12 Mathematics for Finance Assumptions 1.1 to 1.5 as well as the No-Arbitrage Principle extend readily to this case. The forward price F is determined by the No-Arbitrage Principle. In par- ticular, it can easily be found for an asset with no carrying costs. A typical example of such an asset is a stock paying no dividend. (By contrast, a com- modity will usually involve storage costs, while a foreign currency will earn interest, which can be regarded as a negative carrying cost.) A forward position guarantees that the asset will be bought for the forward price F at delivery. Alternatively, the asset can be bought now and held until delivery. However, if the initial cash outlay is to be zero, the purchase must be financed by a loan. The loan with interest, which will need to be repaid at the delivery date, is a candidate for the forward price. The following proposition shows that this is indeed the case. Proposition 1.2 Suppose that A(0) = 100, A(1) = 110, and S(0) = 50 dollars, where the risky security involves no carrying costs. Then the forward price must be F = 55 dollars, or an arbitrage opportunity would exist otherwise. Proof Suppose that F > 55. Then, at time 0: . Borrow $50. . Buy the asset for S(0) = 50 dollars. . Enter into a short forward contract with forward price F dollars and delivery date 1. The resulting portfolio (1, --, -1) consisting of stock, a risk-free position, and a short forward contract has initial value V(0) =0. Then, at time 1: . Close the short forward position by selling the asset for F dollars. . Close the risk-free position by paying = x 110 = 55 dollars. The final value of the portfolio, V(1) = F - 55 > 0, will be your arbitrage profit, violating the No-Arbitrage Principle. On the other hand, if F 0, which once again violates the No-Arbitrage Principle. It follows that the forward price must be F = 55 dollars. Exercise 1.5 Let A(0) = 100, A(1) = 112 and S(0) = 34 dollars. Is it possible to find an arbitrage opportunity if the forward price of stock is F = 38.60 dollars with delivery date 1? Exercise 1.6 Suppose that A(0) = 100 and A(1) = 105 dollars, the present price of pound sterling is S(0) = 1.6 dollars, and the forward price is F = 1.50 dollars to a pound with delivery date 1. How much should a sterling bond cost today if it promises to pay f100 at time 1? Hint: The for- ward contract is based on an asset involving negative carrying costs (the interest earned by investing in sterling bonds). 1.6 Call and Put Options Let A(0) = 100, A(1) = 110, S(0) = 100 dollars and s(1) = 120 with probability p, 80 with probability 1 - p, where 0