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.

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.

Russian options with a finite time horizon

2004 ◽  
Vol 41 (2) ◽  
pp. 313-326 ◽  
Author(s):  
Erik Ekström
Keyword(s):  
Asymptotic Behavior ◽  
Free Boundary ◽  
Boundary Problem ◽  
Optimal Stopping ◽  
Finite Time ◽  
Time Horizon ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Finite Time Horizon ◽  
Russian Option

We investigate the Russian option with a finite time horizon in the standard Black–Scholes model. The value of the option is shown to be a solution of a certain parabolic free boundary problem, and the optimal stopping boundary is shown to be continuous. Moreover, the asymptotic behavior of the optimal stopping boundary near expiration is studied.

scholarly journals On the optimal exercise boundary for an American put option

2001 ◽  
Vol 1 (1) ◽  
pp. 39-45 ◽  
Author(s):  
Ghada Alobaidi ◽  
Roland Mallier
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Asymptotic Expansions ◽  
American Put Option ◽  
Optimal Exercise Boundary ◽  
Put Option ◽  
Exercise Boundary ◽  
A Free Boundary Problem ◽  
The Right ◽  
American Put

An American put option is a derivative financial instrument that gives its holder the right but not the obligation to sell an underlying security at a pre-determined price. American options may be exercised at any time prior to expiry at the discretion of the holder, and the decision as to whether or not to exercise leads to a free boundary problem. In this paper, we examine the behavior of the free boundary close to expiry. Working directly with the underlying PDE, by using asymptotic expansions, we are able to deduce this behavior of the boundary in this limit.

PENALTY AMERICAN OPTIONS

2019 ◽  
Vol 22 (02) ◽  
pp. 1950001
Author(s):  
ZIWEI KE ◽  
JOANNA GOARD
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Numerical Experiments ◽  
Option Value ◽  
Analytic Approximations ◽  
Optimal Exercise Boundary ◽  
Exercise Boundary ◽  
A Free Boundary Problem ◽  
Short Times

We present a new American-style option whereby on the event of exercise before expiry, the holder pays the writer a fee (which will be referred to as a ‘penalty’). The valuation of the option is not straightforward as it involves determining when it is optimal for the holder to exercise the option, leading to a free boundary problem. As most options in the traded markets have short maturities, accurate and fast valuations of such options are important. We derive analytic approximations for the value of the option with short times to expiry (up to [Formula: see text] months) and its optimal exercise boundary. Some properties of the option, such as the put–call relationship, are explored as well. Numerical experiments suggest that our solutions both for the optimal exercise boundary and option value provide very accurate results.

scholarly journals A Simple Numerical Method for Pricing an American Put Option

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Beom Jin Kim ◽  
Yong-Ki Ma ◽  
Hi Jun Choe
Keyword(s):  
Free Boundary ◽  
Numerical Method ◽  
Quadratic Equation ◽  
Numerical Results ◽  
American Put Option ◽  
Optimal Exercise Boundary ◽  
Put Option ◽  
Exercise Boundary ◽  
American Put

We present a simple numerical method to find the optimal exercise boundary in an American put option. We formulate an intermediate function with the fixed free boundary that has Lipschitz character near optimal exercise boundary. Employing it, we can easily determine the optimal exercise boundary by solving a quadratic equation in time-recursive way. We also present several numerical results which illustrate a comparison to other methods.

Russian options with a finite time horizon

2004 ◽  
Vol 41 (02) ◽  
pp. 313-326 ◽  
Author(s):  
Erik Ekström
Keyword(s):  
Asymptotic Behavior ◽  
Free Boundary ◽  
Boundary Problem ◽  
Optimal Stopping ◽  
Finite Time ◽  
Time Horizon ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Finite Time Horizon ◽  
Russian Option

We investigate the Russian option with a finite time horizon in the standard Black–Scholes model. The value of the option is shown to be a solution of a certain parabolic free boundary problem, and the optimal stopping boundary is shown to be continuous. Moreover, the asymptotic behavior of the optimal stopping boundary near expiration is studied.

scholarly journals A Note on Numerical Solution of a Linear Black-Scholes Model

2014 ◽  
Vol 33 ◽  
pp. 103-115 ◽  
Author(s):  
Md. Kazi Salah Uddin ◽  
Mostak Ahmed ◽  
Samir Kumar Bhowmilk
Keyword(s):  
Time Integration ◽  
Weighted Average ◽  
Financial Mathematics ◽  
European Options ◽  
Average Scheme ◽  
Put Options ◽  
Weighted Average Method ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Solution Methods

Black-Scholes equation is a well known partial differential equation in financial mathematics. In this article we discuss about some solution methods for the Black Scholes model with the European options (Call and Put) analytically as well as numerically. We study a weighted average method using different weights for numerical approximations. In fact, we approximate the model using a finite difference scheme in space first followed by a weighted average scheme for the time integration. Then we present the numerical results for the European Call and Put options. Finally, we investigate some linear algebra solvers to compare the superiority of the solvers. GANIT J. Bangladesh Math. Soc. Vol. 33 (2013) 103-115 DOI: http://dx.doi.org/10.3329/ganit.v33i0.17664

scholarly journals Efficacy of Nifty Index Options through BSM Model

2019 ◽  
Vol 8 (3S2) ◽  
pp. 947-949
Keyword(s):  
Options Pricing ◽  
Price Determination ◽  
Strike Price ◽  
Index Options ◽  
Put Options ◽  
Black Scholes Model ◽  
Exercise Price ◽  
Black Scholes ◽  
Long Time ◽  
Optimum Price

Options are one of the products in financial derivatives, which gives the rights to buy and sell the product to an option holder in pre-fixed price which known as the strike price or exercise price at certain periods. Options contract was existed in various countries for long time, but it became very popular among the investors when the Fisher Black, Myron Scholes and Robert Merton were introduced the Black-Scholes Model in the year of 1973. This model was formerly developed by these three economists who were also receiving the Nobel prize for finding this innovative model. This model is mainly used to deal with the theoretical pricing challenge in options price determination. In India the trading in Index Options commenced on 4th June 2001 and Options on individual securities commenced on 2nd July 2001. There are many types in options contracts like stock options; Index options, weather options, real options and etc. This study has mainly been focusing on Nifty 50 index options which are effectively trade at NSE. This paper goes to describe about the importance of options pricing and how the BSM model has effectively used to find the optimum price of the theoretical value of call and put options.

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