2
$\begingroup$

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).

Improve this question
$\endgroup$

1 Answer 1

Reset to default
2
$\begingroup$

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.

Improve this answer
$\endgroup$
4
  • $\begingroup$ when you mention the asset price probability distribution, do you mean, in this case, the underlying futures contracts or the the underlying asset of the futures?? $\endgroup$
    – Joao Serafim
    Aug 16, 2013 at 11:44
  • $\begingroup$ @JoaoSerafim, I meant the asset on which the option is written, which could be a foreign exchange rate, a futures price or an interest rate,...please note that the risk-adjusted probability distribution of the above mentioned product has to be lognormal. $\endgroup$
    – Matt Wolf
    Aug 16, 2013 at 14:00
  • $\begingroup$ I see. I have another question relted to this subject. Let's suppose we have a future contract F in a market where the relation F(t,T)=S(t)er(T−t) doesn't hold. What are the the boundary conditions for the drivation of the Black formula?? $\endgroup$
    – Joao Serafim
    Aug 19, 2013 at 14:17
  • $\begingroup$ I've created a new post for this question. $\endgroup$
    – Joao Serafim
    Aug 20, 2013 at 4:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.