# Black (1976) model: boundary conditions with non-convergence of spot and forward prices

Let's suppose we have a futures contract F in a market where the relation

$$F(t,T)=S(t)e^{r(T−t)}$$ doesn't hold.

What are the the boundary conditions for the derivation of the Black (1976) formula??

• @MattWolf: I probably mislead you with the title. Imagine we have convergence but $F(t,T)=S(t)\exp (r(T−t))$ does not hold. From what was told to me, we can still apply the Black model. Therefore, I want to know how can we specify yhe boundary conditions. – Joao Serafim Aug 20 '13 at 10:41
• I see. But if we have the Black formula $C=\exp(-r(T-t))(FN(d_1)-KN(d_2))$, does this mean we are assuming the relation $F(t,T)=S(t)\exp(r(T−t))$ is verified? or can we get the exact same expression for C without using that relation? – Joao Serafim Aug 20 '13 at 14:11
• @JoaoSerafim please use the math formatting feature of the site it really makes quesations and comments more readable. – SRKX Aug 20 '13 at 15:50
• @SRKX Ok. I will do that. – Joao Serafim Aug 20 '13 at 18:42
• @MattWolf: I'm sorry but I'm getting a little bit confused. I'm going to reformulate my question. For example, in the context of stock markets, the price for an european call option on futures is given by the previous $C$ expression. Now let's imagine we are going to study another market and in this market the relation doesn't hold. Is it possible that the price of the options (in that new market) is still given by the exact $C$ expression? – Joao Serafim Aug 22 '13 at 12:01