Research On Option Pricing And Risk Management Based On Lévy Jump Process And Stochastic Volatility Model

With the rapid development of information technology and the deepening of economic globalization,the electronic trading market based on Internet communication technology has become the main form of financial market.Massive data is properly stored and processed,and the frequency and reliability of the data are improved.Analysis of the high-frequency data contained in the short-term behavior of securities prices and dynamic characteristics of investors is essential so as to improve the options trading strategy,improving risk management capabilities.Since the presentation of B-S option pricing model,how to improve the accuracy of option pricing has become a growing concern.This model assumes that the asset returns follows normal distribution and hedges the option risk through continuous trading.And a large number of empirical researches in financial markets have found that financial data show strong non-normality property,financial asset returns do not obey the normal distribution.Compared to the normal distribution,it displays leptokurtosis and fat tail.There are a number of financial illusions in the financial market that can not be explained by the B-S option pricing model,such as the option volatility smile,the leverage effects between retuurns and volatility,and the volatility clustering.How to rationally construct basic model for underlying asset to accurately price option has important actual background and theoretical sense.Therefore,the study of option pricing in recent years mainly focuses on building alternative models to overcome the defect of B-S option pricing model.Scholars try to construct a Lévy process with independent and identically distributed increments to replace the Brownian motion.The Lévy family of distribution functions can capture the high order moments of financial data,especially the asymmetric characteristics of the returns and the jump of stock index.In order to characterize the stochastic volatility characteristics of asset returns,the square root process of mean return is embedded into the model.Furthermore,a class of tempered stable distribution model is introduced to construct the stochastic volatility model with tempered stable Lévy distribution.The tempered stable Lévy distribution driven stochastic volatility model expands the original stochastic volatility model framework,which can provide a wide range of modeling ideas for derivatives pricing and risk management.At the same time,the stock market risk under the background of financial liberalization has become more and more concerned with the practioner,academia and supervisors.The changes of financial derivative market environments,volatility and other factors will lead to fluctuations in the value of derivatives,and then cause the stock market to fluctuate.Therefore,it is of practical significance to use the new constructed model to evaluate the risk of financial market.We bring the newly constructed tempered stable Lévy distribution driven stochastic volatility model to the risk management field.The developed value at risk(VaR)and conditional value at risk(CVaR)method has become one of the popular issues in financial risk measurement researches.In this paper,considering the leptokurtosis and heteroscedasticity of the stock index time series,we introduce the tempered stable Lévy distribution and the stochastic volatility model in the empirical study of the VaR and CVaR,and then introduce the copula connection function to analyze multi-objective portfolio optimization under Lévy-copula model.Hence,we will combine low-frequency data and high-frequency data for empirical researches,mainly studies the option pricing and risk management issues assuming the underlying asset follows tempered stable Lévy driven stochastic volatility process.Considering that the jump and volatility characteristics of stock index time series are very important for the study of stock option pricing and risk measurement,this paper primarily uses nonparametric test to analyze the path characteristics of asset price process.Then,based on the discrete frame and the continuous frame Lévy stochastic volatility model,the empirical study of European option pricing and American option pricing are carried out.Then,the Lévy stochastic volatility model is used to conduct the risk measurement and the portfolio optimization researches.The details are as follows:(1)Based on the nonparametric statistical method,the spot fluctuation estimation method based on the jump and leverage of financial assets is employed to correct the realized threshold power variation estimation method.The statistical estimator is constructed for screening jump,and a comprehensive study is conducted containing the stochastic fluctuation in the financial asset price process,the finite activity jump and infinite activity jump and other issues.In order to simultaneously incorporate the heteroskedasticity clustering effect of volatility and the asymmetric effect of returns,in this paper,the existing volatility heterogeneous autoregressive prediction(HAR-RV)model is expanded and the error term of the asymmetric heterogeneity autoregressive model is set as GARCH model to examine the complex relationship between the jumping volatility sequence and the continuous volatility sequence.This paper uses the high-frequency data of Shanghai and Shenzhen stock indexes to carry on the empirical research,including the comparisons of jump recognition,jump activity degree and volatility prediction effects.Researches reveal that there are jumps,stochastic volatility and Brownian motion components in the stock dynamic,continuous volatility occupies the main body in the volatility,and the big jump component is relatively small.Among them,the jump composition of infinite activity small jump is larger than that of finite jump.