Academic journal article Journal of International Business Research

A Study on the Sensitivity of Stock Options' Premium to Changes in the Underlying Stock's Dividend Yield

Academic journal article Journal of International Business Research

A Study on the Sensitivity of Stock Options' Premium to Changes in the Underlying Stock's Dividend Yield

Article excerpt

INTRODUCTION

Financial derivatives have been an integral part of the financial markets in the last few decades. Most traders use these derivatives to hedge their existing open positions, reducing their losses whenever the value of the underlying asset should move against their speculative position. Others would use these derivatives to have a determined cash flow in the future, hence reducing the volatility of the overall portfolio.

One of the most popular derivatives today used in the financial markets are options. The unique property of this derivative is that it allows the holder (buyer) of the option to walk away from the contract whenever the situation in the market should become unfavorable while being able to exercise it whenever profitable. The writer (seller) of the option of course is bound to fulfill his end of the bargain regardless of the profit or loss.

The success and popularity of financial options began in 1973 when the Chicago Board Options Exchange (CBOE) came into operations which standardized stock options in terms of maturity and exercise price (Gemmill 1993). During the same year, the option pricing model popularized by Fischer Black and Myron Scholes,

Call(S, t) = SN ([d.sub.1]) - [Xe.sup.-rt]N ([d.sub.2])

Put(S, t) = -SN([-d.sub.1]) + [Xe.sup.-rt]N ([-d.sub.2]) (1)

where

[d.sub.1] = ln(S/X) + (r + 1/2 [[sigma].sup.2])t/[sigma][square root of t] and [d.sub.2] = ln(S/X) + (r - 1/2[[sigma].sup.2])t/[sigma][square root of t] = [d.sub.1] - [sigma][square root of t] (2)

was also publicized which helped in trading of stock options. However one of the biggest shortcomings of the said model at that time was the assumption that the underlying stocks do not pay any dividends. A common adjustment to this model then was to subtract the discounted value of future dividends from the stock price.

A few years later, the model was extended to accommodate more financial market products. This included Fischer Black's extension of the said model to incorporate futures options (1976) and that of Mark Garman and Steven Kohlhagen's extension of the same model to accommodate currency options (1983) and was later on applied to accommodate commodity options as well. Merton later extended the Black-Scholes model to accommodate stock options whose underlying stocks are paying dividends by introducing dividend yield ([delta]) to the formula.

Call(S, q, t) = [Se.sup.-qt]N ([d.sub.1]) - [Xe.sup.-rt]N([d.sub.2])

Put(S, q, t) = -[Se.sup.-qt]N (-[d.sub.1]) + [Xe.sup.-rt]N (-[d.sub.2]) (3)

where,

[d.sub.1] = ln(S/X) + (r - q + 1/2 [[sigma].sup.2])t/[sigma][square root of t] and [d.sub.2] = ln(S/X) + (r - q - 1/2 [[sigma].sup.2])t/[sigma][square root of t] = [d.sub.1] - [sigma][square root of t] (4)

This new model assumes that the dividends are paid out at a continuously compounding rate. While this assumption is imperfect, it is usually a reasonable approximation since it would encompass most types of dividend payment frequencies.

From the development of the Black-Scholes model, finance professionals began to analyze the model, specifically on the sensitivity of the model based on changes of the variables of the underlying asset. This formula (equation 1) is clearly a function of five variables: 1) the price of the underlying asset (S), 2) the exercise price (X), 3) the volatility of the underlying asset ([sigma]), 4) the remaining time until the expiration of the options contract (t) in years or a fraction of years, and 5) the risk-free rate (r).

Options sensitivities are the partial derivatives of the generalized Black-Scholes formula which give the sensitivity of the option price to small movements in the five previously mentioned variables of the underlying asset (Haug 1998). These sensitivities are known as "Greeks" and the most widely used are: 1) sensitivity of the option to changes in the underlying asset (Delta), 2) sensitivity of Delta to the movements of the underlying asset (Gamma), 3) sensitivity of the option to changes in volatility (Vega), 4) change in the option's price to the passage of time or time decay (Theta), and 5) the sensitivity of the option to changes in the risk free rate (Rho). …

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