Pricing Options Under The CEV Diffusion Model By Asymptotic Expansion Method

For a long time academic circle has carried out a series of research for financialderivatives pricing work, scholars have proposed various mathematical models to sim-ulate the stock market. Such as fractional Brownian motion, a good approximation forreturn on assets due to its self-similarity, thick tail and long-term correlation properties.Such as the CEV (Constant Elasticity of Variance) option pricing model,“Volatilitysmile”is made up of price change which is the result of the negative correlation withprice volatility, the CEV option pricing model to separate the negative correlation. Thisprocess has the advantage in asset price fluctuations and price levels related links. Par-tial differential equation method is one of the option pricing method The options andtheir underlying assets in a certain proportion of securities, make the combination yield-s in the short term as the risk-free interest rate, thus can get a matures as the boundaryconditions of the partial differential equation for the option price; the solution of thisequation is the value of the option. So this article is mainly to use asymptotic expansionmethod on studing three kinds of option pricing problem. Main results are as follows:(1)Two value option pricing problem: using asymptotic expansion method to pricetwo value option under the CEV model; discussing the pricing formulas of conver-gence; using numerical experiment to verify the results of the correction.(2)Standard option pricing under fractional Brownian motion: firstly establishedthe model under the CEV model, put the problems into definite solution problem; byusing asymptotic expansion method to solve the partial differential equation to deducethe formula for pricing options value; secondly using numerical experiment to verifythe correction results and discussing the risk-free interest rate affect on the option value.(3)Look back option pricing problem: seting up the CEV model on fractionalBrownian motion; using the hedging principle get the value of partial differential e-quation; using asymptotic expansion method to solve the partial differential equationto deduce the formula for pricing options value; using numerical experiment to verifythe correction results and the hurst index of the impact on the option value, discuss therisk-free interest rate affect on the option value.