(2)Assuming that the innovation stochastic factor follows the non-Gaussian Lévy distribution,we combine the NGARCHSK model that reflects the high order moments of financial assets and the Lévy process that describes the pure jump phenomenon of financial price changes to describe the time-varying nature of the asset returns infinite jump,as well as capturing financial assets returns leptokurtosis and leverage effects.The empirical research on returns time series verifies the superiority of the tempered stable distribution to characterize the peakness and tail.Combined with the higher order moments of volatility,the pricing of the first stock option in China is carried out.Besides,the efficiency of the subordinate process Monte Carlo simulation pricing method is compared with the cosine pricing method of numerical integration.It is found that the non-Gaussian Lévy distribution properly characterizes leptokurtosis of financial data.Among them,the tempered stable model fits the best and accurately captures the financial data fat tail property.(3)In order to capture the leptokurtosis,thick tail,biased characteristic in financial returns distribution and the heteroscedasticity effects and the agglomeration effects in volatility diffusion,as well as jointly describing the infinite jumps in the dynamic evolution of stock price,the classical tempered stable distribution(CTS)in the infinite activity pure jump Lévy distribution is introduced into the the square root CIR model based stochastic volatility model,establishing the tempered stable stochastic volatility(CTSSV)model.The stochastic volatility model driven by the pure jump Lévy distribution(LVSV)framework is reconstructed.By using the characteristic function expression of LVSV model,the formula of European option pricing is deduced by fractional fast Fourier transform(FRFT)method.Due to the large number of model parameters and the difficulty in integrating the objective function,a multi-regional adaptive particle swarm optimization(MAPSO)algorithm is proposed to estimate the LVSV model parameters.Based on the FRFT technique and the MAPSO parameter estimation results,the CTSSV model and the variance gamma stochastic volatility(VGSV)model are used to evaluate the European option in addition to the variance-optimal option hedging of the Hang Seng Index options.We find that compared with the VGSV model,the option pricing and hedging error of the CTSSV model are smaller,and the results are more robust for the modeling and hedging of derivatives.The MAPSO algorithm is used to estimate parameter,and it increases the particle diversity to improve the estimation accuracy.(4)Assuming that the stock price process follows the time-varying tempered stable Lévy process,a new method for American option pricing is proposed.The stochastic time variation is embedded into the normal tempered stable distribution,and the tempered stable stochastic volatility model is constructed.The new model can capture the stochastic variability of stochastic volatility while allowing the infinite active jump of the underlying assets,so it is suitable for reflecting the empirical phenomena in finance,such as leptokurtosis,asymmetry and volatility clustering.The American option is calculated using Fourier-cosine technique,and the improved particle swarm optimization algorithm is adopted to estimate the parameters.In order to demonstrate the validity of the model,the empirical study of American market options is conducted.The empirical study shows that the time-varying tempered stable process has a flexible structure in the American option pricing fitting,which includes both the jumping component and the volatility dynamic.The introduction of the square root time change into the tempered stable distribution can effectively improve the effect of American option pricing.(5)The time series of underlying asset in financial market show high peakness,asymmetry,and heteroscedasticity properties,the stochastic volatility model driven by tempered stable Lévy process(TSSV)is adopted to measure the financial risk and for portfolio revision.By using the analytic characteristic function and the fast Fourier transform(FFT)technique,the analytical form of the probability density function of returns is obtained,and the formula of VaR and CVaR under the TSSV model is derived.Finally,in order to predict the extreme events and market volatility,the Hang Seng index is used to study the risk under TSSV model,and the investment portfolio is constructed based on the risk-adjusted reward risk stock selection criteria.The backtesting analysis of the VaR and CVaR risk prediction of the Hang Seng index shows that the TSSV model has good predictive ability in risk measurement and is suitable for financial risk measurement.(6)Considering the non-linear dependence structure between the multiple underlying asset variables of financial assets in portfolio optimization,the tempered stable distribution is adopted as the marginal distribution,and the copula function is used to describe the correlation between variables.In the context of portfolio optimization,a multi-objective portfolio optimization model with copula dependence structure under tempered stable distribution is proposed,to study the modeling ability of TS distribution coupled with different copula functions.The proposed multi-objective portfolio optimization is designed to maximize reward while minimizing the risk to search the non-dominant Pareto frontier.Then we use three multi-objective optimization algorithms NSGA-?,SPEA-? and MOPSO algorithms to solve the problem of constrained TS copula multi-objective portfolio optimization,and the empirical analysis of China's Shanghai stock index constituent stocks and Shanghai and Shenzhen stock index returns is implemented.The empirical study shows that the financial assets returns do not follow normal distribution and the risk dependence is asymmetric.The multi-objective intelligent optimization algorithm based on particle swarm is suitable for solving the multi-objective portfolio problem of TS-copula type.In the end,the conclusion of the paper is summarized,and the limitation of the research and the future research direction are pointed out.