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 $S_t\ e^{r(T^\prime-t)}$?

  • 3
    $\begingroup$ 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. $\endgroup$
    – Kevin
    Oct 16 '20 at 6:08
  • $\begingroup$ 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. $\endgroup$
    – noob2
    Oct 16 '20 at 8:24
  • $\begingroup$ Where is this proposition taken from please? Does it mention conditioning/filtration etc ? $\endgroup$ Oct 16 '20 at 22:31

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