# American Institute of Mathematical Sciences

July  2011, 10(4): 1205-1224. doi: 10.3934/cpaa.2011.10.1205

## On a general class of free boundary problems for European-style installment options with continuous payment plan

 1 Department of Economics, Faculty of Economics "Federico Caffè", University of Rome III, Via Silvio D'Amico 77, 00145 Rome, Italy

Received  June 2010 Revised  December 2010 Published  April 2011

In this paper we present an integral equation approach for the valuation of European-style installment derivatives when the payment plan is assumed to be a continuous function of the asset price and time. The contribution of this study is threefold. First, we show that in the Black-Scholes model the option pricing problem can be formulated as a free boundary problem under very general conditions on payoff structure and payment schedule. Second, by applying a Fourier transform-based solution technique, we derive a recursive integral equation for the free boundary along with an analytic representation of the option price. Third, based on these results, we propose a unified framework which generalizes the existing methods and is capable of dealing with a wide range of monotonic payoff functions and continuous payment plans. Finally, by using the illustrative example of European vanilla installment call options, an explicit pricing formula is obtained for time-varying payment schedules.
Citation: Pierangelo Ciurlia. On a general class of free boundary problems for European-style installment options with continuous payment plan. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1205-1224. doi: 10.3934/cpaa.2011.10.1205
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