Study On European Option And Arithmetic Average Asian Option Pricing

China’s economy is transforming from high-speed growth to high-quality development.The healthy and orderly development of the financial market is an inevitable requirement of economic construction.It is important to learn western option pricing theory to promote the development of Chinese option market.The flexible options can meet the needs of investors for trading and hedging.Therefore,option pricing receives much attention.Black and Schole established the Black-Scholes model in 1973,which laid the foundation for option pricing.However,the assumptions of the model are strict and do not conform to the actual situation of the market.At the same time,B-S model cannot explain the volatility slope and the volatility smile.The Heston stochastic volatility model was proposed in 1993,which treats the volatility as a stochastic process and has a good pricing effect.The financial market is a complex nonlinear system,and machine learning methods can handle nonlinear problems well.Therefore,in the pricing of European option,in addition to the B-S and Heston models,this thesis introduces the Extreme Learning Machine(ELM),and optimizes the ELM by Particle Swarm Optimization algorithm(PSO).European option pricing research is divided into two chapters.The third chapter introduces the classical model and conducts empirical analysis,and uses ASA-Heston to obtain A-H implied volatility.The fourth chapter introduces the ELM and PSO-ELM models.At the same time,optimizes the volatility parameters with the help of the A-H implied volatility obtained in the third chapter.In order to meet the needs of investors,various new options are created.The third and fourth chapters study the pricing model of European option and optimize the model parameters.The fifth chapter expands the study scope of options,and selects a new type of option——two-asset arithmetic average Asian option for pricing study.However,new options are traded over-the-counter,the market data is difficult to obtain,and machine learning methods are no longer applicable.At the same time,it is difficult to find an analytical solution to the arithmetic average Asian option pricing equation.Therefore,this thesis uses Monte Carlo simulation and improved Monte Carlo simulation for option pricing.The main content of the thesis:The first chapter is the introduction,which introduces the background and significance of this thesis,and sorts out the relevant research and achievements.The second chapter is the theoretical basis,which mainly introduces the relevant concepts and theories.The third chapter studies the pricing of European options based on the B-S model and the Heston model.First,this thesis introduces the theory of the model,and then conducts an empirical analysis of the CSI 300ETF options listed in 2019.The results show that the ASA-Heston model using the Adaptive Simulated Annealing algorithm(ASA)for parameter estimation is more effective.Further,the pricing result of the ASA-Heston model is brought into the B-S model to obtain the A-H implied volatility.The fourth chapter introduces ELM for European option pricing,and then uses PSO to optimize ELM to establish the PSO-ELM model.The result shows that the PSO-ELM model has the smallest pricing error compared with the B-S and ASA-Heston models.In the process of empirical analysis,historical volatility and A-H implied volatility are used as input parameters of ELM and PSO-ELM.The results show that the pricing error of the model under the A-H implied volatility is smaller.The fifth chapter expands the research scope of options and studies the pricing of arithmetic average Asian option.In order to fit the actual situation of the market,this thesis introduces the fractional Brownian motion.Taking the two-asset arithmetic average Asian minimum call option under fractional Brownian motion as an example,its pricing equation is difficult to obtain analytical solution,so this thesis uses the Monte Carlo simulation to solve the option price.At the same time,the Monte Carlo simulation is improved with the help of the analytical solution of the two-asset geometric average Asian minimum call option under fractional Brownian motion,which is used as the control variable to reduce the variance of Monte Carlo simulation.The sixth chapter is the summary part,which mainly analyzes the advantages and disadvantages of this thesis,and tries to explore future research directions.The innovation of the thesis:1.Introduce ELM and PSO-ELM for European option pricing,and conduct empirical analysis on CSI 300ETF options.Finally,this thesis compares the pricing effect of PSOELM with the traditional Black-Scholes model and Heston model.2.Combining the B-S model and the ASA-Heston model,we can obtain the A-H implied volatility.Therefore,we can use the A-H implied volatility to replace the historical volatility to realize the optimization of the volatility parameters of the ELM and PSOELM models.The empirical results show that the PSO-ELM model under the A-H implied volatility are better than other models.The research in this thesis has reference significance for the pricing of European option,and investors can have rational investment according to the predicted price of the model.3.In order to describe the price changes of the underlying asset,this thesis introduces the fractional Brownian motion in the pricing exploration of the arithmetic average Asian option,and takes two-assets arithmetic average Asian minimum call option as an example to study the pricing.However,it is difficult to find an analytical solution to this option pricing equation,so this thesis uses Monte Carlo simulation.At the same time,the analytical solution of the geometric average Asian option is used as the control variable to reduce the variance of the Monte Carlo simulation.This makes the simulation results more stable,and also provides a solution for solving the arithmetic average Asian option price.