In this module, we are going to introduce you to the forward contract and also how to use the no arbitrage principal to price forward contracts. A forward contract gives the buyer the right and also the obligation to purchase a specified amount of an asset. At a specified time capital T, at a specified price capital F, called the forward price, which is set at time T equal to 0. Just to get the mechanics right, here's time T equal to 0. Here's time, the expiration time, capital T. At time T equal to 0, you're specified at price capital F. Which allows you to buy a specified quantity of an asset at that price F, but at time capital T. So you fix the price at time T equal to 0 for a purchase that happens at a later time, capital T. They are many examples of forward contracts. Forward contracts on delivery of a stock with a maturity of six months. Forward contract for sale of gold with a maturity of one year. Forward contract to buy ten million dollars of Euros with a maturity of three months. Forward contract for a delivery of a nine month T bill with a maturity of three months. In all of these cases, you are being quoted a price today to purchase the quantity, which has been specified in the contract at a later time, capital T. For example, in the forward contract for a delivery of a stock with maturity six months, you've been quoted a price today for buying a stock in six months. Our goal in this set of slides is to set the forward price capital F for a forward contract at time T equal to 0 for one unit of an asset with asset price ST at time T and maturity of capital T. Let little fd denote the price or the value at time little t for a long position of forward contract. The value of the forward contract at time capital T is known. F little t at time capital T is going to be the price of the asset at time capital T as sub T. This is the price of the asset minus the forward price. This is the forward price. And why is this exactly the value of the forward contract at time capital T? This is because a long position in the forward, you must purchase the asset at price capital F. You can immediately sell that in the market at a spot price of S capital T, so you make the difference. If the forward price happens to be greater than S capital T, you make a loss. If the forward price happens to be less than S capital T, you make a profit. The forward price capital F is set in such a manner, that at time t equal to 0, the value of the price of the forward contract f sub 0 is exactly equal to 0. F0, this is the price at time t equal to 0, and that must be set equal to 0. Which means that you're indifferent between taking either a long position or buying the contract or taking a short position or selling the contract. The buyers and sellers are indifferent about taking any position. Just to remind you, little f0 is going to relate to the price of the value of the contract. Capital F would relate to the forward price. This is the full price that is set at time t equal to 0. But the purchase is going to happen at time t equal to capital T. We're going to use the no arbitrage principle to set the forward price F. In order to construct this price, I have to get into a concept called short selling. Short selling is the selling of shares in a stock that the seller doesn't own. What happens is that the seller borrows the shares from a broker. The shares come from the brokerage's own inventory. The shares are sold and the proceeds are credited to the seller's account. However, sooner or later, the seller must close the short by buying back the shares. This is also called covering. So you sell the shares in the market by borrowing it from the broker. And then later on, you buy back from the market and give it back to the broker. Why would one be interested in short selling? Because one feels that the price of the asset is going to drop. If the price of the asset drops, then you'd make a profit by short selling. If the price of the asset goes up then a short sale results in a loss. The short positions can be very risky and have to be handled very carefully. The price can only drop to zero, so the potential profit that you can get from a short position is bounded. But the price can increase to arbitrary large values, and therefore the potential loss from a short position is unbounded. So if you're taking short positions, one has to be very careful about how you manage those short positions and the risks associated with those short positions. We are going to use short selling as a technique for pricing formulas. Suppose the asset has no intermediate cash flows. For example, dividends, or storage costs, and so on. Construct the following portfolio. You buy the forward contract, short sell the underlying, and lend S0 up to time capital T. And let's look at one of the cash flows associated with this. If you buy the contract, you can get into the contract by paying nothing because f0 is equal to 0. This is how it's constructed at time capital T. What are the cash flows associated with it? It's equal to the price or the value of the contacted time capital T, which is S capital T minus F. You short sell the asset and buy back at time capital T. What happens when you short sell the asset? You get S0 into your account because you sold it. You immediately, the funds get credited to your account. You buy it back at capital T and therefore you have a cash flow minus S capital T happening at time capital T. Notice as zero is a price today, you know exactly what it's going to be. S capital T is a random quantity. It's some quantity that is going to be there in the future. Why did I construct this particular object in my portfolio? I put that because now this ST and that