Valuation of European continuous-installment options

Abstract

This paper is concerned with the valuation of European continuous-installment options where the aim is to determine the initial premium given a constant installment payment plan. The distinctive feature of this pricing problem is the determination, along with the initial premium, of an optimal stopping boundary since the option holder has the right to stop making installment payments at any time before maturity. Given that the initial premium function of this option is governed by an inhomogeneous Black–Scholes partial differential equation, we can obtain two alternative characterizations of the European continuous-installment option pricing problem, for which no closed-form solution is available. First, we formulate the pricing problem as a free boundary problem and using the integral representation method, we derive integral expressions for both the initial premium and the optimal stopping boundary. Next, we use the linear complementarity formulation of the pricing problem for determining the initial premium and the early stopping curve implicitly with a finite difference scheme. Finally, the pricing problem is posed as an optimal stopping problem and then implemented by a Monte Carlo approach.

A finite element method for pricing of continuous-installment options under a Markov-modulated model: existence, uniqueness, and stability of solutions

“…In the numerical valuation of the European continuous-installment options, we consider following papers: Using Laplace-Carson transform, Kimura [23] solved the integral representation arising from the pricing problem of European continuous-installment call and put options written on assets with dividends by PDE approach and obtained a closed form for the stopping boundary. Ciurlia [10] priced European continuous-installment options with constant continuous dividend under Black-Scholes model and applied Monte Carlo approach to the integral representation for both the initial premium and the optimal stopping boundary. Recently Beiranvand et al [3] applied penalty method for pricing problem of European continuous-installment option written on the underlying asset which pays constant dividend.…”

Numerical approach for coupled systems resulting from pricing of derivatives: Modeling and pricing of installment options

“…In the numerical valuation of the European continuous‐installment options, we consider the following papers: Using Laplace–Carson transform, Kimura 12 solved the integral representation arising from the pricing problem of European continuous‐installment call and put options written on assets with dividends by PDE approach and obtained a closed‐form for the stopping boundary. Ciurlia 13 priced European continuous‐installment options with constant continuous dividend under the Black–Scholes model and applied the Monte Carlo approach to the integral representation for both the initial premium and the optimal stopping boundary. Recently, Beiranvand et al 14 applied a penalty method for the pricing problem of the European continuous‐installment option written on the underlying asset which pays a constant dividend.…”