Vulnerable Options Pricing Under Uncertain Volatility Model And The Double Heston Model

In recent years,China’s financial market has made great progress.With more and more financial derivatives entering China’s financial market,the transaction scale is gradually expanding at the same time,the over-the-counter market(OTC)appears.When options are traded in OTC markets,as for there is no unified and centralized trading venue and trading system in these markets,so it is impossible to supervise the short positions at the expiration of options,which will cause option holders to bear both market risks and credit risks.Vulnerable option is a kind of financial derivative,which has the character of credit risk,and it has aroused the research interest of many scholars at home and abroad.Initially,most scholars’ research on the pricing of vulnerable options is based on the assumption that volatility is constant,but this does not conform to the actual situation.Therefore,a large number of scholars have studied the pricing problem of fragile options under different volatility models.The uncertain volatility model and the double Heston stochastic volatility model have their own advantages:the general stochastic volatility model has more state variables,but the uncertain volatility model just overcomes this shortcoming;The double Heston stochastic volatility model has more flexibility in simulating the term structure of volatility than Heston stochastic volatility model,and can better fit the option price empirically.Therefore,this paper will study the pricing problem of vulnerable options under the two models.When we study the pricing problem of vulnerable options under the uncertain volatility model,assuming that the volatility of the underlying asset is within a small range,the volatility of the counterparty asset is certain.By applying additional conditions to the boundary conditions,the Black-Scholes-Barenblatt equation was decomposed into two Black-Scholes-like equations,and the approximate solution of the completely nonlinear partial differential equation satisfying the vulnerable option price under the uncertain volatility model was obtained.When we study the pricing problem of vulnerable options under the double Heston stochastic volatility model,we assume that the underlying assets and counterparty assets obey the double Heston stochastic volatility model,and assume that the liabilities of the company are random.Based on this model,we derive the pricing formula of vulnerable options under special circumstances.This paper mainly includes the following two parts:the first part is to study the pricing of vulnerable options under the uncertain volatility model;The second part studies the pricing problem of vulnerable options under the double Heston stochastic volatility model.