Pricing Vulnerable Options Under Lévy Process

Options are an important financial derivative tool in the financial market,which can better meet the needs of investors in hedging,risk aversion and investment.In recent years,under the context of the growing global OTC market,the scale of China’s OTC options market has gradually expanded.Since OTC options market has no corresponding regulatory authority,more flexible trading features make OTC options holders often face the credit risk of counterparties.Many scholars call options with credit risk vulnerable options.Aiming at the pricing problem of vulnerable option,how to construct the pricing model scientifically and reasonably is an important research topic in financial mathematics.Based on the existing research results,this paper describes the credit risks of counterparties through structured model,establishes two kinds of vulnerable option pricing models in line with market performance.For a start,considering the uncertainty of market interest rate and the fluctuation of asset price,this paper establishes a vulnerable option pricing model with stochastic interest rate under Lévy process.In this model,the Vasicek interest rate model is used to represent the risk-free interest rate of the market,and the compensated Poisson random measure is used to describe the jumping activity in the model.For the model proposed,the martingale method is used to derive the option pricing formula.Since the model proposed in this paper is considered in the incomplete financial market,it is known from the fundamental theorem of asset pricing that the equivalent martingale measure is not unique.Therefore,this paper adopts the Esscher transform to determine an optimal equivalent martingale measure,and then deduce the pricing formula.Finally,numerical experiments are used to compare the vulnerable option value with BS model,Klein model and the Merton model,and the influence of interest rate parameters and jump parameters on option prices is analyzed,and these numerical experiments demonstrate the correctness of the model and calculation.The next,comprehensively consider the correlation between asset prices under different market conditions and the jump risk of asset prices,this paper establishes a European vulnerable option pricing model with Markov switching under the Lévy process.This model is similar to the previous model,except that the risk-free rate of the market,the rate of return and volatility of risky assets depend on the unobservable market economic state governed by a continuous-time Markov chain.Using the same solution as the previous study,the option price formula of the model is derived in a structured framework,the advanced nature of the model is demonstrated by some special cases,and finally the two-state transition is simulated by numerical experiments.In brief,this paper discusses the option pricing under Lévy process,establishes two hard vulnerable option pricing model,and the pricing formulas of vulnerable European options are derived.In addition,since the market under the Lévy process-driven option pricing model is incomplete,this paper uses the Esscher transform to determine an optimal equivalent martingale measure.These results are not only useful in theory but also significant in practice for pricing option in the financial market.