Numerical Solution Of Black-Scholes-Barenblatt Option Pricing Model And Its Application

In the study of option pricing,option pricing and risk measurement are the two most important concerns.This thesis mainly studies a derivative pricing model with volatility uncertainty.In this case,the volatility is not exactly known,but within a range,and the stock price process no longer follows the traditional lognormal distribution.In this thesis,G-normal distribution and G-Brownian motion are introduced into the model,and black-Scholes-Barenblatt option pricing equation(B-S-B option pricing equation)is established.In specific pricing process,this thesis adopts the numerical algorithms to solve equations,trigeminal tree is one of the most commonly used method in option pricing,followed by considering the equation and parabolic partial differential equation similar,therefore,first of all to parabolic partial differential equation,for example,respectively introduces the forward difference and backward difference scheme,And then apply it to the B-S-B pricing equation.Because the general backward difference scheme is obtained by solving linear algebraic equations,but in B-S-B equation,the iterative coefficients are constantly changing and cannot be given directly,so the application of backward difference cannot be solved.Therefore,an improved semi-implicit backward difference scheme is introduced in this thesis.Based on Fourier method and von Neumann criterion of finite difference schemes,the stability conditions of the difference scheme of B-S-B pricing equations are given:1.If Forward difference scheme meet τ≤1/(σ2/h2+r),this format is stable;2.The semi-implicit backward difference scheme is constant stable.G-VaR is a worst-case value at risk(VaR),defined as a measure of risk involving model uncertainty.In this thesis,we give the definition of G-VaR under nonlinear expectation,which can be calculated by an explicit formula.Under G-normal distribution,the estimated value of G-VaR of logarithmic return of stock is in a range.According to the property of G-Brownian motion and comparison theorem,the range of G-VaR of logarithmic return of stock is discussed from definition and equation:Finally,the thesis calculates the price of different kinds of options,including European option,Binary option,Bull spread and Butterfly spread,and analyzes the error of each algorithm,which proves that these algorithms are feasible.B-S-B pricing equation can effectively narrow the buying and selling range of the price,make the pricing more reasonable,and increase the liquidity of the market.Based on this equation,G-VaR sizes under different volatility intervals are calculated,which can better help managers to avoid risks in the market and has strong application value.