Acc 403- Financial Mathematics

Introduction to Present-, Forward and Futures Prices Assume that we want to buy a number of coffee beans with delivery in nine months. However, we are concerned about what the (spot) price of coffee beans might be then, so we draw up a contract where we agree on the price today. There are now at least three ways in which we can arrange the payment: 1) we pay now, in advance. We call this price the present price of coffee beans with delivery in nine months' time and denote it by P. Note that this is completely different from the spot price of coffee beans, i.e., the price of coffee for immediate delivery. 2) we pay when the coffee is delivered, i.e., in nine months' time. This price is the forward price, which we denote by G. 3) we might enter a futures contract with delivery in nine months time. A futures contract works as follows: Let us denote the days from today to delivery by the numbers 0, 1,... ...,T, so that day 0 is today, and day T is the day of delivery. Each day n a futures price Fn of coffee beans with delivery day T is noted, and FT equals the prevailing spot price of coffee beans day T. The futures price Fj is not known until day j; it will depend on how the coffee bean crop is doing, how the weather has been up to that day and the weather prospects up till day T, the expected demand for coffee, and so on. One can at any day enter a futures contract, and there is no charge for doing so. The long holder of the contract will each day j receive the amount Fj −Fj−1 (which may be negative, in which he has to pay the corresponding amount,) so if I enter a futures contract at day 0, I will day one receive F1 −F0, day two F2 − F1 and so on, and day T, the day of delivery, FT − FT −1. The total amount I receive is thus FT − F0. There is no actual delivery of coffee beans, but if I at day T buy the beans at the spot price FT , I pay FT , get my coffee beans and cash the amount FT − F0 from the futures contract. In total, I receive my beans and pay F0, and since F0 is known already at day zero, the futures contract works somewhat like a forward contract. The difference is that the value FT − F0 is paid out successively during the time up to delivery rather than at the time of delivery.