2 Proportional carrying charges Suppose that u foiward contr

2. (Proportional carrying charges Suppose that u foiward contrac on an asset is written at time zero and there are M periods unti delivery Suppose that the camying chaige in period k is qSck), where Stk) is the spot price of the aset in period k Show that the forward price is d(0, M) FORWARDS, FUTURES, AND SWAPS (H?nt Consider a portfolio that pays all carrying costs by selling a fraction of the asset as required Let the number of units of the asset held at time k be (k) and find .r(M) ?n ierms of x (0)

Solution

To prove the above, we would assume that we are at the sellers side of forward contract and hence we short the position in the forward contract. We enter into contract where we would be selling (1-q)^M commodity at the end of M periods for price F.

Now at time zero we take loan for S(0) to purchase one unit of the commodity. We need to pay carrying charges qS(k) in each period k, so in each period we sell a fraction q of these commodities holdings.

We would follow the following procedure

In period k=0, we would have one commodity remaining, we would sell a fraction q of these commodities to obtain cash to pay the carrying cost qS(0). Than in period k=1 we would have (1-q) commodity remaining and so we would sell these commodities to obtain cash to pay the carrying cost of q(1-q)S(1). In period k=2 we would have (1-q)(1-q)=(1-q)^2 commodity remaining and as usual we would sell these commodities to obtain cash to pay the carrying charge q(1-q)^2S(2)

As above at period k=M-1, we would have (1-q)^M-1 commodity remaining. We sell a fraction and pay carrying cost of q(1-q)^M-1S(M-1), In period k=M we have (1-q)^M commodity remaining.

After this we would than deliver our commodity holdings to receive F(1-q)^M and repay our loan by paying S(0)/d(0,M). Total profits from these transactions would be

F(1-q)^M-S(0)/d(0,M) and to avoid the arbitrage theis profit must be zero. Hence

F= (1-q)^M*S(0)/d(0,M)