Data Analysis and Related Applications 3

7Properties of American Options Under a Semi-Markov Modulated Black-Scholes Model

In this research, we consider the valuation of the American option when the asset price dynamics follow a geometric Brownian motion driven by a varying economic situation. We assume that the economic situation transits based on a semi-Markov chain, where the option will be evaluated at each epoch of economic situation transition. The decision-maker can decide whether to early exercise or hold the option, based on the asset price, economic situation and holding time. The decision-making problem is formulated using a semi-Markov decision process. Based on simulation results, some properties on the optimal exercise regions and the monotonicity of option prices are discussed. In addition, the conditions of the transition probability matrix for maintaining the monotonicity of the option price are examined.

7.1. Introduction

A financial option is a derivative traded in a market which gives its holder a right, but not the obligation, to buy or sell an underlying asset at a predetermined strike price. If the holder has the right to buy, it is called the call option, and if the holder has the right to sell, it is a put option. Options have a maturity that enables the exercising of rights. Typically, they are divided into European options that can be exercised only on their maturity, and American options that can be exercised at any time before their maturity.

Option prices mainly fluctuate depending ...