The Derivatives-on-Line Pages

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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

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

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) $.

next up previous contents
Next: Coupons, Dividends and FX Up: Futures and Forwards Previous: Short selling   Contents
Kyriakos 2003-03-17