Journal of Prime Research in Mathematics

# Continuity estimate of the optimal exercise boundary with respect to volatility for the american foreign exchange put option

Nasir Rehman
Department of Mathematics and Statistics, Allama Iqbal Open University, Islamabad, Pakistan.
Sultan Hussain
Department of Mathematics, COMSATS Institute of Information Technology, Abbotabad,
Pakistan.
Malkhaz Shashiashvili
Andrea Razmadze Mathematical Institute, Tbilisi, Georgia.

$$^{1}$$Corresponding Author: nasirzainy1@hotmail.com

### Abstract

In this paper we consider the Garman-Kohlhagen model for the American foreign exchange put option in one-dimensional diffusion model where the volatility and the domestic and foreign currency risk-free interest rates are constants. First we make preliminary estimate regarding the optimal exercise boundary of the American foreign exchange put option and then the continuity estimate with respect to volatility for the value functions of the corresponding options. Finally we establish the continuity estimate for the optimal exercise boundary of the American foreign exchange put option with respect to the volatility parameter.

#### Keywords:

Foreign exchange option, optimal exercise boundary, value function, volatility.