minus ST cancel each other. Remember what we did for the floating rate bonds. We removed the randomness and that's we are doing here too. We are removing the randomness. You lend S0 up to time capital T, what happens now? Minus S0 is the cash flow at time T equal to 0, S0 divided by d(0, T) is the cash flow at time capital T. This is just a fancy way of writing, S0 times 1 plus r to the power capital T. But because now the interest rate could have a term structure associated with it, instead of writing it as 1 plus r to the capital T, I'm just using the discount. And that allows me to give you a more general result, which doesn't rely on flat interest rates. What is the net cash flow? I paid nothing for this portfolio. What is the cash flow at time capital T? It's S0 divided by d(0, T) minus F. The portfolio has a deterministic cash flow at time capital T and the cost is equal to 0, therefore, discounted, you should get price V equal to 0. So S0 divided by D0 T minus F times D0 T. Discounted back you end up getting equal to 0 and therefore F is nothing but S0 divided by D0 T. Now why is F strictly greater than the stock price? It's because of cost of carry. You are, you're not committing your cash for up to time capital T. The opportunity cost associated with the, that cash is essentially 1 over d(0,T), and that's why the F, which is the forward price, which is the price at which you can purchase an asset sometime in the future, is greater than the spot price. Here's an example. A forward contract on a non-dividend paying stock that matures in six months. The current stock price is $50. The six-month interest rate is 4% per annum. Assuming semi-annual compounding. Why semi-annual? Because the maturity is six months, so I'm taking, sort of, the largest period that is associated with the problem. What is the discount rate? It's going to be 1 plus the annual interest rate divided by 2. The discount rate turns out to be 0.9804 and therefore, F, which is the forward price is going to be the spot price, 50 divided by 0.9804 and turns out to be $51. Now what we want to do is construct the value of the forward at some time T strictly greater than 0. Remember this is the value in the price of a forward contract. We already know that at time T equal to 0, F sub 0 is equal to 0. At time T equal to capital T, F sub t is going to be s capital T minus F. The spot price at the expiration, capital D minus the forward price. Now we want to construct F sub little t, which for time t which is not 0 or capital T. And we could do this using two different forward contracts. So, let f 0 denote the forward price at time 0 for delivery at time capital T. So, here's what is going to happen. Here's my time t. Here's time 0. Here time little t. Here time capital T. So F0 is a forward strike price that is negotiated here for delivery at time capital T. Ft is a forward price negotiation at time little t also for delivery or expiration capital T. Now we are going to again construct a portfolio. We are going to short this contract. I'm going to short the Ft contract. What are the cash flows associated with that? At time t equal to little t, because this contract has been struck at time t equal to little t, I don't have to pay anything to take the short position. At time T equal to capital T, I get F sub little t minus St, notice, this is the worst. Because I take a short position. I long, go long on a contract F0. If I had struck this contract at time 0, I would have paid nothing for it. Because the expiration at time capital T, but the contract was struck at time T equal to 0. However, now I'm buying this contract at time little t. Its value, or its fair price is F sub little t. And therefore, I have to pay that amount in order to get it. So with the cash flow at time little t is going to be minus Ft associated with this contract. The cash flow at time capital T is going to be s capital T minus F0. F0 is the forward price associated with the contract that was written at time 0 for expiration at time capital T. If you take the difference, ST cancels. F little t minus f0 is a deterministic quantity minus Ft is a deterministic quantity, and therefore you end up getting that the price, ft, must be just a discounted value of the cash flow in the future. F capital T minus F0 times the discount rate over the window little t to capital T. Let me review what happened here. We're constructing a portfolio using two different forward contracts. The forward contract that I'm call, calling f sub 0, is a contract that was written at time 0, for delivery at time capital T. If I had gotten into this contract time T equal to 0, I would have not paid anything. However, in my portfolio I'm getting into this contract at time little t. And therefore, I have to pay the value or price of the contract at little t which is F sub little t. I have another forward contract which was constructed at time little t for expiration of capital T. And for this contract, I know that the value is equal to 0 so I can get into this contract or I could sell this contract at price equal to 0. The payoff from this contract is going to be something different. It's going to depend on the forward price constructed at time little t. I take both of these into my portfolio and I do this in order to cancel out the random quantity S capital T. The spot price of the underlying asset on which these forward contracts are written. When you do the net cash flow, you end up getting a deterministic quantity, and I know how to price that. And that's how I end up computing the value of a forward contract at some time T in the future, which is not the time at which it was struck. And it's not the time at which it's expired.
