Black (1976) Option Pricing Formula

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The Black model sometimes known as the Black model is a variant of the Black—Scholes option pricing model. Its primary applications are for pricing options on future contractsbond optionsInterest rate cap and floorsand swaptions. It was first presented in a paper written by Fischer Black in Commodity option pricing formula model can be generalized into a class of models known as log-normal forward models, also referred to as LIBOR market model.

The Black formula is similar to the Black—Scholes formula for valuing stock options except that the spot price of the underlying is replaced by a discounted futures price F. Note that T' doesn't appear in the formulae even though it could be greater than T. This is because futures contracts are marked to market and so the payoff commodity option pricing formula realized when the option is exercised.

The difference in the two cases is clear from the derivation below. The Black formula is easily derived from the use of Margrabe's formulawhich in turn is a simple, but clever, application of the Black—Scholes formula. The payoff of the call option on the futures contract is max 0, F T - K. Then the call option is exercised at time T when the first asset is worth more than K riskless bonds. The assumptions of Margrabe's formula are satisfied with these assets.

The only remaining thing to check is that the first asset is indeed an asset. This can be seen by considering commodity option pricing formula portfolio formed at time 0 by going long a forward contract with delivery date T and short F 0 riskless bonds note that under the deterministic interest rate, the forward and futures prices are equal so there is no ambiguity here. Then at any time t you can unwind your obligation for the commodity option pricing formula contract by shorting another forward with the same delivery date to get the commodity option pricing formula in forward prices, but discounted to present value: From Wikipedia, the free encyclopedia.

This article includes a list of referencesrelated reading or external linksbut its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. March Learn how and when to remove this template message. Energy derivative Freight derivative Inflation derivative Property derivative Weather commodity option pricing formula. Retrieved from " https: Options finance Financial models. Articles lacking in-text citations from March All articles lacking in-text citations.

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