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very similar to the pricing formula for an American option. Note, however, that equation (1.2) does not follow directly from standard theory for American contracts, the reason being that the futures price is not a price in the technical sense. The futures price process instead plays the role of the cumulative dividend process
for the futures contract, which in turn can be viewed as a price-dividend pair, with spot
price identically equal to zero. Furthermore,we prove that the optimal delivery policy ˆ τ (t), for a short contract entered at t, is given by
ˆ τ = inf{t ≥ 0; F(t, T) = Xt(1.3) }.
We also study some special cases and show the following.
If the underlying X is the price of a traded financial asset without dividends, then
it is optimal to deliver immediately, so ˆ τ (t) = t and thus
(1.4) F(t, T) = Xt .
If the underlying X has a convenience yield which is greater than the short rate,
then the optimal delivery strategy is to wait until the last day. In this case we thus
have ˆ τ (t) = T and
F(t, T) = EQ[XT |F(1.5) t ],
whichwe recognize from(1.1) as the classical formula for futures contracts without a timing option.Option elements of futures contract have also been studied earlier. The quality option is discussed in detail in Gay and Manaster (1984), and the wild card option is analyzed in Cohen (1995) and Gay and Manaster (1986). The timing option is (among other topics) treated in Boyle (1989) but theoretical results are only obtained for the special case when
X is the price process of a traded underlying asset. In this setting, and under the added assumption of a constant short rate, the formula (1.4) is derived.The organization of the paper is as follows. In Section 2, we set the scene for the financial market. Note that we make no specific model assumptions at all about market completeness or the nature of the underlying process, and our setup allows for discrete
ON THE TIMING OPTION IN A FUTURES CONTRACT 269
as well as continuous time models. In Section 3, we derive a fundamental equation, the solution of which will determine the futures price process. We attack the fundamental equation by first studying the discrete time case in Section 4.1, and prove the main formula
(1.2). In Section 4.2, we prove the parallel result in the technically more demanding continuous time case. We finish the main paper by some concrete financial applications, and in particular we clarify completely under which conditions the futures price process,including an embedded timing option, coincides with the classical formula (1.1). At the other end of the spectrum, we also investigate under which conditions immediate delivery is optimal.
2. SETUP
We consider a financial market living on a stochastic basis (,F, F, Q), where the filtration F = {Ft}0≤t≤T satisfies the usual conditions.We allow for both discrete and continuous time, so the contract period is either the interval [0, T] or the set {0, 1, . . . , T}.
To set the financial scene we need some basic assumptions, so for the rest of the paper we assume that there exists a predictable short rate process r, and a corresponding money account process B. In continuous time B has the dynamics
(2.1) dBt = rtBt dt.
In the discrete time case, the short rate at time t will be denoted by rt+1 so the bank account B has the dynamics
Bt+1 (2.2) = (1 + rt+1)Bt .In this case the short rate is assumed to be predict