An Efficient Exercise Boundary Search Algorithm for Valuing American Options

AbstractThis paper proposes an alternative characterization of the early exercise premium that is valid for any Markovian and diffusion underlying price process as well as for any parameterization of the exercise boundary. This new representation is shown to provide the best pricing alternative available in the literature for medium- and long-term American option contracts, under the constant elasticity of variance model. Moreover, the proposed pricing methodology is also extended easily to the valuation of American options on defaultable equity and possesses appropriate asymptotic properties.

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A NEW MONTE CARLO METHOD FOR AMERICAN OPTIONS

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024904002554 ◽

2004 ◽

Vol 07 (05) ◽

pp. 591-614 ◽

Cited By ~ 6

Author(s):

G. N. MILSTEIN ◽

O. REIß ◽

J. SCHOENMAKERS

Keyword(s):

Monte Carlo Simulation ◽

Monte Carlo ◽

Monte Carlo Method ◽

Boundary Value Problem ◽

Numerical Experiments ◽

American Options ◽

American Option ◽

Parabolic Boundary ◽

Black Scholes ◽

Exercise Boundary

We introduce a new Monte Carlo method for constructing the exercise boundary of an American option in a generalized Black–Scholes framework. Based on a known exercise boundary, it is shown how to price and hedge the American option by Monte Carlo simulation of suitable probabilistic representations in connection with the respective parabolic boundary value problem. The method presented is supported by numerical experiments.

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Evaluating approximations to the optimal exercise boundary for American options

Journal of Applied Mathematics ◽

10.1155/s1110757x02000268 ◽

2002 ◽

Vol 2 (2) ◽

pp. 71-92 ◽

Cited By ~ 7

Author(s):

Roland Mallier

Keyword(s):

Monte Carlo ◽

Monte Carlo Methods ◽

Asymptotic Approximation ◽

Series Solution ◽

American Options ◽

Expected Value ◽

Series Solutions ◽

Optimal Exercise Boundary ◽

Approximate Location ◽

Exercise Boundary

We consider series solutions for the location of the optimal exercise boundary of an American option close to expiry. By using Monte Carlo methods, we compute the expected value of an option if the holder uses the approximate location given by such a series as his exercise strategy, and compare this value to the actual value of the option. This gives an alternative method to evaluate approximations. We find the series solution for the call performs excellently under this criterion, even for large times, while the asymptotic approximation for the put is very good near to expiry but not so good further from expiry.

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On a Free Boundary Problem for American Options Under the Generalized Black–Scholes Model

Mathematics ◽

10.3390/math8091563 ◽

2020 ◽

Vol 8 (9) ◽

pp. 1563

Author(s):

Jung-Kyung Lee

Keyword(s):

Free Boundary ◽

Boundary Problem ◽

American Options ◽

Closed Form Solution ◽

Numerical Results ◽

Put Options ◽

Optimal Exercise Boundary ◽

Black Scholes Model ◽

Black Scholes ◽

Exercise Boundary

We consider the problem of pricing American options using the generalized Black–Scholes model. The generalized Black–Scholes model is a modified form of the standard Black–Scholes model with the effect of interest and consumption rates. In general, because the American option problem does not have an exact closed-form solution, some type of approximation is required. A simple numerical method for pricing American put options under the generalized Black–Scholes model is presented. The proposed method corresponds to a free boundary (also called an optimal exercise boundary) problem for a partial differential equation. We use a transformed function that has Lipschitz character near the optimal exercise boundary to determine the optimal exercise boundary. Numerical results indicating the performance of the proposed method are examined. Several numerical results are also presented that illustrate a comparison between our proposed method and others.

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ALTERNATIVE RANDOMIZATION FOR VALUING AMERICAN OPTIONS

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595910002624 ◽

2010 ◽

Vol 27 (02) ◽

pp. 167-187 ◽

Cited By ~ 9

Author(s):

TOSHIKAZU KIMURA

Keyword(s):

Numerical Experiments ◽

Recursive Algorithm ◽

American Options ◽

Random Variable ◽

Option Value ◽

Early Exercise ◽

Maturity Date ◽

Exercise Boundary ◽

Early Exercise Boundary ◽

Randomization Method

This paper deals with randomization methods for valuing American options written on dividend-paying assets, which are based on the idea of treating the maturity date as a random variable. In the randomization method introduced by Carr in 1998, he used the Erlangian distributed random variable to develop a recursive algorithm starting from the so-called Canadian option with an exponentially distributed random maturity. The purposes of this paper are (i) to provide much simpler pricing formulas for the Canadian option; (ii) to interpret the Gaver–Stehfest method developed for inverting Laplace transforms as an alternative randomization method in the context of valuing American options; and (iii) to evaluate the performance of the Gaver–Stehfest method in details with theoretical and numerical views. Numerical experiments indicate that the Gaver–Stehfest method works well to generate accurate approximations for the early exercise boundary as well as the option value.

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The Sensitivity of American Options to Suboptimal Exercise Strategies

Journal of Financial and Quantitative Analysis ◽

10.1017/s002210901000058x ◽

2010 ◽

Vol 45 (6) ◽

pp. 1563-1590 ◽

Cited By ~ 12

Author(s):

Alfredo Ibáñez ◽

Ioannis Paraskevopoulos

Keyword(s):

Risk Aversion ◽

American Options ◽

American Option ◽

Second Order ◽

Optimal Exercise Boundary ◽

Misspecified Model ◽

Exercise Price ◽

Exercise Boundary ◽

The Cost ◽

Suboptimal Behavior

AbstractThe value of American options depends on the exercise policy followed by option holders. Market frictions, risk aversion, or a misspecified model, for example, can result in suboptimal behavior. We study the sensitivity of American options to suboptimal exercise strategies. We show that this measure is given by the Gamma of the American option at the optimal exercise boundary. More precisely, “ifBis the optimal exercise price, but exercise is eitherbrought forward whenordelayed untila priceB̃has been reached, the cost of suboptimal exercise is given by ½ ×Γ(B) × (B−B̃)2, whereΓ(B) denotes the American option Gamma.” Therefore, the cost of suboptimal exercise is second-order in the bias of the exercise policy and depends on Gamma. This result provides new insights on American options.

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