many classical models of option pricing,among which the Black-Scholes-Merton option-pricing model is the most classical one.However,the volatility of stock price is assumed to be constant in this model,while the actual volatility of the subject matter is underestimated.Therefore,there is a significant deviation between the calculated option price and the actual market price.For the option of an underlying asset,they have the same expiration date but different strike prices.The more the strike price deviates from the spot price,the greater the implied volatility will be,and the "volatility smile" existed.Therefore,some scholars studied the volatility problem and found that the stochastic volatility was more consistent with the actual situation through empirical research.In order to obtain more accurate option prices,many scholars began to study the impact of volatility changes on option prices,and put forward some improved models,among which the continuous stochastic volatility model is more widely used.For example,Elias Stein and Jeremy Stein(1991)proposed the Stein-Stein model by assuming that volatility conforms to the mathematical Ornstein-Uhlenbeck process.Steven Heston(1993)assumed that the square root of volatility conforms to a stochastic process and proposed the Heston model.Alòs has been studying the stochastic volatility model and working on obtaining a new option price decomposition formula for a long time.In this regard,she used the classical Ito formula to expand prices around the Black-Scholes-Merton formula and described the option prices as a classical Black-Scholes-Merton formula plus a correlation term and a term due to the volatility of the volatility.Thus,she obtained the option price decomposition formula in the Heston volatility framework.Referring to the method of Alòs,this paper deduces the option price decomposition formula of Stein-Stein model,and adds European call option into the calculation.Therefore,we obtain a more specific European call option price decomposition formula in the Stein-Stein model.