Statistical Inference On Some Stochastic Volatility Models

In Financial Mathematics, The important contribution of the Black-Scholes model is based on the assumption that the volatility is constant; the option price is related to the underlying asset stock price.But in fact, the constant volatility can not simulate the real financial market changes, so the generalization of the Black-Scholes model is taking the original constant volatility as a random process. In this paper, we mainly study the statistical inference of the stochastic volatility model,and give the estimation of the parameters of the stochastic volatility model.This paper first introduces the two kinds of classical model parameters estimation method, Stein-stein model can be obtained directly from the transition density function, maximum likelihood function is obtained by transferring the product of the density function, based on the derivative of the likelihood function to obtain the estimated parameter expressions. And for the Heston model to get the true maximum likelihood function, therefore, Nowman discrete method get approximation of the likelihood function, to approximate the derivative of the likelihood function parameters approximate values.The main work of this paper is to estimate the parameters of the 3/2 model by statistical inference based on the existing parameter estimation methods. Refer to the above method of statistical inference,3/2 model can not get the true likelihood function, approximate likelihood function are obtained by Euler method[ ]21 1 11 2 3/2 3111()(,) exp22 i i i i i i ii V V a b V VV V c Vc Vpqddpdd- - ----ì - - -ü? ?= í- y???t.The approximate differential likelihood function, and the function is equal to zero, obtained the parameters to be estimated as follows( ) ( )( )2 11 1 1 1 11 1 11 21 11 1?n n n i i i i i i ii i i n n i ii i V V V V n V V Va V V n d- -- - - - -= = =-- -= =- - -=-? ? ?? ?,( ) ( )( )2 1 11 1 1 1 11 1 11 21 11 1?n n n i i i i i i ii i i n n i ii i n V V V V V V Vb V V n d- - -- - - - -= = =-- -= =- - -=-? ? ?? ?,[ ]221 1 131 11()?n i i i i i i V V a b V Vcn Vdd- - -= -- - -= ?.In the process of numerical simulation, the sample data can not be obtained in the real financial market; therefore, by means of the Girsanov theorem to measure the transformation, the new measure of the standard Brown motion expression is as follows0 t t t t W W dsV??? ? ?.After the proof and inference can be used to express the volatility of the stock price, then the sample data can be obtained from the stock price volatility of the sample data? ?212i i i i n S SVTS???.