Oct
22
Modern option pricingtechniques are often considered among the most mathematically complexof all applied areas of finance. Financial analysts have reached thepoint where they are able to calculate, with alarming accuracy, thevalue of a stock option.Most of the models and techniques employeed bytoday's analysts are rooted in a model developed by Fischer Black andMyronScholes in 1973. This paper examines the evolution of optionpricing models leading up to and beyond Black and Scholes' model.
Inorder to understand the model itself, we divide it into two parts.Thefirst part, SN(d1), derives the expected benefit from acquiring astockoutright. This is found by multiplying stock price [S] by thechange inthe call premium with respect to a change in the underlyingstock price[N(d1)]. The second part of the model, Ke(-rt)N(d2), givesthe presentvalue of payingthe exercise price on the expiration day. The fairmarket value of thecall option is then calculated by taking thedifference between thesetwo parts.
C = theoreticalcall premium
S = current stockprice
t = time untiloption expiration
K = optionstriking price
r = risk-freeinterest rate
N =cumulativestandard normal distribution
e = exponentialterm
s = standarddeviation of stock returns
ln =natural logarithmInorder to understand the model itself, we divide it into two parts.Thefirst part, SN(d1), derives the expected benefit from acquiring astockoutright. This is found by multiplying stock price [S] by thechange inthe call premium with respect to a change in the underlyingstock price[N(d1)]. The second part of the model, Ke(-rt)N(d2), givesthe presentvalue of payingthe exercise price on the expiration day. The fairmarket value of thecall option is then calculated by taking thedifference between thesetwo parts.
