Abstract
For the superreplication problem with discrete time, a guaranteed deterministic formulation is considered: the problem is to guarantee coverage of the contingent liability on sold option under all admissible scenarios. These scenarios are defined by means of a priori defined compacts dependent on price prehistory: the price increments at each point in time must lie in the corresponding compacts. In a general case, we consider a market with trading constraints and assume the absence of transaction costs. The formulation of the problem is game theoretic and leads to the Bellman–Isaacs equations. This paper analyses the solution to these equations for a specific pricing problem, i.e., for a binary option of the European type, within a multiplicative market model, with no trading constraints. A number of solution properties and an algorithm for the numerical solution of the Bellman equations are derived. The interest in this problem, from a mathematical prospective, is related to the discontinuity of the option payoff function.
Highlights
The guaranteed deterministic approach is closely related to a class of market models called interval models in [3], especially to the ideas and results of Kolokoltsov published in [3] (Chapters 11–14), including the independent discovery of the game-theoretic interpretation of risk-neutral probabilities under the assumption of no trading constraints; we find this interpretation to be quite important from an economic point of view
They are assumed to lie in the given compacts that depend on the prehistory of the prices (such a model is an alternative to the traditional probabilistic market model)
This paper considers the problem of superhedging pricing of a binary option (European type) for a multiplicative one-dimensional market model, under the assumption of no trading constraints
Summary
The present paper joins a series of publications (in particular, [9] describes the market model in detail and provides a literature review) [9,10,11,12,13,14,15] that develop a financial market model consistent with an uncertain deterministic price evolution with discrete time: asset prices evolve deterministically under uncertainty described using a priori information about possible price increments They are assumed to lie in the given compacts that depend on the prehistory of the prices (such a model is an alternative to the traditional probabilistic market model (in our proposed deterministic approach, the reference probability measure is not initially set, as it is supposed in the probabilistic approach, see, e.g., [16])). The interest to this problem is caused by the fact that the payout function is discontinuous, and the results concerning the case of continuous payout functions given in [12, 13] are not applicable here
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