Financial Derivatives/Basic Derivatives Contracts – Forwards

Spot markets allow the purchase and sale of an asset today. By contract a forward contract specifies the price at which an asset can be purchased or sold at some future date. Although a forward contract is classified as a derivative in many markets it is difficult to distinguish between the underlying and the forward contract. Large trading volumes in OTC forwards can in fact make them more significant than spot markets.

A forward contract does not require upfront payment. It is simply the purchase or sale of an asset at some future date at a fixed price (the forward price). Therefore the assumption is that the forward price reflects the value of this asset on this date. If this assumption is based on a market view, characterising a forward contract as a derivative is misleading.

The primary reason for the classification of a forward contract as a derivative is that in many cases its price can be derived through a no-arbitrage argument that relates the forward price of an asset to its spot price. For assets like oil this is not possible; given the spot price of a barrel of oil it is not possible to construct an arbitrage argument that relates it to the forward price. In the oil markets forwards or futures are effectively the underlying and cannot be understood as derivatives. In these markets the forward price of oil is similar in nature to the price of a stock: it reflects the current consensus of the market and has nothing to do with risk-neutral valuation.

In financial markets forwards can be determined through a no-arbitrage argument. Consider for example a forward on the USD vs EUR exchange rate. If today one euro can be exchanged for 1.3 dollars (FX_spot) then in order to determine the forward exchange rate one year from now we can look at the following set of trades,

• We buy a one year forward that guarantees an exchange rate of FX_oneyear dollars per euro.
• We borrow one dollar today.
• We exchange it for (1/1.3) euros and invest this amount in a deposit account.
• After one year we withdraw the principal and the interest earned and exchange them into dollars at FX_oneyear.

The net cashflow of this trade at expiry is,

In the absence of arbitrage opportunities the net cashflow of this trade should be zero and therefore,

Another example is a forward contract on a zero coupon one year bond , one year from now. Given the price of a one year bond P_1year and a two year bond P_2year we look at the following set of trades,

• Sell a one year zero coupon bond one year from now at forward price P_1,1.
• Buy a one year zero coupon bond today.
• Sell a two year zero coupon bond today.

Since P_1year > P_2year we must borrow the difference. After one year we receive $1 from the one year bond and pay interest and principal on the amount borrowed. The two year bond has one year to maturity and we transfer it to the buyer of the forward in return for P_1,1. Therefore the net cashflow in one year is,

− (P_1year − P_2year)(1 + r_1year) + 1 + P_1,1

In the absence of arbitrage opportunities this cashflow must be zero. Since,

we conclude that,

P_1,1 = P_2year / P_1year

It is interesting to note that the formula,

is based on a "no-arbitrage" argument itself and the one year bond can be viewed as the "forward contract" for one dollar received in one year. Given the value of r_1year, if the price of the one year bond was different from 1 / (1 + r_1year) one could sell a one-year bond at a price P ^* > P_1year. At expiry one dollar would be paid to the buyer of the bond but since the proceeds from the sale would have earned P ^* (1 + r_1year) they would cover this payment and leave a clear profit. Only if P ^* = 1 / (1 + r_1year) = P_1year the condition of no arbitrage holds.