Derivatives: Forwards Options Put-Call Parity Floors and Caps Swaps
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"Forwards"

Consider these scenarios: A farmer wishes to fix the sale price of his crops in advance, an importer arranges to buy foreign currency at a fixed rate in the future, a fund manager who wants to sell stocks for a price known in advance. What can the farmer, the importer or the fund manager do to address his problem?

Each one of them should enter into a forward contract to become independent of the unknown future price of his risky asset (the crops, the foreign currency, the stocks).

So, what exactly is a forward contract? Here comes the definition:

Definition (Forward Contract)
The holder of a long (short) forward contract has an agreement to buy (sell) an asset at a certain time in the future for a certain price, which is agreed upon today. The buyer (or seller) in a forward contract:
  • acquires a legal obligation to buy (or sell) an asset (known as the underlying asset)
  • at some specific future date (the expiration date)
  • at a price (the forward price) which is fixed today.
All this new vocabulary related to the forward contract should become clear by looking at an example:

Example (Forward Contract)
Our farmer wants to lock in a price of $ 50 for a load of crops. He is afraid that the price will drop and he will get less than his $ 50. So he entered into a forward contract (on the underlyer "load of crops") with the forward price of $ 50 and the expiration date 6 months from now (when harvesting begins).

Actually he is taking a short forward position because he benefits when his underlying asset (the load of crops) goes down in price: If the price drops to $ 40, he still has the right to charge $ 50 for his load. The buyer, e.g. a bakery has the long forward position (The bakery profits when the price of crops goes up) and looses in this scenario.

What happens if the price of crops goes up to $ 60. Here the farmer looses (he has the obligation to sell the crops for $ 50 when the market price is $ 60). On the other hand, the bakery wins for having to pay only $ 50 for the crops who are worth $ 60 now.



You can see that a forward contract is essentially a bet: The farmer bets that the price of the underlyer will not be higher than $ 50, the bakery has the opposite view.

Let us define a new term "payoff of the forward". It's what the forward contract is worth for each party at a moment in time.

In our example we have:



Let us denote the time when the forward contract agreement is achieved by 0, the expiration date by T, and the forward price by F(0,T). The time t market price of the underlying asset will be denoted by S(t). At expiration the party with a long forward position will benefit if F(0,T) < S(T). (The party can buy the underlying asset for F(0,T) and sell it for the higher market price S(T), making an instant profit of S(T) - F(0,T). Meanwhile the party holding a short forward position will suffer a loss of S(T) - F(0,T) because they will have to sell below the market price. If F(0,T) > S(T), then the situation is reversed.

If the simultaneous purchase and sale of an asset in order to profit from a difference in the price. leads to a difference of 0 (this is called "No-Arbitrage", you also can say that there is "no free lunch") then we can compute the prices of different forwards right away.

But at first we introduce some notation. Consider a forward contract established today at time 0. The contract expires at time T (an integer number presenting number of years). The forward price, which is the price agreed upon today, is denoted as F(0,T). The price of the underlying asset, known as the spot price, is S0 today, and ST at the expiration time and in-between at any point in time t St. The risk-free interest rate is rc, compounded annually.

Fair Forward Price on an Asset when there are no Cash Flows on the asset during the Life of the Contract

There are two alternative ways of owning an asset at time T:
  1. buy the asset now and hold it for the timeT (Alt.1)
  2. take a long position in a forward contract on the asset, that expires at time T (Alt.2)
B/c we have a "No-Arbitrage" situation, the price of 1. , which is S0 accumulated to time T, or S0(1+r)T, must be equal to the price of 2. , which is F(0,T). That leads us to the fair forward at time T price of an asset which pays no dividends:







The same thinking leads us to the fair price of a forward contract at any point in time t prior to the expiration of the forward contract. We only have to change the accumulation: You don't start the accumulation at time 0 but at time t and apply it to the spot price at t:



Fair Forward Price on an Asset when there are Cash Flows on the asset during the Life of the Contract

Most assets generate cash inflows or outflows (e.g. most stocks pay dividends). And that means that the formula (F.1) for the Forward Price doesn't hold anymore: (Alt.1) gets better comparing to (Alt.2) b/c only the holder of the stock gets the corresponding dividend.

In the discrete case we have to substract the present value of the dividends from the spot price S0. We get



In the continuous case equation (F.3) gets to



Now suppose the underlying asset is a commodity that incurs storage costs that are denoted by gamma(i)( gamma(i) is the storage cost for storing commodity i). Then the same thoughts as above leads us to the formula