e can now give formal definition of the (continuous time) contract. See below for the discrete time modification.
DEFINITION 2.1. A futures contract on X with final delivery date T, including an embedded timing option, on the interval [0,T], with continuous resettlement, is a financial contract satisfying the following clauses. ON THE TIMING OPTION IN A FUTURES CONTRACT 271
At each time t ∈ [0,T] there exists on the market a futures price quotation denoted by F(t,T). Furthermore, for each fixed T, the process t −→ F(t, T) is a semimartingale w.r.t. the filtration F. Since T will be fixed in the discussion below, we will often denote F(t,T) by Ft.
The holder of the short end of the futures contract can, at any time t ∈ [0,T], decide whether to deliver or not. The decision whether to deliver at t or not is allowed to be based upon the information contained in Ft .
If the holder of the short end decides to deliver at time t, she will pay the amount Xt and receive the quoted futures price F(t,T).
If delivery has not been made prior to the final delivery date T, the holder of the short end will pay XT and receive F(T,T).
During the entire interval [0, T] there is continuous resettlement as for a standard futures contract. More precisely; over the infinitesimal interval [t, t + dt] the holder of the short end will pay the amount
dF(t, T) = F(t + dt, T) − F(t, T).
The spot price of the futures contract is always equal to zero, i.e., you can at any time enter or leave the contract at zero cost.
The cash flow for the holder of the long end is the negative of the cash flow for the short end.
The important point to notice here is that the timing option is only an option for the holder of the short end of the contract. For discrete time models, the only difference is the resettlement clause which then says that if you hold a short future between t and t + 1,you will pay the amount F(t + 1,T) − F(t,T) at time t + 1.
Our main problem is the following.PROBLEM 2.1. Given an exogenous specification of the index process X, what can be
said about the existence and structure of the futures price process F(t,T)?
3. THE FUNDAMENTAL PRICING EQUATION
We now go on to reformulate Problem 2.1 in more precise mathematical terms, and this will lead us to a fairly complicated infinite dimensional system of equations for the determination of the futures price process (if that object exists). We focus on the continuous time case, the discrete time case being very similar.
3.1. The Pricing Equation in Continuous Time
For the given final delivery date T, let us consider a fixed point in time t ≤ T and discuss the (continuous time) ..............................
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