Numerical Methods For Unconditional Stability Of Non-Arbitrage Liquid Market Model

With the variability brought by the entry of options into the market,investors are urgently concerned about how to choose the proportion of the investment portfolio in order to obtain risk-free returns,that is,the pricing of financial derivatives(futures options,securities,etc.).Therefore,this paper studied an option pricing model with fixed volatility in a non-arbitrage liquid market,which is based on the two preconditions of the"risk-free arbitrage principle"and an idealized financial market.In this paper,the original Black-Scholes model that does not pay dividends is converted into a nonlinear diffusion equation through a variable transformation.Then the second-order partial derivative and the first-order partial derivative in the nonlinear equation are approximated by the central difference quotient and the first-order backward difference quotient respectively,thus establishing an implicit finite difference scheme.According to the properties of a class of tridiagonal matrices and a series of theorems and lemmas,several conclusions can be proved:(1)the implicit scheme is uniquely solvable;(2)the option Gamma is non-negative;(3)the numerical solution of the scheme is non-negative and monotonically non-decreasing;(4)the scheme can achieve unconditional stability and compatibility.Finally,it is verified by numerical simulation that the implicit difference scheme proposed in this paper can achieve the accuracy of first-order convergence in the time direction and second-order convergence in the space direction,namely,O(h~2+k).The numerical solution and the option Gamma of the difference scheme are both non-negative,and the numerical solution of the scheme is monotonically non-increasing with respect to the spatial direction.