The evaluation of two vulnerable powered options under a general stochastic volatility jump diffusion model using a Laplace–Fourier transform

This paper investigates the pricing problem of two vulnerable powered options under the condition that the underlying assets price mechanism satisfies general stochastic volatility, stochastic interest rate, and stochastic jump. Specifically, a Laplace transform is first applied to simplify the payoff function of the vulnerable powered options into the product form of key term and beta function. Second, the total differentiation technique of multivariate functions is implemented to approximate the nonlinear partial differential equation group into a linear equation group, and the affine structure method is adopted to obtain the approximate joint characteristic function of log-price. Third, under the inverse Fourier transform, the approximate semi-analytical solutions of the key term are derived. Finally, the combination algorithm of discrete Laplace transform and Fourier transform is constructed to obtain the asymptotic solutions of two vulnerable powered options, and the effectiveness and stability of the pricing method are verified by numerical examples. Experimental results show that the model constructed in this paper can well reflect the laws of the real market, and the proposed method can deal with the pricing problem of vulnerable powered options efficiently.