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The timing option embedded in a futures contract allows the short position to decide when to deliver the underlying asset during the last month of the contract period. In this paper we derive, within a very general incomplete market framework, an explicit model independent formula for the futures price process in the presence of a timing option. We also provide a characterization of the optimal delivery strategy, and we analyze some concrete examples.

1. INTRODUCTION
In standard textbook treatments, a futures contract is typically defined by the properties of zero spot price and continuous (or discrete) resettlement, plus a simple no arbitrage condition at the last delivery day. If the underlying price process is denoted by Xt and the futures price process for delivery at T is denoted by F(t,T) this leads to the well known formula
F(t, (1.1) T) = EQ[XT |Ft ], 0 ≤ t ≤ T,
where Q denotes the (not necessarily unique) risk neutral martingale measure.In practice, however, there are a number of complicating factors which are ignored in the textbook treatment, and in particular it is typically the case that a standard futures contract has several embedded option elements. The most common of these options are
the timing option, and the end-of-the-month option, the quality option, and the wild card option. All these options are options for the short end of the contract, and they work roughly as follows.

The timing option is the option to deliver at any time during the last month of the contract.

The end of the month option is the option to deliver at any day during the last week of the contract, despite the fact that the futures price for the last week is fixed on the first day of that week and then held constant. Support from the TomHedelius and JanWallander Foundation is gratefully acknowledged. Both authors are grateful to B. N¨aslund, J. Kallsen, C. Kuehn, an anonymous associate editor, and an anonymous referee for a number of very helpful comments and suggestions.Manuscript received March 2005; final revision received November 2005. Address correspondence to Tomas Bj¨ork, Department of Finance, Stockholm School of Economics, Box 6501, SE-113 83 Stockholm, Sweden; e-mail: tomas.bjork@hhs.se.
C  2007 The Authors. Journal compilation C 
2007 Blackwell Publishing Inc., 350 Main St., Malden, MA 02148,
USA, and 9600 Garsington Road, Oxford OX4 2DQ, UK.
267
268 F. BIAGINI AND T. BJO¨ RK

The quality option is the option to choose, out of a prespecified basket of assets,
which asset to deliver.

The wild card option is, for example for bond futures, the option to initiate delivery between 2 P.M. and 8 P.M. in the afternoon during the delivery month of the contract. The point here is that the futures price is settled at 2 P.M. but the trade in the underlying bonds goes on until 8 P.M.The purpose of the present paper is to study the timing option within a very general framework, allowing for incomplete markets, and our goal is to investigate how the general formula (1.1) has to be modified when we introduce a timing option element. Our main result is given in Theorems 4.2 and 4.3 where it is shown that, independently of any model assumptions, the futures price in the presence of the timing option is given by the formula
F(t, T) = inf
t≤τ≤T
EQ[Xτ |F(1.2) t ],
where τ varies over the class of optional stopping times, and inf denotes the essential infimum. This formula is of course 本论文由英语论文网提供整理，提供论文代写，英语论文代写，代写论文，代写英语论文，代写留学生论文，代写英文论文，留学生论文代写相关核心关键词搜索。