Modeling and Pricing European-Style Continuous-Installment Option Under the Heston Stochastic Volatility Model: A PDE Approach

Abstract

Installment options, as path-dependent contingent claims, involve paying the premium discretely or continuously in installments, rather than as a lump sum at the time of purchase. In this paper, we applied the PDE approach to price European continuous-installment option and consider Heston stochastic volatility model for the dynamics of the underlying asset. We proved the existence and uniqueness of the weak solution for our pricing problem based on the two-dimensional finite element method. Due to the flexibility to continue or stop paying installments, installment options pricing can be modeled as an optimal stopping time problem. This problem is formulated as an equivalent free boundary problem and then as a linear complementarity problem (LCP). We wrote the resulted LCP in the form of a variational inequality and used the finite element method for the discretization. Then the resulting time-dependent LCPs are solved by using a projected successive over relaxation iteration method. Finally, we implemented our numerical method. The numerical results verified the efficiency and accuracy of the proposed numerical method.