Positivity-preserving Numerical Schemes Of Generalized Black-Scholes Models

The classical Black-Scholes model plays an important role in financial markets,but the assumption that underlying asset price change continuously and assets transaction without transaction costs and with static volatility does not match the actual situation. Positivity-Preserving numerical schemes of some generalized Black-Scholes models caused by the underlying asset price jumps and transaction cost and stochastic volatility was studied, and reliability of the corresponding numerical algorithms was validated from the theory and numerical examples in this paper. The main results were as follows:(1) Option pricing model with transaction costs and its numerical solution under jump diffusion process was studied. Partial integro-differential equation(PIDE) satisfied by the option value was derived by delta–hedge method, which is a nonlinear Black-Scholes equation with an infinite integral, and it is difficult to obtain the analytic solution. Transforming it into a nonlinear diffusion equation with integral by variable substitution. and an unconditionally stable and positive difference scheme was constructed based on nonstandard approximation of the second partial derivative and double-discrete strategy. Monotonicity, stability and consistency was proved theoretically. The scheme also is not unconditionally consistency, but we presented a method to reduce the non-consistency items. Numerical examples were presented to illustrate the effectiveness of the scheme, and the bid price and ask price were analyzed in these examples, the results showed that the scheme relaxed the step requirements, reduced the calculation compared to the standard difference scheme.(2) A positivity-preserving numerical scheme for the solution of European and American option pricing model with stochastic volatility is proposed in this paper. This scheme is based on a nonstandard approximation of the first and second partial derivatives. The scheme is not only unconditionally positive and stable, but also allows us to solve the discrete equation explicitly. At last, the numerical results for European call option and American put option are compared to the standard finite difference scheme,and it turns out that the proposed scheme is efficient and reliable.