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.

In this work, the analytical pricing of maximum and minimum options under the integrated scenario of interdependent stochastic volatility and jump, and the stochastic interest rate, were investigated by means of the composite Mellin transform approach. The analytic expressions of Mellin transform functions of the price of the maximum put option, minimum call option, and the exchange option were derived by different partial differential-integral equations (PDIEs). Meanwhile, the explicit price of other maximum and minimum options were obtained by means of the payoff decomposition technique and the parity relation of max-min and exchange options. In addition, the convergence of solutions of PDIEs was further demonstrated by transform techniques and decomposition skills. The simulation of the price process of two underlying assets was given to present the effectiveness and uniqueness of the proposed model. Finally, numerical analysis was implemented to examine the accuracy of the PDIE method and the validity of key parameters.