Convex Duality with Transaction Costs

Abstract

Abstract. Convex duality for two two different super-replication problems in a continuous time financial market with proportional transaction cost is proved. In this market, static hedging in a finite number of options, in addition to usual dynamic hedging with the underlying stock, are allowed. The first one of the problems considered is the model-independent hedging that requires the super-replication to hold for every continuous path. In the second one the market model is given through a probability measure P and the inequalities are understood P almost surely. The main result, using the convex duality, proves that the two super-replication problems have the same value provided that P satisfies the conditional full support property. Hence, the transaction costs prevents one from using the structure of a specific model to reduce the super-replication cost.

Arbitrage and Hedging in Model-Independent Markets with Frictions

“…In both cases a version of the Fundamental Theorem of Asset Pricing is derived from the superhedging duality. The continuous time case with a single risky asset is investigated in [DS15].…”

Super-replication with transaction costs under model uncertainty for continuous processes

“…In [6], the authors introduced a fictitious market without transaction cost using a randomization technique, then the results of [7] were applied. In continuoustime markets with transaction costs, however, the only results we know of are those of [25]. They establish, under technical conditions, that the pathwise superhedging price is the same as the superhedging price for a continuous price process satisfying the conditional full support property.…”

Super‐replication in fully incomplete markets

“…We note that the above two setups were studied recently for the case where hedging of the stock is subject to proportional transaction costs; see Dolinsky () for the game options setup and Dolinsky and Soner () for semistatic hedging of European options. In these two papers, it was shown that if the logarithm of the discounted stock price process satisfies the conditional full support (CFS) property, then the super‐replication price coincides with the model‐free super‐replication price.…”