The Black Model
In 1976 Fischer Black made some minor modifications to the Black Scholes model to adapt its use for evaluating options on futures contracts. The model takes into consideration the fact that there are no financing costs related to a futures contract. This results in a lower option price than for a similar option on equity that does not pay a dividend. The reason is as follows: If you were to replicate a call option using a portfolio of stock and a risk free bond, the Black Scholes model assumes that you finance the stock purchase at the risk free rate. The cost of this loan is embedded in the price of the option. Since the financing cost for a futures contract is zero, the option no longer has to include this premium.
The following assumptions apply to the Black model:
- the option can only be exercised on the expiry date (European style);
- the underlying instrument does not pay dividends;
- there are no taxes, margins or transaction costs;
- the risk free interest rate is constant;
- the price volatility of the underlying instrument is constant; and
- the price movements of the underlying instrument follow a lognormal distribution.
The Black Formula
where
= 
= the theoretical value of a call
= the theoretical value of a put
= the price of the underlying futures contract
= the exercise price
= the time to expiration in years
= the annual volatility
= the risk free interest rate
= the base of the natural logarithm
= the natural logarithm
= the cumulative normal density function
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