Agency Costs, Risk Management, and Capital Structure

This 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.

Valuation of European continuous-installment options

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.

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

In order to pricing American style continuous-installment options,hedging was used to establish a Partial Differential Equation(PDE) model for call options.The PDE in the model is a nonhomogeneous Black-Scholes equation.The nonhomogeneous term is the installment rate.In the financial field,the problem can be expressed as a free boundary problem with two free boundaries: the optimal stop boundary and the optimal exercise boundary.Then the PDE model can be transformed into a variation inequality problem.By using the finite-difference method,some numerical results were obtained and displayed.The optimal stop boundary and the optimal exercise boundary were calculated and analyzed with different parameter values.The results show that the optimal stopping boundarg of call option is monotone and not decreasing,while the optimal exercise boundary is monotone and not creasing.

Valuation of American continuous-installment options

In 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.

An integral representation approach for valuing American-style installment options with continuous payment plan

A perpetual continuous-installment option is an infinite maturity option in which the premium is paid continuously instead of upfront. 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 continuousinstallment call option.

Caperdoni Camilla

Papers

A note on the pricing of perpetual continuous-installment options