Statistical Inversion Of Drift Rate Based On Extended Black-Scholes Model

The Black-Scholes option pricing model created by Black and Scholes in 1973 is one of the most important achievements in contemporary financial theory.However,due to the existence of specific assumptions in this model,the theoretical price of the financial market obtained in some cases is inconsistent with the actual price.Inconsistent,there are many limitations in the application.Studies have shown that if the Black-Scholes model is applied to the probability of default estimation,considering the underlying asset S as the value of the company,a difference between the drift rate ? and the risk-free rate r called the risk premium must be considered to price the corporate bond u(t,S).Merton proposed the"interest rate risk structure" theory based on corporate bond pricing,and the deduced corporate bond pricing model has an isomorphic relationship with the Black-Scholes equation model,that is,the extended Black-Scholes model.This article is based on the extended Black-Scholes model to quantitatively study corporate bonds.Different from general option pricing,in the model corresponding to corporate bonds,there is a difference called risk premium between the drift rate ? and the risk-free interest rate r,and the arbitrage opportunity exists in the risk premium.Therefore,the inversion study of drift rate can help us discover possible arbitrage opportunities,and thus produce an inverse problem worthy of discussion.This article mainly focuses on the research of numerical inversion algorithm of drift rate ?.Under the assumption that the drift rate to be inverted is a more general polynomial or trigonometric function and the given prior condition is Gaussian distribution,an improved Markov chain Monte Carlo method based on Metropolis-Hastings criterion is given,which realizes the Numerical inversion of drift rate parameters.Starting from the extended Black-Scholes model,this paper first explains the background and reasons of the drift rate ? solving problem,then introduces some basic knowledge and improvement methods of the numerical inversion algorithm,and finally performs numerical inversion based on the simulated data.Through several numerical examples,it is found that the improved MCMC-MH algorithm is effective.Compared with the traditional method,it can perform numerical inversion for inverse problems with more unknown parameters and larger errors.At the same time,the inversion results can show more probability distribution information of parameters corresponding to the inverse problem.