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Abstract
Financial options typically incorporate times of exercise. Alternatively, they embody set-up costs or indivisibilities. Such features lead to planning problems with integer decision variables. Provided the sample space be finite, it is shown here that integrality constraints can often be relaxed. In fact, simple mathematical programming, aimed at arbitrage or replication, may find optimal exercise, and bound or identify option prices. When the asset market is incomplete, the bounds stem from non-linear pricing functionals.
†Dedicated to H. Th. Jongen on the occasion of his 60th birthday.
Acknowledgements
Thanks are due Finansmarkedsfondet and the Arne Ryde Foundation for support–and a referee for very helpful criticism and comments.
Notes
†Dedicated to H. Th. Jongen on the occasion of his 60th birthday.
Notes
1. Constancy of p bc across c ∈ 𝒞(n) points to b as locally riskless at node n ∉ 𝒩 T .
2. One may interprete p bn as the face value of a zero-coupon bond that matures at node n. Thus, the spot rate (p bc −p bn )/p bn , c ∈ 𝒞(n), of a predictable bond is perfectly known at node n. The mapping n ↦ δ n = p b0/p bn is often called the term structure. It may well be random Citation29.
3. See for instance the excellent text Citation11.
4. Customary but weaker definitions of arbitrage require that θ be self-financing in that G n (θ) = 0 for all n ≠ 0 (or for all n); see, e.g. Citation18.
5. For a relation to utility maximization see Citation4 and references therein.
6. As an example consider purchase of a physical asset, before or at the expiration of a lease.
7. One may envisage that continuous components, if any in x, have already been optimized away, leaving a reduced objective. The remaining variables could correspond to stopping times.