able, i.e., rt is Ft−1-measurable (rt is
known already at t − 1) for all t, with the convention F−1 = F0.The market is assumed to be free of arbitrage in the sense the measure Q above is a martingale measurew.r.t the money account B for the given time horizon. Note thatwe do not assume market completeness. Obviously, if the market is incomplete, the martingale measure Q will not be unique, so in an incomplete setting the pricing formulas derived below will depend upon the particular martingale measure chosen. We discuss this in more detail in Section 5. We will need a weak boundedness assumption on the short rate.
ASSUMPTION 2.1. For the rest of the paper we assume the following.
In the continuous time case we assume that the interest rate process is predictable, and that there exists a positive real number c such that rt (2.3) ≥ −c,
with probability one, for all t.
Defining the money account as usual by B by Bt = exp( t 0 rs ds) we assume that (2.4) EQ[BT] < ∞.
In the discrete time case we assume the interest rate process is predictable, and that there exists a positive real number c such that 1 + rn (2.5) ≥ c, with probability one, for all n. 270 F. BIAGINI AND T. BJO¨ RK
REMARK 2.1. We note that the if we define C byCt = sup t≤τ≤TEQ[Bτ |F(2.6) t ],where τ varies over the class of stopping times, then the inequality EQ[BT] < ∞ easily implies(2.7) Ct < ∞Q − a.s. for all t ∈ [0,T].
Within this framework we now want to consider a futures contract with an embeddedtiming option.ASSUMPTION 2.2. We assume the existence of an exogenously specified nonnegative
adapted cadlag process X. The process X will henceforth be referred to as the index process,and we assume that
(2.8) EQ[Xt ] < ∞, 0 ≤ t ≤ T.The interpretation of this assumption is that the index process X is the underlying
process on which the futures contract is written.Forobvious reasonswewant to include contracts like commodity futures, index futures,
futures with an embedded quality option, and also futures on a nonfinancial index like a weather futures contract. For this reason we do not assume that X is the price process of a traded financial asset in an idealized frictionless market. Typical choices of X could thus be one of the following.
Xt is the price at time t of a commodity, with a nontrivial convenience yield.
Xt is the price at time t of a, possibly dividend paying, financial asset.
Xt = min{S1 t , . . . , Sn t} where S1 t , . . . , Snt are price processes of financial assets (for example stocks or bonds). This setup would be natural if we have an embedded quality option.
Xt is a nonfinancial process, like the temperature at some prespecified location.We now want to define a futures contract, with an embedded timing option, on the underlying index processX over the time interval [0,T]. If, for example,we are considering aU.S. interest rate future, this means that the interval [0, T] corresponds to the last month of the contract period. Note that we thus assume that the timing option is valid for the entire interval [0, T]. The analysis of the futures price process for times prior to the timing option period, is trivial and given by standard theory. If, for example, we let the timing option be active only in the interval [T0,T], then we immediately obtain F(t, T) = EQ[F(T0, T) |F(2.9) t ], 0 ≤ t ≤ T0,where F(T0,T) is given by the theory developed in the present paper. W
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