The forward price

In the remaining of this chapter we will maintain a number of assumptions:

1. There are no transaction costs;

2. The underlying security is traded in a market --thus excluding for example weather derivatives;

3. All investors are subject to the same tax rates;

4. The interest rate is them same for borrowing and lending; and

5. All arbitrage opportunities are immediately exploited.

Now suppose that today [time $t$] there is a forward contract on an asset that matures at time $\tau$. Let the spot price be denoted by $S\left( t\right)$ and the forward price by $F\left( t,\tau \right)$. The interest rate is considered constant and equal to $r$. We will conjecture that

$$F\left( t,\tau \right) =S\left( t\right) e^{r\left( \tau -t\right) }$$.

Suppose that an investor buys one unit of the asset, paying $S\left( t\right)$, and at the same time sell a forward contract, fixing the time of delivery at time $\tau$ and the price at $F\left( t,\tau \right)$. Since there is no risk involved in this strategy, the price $S\left( t\right)$ must be equal to the present value of the investment which is $F\left( t,\tau \right) e^{-r\left( \tau -t\right) }$. This will lead to the relationship above.

If the relationship does not hold, arbitrage opportunities arise. If $%% F\left( t,\tau \right) >S\left( t\right) e^{r\left( \tau -t\right) }$ the above strategy --buying the asset and shorting a forward, would generate a riskless profit. An investor could borrow the amount $S\left( t\right)$ at the risk free rate $r$ in order to initialize the transaction. This would still result into a riskless profit without any initial investment, the definition of an arbitrage opportunity.

If $F\left( t,\tau \right) <S\left( t\right) e^{r\left( \tau -t\right) }$, an investor would short sell the asset and buy a forward on it, an arbitrage. If short selling is not feasible, there would still be investors that currently hold the asset who would exploit this arbitrage opportunity. A person that holds the asset at time $t$ would sell it at $S\left( t\right)$, long a forward at $F\left( t,\tau \right)$, put the money in the bank for a period $\tau -t$ and use the proceedings $S\left( t\right) e^{r\left( \tau -t\right) }$ to repurchase the asset. She has started with one unit of the asset and has ended with a unit of the asset plus a riskless profit of $%% S\left( t\right) e^{r\left( \tau -t\right) }-F\left( t,\tau \right)$.