Abstract

We pursue a robust approach to pricing and hedging in mathematical finance. We consider a continuous-time setting in which some underlying assets and options, with continuous price paths, are available for dynamic trading and a further set of European options, possibly with varying maturities, is available for static trading. Motivated by the notion of prediction set in Mykland (Ann. Stat. 31:1413–1438, 2003), we include in our setup modelling beliefs by allowing to specify a set of paths to be considered, e.g. superreplication of a contingent claim is required only for paths falling in the given set. Our framework thus interpolates between model-independent and model-specific settings and allows us to quantify the impact of making assumptions or gaining information. We obtain a general pricing–hedging duality result: the infimum over superhedging prices of an exotic option with payoff G is equal to the supremum of expectations of G under calibrated martingale measures. Our results include in particular the martingale optimal transport duality of Dolinsky and Soner (Probab. Theory Relat. Fields 160:391–427, 2014) and extend it to multiple dimensions, multiple maturities and beliefs which are invariant under time-changes. In a general setting with arbitrary beliefs and for a uniformly continuous G, the asserted duality holds between limiting values of perturbed problems.

Highlights

  • Two approaches to pricing and hedging The question of pricing and hedging of a contingent claim lies at the heart of mathematical finance

  • While very appealing at first, it has been traditionally criticised for producing outputs which are too imprecise to be of practical relevance

  • “To do so, more structure must be added to the problem through additional assumptions at the expense of losing some agreement.”1 Typically this is done by fixing a filtered probability space (Ω, F, (Ft )t≥0, P) with risky assets represented by some adapted process (St )

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Summary

Introduction

Two approaches to pricing and hedging The question of pricing and hedging of a contingent claim lies at the heart of mathematical finance. All the above works on the model-independent approach stay within Merton [38]’s “universally accepted” setting and analyse the implications of incorporating the ability to trade some options at given market prices for the outputs, namely prices and hedging strategies of other contingent claims. We amend this setup and allow expressing modelling beliefs. The proofs of two auxiliary results are relegated to the Appendix

Traded assets
Beliefs
Trading strategies and superreplication
Market models
Main results
General duality
Martingale optimal transport duality
Auxiliary results and proofs
Pricing–hedging duality without constraints
A discrete-time approximation through simple strategies
A countable class of piecewise constant functions
A countable probabilistic structure
Duality for the discretised problems
Discretisation of the primal
Full Text
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