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. -

Ciurlia, P. (2011). On a general class of free boundary problems for European-style installment options with continuous payment plan. COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 10, 1205-1224 [10.3934/cpaa.2011.10.1205].

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

#### 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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2011
Ciurlia, P. (2011). On a general class of free boundary problems for European-style installment options with continuous payment plan. COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 10, 1205-1224 [10.3934/cpaa.2011.10.1205].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11590/124909`
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