Duality and martingales: a stochastic programming perspective on contingent claims

Mixed-integer second-order cone programming for lower hedging of American contingent claims in incomplete markets

“…The key to turn this situation around is via duality. Before we can carry on, we need the following definition (see [16] for details).…”

“…. , T as in [16]. We further assume the market evolves as a discrete, non-recombinant scenario tree (hence, suitable for incomplete markets) in which the partition of probability atoms ω ∈ generated by matching path histories up to time t corresponds one-to-one with nodes n ∈ N t at level t in the tree.…”

“…It can be lifted to the nodes of a partition N t of if each level set {X −1 (a) : a ∈ R} is either the empty set or is a finite union of elements of the partition. In other words, X can be lifted to N t if it can be assigned a value on each node of N t that is consistent with its definition on , [16]. This kind of random variable is said to be measurable with respect to the information contained in the nodes of N t .…”

Pricing American contingent claims by stochastic linear programming

“…It can be lifted to the nodes of a partition N t of if each level set {X À1 (a) : a 2 R} is either the empty set or is a finite union of elements of the partition. In other words, X can be lifted to N t if it can be assigned a value on each node of N t that is consistent with its definition on [8]. This kind of random variable is said to be measurable with respect to the information contained in the nodes of N t .…”

“…Mathematical programming tools, especially stochastic programming (see [12] for a recent survey) are becoming increasingly useful as an entry point to studying the specialized methods of mathematical finance [5,8,9]. In this note, we are interested in the pricing of American Contingent Claims (ACC) as well as their special cases, in a multi-period, discrete time, discrete state space framework.…”