Option Pricing under Risk-Minimization Criterion in an Incomplete Market with the Finite Difference Method

Abstract

We study option pricing with risk-minimization criterion in an incomplete market where the dynamics of the risky underlying asset is governed by a jump diffusion equation with stochastic volatility. We obtain the Radon-Nikodym derivative for the minimal martingale measure and a partial integro-differential equation (PIDE) of European option. The finite difference method is employed to compute the European option valuation of PIDE.

Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations

“…Therefore we need to use a numerical approximation. To obtain an approximation of the option value, one can compute a solution of the BS equations (1) and (2) using a finite difference method (FDM) [2][3][4][5][6][7][8], finite element method [9][10][11], finite volume method [12][13][14], a fast Fourier transform [15][16][17], and also their optimal BC [18].…”

Pricing of American Put Option under a Jump Diffusion Process with Stochastic Volatility in an Incomplete Market

“…It was also found that, under the minimal martingale measure, there exists a unique risk-minimizing strategy hedging of contingent claims in complete market [10]. The criterion under the minimal martingale measure is thus referred to as riskminimization criterion (see [11]). According to this, it is possible to price option with risk-minimization criterion.…”

Equilibrium Asset and Option Pricing under Jump-Diffusion Model with Stochastic Volatility

“…Due to drawbacks of the Black-Scholes model which cannot explain numerous empirical facts such as large and sudden movements in prices, heavy tails, volatility clustering, the incompleteness of markets, and the concentration of losses in a few large downward moves, many option valuation models have been proposed and tested to fit those empirical facts. Jump-diffusion models with stochastic volatility could overcome these drawbacks of the Black-Scholes model in [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Based on those advantages, in this paper, we focus on studying the jump-diffusion model with stochastic volatility.…”