Abstract
The main aim of this article is to show the role of structural stability in financial modelling; that is, a specific “no-arbitrage” property is unaffected by small perturbations of the model’s dynamics. We prove that under the structural stability assumption, given a convex compact-valued multifunction, there exists a stochastic transition kernel with supports coinciding with this multifunction and one that is strong Feller in the strict sense. We also demonstrate preservation of structural stability for sufficiently small deviations of transition kernels for different probability metrics.
Highlights
PreliminariesA detailed analysis of the relation between different “no-arbitrage” notions in the framework of robust modelling of financial markets in discrete time is presented in [12].) and geometric criteria (in terms of convex hulls of Kt(·)) are obtained
Ere are ad hoc definitions of the robust no-arbitrage property, for example, that by the authors of [3], who considered a case with nontradable options quoted with bid-ask spreads
Sometimes, such properties are used implicitly, as in [5], who considered a continuous-time model of a financial market with primary assets, options that are nontradable except at the initial time moment, and continuously traded European options. e general duality results of that paper exploited its Assumption 3.1 concerning these options, requiring that calibrated martingale measures exist under arbitrarily small perturbations of the initial prices. us, depending on the context, structural stability can be formalised in various ways
Summary
A detailed analysis of the relation between different “no-arbitrage” notions in the framework of robust modelling of financial markets in discrete time is presented in [12].) and geometric criteria (in terms of convex hulls of Kt(·)) are obtained. We call this model “realistic” if there exist mixed market strategies Pt(x, ·) representable as Feller transition kernels, ( is can be interpreted as a smooth version of conditional distributions Xt, given prehistory Xt−1 x.) satisfying the consistency condition, which is a relation between stochastic and deterministic models: supp Pt(x, ·) Ft(x), x ∈ Bt−1 for t 1, . We call this model “realistic” if there exist mixed market strategies Pt(x, ·) representable as Feller transition kernels, ( is can be interpreted as a smooth version of conditional distributions Xt, given prehistory Xt−1 x.) satisfying the consistency condition, which is a relation between stochastic and deterministic models: supp Pt(x, ·) Ft(x), x ∈ Bt−1 for t 1, . . . , N. (2)
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