Black’s Model Does Not Require the Spot–Forward Carry Relation

Summary

The note explains that the Black model does not depend on the standard spot-to-forward relationship in which the forward price equals spot grown at the risk-free rate. Currency futures are given as an example where borrowing and lending in the two currencies produce a different relationship. The key condition described for applying Black to European futures options is that the asset price distribution, when weighted by the pricing kernel, is log-normal.

This condition does not require the asset price distribution by itself to be log-normal. The answer points to applications involving European options on stocks, bonds, and some interest-rate products, and describes the Libor Market Model as a special case of the Black framework. The note is a brief conceptual explanation rather than a derivation: it gives no assumptions in detail, proofs, or calibration procedure. Its claims concern whether the pricing framework can apply, not whether any particular market’s prices or dynamics satisfy the required condition.

Key ideas

• Black pricing need not rely on the standard spot–forward carry relationship.
• For European futures options, the stated requirement is log-normality of the asset distribution weighted by the pricing kernel.
• The asset price distribution alone need not be log-normal.
• The note identifies stock, bond, and some interest-rate options as possible applications.

Tags

# Black (1976) model: relationship between spot and forward prices


# Black (1976) model: relationship between spot and forward prices












Does the Black (1976) model require the existence of the relation $F(t,T)=S(t)e^{r(T−t)}$?

I studied the derivation of the Black-Scholes formula. However, although I know the Black formula, I've never studied its entire derivation process. And probably the easiest way to justify the formula is using that relationship (something that motivates the previous question).

## Answer by Matt Wolf (score 2)

https://quant.stackexchange.com/a/8777

No, the Black model does not require this relationship to hold. For example, futures on currencies exhibit a different relationship between the future and spot price because of the interest debit/credit nature of currency borrowed/lent. However for the Black model on European futures options to hold the following condition has to be met:

The product of the asset price probability distribution and the pricing kernel has to be log-normal.

It can be shown that the asset price probability distribution does not necessarily have to be log-normal but the above has to hold to price, for example specific stock options (such as European spot options), bond options (where bond prices are log-normal), and some of the interest rate options (where such interest rates follow a MSS-BGM process). The Libor Market Model (LMM) is actually just a special case of the Black model.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.