Valuation of American Continuous-Installment Options

Abstract

We present three approaches to value American continuous-installment calls and puts and compare their computational precision. In an American continuous-installment option, the premium is paid continuously instead of up-front. At or before maturity, the holder may terminate payments by either exercising the option or stopping the option contract. Under the usual assumptions, we are able to construct an instantaneous riskless dynamic hedging portfolio and derive an inhomogeneous Black-Scholes partial differential equation for the initial value of this option. This key result allows us to derive valuation formulas for American continuous-installment options using the integral representation method and consequently to obtain closed-form formulas by approximating the optimal stopping and exercise boundaries as multipiece exponen- * We are grateful to Manfred Gilli, Henri Loubergé and Evis Këllezi for encouragements, suggestions and remarks. 1 tial functions. This process is compared to the finite-difference method to solve the inhomogeneous Black-Scholes PDE and a Monte Carlo approach.

A variational inequality arising from American installment call options pricing

“…10) Then we see that the value of the American continuous-installment call option C (S, t) satisfies (A.6), (A.8)-(A.10), that is (1.1). Moreover, from the smooth fit conditions [9], we know that C , ∂ S C are continuous.…”

“…There are some papers about install options, such as [3,6,10]. Particularly, there are a free boundary model in [10] and some numerical results about the model.…”

“…Particularly, there are a free boundary model in [10] and some numerical results about the model. In the following, we deduce a parabolic variational inequality model.…”

Efficient Numerical Valuation of Continuous Installment Options

“…There only exists a few related works. Ciurlia and Roko [6] studied the American case applying the multipiece exponential function (MEF) method to derive an integral form of the option's value. Their applied technique suffers from a serious drawback, since the MEF method generates a discontinuity in the optimal stopping and early exercise boundaries.…”

Valuation of European Installment Put Option: Variational Inequality Approach

“…There are some papers about installment options, such as [5,6,9,12], in which the authors develop some models and consider numerical analysis. Moreover, we had considered the free boundary problems about American installment call option in [14] and European installment call option in [15].…”