scholarly journals Analysis of the Optimal Exercise Boundary of American Options for Jump Diffusions

2009 ◽  
Vol 41 (2) ◽  
pp. 825-860 ◽  
Author(s):  
Erhan Bayraktar ◽  
Hao Xing
Keyword(s):  
American Options ◽  
Jump Diffusions ◽  
Optimal Exercise Boundary ◽  
Exercise Boundary

scholarly journals Evaluating approximations to the optimal exercise boundary for American options

2002 ◽  
Vol 2 (2) ◽  
pp. 71-92 ◽  
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.

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

Mathematics ◽  
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.

The Sensitivity of American Options to Suboptimal Exercise Strategies

2010 ◽  
Vol 45 (6) ◽  
pp. 1563-1590 ◽  
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.

A NOVEL NUMERICAL APPROACH OF COMPUTING AMERICAN OPTION

2002 ◽  
Vol 13 (05) ◽  
pp. 685-693 ◽  
Author(s):  
Jiwu SHU ◽  
Yonggeng GU ◽  
Weimin ZHENG
Keyword(s):  
Numerical Methods ◽  
American Options ◽  
Closed Form Solution ◽  
Numerical Approach ◽  
American Option ◽  
Form Solution ◽  
American Option Pricing ◽  
Put Options ◽  
Optimal Exercise Boundary ◽  
Exercise Boundary

It is well acknowledged that the European options can be valued by an analytic formula, but situation is quite different for the American options. Mathematically, the Black-Scholes model for the American option pricing is a free boundary problem of partial differential equation. This model is a non-linear problem; it has no closed form solution. Although approximate solutions may be obtained by some numerical methods, but the precision and stability are hard to control since they are largely affected by the singularity at the exercise boundary near expiration date. In this paper, we propose a new numerical method, namely SDA, to solve the pricing problem of the American options. Our new method combines the advantages of the Semi-analytical Method and the Sliced-fixed Boundary Finite Difference Method while overcomes demerits of the two. Using the SDA method, we can resolve the problems resulted from the singularity near the optimal exercise boundary. Numerical experiments show that the SDA method is more accurate and more stable than other numerical methods. In this paper, we focus on the American put options, but the proposed method is also applicable to other types of options.

Perpetual Options and the Leland Model

2017 ◽  
Author(s):  
Kerry E. Back
Keyword(s):  
Cash Flows ◽  
Hitting Time ◽  
Optimal Time ◽  
Trade Off ◽  
Optimal Amount ◽  
Debt Burden ◽  
Optimal Exercise Boundary ◽  
Exercise Boundary

Perpetual options are time‐independent, so the fundamental PDE is actually an ODE. The optimal exercise boundary can be found by directly optimizing over the boundary or by using smooth pasting. The chapter explains the pricing of perpetual calls, perpetual puts, securities that pay a given amount at a hitting time, securities that pay at a hitting time but are knocked out if another boundary is hit first, and securities that pay cash flows continuously prior to a hitting time. The valuation results are applied to analyze the optimal bankruptcy time of a firm with a given debt burden, the optimal amount of debt for the firm, and the optimal time to take on more debt when debt is perpetual (Leland’s model of the trade‐off theory). Finite maturity debt is briey discussed.

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