Research On Vulnerable Option Pricing Under Jump Diffusion

With the acceleration of financial globalization,the sudden events occur more frequently.Structural changes often occur in financial time series due to policy changes and sudden critical social events.In particular,once the structural changes occur,a pure diffusion model can no longer provide a good fit to the financial data such as stock returns and interest rates.The complexity of the financial market makes it difficult to analyze if only traditional financial theories are used.In addition,with the continuous development of the financial market,more and more options are traded in the OTC market,and the scale of the transaction is also growing.The credit risk cannot be ignored in the transaction process.At the same time,with the increasing frequency of international transactions,especially for large import and export countries like China,exchange rate changes have a crucial impact on a country’s national economy and foreign economic relations.Because the exchange rate is an important lever for the regulation of the foreign trade market.It guides the flow of international capital.Therefore,in transaction process of financial products,in addition to the jump risk of the product price itself,the credit risk and the exchange rate risk cannot be ignored.Based on the above analysis and existing research,this paper has carried out a series of researches under the jump diffusion.The research results have improved the theoretical system of vulnerable option pricing,and deepened people’s understanding of finance market.The understanding has important theoretical and practical significance.The specific innovations and research contents are summarized as follows:(1)In the option market,credit risk is introduced,the vulnerable option pricing model under fuzzy environment is constructed based on fuzzy set theory and binomial tree option pricing theory,and then a numerical comparison analysis is shown.Options with credit risk are called vulnerable options.Most of existing studies on vulnerable option pricing have assumed that investors have complete and accurate information about firm value and recovery rate.However,in the actual financial market,investors usually cannot directly observe the true value of assets.As a result,everyone may have a subjective judgment on the reliability of corporate accounting data and the recovery rate of asset value in default.Therefore,we use the fuzzy set theory proposed by Zadeh to characterize this uncertainty.Based on the Klein’s model framework,the relevant variables and parameters are fuzzified,and the vulnerable option pricing problem is studied in a fuzzy environment.At the same time,by fully using the advantages of binomial tree option pricing model and discretizing the relevant variables,we respectively obtain the pricing formula of European vulnerable option and American vulnerable option under fuzzy environment.In addition,the crisp binomial tree pricing models for vulnerable option are shown.Finally,we compare the results of the proposed model with those of the classical Klein’s model.The findings suggest that the precise information assumption about the firm value and recovery rate in Klein’s model may lead to underestimate the price of vulnerable.This study can provide insights for future research on defaultable options pricing under imprecise market information.(2)In the option market,extending the above study to continuous time situations,the European vulnerable option pricing model with credit risk is constructed under fixed liabilities and stochastic liabilities,and the pricing formulas of the call and put option are derived.Existing researches on vulnerable options are mostly under pure diffusion.This paper studies the pricing of European vulnerable options under the jump diffusion process.The vulnerable options are priced in a structured way,and the Poisson jump distribution is introduced to construct the European vulnerable option pricing model under the two cases which the firm debt are fixed and random,and the corresponding call and put option price formula is given.Studies have shown that the value of the vulnerable options is affected by jumps in the underlying stock prices,firm value,and firm debt.In particular,in most cases,the value of options with stochastic debt is less than that of the corresponding options with fixed debt,implying that the change in the debt could increase the likelihood of a default and affect the value of the corresponding options.Moreover,the Klein’s model,which does not consider the risk of jumping,tends to overestimate or underestimate the value of the option.Therefore,compared with the Klein’s model,our proposed model has better practical application value and further improves the theory of option pricing.(3)Based on the previous research,credit risk and exchange rate risk are introduced in the option pricing problem.And the European exchange rate option pricing model with exchange rate risk and credit risk is constructed,and the European vulnerable option pricing formula is derived.Exchange rate options with credit risk are a new type of cross-currency option in the financial market.This exchange rate option has two risks: exchange rate risk and credit risk.This paper further expands the previous research,comprehensively considers the stock market and foreign exchange market,introduces exchange rate risk in the option pricing problem,and applies the structured method to link the final execution of option with credit risk to the counterparty’s corporate value and liabilities.Then the European exchange rate option pricing model with exchange rate risk and credit risk is constructed,and the pricing formulas of the European vulnerable call and put option under fixed debt and stochastic liability are derived.Finally,a numerical example analysis is showed.The results show that the impact of exchange rate fluctuations on option prices cannot be ignored,especially during the “One Belt,One Road” development strategy period.Our findings will can provide financial investors with decision-making reference,and further enrich the financial asset pricing theory.(4)Considering the jump risk,the jump diffusion model is constructed based on the SGT distribution.Due to the complexity of financial markets and the frequent occurrence of emergencies,many scholars have found that classical pure diffusion models cannot describe the changes of financial time series well.Jumping risk has become the focus of scholars in research.In order to better describe the jumping phenomena(such as peaks,thick tails and skewed features)which exhibited by financial time series,this paper introduces generalized skewed t-distribution(SGT)to describe these characteristics and builds two jump diffusion models based SGT distribution,namely the BS-SGT model and the Kou-SGT model.An empirical analysis is showed.First,the result based on the yield distribution shows that the model considering the SGT distribution is superior to its corresponding basic model.It implies that the model with SGT distribution can better characterize the peaks,thick tails and skewed features in the financial market.In particular,the Kou model is significantly better than the BS model,which is consistent with Kou’s findings.Second,the implied volatility prediction results show that the model with SGT distribution has relatively high prediction accuracy.Our findings can be used to explore the movements of underlying asset price,company value and liability related to vulnerable options,and provide direction for further expansion of vulnerable option pricing research in the future.It can further improve the vulnerable option pricing theory system and provide some decision-making reference for financial institutions and investors in practice.