Vulnerable Option Pricing Under CEV Jump-diffusion Process

The over-the-counter options trading market has developed rapidly due to its diverse varieties and flexible forms.However,when the shorts are unable or unwilling to fulfill the contract,the longs are extremely vulnerable to suffer from default risk.Therefore,evaluating the value of options with credit risk has both theoretical and practical significance.It is constant variance elasticity model(CEV model)that is a good alternative to the Black-Scholes option pricing model,which explains the"smile"phenomenon of implied volatility at a certain level.The jump diffusion model can well describe the jump behavior of financial assets price due to the stimulation of newly arrived information.This paper studies the pricing of vulnerable options based on the CEV jump-diffusion process.The main results are as follows:(1)Research on the vulnerable option pricing problem based on the CEV jump-diffusion process.On the basis that the price of the underlying asset and the value of the counterparty’s asset obeyed the geometric Brownian motion,an elasticity factor is introduced to correct the volatility of the two,and the jump process is decomposed into two parts:a common jump and an individual jump.Applying the principle of no arbitrage to construct a risk-free investment portfolio,a mathematical model satisfied by the value of vulnerable option is obtained,which is a backward integral-differential equation.We use variable substitution method and Taylor expansion technique to convert integral-differential equations into partial differential equations firstly;then the finite difference method is used to transform the partial differential equation model into an algebraic equation,and the numerical algorithm is designed;finally,the numerical experiments are carried out on European and American vulnerable put options.The results show that the put option value is negatively correlated with the price of the underlying asset,and positively correlated with the value of the counterparty’s asset,the strike price and the expiry date;the value of American vulnerable put option is not lower than that of European vulnerable put option;the impact of the jump-related parameters of the price of the underlying asset is significantly greater than the impact of the jump-related parameters of the value of the counterparty’s asset.(2)Research on the vulnerable option pricing problem that paying transaction costs based on the CEV jump-diffusion process.Firstly,combined with Leland’s thoughts,the vulnerable option pricing problem that paying a fixed proportion of transaction costs is studied,and the partial differential equation model of option pricing and its difference format are given.Secondly,transaction costs may change with the change of the underlying asset transaction shares in the actual transaction process.This article assumes that the transaction costs ratio was h(v _t)(28)a-b v_t(29)0.Hence,on this basis,a vulnerable option pricing model with monotonic proportional transaction payment under the CEV jump diffusion process is established.Then we discretize the model and construct an implicit difference scheme.At the same time,the second-order Gear formula and the linear interpolation are introduced to extend the boundary area.Results of the numerical experiment show that the value of European vulnerable put option is negatively correlated with the fixed transaction costs ratio,the monotonic transaction costs parameter a,and the company’s liabilities,and positively correlated with the monotonic transaction cost parameter b.What’s more,the results of European vulnerable put option pricing based on the Klein model,the CEV model,the CEV model with transaction costs,the CEV jump-diffusion model,the CEV jump-diffusion model with transaction costs are respectively studied.Through comparative analysis,we can see that the model constructed in this paper is improved the accuracy and flexibility of options pricing effectively.The paper has 14 pictures,5 tables,and 82 references.