Fischer Black & Myron Scholes are 2 economists, who in 1973 published a paper which redefined finance and derivatives, with "The Pricing of Options & Corporate Liabilities" featured in the Journal of Political Economy in May of that year. While the framework was developed earlier in the century, the piece is arguably one of the most important papers within finance theory and is used every day to price various derivatives, including options on commodities, financial assets and even pricing of employee stock options.
Following loosely on a PhD thesis written by University of Chicago student James Boness, they developed an analytical model which we now know as The Black-Scholes Pricing model, a widely used closed form solution to price European vanilla options and eventually extended as a framework to price other option types.
The Assumptions Underlying the Model
1. No dividends are paid out on the underlying stock during the option life. 2. The option can only be exercised at expiry (European characteristics) 3. Efficient markets (Market movements cannot be predicted) 4. Commissions are non-existent 5. Interest rates do not change over the life of the option (and are known) 6. Stock returns follow a lognormal distribution
The Model (Non-Dividend)
The basic inputs to price a European option on a non-dividend paying stock is as follows:
S = Underlying stock price X = Strike price r = Risk free rate of interest V = Volatility T-t = Time to maturity
We can then apply these input variables into the following set of equations to derive the price for a European call option on a non-dividend stock:
And for a European put option on a non-dividend stock:
Where N(d1) and N(d2) are the cumulative normal distribution functions for d1 and d2, which are defined as:
d2 can be further simplified as:
By means of substitution.
In order to compute the cumulative normal distribution function, we can consider the partial derivative of N(x).
We then apply the terms and to the equation and we obtain the solutions to the terms (as defined above). We will shortly discuss the Partial differential equations resulting in the Black-Scholes equation and its greeks.
The Model (Dividend Paying) - Merton (1973)
For a dividend paying stock, we can alter the standard Black-Scholes model to incorporate an annual dividend yield (extended by Merton in 1973) and include the term "d" (no-subscript) as being the dividend yield per year.
The value of a call option can be calculated as:
Where d1 and d2 equals:
And similarly, the non-dividend version of the model, we can simplify d2 as being:
The value of a put can be calculated using the put-call parity (for non dividend paying options):
or for dividend paying options:
Or with the full formula:
The work done by Black & Scholes in the 70's made way for further pricing of derivatives and in particular, exotic options. The Black-Scholes partial differential equation also enabled derivation of the 'greeks' of option pricing.
The Black-Scholes model today is used in everyday pricing of options and futures and almost all formulas for pricing of exotic options such as barriers, compounds and asian options take their foundation from the Black-Scholes model.
Additional/Useful List of resources
Papers:
Black, F. & Scholes, M. "The Pricing of Options & Corporate Liabilities" The Journal of Political Economy (May '73) Hull, J. "Options, Futures & Other Derivatives" 5th Edition 2002 - Chapter 12 Merton, R. "Theory of Rational Option Pricing" Bell Journal of Economics & Management (June '73)