# Price of a Forward Contract

I have the following,

Let $${F_t,t\geq0}$$ be the price process of the forward contract on the risky asset with maturity $$T' > 0$$. Since interest rates are deterministic, we have

$$F_t=S_t\ e^{r(T^\prime-t)}\ =F_0\ e^{-\frac{1}{2}\sigma^2t+\sigma B_t}$$

Why do we have $$-\frac{1}{2}\sigma^2t+\sigma B_t$$ in the exponent? Where did it come from? If the price of the underlying $$S_t$$ is stochastic, why don't we have a stochastic term in the middle equation $$S_t\ e^{r(T^\prime-t)}$$?

• In the BS model, we assume $dS=rSdt+\sigma SdB_t$. Itô’s Lemma implies $S_t=S_0e^{\left(r-0.5\sigma^2\right)t+\sigma B_t}$. That’s just true if the stock price follows a geometric Brownian motion. Oct 16 '20 at 6:08
• Once you substitute $S_t$ with the expression KeSchn gave, and then rewrite $S_0$ in terms of $F_0$ you will have your result. Oct 16 '20 at 8:24
• Where is this proposition taken from please? Does it mention conditioning/filtration etc ? Oct 16 '20 at 22:31