Implied Volatility Under The CEV Model

In the risky assets problems such as option pricing and risk management, deter-mination of volatility is one important part.The volatility,which is a measure of the extent of the underlying asset price changes in a period of time, is usually defined as the standard deviation of returns of the underlying assets within one year.According to different specific meanings and levels of difficulty in getting it,the volatility can be classified into different kinds,such as:historical volatility,which is based on histor-ical data of the asset price changes and counted for the standard deviation of these changes'natural logarithm.This is the most simple method,but usually the error is much bigger.If you can accurately predict volatility and get a function of time and the underlying assets price:σ=σ(S, t),this is the local volatility.However,the future data is not available,the changes in option prices during options trading reflect partici-pants'expectation of changes of the underlying asset price.Therefore,we can derive the volatility through the market option prices,which is called implied volatility.Now we focus on the implied volatility. Deriving the implied volatility is essentially an inverse problem. We first introduce three important work of predecessors in this field.The first job is given by Dupire who brought forward the inverse problem of solving implied volatility.Basing on the existing conclusions about parabolic equations and conjugate equation theory,he originally transformed the option pricing equations of S, t into equations of two independent variables,the strike price K and expiration date T.From this equation he got the formly solution sigma(K, T),then he used option prices for different striking price K and different expiry date T to obtain volatility by numerical methods.The initial model is not perfect, and the numerical method is unstable.However using the observed option market prices to get the volatility and forecast changes of underlying price,this idea is of great significance.The second job is given by Isakov.He followed Dupire's idea, and gave the op- tion pricing equation with the dual equation theory. Then he gave the mathematical description of solving implied volatility and improved the theoretical analysis of this problem,giving a rigorous mathematical proof of the uniqueness and stability.He also gave a stable numerical method. The proof and stability conditions in which to explore the later method for solving the implied volatility problem was a good foreshadowing.The third study, is given by Jiang LiShang and others. Since there has been a significant development in optimization theory in the past ten years,with Isakov's uniqueness and stability analysis,Jiang LiShang developed the optimization method to get implied volatility and gave the existence and uniqueness theorem about this optimal control problem.According to the analysis of Dupire,the volatility function of S, t can be trans-formed into that of the maturity date T And the strike price K:σ=sigma(K, T).For a fixed T,the relationship between the implied volatilityσand the strike price K is usually described as a smile or a skrew.Jiang LiShang,et al([20])carried out numerical experiments on the volatility flat (rare),smile and skrew.The results showed that the optimization method is stable.As a special case of the implied volatility model,the CEV model can well describe the "skrew" situation,and is effective in some specific market.However the CEV model has been widely used in recent years([8][37][40],etc.).Based on the results of an article in 2008[21],using the CEV model can describe the law of oil price fluctuations very well, so we will start with oil-related products market.For the purpose of hedging it requires that the product can be traded in both stock and options markets.The CEV model, when it comes to shorter maturity options,has a greater advantage than the BS model[36].Considering all these factors,we have collected USO daily stock prices from 2009, July 20th to August 21st,and the European call option prices to get the volatility.USO and the price of international crude oil market is basically the same trend and the risk of bankruptcy is relatively small.Assuming that the stock{St}t≥0 follows whereβis a constant,{Wt}t>ois the Brownian motion. Assuming that the European call option price is V,the strike price is K,the maturity date is T.Using△-hedging to get option pricing equation under the CEV model The essence for solving implied volatility under the CEV model is, identifying the value ofσ0,βinσ(S,t)=σ0Sβ/2-1,so that the solution of (1) fits the current market prices of options at t*(0≤t*＜T) for different strikes, K, and/or maturities T.Use Dupire's idea to get equations about K,T Let ThenTo satisfy the existence and uniqueness conditions ([20]),the model needs to be modified. Setα1,α0 respectively are the upper and lower bounds of a(y).In addition,do the boundedness truncated and set a(y) satisfying the smooth condition,then note it as A'(Related in§3).Now we can get the equationWhenεis small enough,especially when it is far smaller than the width of the grid,the impact to the numerical result is negligible.After this change,we knowσ(Κ) = (2a(y))^1/2 by (3),a(y) is bounded,so the volatility is bounded. Then we can get Isakov,Jiang LiShang and others' conclusions extended to the CEV model.Theory The corresponding optimal control problem to equations (5) find a a∈A' satisfying where, N is a regularized parameter. V(·,·) satisfying (6). The optimal control exists and is unique.The optimal control problem can be transformed into elliptic variational inequality, then iteratively solved using numerical methods. In the numerical experiments,we use the underlying asset price at t* (2009.7.20) and the option price V*(K) to obtain a(y) iteratively,then we can derive implied volatility byσThe result shows the implied volatility derived from this method consistents with the market data,and the process is robust.When you select the appropriate effective range of asset prices, the numerical results is close to the real.