Decomposition Method And Application Of European Option Pricing

With the increase in the complexity and uncertainty of financial markets and financial derivatives,the pricing issue of financial derivatives represented by options continues to arouse the attention of academia and practice.Since the listing of China's first stock option SSE 50 ETF on February 15 2015,the Chinese stock market has opened a new era of options.And the research on option pricing decisions with Chinese characteristics has become a hot area for many scholars at home and abroad.As an indispensable part of the financial market,the option market provides a strategic method for strengthening the risk management of financial products,improving the efficiency of various resource allocations and optimizing the investment portfolio of the financial market.In option trading,price has always been a hot issue in research and discussion.Establishing an effective option pricing model can provide a basis for financial market transactions,which is of great significance to its price stability and healthy development.Therefore,research on option pricing models and methods has important theoretical significance and application value.This article mainly studies the pricing methods and applications of several types of European options.It uses partial differential equations as basic methods to analyze the pricing problems of European options,and studies the decomposition methods and applications of European options pricing.The article is divided into eight chapters.The first chapter introduces the research background and significance,analyzes the latest research progress of the use of options,and introduces the main research content and methods of the thesis,the main contributions and innovations.The second chapter is the literature review section,which summarizes the option pricing methods and options applications.It respectively reviews the time-homogeneous model and the non-timehomogeneous model in the diffusion process,the 0)? process model in the jumpdiffusion process to describe the non-Gaussian characteristics,and singularity Barrier options in options are reviewed in the literature on pricing models and other aspects,and sorts out the logical relationships and limitations of previous literatures on option pricing research.The third chapter introduces factors such as time-varying conditions and leverage effects to construct a risk-neutral model under the condition 0)?process.First it establishes the basic framework of the 0)? process.From the three characteristic features of 0)? measure,namely jump rate?d(x),drift rate ? and diffusion rate ?,the definite characteristic function form of 0)? process is obtained.Then the harmonic steady state process(TS)representing the infinite jump 0)?process is analyzed,focusing on the classical harmonic steady state(CTS)and downhill harmonic steady state(RDTS).Finally,based on the ARMA-GARCH model,the three variables of time-varying volatility and drift rate,as well as the leverage effect will be added to construct the conditional 0)? process.Chapter 4 discusses the application of the option pricing model of 0)? process based on the air quality index AQI.First,the stochastic differential equation of the O-U average recovery model with 0)?process is obtained to simplify the random disturbance of uncertain terms,and the time series model of AQI is obtained.Secondly,the multi-step binary tree model is used to calculate the option pricing formula from the single-step ladder option price,and then the AQI index is used to obtain the binary tree pricing model of call options and put options at different periods and different execution prices.Then the parameters of the AQI O-U model are estimated.Chapter 5 constructs the CEV option pricing model and time-varying CEV option pricing model based on the hyperbolic absolute risk aversion utility.The former assumes that investors obey the hyperbolic absolute risk aversion utility function and solve the optimal dynamic asset allocation problem when fully hedged with random capital flows under the CEV model.The latter assumes that stock price changes follow the modified fractional Brownian motion.Compared with the standard Brownian motion,the modified fractional Brownian motion has persistence,that is,"long-range correlation." Chapter 6 takes the mathematical models of falling knock-out call options and falling knock-out put options as examples,and introduces a hypergeometric stochastic volatility model for stochastic volatility and drift terms.Using the analyzability of the hypergeometric model,a display pricing formula for double integral numerical calculation is obtained.The seventh chapter is based on the principle of differential dynamics,deducing the price trend model of financial derivatives and options.Due to the impact of external shocks,the price of financial derivatives is approximated by a pseudo-cyclic change of ?.By substituting the initial conditions,the fluctuation period and amplitude of the price of financial derivatives can be determined.By transforming the first-order non-homogeneous non-homogeneous equations with appropriate variables and converting them into Abel's type I equations,the general solution of the equation is obtained.Chapter 8 summarizes the application of European option method in reality,and proposes the application of partial differential equation method in option pricing research.And combined with the actual analysis of the application prospects of option pricing theory in my country's financial market.The main conclusions drawn in this article:(1)It can be found through numerical verification of market indexes such as the Shanghai Composite Index(SZZZ),Shenzhen Component Index(SZCZ),Shanghai and Shenzhen 300(H&S300)and S&P500 Index(S&P)There are widespread thick tails,spikes and left deviations in financial markets.In terms of the numerical results of skewness and kurtosis,the results of harmonic steady state are closer to the theoretical values.At the same time,the coefficient of leverage coefficient ? is positive,and the result of t value is significant.The local martingale condition 0)? process model established in this paper adds time-varying conditions and leverage effects,which can dynamically describe the asymmetric infinite pure jump process.The theory and methods of option pricing can better reflect the future trend of the financial market,and have more Good fit and universality.(2)The CEV option pricing model based on the hyperbolic absolute risk aversion utility converts the complicated three-dimensional nonlinear partial differential equations into solveable parabolic partial differential equations,and converts the optimal allocation problem of the non-self-financing portfolio into selffinancing.The combined optimal allocation problem simplifies the problem,thereby obtaining a more universal risk-neutral CEV model.The time-varying CEV option pricing model starts with the "long-range correlation" feature of the financial market and establishes a European-style pricing equation that is more in line with the actual market situation.(3)For the stochastic volatility and drift terms,taking the mathematical models of falling knock-out call options and falling knock-out put options as an example,a hypergeometric stochastic volatility model is obtained.This model avoids the general loss.The first-order approximation developed by it is simple,effective and fast in calculation,and is suitable for practical applications.(4)Based on the principle of differential dynamics,a price trend model of financial derivatives and options is derived.In the three cases and,use low,medium and high frequency data to analyze the general equation of option price trend.The results of the SSE 50 ETF numerical simulation verified the results of the option price trend,and found that the option model has a more obvious pulsating trend in the numerical test.The main contributions of this paper are:(1)The 0)? process option pricing model established under the partially equivalent martingale measure.By introducing two time-varying variables,instantaneous variable volatility and time-varying drift rate,and at the same time introducing leverage effect variables,the resulting model can dynamically describe the asymmetric infinite pure jump process.Because of the jumping and non-Gaussian characteristics of the financial market,the theory and method of option pricing in this model can better reflect the actual situation and future trends of the financial market.(2)Establish a CEV model based on the assumption that investors are hyperbolic absolute risk aversion(HARA).The contribution of this model is to solve the problem of optimal dynamic asset allocation when hedging random capital flows.And because of the relative relative risk aversion(CRRA)utility function,the constant absolute risk aversion(CARA)utility function,and the quadratic utility function are all special cases of HARA.Therefore,the method used in this model is more general.(3)Introduce stochastic volatility and non-zero drift factors to establish a hypergeometric stochastic volatility model.This model uses European vanilla put options and falling knock-out put options as examples to avoid the general loss generated when dealing with barrier options.At the same time,because the stochastic volatility model generally does not have an explicit solution,this model uses the PDE regular perturbation method of Rúben et al.(2018)to carry out series expansion derivation of the approximation of the hypergeometric model,which provides some research for the study of asymptotic approximation.Ideas and references.(4)The price change trend model of financial derivatives and options established in this paper decomposes the movement law of options into price trend changes and their fluctuations,which can better explain the change process of option prices and the influencing factors with general fluctuation characteristics.