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

Pierangelo Ciurlia

doi:10.3934/cpaa.2011.10.1205

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2011, Volume 10, Issue 4: 1205-1224. Doi: 10.3934/cpaa.2011.10.1205

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

Pierangelo Ciurlia^1,

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

Received: June 2010

Revised: December 2010

Published: July 2011

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Abstract

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.

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Mathematics Subject Classification: Primary: 91B28, 35K20; Secondary: 42B10.

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