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On a general class of free boundary problems for European-style installment options with continuous payment plan
No full textIn 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.
Valuation of European continuous-installment options
No full textThis 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.
An integral representation approach for valuing American-style installment options with continuous payment plan
No full textIn this paper, we present an integral equation approach for the valuation of American-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 system of coupled recursive integral equations for the pair of free boundaries 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 American vanilla installment call options, an explicit pricing formula is obtained for time-varying payment schedules.
A note on the pricing of perpetual continuous-installment options
No full textA perpetual continuous-installment option is an infinite maturity option in which the premium is paid continuously instead of up-front. The holder has the right to terminate payments at any time by either exercising the option or dropping the option contract. Within the standard Black-Scholes framework, the perpetual continuous-installment option pricing problem is discussed and solved as a free boundary problem for a parabolic inhomogeneous ordinary differential equation. The closed-form solution obtained for the special case of a non-dividend paying asset gives the
possibility to observe some analytical properties of the initial premium and the optimal boundaries for the perpetual continuous-installment call option.
Metodologia generale per il calcolo dell'elemento di aiuto nei regimi di garanzia
No full textL’articolo, nell’ambito della valutazione degli aiuti di Stato concessi sotto forma di garanzia, mostra come calcolare in modo univoco l’elemento di aiuto nelle garanzie su prestiti fornite nel quadro di un regime di garanzie pubbliche a favore di piccole e medie imprese. Con specifico riferimento al Fondo di Garanzia per le PMI (Legge n. 662/1996), il principale organismo nazionale che facilita l’accesso al credito concedendo garanzie sui prestiti bancari, la finalità del presente lavoro è quella di sviluppare una nuova metodologia generale per il calcolo dell’Equivalente Sovvenzione Lordo (ESL) che, alla luce delle più recenti disposizioni normative, possa essere applicabile a tutte le tipologie previste di aiuto sotto forma di garanzia nonché consentire ulteriori sviluppi analitici
A note on the pricing of perpetual continuous-installment options
No full textA perpetual continuous-installment option is an infinite ma-
turity option in which the premium is paid continuously instead of up-
front. The holder has the right to terminate payments at any time by
either exercising the option or dropping the option contract. Within the
standard Black-Scholes framework, the perpetual continuous-installment
option pricing problem is discussed and solved as a free boundary prob-
lem for a parabolic inhomogeneous ordinary differential equation. The
closed-form solution obtained for the special case of a non-dividend pay-
ing asset gives the possibility to observe some analytical properties of the
initial premium and the optimal boundaries for the perpetual continuous-
installment call option
Esercizi di algebra lineare e sistemi di equazioni lineari con applicazioni all’economia
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