Numerical Methods For Option Pricing Under Contagion Risk Models

The default of one company in the financial market will have direct impacts on the asset value of other related companies.This has been shown clearly by the bankrupt of Lehman Brothers led to a sharp falls in stock prices of other investment banks during the global financial crisis of 2007–2008.Because the conventional dependence modeling of assets using covariance matrix can not capture sudden market co-movements,the contagion risk has become an important topic for domestic and foreign scholars.Option pricing is the core of the option contract design.However,classical option pricing models do not consider the impact of contagion risk on option price.On the other hand,there is usually no precise analytic pricing formula for the options under the contagion risk model,so it is necessary to exploit appropriate numerical methods to study the option pricing problem in this model.Therefore,this thesis discusses the pricing of European options under the framework of contagion risk model,and mainly focuses on the numerical methods and convergence problems involved in the option pricing.The mainly work of this thesis includes the following aspects:(1)For the research of option pricing with counterparty risk,we primary consider the European option pricing under dynamic elasticity of variance model with counterparty risk.We derive the price formula of the European call option at the initial moment under this model based on the risk-neutral pricing theory and the probability density function of the counterparty’s default time.Two kinds of option value partial differential equations and integral term included in the option price formula are solved by implicit difference method and trapezoidal formula,respectively.Since the boundary error will be propagated in the computational grid of the partial differential equation when calculating the connected partial differential equation and the calculation error of the partial differential equation will accumulate due to the integral term in the option price formula,we provides a rigorous theoretical analysis for the convergence of these numerical schemes.Some numerical examples are carried out to confirm the theoretical results,illustrate the effective of the numerical methods,and discuss the influence of model parameters on the option price and give the corresponding economic explanation.(2)In order to study the risk infection mechanism between risk assets,the counterparty is introduced into the financial market to establish the looping contagion risk model and transform it into risk neutral measure through measure transformation.Then we apply the risk-neutral pricing theory to derive the partial differential equations of European option under the condition that defaultable stock occurs or does not occur in the life of this option and assign the appropriate boundary conditions for these partial differential equations according to the economic implication of the model,and further combined with the probability density function of default time for defaultable stock to obtain the risk neutral price formula of European options at the initial moment.Because the two types of partial differential equations for European call option with assets subject to looping contagion risk are two-dimensional,we proposed an alternating direction implicit difference scheme to discretize these partial differential equations.In addition,the Monte Carlo method and the trapezoidal formula are used to calculate the probability density function of the default time and the integral term in the option price formula,respectively,to obtain the approximate value of option price.Numerical results show that these alternating direction implicit difference schemes have first-order convergence rates in time and second-order convergence rates in space.(3)To discuss the impact of contagion risk on option pricing during financial crisis,we first proposed a new option pricing model which the underlying asset exposed to counterparty risk and its price dynamics is coupled to a post-crash market index.Secondly,we derive the risk-neutral pricing formula of European call option at the initial time under this stock price model due to the risk neutral pricing theory and the probability density function of the counterparty default time.Because this price formula involves not only the calculation of integrals but also the numerical solution of partial differential equations,the trapezoidal formula and crank-nicolson difference scheme are developed to obtain the option price,respectively.Moreover,we also strictly proved that these numerical scheme have second-order convergence rates both in time and space.Finally,we also present some numerical examples to verify the convergence results and analyze the influence of model parameters on option price and illustrate its economic meaning.(4)In term of option pricing with multi-defaults contagion risk,we first develop the conditional density approach of default to derive the analytical pricing formulas for European call option and put option at the initial time under the GBM model with multi-defaults risk.Then the GBM model with multidefaults risk is extended to the CEV model with multi-defaults risk,and the risk-neutral price formulas of European option under this stock price model at the initial time are derived by using partial differential equation method.Moreover,we respectively adopt implicit difference method and Crank-Nicolson difference method to solve the initial-boundary value problem included in this option price formula,and then employ the trapezoidal formula calculate the integral term in the option price formula to obtain the approximate value of the option price.Besides,some numerical examples are given to compare the accuracy of the two difference methods and illustrate the necessity of introducing the default of the underlying assets into the market model by analyze the sensitivity of default intensity to option price.In summary,we use the finite difference method systematically investigate the European option pricing problem under the framework of contagion risk model from the perspective of partial differential equation,and deeply analyze the convergence of the finite difference method in option pricing.The time and space convergence rates of the implicit difference method and CrankNicolson difference method are rigorously obtained.However,the convergence of the alternating direction implicit difference method is only demonstrated by numerical examples.