Study On The Portfolio And Derivative Pricing Models With Entropy

Portfolio and option pricing are two major topics of modern mathematical finance theory.Classical portfolio research is usually based on the Markowitz mean variance model.And classical option pricing problem is usually based on non-arbitrage pricing principle.This paper mainly deals with two problems from different perspectives :one is studying the portfolio model by using entropy to measure portfolio risk and considering transaction cost;another is the research of power option pricing and power polynomial option pricing based on the motion of share prices,which is described by Tsallis entropy distribution and jump diffusion process.In the study of portfolio investment risk measurement model,using entropy to measure investment risk and dividing the entropy into two parts—mutual information and conditional information entropy respectively by drawing ideas from the risk decomposition of William Sharpe's single index model.The mutual information entropy of one stock represents the system risk and the conditional information entropy of one stock represents the non-systematic risk.Thus a Mean-Entropy model can be established.On this basis,building a new model,which is suit to the realities of securities market in China and takes transaction costs into consideration.Further more,selecting some representational stocks in different industries from SSE 50 index to conduct an empirical study.On the one hand,considering the model's effective frontier at the same rate of return,we find the model underestimates portfolio risk obviously if not taking transaction costs into consideration;On the other hand,we can conclude that the entropy model could help us to find out a portfolio with lower risk when compared with traditional MV model.As for option pricing model,we use the maximum Tsallis entropy distribution and a renewal jump process to describe the law of motion of asset prices.In the risk-neutral condition,the pricing formulas of power options based on Tsallis distribution and jump-diffusion process can be obtained by using the stochastic differential method and martingale.And the pricing formulas of polynomial options could be derived.