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\textbf{Question}

Give a short explanation of what is involved in a FORWARD CONTRACT and
explain briefly the major differences between a forward contract and a
future. Denoting the fair value of a forward contract by $F$, show
that
$$F = S(t_0) e^{r(T-t_0)}$$
where $t_0$ is the date on which the forward contract was greed, $T$
is the delivery date, $r$ is the interest rate and $S$ is the price of
the underlying asset.

Now suppose that we wish to value a forward contract using the
Black-Scholes equation
$$V_t + \frac{1}{2} \sigma^2 S^2 V_{SS} + r SV_S -rV =0.$$

Show that this equation has solutions of the form
$$V(S,t) + AS + Bf(t)$$
where $A$ and $B$ are constants and $f$ is to be determined. By using
this solution along with the condition that must be satisfied at $t=T$
show that
$$V(S,t)+S - Fe^{-r(T-t)}$$
and hence again show that
$$F = S(t_0)e^{r(T-t_0)}.$$

An OPTION ON A FUTURE has a value $V(F,t)$ where
$F=Se^{-r(T-t)}$. Show from the Black-Scholes equation that $V$
satisfies
$$V_t + \frac{1}{2}\sigma^2 F^2 V_{FF} - rV = 0.$$  

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\textbf{Answer}

A FORWARD CONTRACT is an agreement between two parties in which one
agrees to but a specific asset from the other at a given agreed
``forward price'' at a specific ``delivery date''. No money changes
hands until the delivery date and at that date, BOTH parties are
committed to doing the deal.

A future is just the same as a forward contract, except
\begin{description}
\item{(a)}
rather than being set up between 2 disparate parties, futures are
traded on specific exchanges and are subject to the rules and customs
of that exchange
\item{(b)}
the payoff (from whoever to whoever) is evaluated and paid at regular
intervals, rather than just at the delivery date.
\end{description}

Now let $F$ denote the fair value of a forward contract. Then if the
contract was agreed at $t_0$ the party who must deliver the asset at
$T$ can buy the asset (cost $S(t_0)$) having borrowed the money to do
so from the bank.

To borrow $S(t_0)$ for a period $T-t_0$ at interest rate $r$ costs
$S(t_0) e^{r(T-t_0)}$ and thus
$$F=S(t_0) e^{r(T-t_0)}$$

Now consider Black-Scholes
$$V-t+ \frac{1}{2} \sigma^2 S^2 V_{SS} + rSV_S -rV=0.$$
Try for solution
$$V(S,t)=AS+Bf(t)$$
\begin{eqnarray*}
\Rightarrow Bf' +\frac{1}{2} \sigma^2 S^2(0) +rSA - r(AS+Bf) & = & 0\\
Bf'+ rSA -rAS - rBf & = & 0 \\ \Rightarrow f' & = & rf.
\end{eqnarray*}  

Then $f=ke^{rt}$ where k is a constant (which can obviously be
absorbed into B).

Thus
$$V(S,t) = AS+Be^{rt}$$
Now at $r=T$ (expiry) we must have $V(S,T)=S-F$ (the value must be
what it's worth now minus what we paid for it.
$$\Rightarrow AS+Be^{eT} = S-F$$
This must be true for ANY $S$ $\Rightarrow A=1$, $Be^{rT}=-F$
\begin{eqnarray*}
\Rightarrow B & = & -Fe^{-rT}\\
\Rightarrow V(S,t) & = & S - Fe^{-r(T-t)}
\end{eqnarray*}

Now at $t_0$ the value is $0$ (no money is paid until expiry)

$\Rightarrow V(S,t_0) = 0 = S(t_0) -Fe^{-r(T-t_0)}$

and so again
$$F=S(t_0)e^{r(T-t_0)}$$

Now $V=V(F,t)$ where $F=Se^{r(T-t)}$
\begin{eqnarray*}
\Rightarrow V_t & \to & V_t +F_tV_F = V_t -rFV_F\\
V_S & \to & V_tt_s+ V_FF_S = e^{r(T-t)}V_F\\
V_{SS} & \to & e^{2r(T-t)}V_{FF}
\end{eqnarray*}

So Black-Scholes becomes
\begin{eqnarray*}
V_t+ rFV_F + \frac{\sigma^2}{2}S^2 e^{2r(T-t)} V_{FF} + rS
e^{r(T-t)}V_F -rV & = & 0\\
V_t - rFV_F + \frac{\sigma^2}{2} F^2 V_{FF} +rFV_F -rV & = & 0
\end{eqnarray*}
$$\Rightarrow V-t +\frac{\sigma^2}{2}F^2V_{FF} -rV = 0$$


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