Super-Hedging and Arbitrage Pricing in Markets with Transaction Costs and Trading Constraints

The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to reality. Even if, they are certainly not enforced by real world arbitrage because of transaction costs and trading constraints. Recently, results on several aspects of the bounds implied by the absence of arbitrage in realistic models have been established by Dermody and Rockafellar (1991); El Karoui and Quenez (1992); Cvitanic and Karatzas (1993); Dermody and Rockafellar (1995); Jouini and Kallal (1995b,a), and Cvitanic and Karatzas (1996). Using these results and contributing some new ones, this paper attempts to put together a \realistic arbitrage pricing theory" in a discrete-time market like that in (Harrison and Pliska; 1981). Under transaction costs and trading constraints, several aspects change fundamentally compared to the classical theory. The 1-to-1 relation between price functionals and martingale measures can be maintained only if there is a numeraire without transaction costs and trading constraints. A version of the fundamental theorem of nance (present value principle) can be maintained only of the set of admissible trading strategies is a cone and transaction costs are linear. Instead of looking for exact solutions to the present value equation (martingale measures) one should look for approximate solutions to the present value equation, with the metric depending on the structure of the transaction costs.