Market with Markov jump volatility II: Algorithm of derivative fair price calculation

The second part of the series is devoted to the numerical realization of the fair derivative price in the case of the incomplete financial market with a Markov jump stochastic volatility. The concept of the market price of risk applied to this model by Runggaldier (2004) allows derivation of a system of partial differential equations describing the temporal evolution of the derivative price as a function of the current underlying price and the implied stochastic volatility. This system represents a generalization of the classic Black–Sholes equation. By contrast with this classic version, the proposed system of equations does not permit an analytical solution. The paper presents an approximate analytical method of the fractional steps. The author equips the temporal axis with a grid, then the author approximates the required solution as a combination of the solution to the classical heat equation and the system of the ordinary linear differential equations. The paper contains the results of the numerical experiments illustrating the properties of both the Black–Sholes generalization and the joint evolution of the derivative and corresponding underlying prices.