In 1973, Fischer Black and Myron Scholes published their groundbreaking paper the pricing of options and corporate liabilities. Not only did this specify the first successful options pricing formula, but it also described a general framework for pricing other derivative instruments. That paper launched the field of financial engineering. Black and Scholes had a very hard time getting that paper published. Eventually, it took the intersession of Eugene Fama and Merton Miller to get it accepted by the Journal of Political Economy. In the mean time, Black and Scholes had published in the Journal of Finance a more accessible (1972) paper that cited the as-yet unpublished (1973) option pricing formula in an empirical analysis of current options trading. The Black-Scholes (1973) option pricing formula prices European put or call options on a stock that does not pay a dividend or make other distributions. The formula assumes the underlying stock price follows a geometric Brownian motion with constant volatility. It is historically significant as the original option pricing formula published by Black and Scholes in their landmark (1973) paper. Values for a call price c or put price p are:
where:
Here, log denotes the natural logarithm, and:
Consider a European call option on 100 shares of non-dividend-paying stock ABC. The option is struck at USD 55 and expires in .34 years. ABC is trading at USD 56.25 and has 28% (that is .28) implied volatility. The continuously compounded risk free interest rate is .0285. Applying formula [1], the option's market value per share of ABC is USD 4.56. Since the call is for 100 shares, its total value is USD 456. Of this, USD 125 is intrinsic value, and USD 331 is time value. The Greeks—delta, gamma, vega, theta and rho—for a call are:
where
Note that gamma formulas [6] and [11]are identical for puts and calls, as are vega formulas [7] and [12].
copyright © Glyn A. Holton, 1996, 2010 |
s = the price of the
denotes the standard
normal probability density function. For a put, the Greeks are:
