A challenge in pricing options on commodities is non-randomness in the evolution of many commodity prices. For example, the spot price of an agricultural product will generally rise prior to a harvest and fall following the harvest. Natural gas tends to be more expensive during Winter months than Summer months. Because of such non-randomness, many spot commodity prices cannot be modeled with a geometric Brownian motion, and the Black-Scholes (1973) or Merton (1973) models for options on stocks do not apply.
In 1976, Fischer Black published a paper addressing this problem. His solution was to model forward prices as opposed to spot prices. Forward prices do not exhibit the same non-randomness of spot prices. Consider a forward price for delivery shortly after a harvest of an agricultural product. Prior to the harvest, the spot price may be high, reflecting depleted supplies of the product, but the forward price will not be high. Because it is for delivery after the harvest, it will be low in anticipation of a drop in prices following the harvest. While it is not reasonable to model the spot price with a Brownian motion, it may be reasonable to model the forward price with one. Black’s (1976) option pricing formula reflects this solution, modeling a forward price as an underlier in place of a spot price. The model is widely used for modeling European options on physical commodities, forwards or futures. It is also used for pricing interest rate caps and floors. The model is popularly known as Black ’76 or simply Black’s model.
Values for a call price c or put price p are:
[1]
[2]
where:
[3]
[4]
Here, log denotes the natural logarithm, and:
- f = the current underlying forward price
- x = the strike price
- r = the continuously compounded risk free interest rate
- t = the time in years until the expiration of the option
- σ = the implied volatility for the underlying forward price
- Φ = the standard normal cumulative distribution function.
The Greeks—delta, gamma, vega, theta and rho—for a call are:
[5]
[6]
[7]
[8]
[9]
where ϕ denotes the standard normal probability density function and c is call price [1]. For a put, the Greeks are:
[10]
[11]
[12]
[13]
[14]
where p is put price [2].
Note that gamma formulas [6] and [11] are identical for puts and calls, as are vega formulas [7] and [12].
References
- Black, Fischer (1976). The pricing of commodity contracts, Journal of Financial Economics, 3, 167-179.
- Black, Fischer and Myron S. Scholes (1973). The pricing of options and corporate liabilities, Journal of Political Economy, 81, 637-654.
- Merton, Robert C. (1973). Theory of rational option pricing, Bell Journal of Economics and Management Science, 4 (1), 141-183.