scholarly journals Transformation Methods for Evaluating Approximations to the Optimal Exercise Boundary for Linear and Nonlinear Black-Scholes Equations

2008 ◽  
Author(s):  
Daniel Sevcovic
Keyword(s):  
Optimal Exercise Boundary ◽  
Black Scholes ◽  
Exercise Boundary ◽  
Linear And Nonlinear ◽  
Transformation Methods

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.

scholarly journals Asymptotic analysis of American call options

2001 ◽  
Vol 27 (3) ◽  
pp. 177-188 ◽  
Author(s):  
Ghada Alobaidi ◽  
Roland Mallier
Keyword(s):  
Asymptotic Expression ◽  
Financial Derivatives ◽  
European Options ◽  
Call Options ◽  
Early Exercise ◽  
Optimal Exercise Boundary ◽  
Black Scholes ◽  
Exercise Boundary ◽  
A Free Boundary Problem ◽  
The Right

American call options are financial derivatives that give the holder the right but not the obligation to buy an underlying security at a pre-determined price. They differ from European options in that they may be exercised at any time prior to their expiration, rather than only at expiration. Their value is described by the Black-Scholes PDE together with a constraint that arises from the possibility of early exercise. This leads to a free boundary problem for the optimal exercise boundary, which determines whether or not it is beneficial for the holder to exercise the option prior to expiration. However, an exact solution cannot be found, and therefore by using asymptotic techniques employed in the study of boundary layers in fluid mechanics, we find an asymptotic expression for the location of the optimal exercise boundary and the value of the option near to expiration.

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.

scholarly journals Optimal Exercise Boundary of American Fractional Lookback Option in a Mixed Jump-Diffusion Fractional Brownian Motion Environment

2017 ◽  
Vol 2017 ◽  
pp. 1-17 ◽  
Author(s):  
Zhaoqiang Yang
Keyword(s):  
Brownian Motion ◽  
Stock Price ◽  
Jump Diffusion ◽  
Fundamental Solutions ◽  
Early Exercise ◽  
Optimal Exercise Boundary ◽  
Lookback Options ◽  
Exercise Price ◽  
Lookback Option ◽  
Exercise Boundary

A new framework for pricing the American fractional lookback option is developed in the case where the stock price follows a mixed jump-diffusion fraction Brownian motion. By using Itô formula and Wick-Itô-Skorohod integral a new market pricing model is built. The fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions. The explicit integral representation of early exercise premium and the critical exercise price are also given. Numerical simulation illustrates some notable features of American fractional lookback options.

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