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If the initial value of a forward contract is zero, surely the forward price used would be the spot price at the time the contract was created?

However, my notes tell me that the forward price F, at $t = 0$ for delivery at time $t = T$, is given by $F = e^{rT}S_{0}$ where r is the risk-free rate and $S_{0}$ is the spot price at $t = 0$.

Why is this?

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it's easiest to see in terms of replication. The pay-off of a forward contract is $$ S_T - K. $$ We can replicate this precisely and statically by buying one unit of stock, $S_0,$ and $Ke^{-rT}$ riskless bonds growing at rate $r.$

So its value today is $$ S_0 - Ke^{-rT}. $$ This has zero value if and only if $K= S_0 e^{rT}.$

This value is then called the forward price since it makes the forward contract have zero value.

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  • $\begingroup$ That makes a lot of sense, apart from the fact that $S_{0}$ is replicated by $Ke^{-rT}$. Could you please explain why this is? Thanks. $\endgroup$
    – M Smith
    Aug 11 '15 at 10:47
  • $\begingroup$ $S_T$ is replicated by one stock at cost $S_0$, $K$ is replicated by $Ke^{-rT}$ riskless bonds. $\endgroup$
    – Mark Joshi
    Aug 11 '15 at 11:25
  • $\begingroup$ So are all risk less bonds compounded continuously? $\endgroup$
    – M Smith
    Aug 11 '15 at 12:23
  • $\begingroup$ the argument only needs a zero coupon bond with expiry $T$, $r$ is then defined to makes its price $e^{-rT}.$ What happens between 0 and $T$ is irrelevant to the argument. $\endgroup$
    – Mark Joshi
    Aug 11 '15 at 21:16
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When you buy a forward you don't have to invest any money, so that's to your advantage in a world of positive interest rates. To charge you the same as the spot rate would be unfair, you would be "getting something for nothing", that is why the appropriate price for a forward is higher. It takes the interest rate into account, balancing things out. In equilibrium you are indifferent between a cheaper spot and a more expensive forward that frees up your money (which you can invest elsewhere).

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  • $\begingroup$ Forgive me for being dense, but I'm still struggling to understand how you're getting something for nothing. Surely prices have an equal chances of going up and down (in general) so someone who believed prices would fall would happily sell a forward contract at the spot price? $\endgroup$
    – M Smith
    Aug 10 '15 at 13:48
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    $\begingroup$ A forward is a derivative. Prices can go up or down, but the issuer of the derivative can completely hedge this risk, so it does not enter into the pricing. The pricing is determined by "cash and carry arbitrage", which I encourage you to look up. It is based in the spot price and the interest rate. $\endgroup$
    – noob2
    Aug 10 '15 at 14:04
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    $\begingroup$ @user2910074 Forward is priced by static hedging. I can easily buy the asset now and hold it until the maturity. Discounted back must be the price of the forward. $\endgroup$
    – SmallChess
    Aug 10 '15 at 14:08
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The forward price $F$ for a forward contract, determined at the contract inception time today, is the price that the holder will pay at maturity $T$ to buy the underlying equity. Then the payoff, at maturity $T$, of the forward contract is given by \begin{align*} S_T-F. \end{align*} The present value of the contract is then \begin{align*} e^{-rT} \mathbb{E}\big(S_T-F\big) = S_0 - e^{-rT} F. \end{align*} As the forward price $F$ is determined so that the value of the forward contract is zero, we have \begin{align*} F= S_0 e^{rT}. \end{align*}

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