Valuation of American Continuous-Installment Options
01 Feb 2005-Computing in Economics and Finance (Kluwer Academic Publishers)-Vol. 25, Iss: 1, pp 143-165
TL;DR: In this paper, the authors presented three approaches to value American continuous-installment options written on assets without dividends or with continuous dividend yield, and derived closed-form formulas by approximating the optimal stopping and exercise boundaries as multipiece exponential functions, which is compared to the finite difference method to solve the inhomogeneous Black-Scholes PDE and a Monte Carlo approach.
Abstract: We present three approaches to value American continuous-installment options written on assets without dividends or with continuous dividend yield. In an American continuous-installment option, the premium is paid continuously instead of up-front. At or before maturity, the holder may terminate payments by either exercising the option or stopping the option contract. Under the usual assumptions, we are able to construct an instantaneous riskless dynamic hedging portfolio and derive an inhomogeneous Black--Scholes partial differential equation for the initial value of this option. This key result allows us to derive valuation formulas for American continuous-installment options using the integral representation method and consequently to obtain closed-form formulas by approximating the optimal stopping and exercise boundaries as multipiece exponential functions. This process is compared to the finite difference method to solve the inhomogeneous Black--Scholes PDE and a Monte Carlo approach.
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TL;DR: In this paper, the authors derive a closed-form solution to the value of an option in the Black-Scholes model and prove that the limiting case of an Instalment option with a continuous payment plan is equivalent to a portfolio consisting of a European Vanilla option and an American Put on this Vanilla option with time-dependent strike.
Abstract: In Foreign Exchange Markets Compound options (options on options) are traded frequently. Instalment options generalize the concept of Compound options as they allow the holder to prolong a Vanilla Call or Put option by paying instalments of a discrete payment plan. We derive a closed-form solution to the value of such an option in the Black-Scholes model and prove that the limiting case of an Instalment option with a continuous payment plan is equivalent to a portfolio consisting of a European Vanilla option and an American Put on this Vanilla option with a time-dependent strike.
32 citations
TL;DR: Valuing European continuous-installment options written on dividend-paying assets in the standard Black-Scholes-Merton framework using the Laplace transform approach, which results in explicit Laplace transforms of the initial premium as well as its Greeks, which include the transformed stopping boundary in a closed form.
28 citations
Cites background or methods from "Valuation of American Continuous-In..."
...From the standard argument of constructing the hedged portfolio consisting of one option and an amount −∂V ∂S of the underlying asset, we see that the initial premium V satisfies an inhomogeneous PDE ∂V ∂t + 1 2 σ2S2 ∂2V ∂S2 + (r − δ)S∂V ∂S − rV = q. (2.2) See Ciurlia and Roko (2005) for details....
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..., by the MEF method (Ju, 1998) just as in Ciurlia and Roko (2005). In this paper, however, we use an alternative approach based on Laplace transforms, which generates the transformed stopping boundary in a closed form; see Equation (3....
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...…matching condition, we also see that the optimal stopping boundary (St)t∈[0,T ] satisfies the integral equation c(t, St) − q ∫ T t e−r(u−t)Φ ( d−(St, Su, u − t) ) du = 0, (2.13) which can be solved numerically for (St)t∈[0,T ], e.g., by the MEF method (Ju, 1998) just as in Ciurlia and Roko (2005)....
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...For continuous-installment options, Ciurlia and Roko (2005) analyzed the American case approximately by applying the multipiece exponential function (MEF) method to an integral representation of the initial premium....
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TL;DR: This paper considers a parabolic variational inequality arising from European continuous installment call options pricing and proves the existence and uniqueness of the solution and obtains regularity and the bounds of the free boundary as $\tau=T-tarrow+\infty$.
Abstract: In this paper we consider a parabolic variational inequality arising from European continuous installment call options pricing and prove the existence and uniqueness of the solution to the problem. Moreover, we obtain $C^\infty$ regularity and the bounds of the free boundary, as well as the limit of the free boundary as $\tau=T-t\rightarrow+\infty$. Eventually we show its numerical result by the binomial method.
