Vulnerable Option Pricing Under Stochastic Interest Rate And Stochastic Volatility Model With Markov Regime-switching

As an important financial derivative,option is widely used in hedging and risk management,which plays a positive role in enhancing the risk resistance capacity and maintaining the stability of financial market.Options are generally traded on the exchanges or the over-the-counter market.But different with the exchanges,there is no special agency to urge the option's seller to assume the obligation in the over-the-counter market,so that the option's holder is always vulnerable to counterparty default risk.And the risk of over-the-counter trading also increases because of the lack of a unified trading system and supervision.In recent years,with the advance of economic globalization and financial innovation,the scale of the over-the-counter market has shown explosive growth.Therefore,under the current market environment,it is important to study the options with credit default risk,which is beneficial to the investment decision of investors and the stability of the financial market.Options exposed to the default risk are called vulnerable options.It is one of the core contents in the field of financial mathematics and financial engineering to construct a reasonable market model and derive an analytical pricing formula of vulnerable options.In the classic Black-Scholes option pricing model,it is assumed that both the volatility and the interest rate are constants.However,a large number of empirical studies show that this assumption is not consistent with the real financial market.Although the Heston stochastic volatility model has some advantages compared with Black-Scholes model,it still can not describe the structural changes of the economic state,and the Markov regime-switching model can overcome this question.Therefore,if we combine the Markov regime switching with the Heston stochastic volatility and the stochastic interest rate model,the hybrid model will be more suitable to the real financial market,and the research under this model will be more reasonable and more valuable.In this paper,we will study the pricing of European sensitive option and sensitive extreme option under the hybrid model.In this paper,we first provide the Markov regime-switching model with stochastic volatility and stochastic interest rate,where the prices of underlying asset and the option writer's asset are subject to Heston stochastic volatility model,and the mean-reversion level of volatility satisfies a continuous-time Markov process with finite-state space,and dynamics of interest rate is modelled by a liner combination of a stochastic volatility and a non-negative constant.Then,by using the Feynman-Kac theorem,the technique of partial differential equation and the law of iterated expectation,we can conduct the discounted joint characteristic function,and derive the closedform pricing formula for vulnerable options and vulnerable extreme options based on the Esscher transform and the Fourier inversion transform technique.Finally,we provide some numerical experiments to implement sensitivity analysis and examine the implication of regime-switching on option price and Delta values using the FFT algorithm and the Monte Carlo simulation.The Markov regime-switching model with stochastic volatility and stochastic interest rate is established in order to close the financial market and get the more reasonable option prices.For simplicity,we consider the Markov chain with two states in our numerical experiments,and the results shows that the option prices under our model are more accurate the the Heston model.So,when investors trade in the over-the-counter market,they should avoid the risk of credit default and also pay attention to the structural changes of the economics state.And the regulators should also consider the impact of the change of economic states on the market when making relevant policies.