This work considered strong convergence of the Euler-Maruyama (EM) method for a stochastic volatility jump-diffusion model (SVJD model, for short). In this model, the underlying asset price follows a jump-diffusion geometric Brownian motion with stochastic volatility, and the volatility process obeys a mean-reverting square root process with Poisson jumps. As preliminary results, the existence and uniqueness of nonnegative solutions for the SVJD model was shown by means of Tanaka's formula and the comparison theorem. Also, some moment properties of the solution to the SVJD model were given. In view of unavailability of an explicit solution for the SVJD model, we used the EM method to approximate the exact solution and proved strong convergence of the EM approximation in the $ L^2 $ sense. In addition, the EM approximation for the SVJD model was applied to approximately compute expected payoffs of a European option and a barrier option. Finally, simulations were presented to verify the theoretical analysis.
Citation: Weiwei Shen, Yan Zhang. Strong convergence of the Euler-Maruyama method for the stochastic volatility jump-diffusion model and financial applications[J]. AIMS Mathematics, 2025, 10(5): 12032-12054. doi: 10.3934/math.2025545
This work considered strong convergence of the Euler-Maruyama (EM) method for a stochastic volatility jump-diffusion model (SVJD model, for short). In this model, the underlying asset price follows a jump-diffusion geometric Brownian motion with stochastic volatility, and the volatility process obeys a mean-reverting square root process with Poisson jumps. As preliminary results, the existence and uniqueness of nonnegative solutions for the SVJD model was shown by means of Tanaka's formula and the comparison theorem. Also, some moment properties of the solution to the SVJD model were given. In view of unavailability of an explicit solution for the SVJD model, we used the EM method to approximate the exact solution and proved strong convergence of the EM approximation in the $ L^2 $ sense. In addition, the EM approximation for the SVJD model was applied to approximately compute expected payoffs of a European option and a barrier option. Finally, simulations were presented to verify the theoretical analysis.
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Weiwei Shen, Yan Zhang. Strong convergence of the Euler-Maruyama method for the stochastic volatility jump-diffusion model and financial applications[J]. AIMS Mathematics, 2025, 10(5): 12032-12054. doi: 10.3934/math.2025545
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