A Study On The Pricing Of Vulnerable European Option In The Incomplete Financial Market

In recent years,with the development of economic globalization and financial innovation,the market scale of OTC options has expanded rapidly.In the first half of 2018,the notional amounts outstanding of global OTC options contracts was close to $64.4 trillion.Owing to the advantages of diversified products and trading forms,OTC options can better meet the needs of investors in hedging,risk aversion and investment.In 2013,since the issuance of OTC option business in China,the size of the China's OTC option market has shown an explosive growth.In November 2018,Nominal principal of the end-of-month stock of OTC options in China reached 265 billion yuan,and the proportion of OTC options trading in the OTC derivatives market was about 83.3%.Unlike the options traded in regulated exchanges,In the OTC markets,there is no third party to guarantee all payments required by a contract.Thus,the holders of OTC options are exposed to counterparty credit risk.In the global financial crisis of 2007-2008,many large financial institutions went bankrupt or were on the verge of collapse,such as Lehman Brothers and American International Group.Since then,the counterparty credit risk of OTC options has attracted more and more attention from participants in OTC markets and financial regulators.Therefore,when pricing OTC options,the credit of the counterparty must be taken into account.Many researchers refer to options subject to credit risk as vulnerable options.The study of pricing vulnerable option is not only of great theoretical significance,but also has important practical significance to the financial market.For the pricing problem of vulnerable options,it is an important research topic in financial engineering to construct a reasonable pricing model and use option pricing theory and stochastic control theory to obtain the price of vulnerable options.A large number of empirical studies show that the return distribution of financial assets does not satisfy the normal distribution,but presents patterns of leptokurtosis and heavy tails.In the existing literature on the pricing of vulnerable options,Interest rates and asset volatility are not constant,but they vary significantly through time.In addition,major market news and the adjustment of national policies have caused jump fluctuations in financial assets.Based on the structural form model and the reduced form model,in imperfect financial markets,this paper extends the existing option pricing models and establishes more reasonable pricing models for vulnerable European options.Then,the price of vulnerable European options is deduced by using relevant pricing theory and numerical methodsunder different economic environment.Firstly,in view of time-varying patterns of interest rate and the volatility of asset price,the pricing model of vulnerable European options with stochastic volatility and stochastic interest rates is established.We consider the price process of the underlying asset follows the GARCH diffusion model with stochastic interest rate,and the stochastic interest rate satisfies the classical Vasicek interest rate model.Based on this model,the joint characteristic function of the underlying log-asset price and the counterparty's log-asset value is derived explicitly.We obtain an approximate solution for the vulnerable European option price via means of Fourier transform.In addition,The Greeks of vulnerable option price are derived explicitly.Besides,the approximate solution of vulnerable option price can be quickly computed by using the fast Fourier transform(FFT)algorithm.The results of Monte Carlo simulations indicate that FFT is accurate,fast and easy to implement.The results show that: the price of vulnerable European options is higher than that of Klein(1996)for in-price options,while for out-of-price options,the vulnerable option prices of the proposed model are smaller;the higher long-run mean of the underlying asset price's instantaneousvariance,the higher vulnerable option price;the value of vulnerable European option is an increasing function of the long-run meanof the stochastic interest rate.Secondly,according to jump pattern of underlying assets and counterparty's asset,this paper investigates the pricing issue of vulnerable European options under a jump-diffusion model with stochastic volatility.The price processes of the underlying asset value and counterparty's asset value follow two correlated exponential Lévy processes with stochastic volatility,and the stochastic volatility is divided into the long-term and short-term volatility.A mean-reverting process is introduced to describe the common long-term volatility risk in underlying asset price and counterparty's asset value.The short-term fluctuation of stochastic volatility is governed by a mean-reverting process.Based on the proposed model,we derive a closed-form solution for the vulnerable European option price by using the Fourier inversion formula for distribution functions.The research results show that: the higher long-run mean of common long-term variance of underlying asset and counterparty's asset have positive effect on the vulnerable option price,the higher vulnerable option price;when the jump components of underlying asset price and counterparty's asset value follow the compound Poisson process,the value of option price increases with thejump intensity of underlying asset price and decreases with thejump intensity of counterparty's asset value.Thirdly,based on the structural model,this paper considers the vulnerable European options with risky collateral.The price processes of underlying assets,risky collateral and counterparty assets follow three stochastic volatility models with stochastic interest rates,respectively.The interest rate satisfies the Vasicek model.The price of vulnerable European option is represented as the difference between a European option without credit risk and a spread option.By using the Fourier inverse transform,we obtain a closed-form solution of European option without credit risk.For spread options,the numerical results are provided by Monte Carlo simulation.The results show that the higher the correlation coefficient between underlying assets and risky collateral,the higher the price of vulnerable European options;the price of vulnerable European options is an increasing function of the initial value of risky collateral;the stochastic volatility of underlying assets has a positive effect on the price of vulnerable European option.Finally,this paper considers the common jump risk between the underlying asset and the intensity of default.The dynamics of underlying asset follows a jump-diffusion process with stochastic volatility and stochastic interest,and the stochastic interest rate follows the CIR model.Besides,the intensity of default is governed by an affine diffusion process with jumps.To describe common jump risk between the underlying asset and the intensity of default,a common jump process is introduced.Based on the reduced form model,the pricing model of vulnerable European options is established.Furthermore,the explicit expression of vulnerable European options is derived.The results reveal that: the price of vulnerable options is an increasing function of theindividual jump intensity of underlying assets;the greater the idiosyncratic jump intensity of default intensity,the smaller the price of vulnerable European options;the impact of the common jump intensity of underlying assets and default intensity on the price of vulnerable options depends on the recovery rate.When the recovery rates very small,the price of vulnerable European option is a decreasing function of common jump intensity;When the recovery rates relatively,the price of vulnerable European option is an increasing function of common jump intensity.