24 citations
TL;DR: In this article, a parabolic variational inequality with two free boundaries arising from American continuous-installment call options pricing is considered and the existence and uniqueness of the solution to the problem is proved.
18 citations
01 Jan 2009
TL;DR: In this article, the perpetual continuous-installment option pricing problem is discussed and solved as a free boundary problem for a parabolic inhomogeneous ordinary differential equation, and 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 PLS call option.
Abstract: 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.
14 citations
References
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12 Sep 2011
TL;DR: In this paper, the authors deduced a set of restrictions on option pricing formulas from the assumption that investors prefer more to less, which are necessary conditions for a formula to be consistent with a rational pricing theory.
Abstract: The long history of the theory of option pricing began in 1900 when the French mathematician Louis Bachelier deduced an option pricing formula based on the assumption that stock prices follow a Brownian motion with zero drift. Since that time, numerous researchers have contributed to the theory. The present paper begins by deducing a set of restrictions on option pricing formulas from the assumption that investors prefer more to less. These restrictions are necessary conditions for a formula to be consistent with a rational pricing theory. Attention is given to the problems created when dividends are paid on the underlying common stock and when the terms of the option contract can be changed explicitly by a change in exercise price or implicitly by a shift in the investment or capital structure policy of the firm. Since the deduced restrictions are not sufficient to uniquely determine an option pricing formula, additional assumptions are introduced to examine and extend the seminal Black-Scholes theory of option pricing. Explicit formulas for pricing both call and put options as well as for warrants and the new "down-and-out" option are derived. The effects of dividends and call provisions on the warrant price are examined. The possibilities for further extension of the theory to the pricing of corporate liabilities are discussed.
9,635 citations
TL;DR: In this paper, a new approach for approximating the value of American options by simulation is presented, using least squares to estimate the conditional expected payoff to the optionholder from continuation.
Abstract: This article presents a simple yet powerful new approach for approximating the value of American options by simulation. The key to this approach is the use of least squares to estimate the conditional expected payoff to the optionholder from continuation. This makes this approach readily applicable in path-dependent and multifactor situations where traditional finite difference techniques cannot be used. We illustrate this technique with several realistic examples including valuing an option when the underlying asset follows a jump-diffusion process and valuing an American swaption in a 20-factor string model of the term structure.
2,612 citations
2,605 citations
09 May 2001
TL;DR: In this article, a simple yet powerful new approach for approximating the value of American options by simulation is presented, based on the use of least squares to estimate the conditional expected payoff to the optionholder from continuation.
Abstract: This article presents a simple yet powerful new approach for approximating the value of American options by simulation. The key to this approach is the use of least squares to estimate the conditional expected payoff to the optionholder from continuation. This makes this approach readily applicable in path-dependent and multifactor situations where traditional finite difference techniqes cannot be used. We illustrate this technique with several realistic examples including valuing an option when the underlying asset follows a jump-diffusion process and valuing an American swaption in a 20-factor string model of the term structure.
2,602 citations
TL;DR: In this article, the authors applied the technique for valuing compound options to the risky coupon, bond problem and derived a formula which contains n-dimensional multivariate normal intecjrals.
Abstract: This paper applies the technique for valuing compound options to the risky coupon, bond problem. A formula is derived which contains n-dimensional multivariate normal intecjrals. It is shown that, for some compound option problems, the special correlation structure allows an application of an integral reduction which may simplify the numerical evaluation. The effects of various indenture restrictions on the formula are discussed, and a new formula for evaluating subordinated debt is presented.
901 citations
"Valuation of American Continuous-In..." refers methods in this paper
...Their analysis is restricted to European-style installment options, which allows for an analogy with compound options, previously considered in Geske (1977) and Selby and Hodges (1987)....
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