Hedging of Claims with Physical Delivery under Convex Transaction Costs

Abstract

Abstract. We study superhedging of contingent claims with physical delivery in a discrete-time market model with convex transaction costs. Our model extends Kabanov's currency market model by allowing for nonlinear illiquidity effects. We show that an appropriate generalization of Schachermayer's robust no-arbitrage condition implies that the set of claims hedgeable with zero cost is closed in probability.Combined with classical techniques of convex analysis, the closedness yields a dual characterization of premium processes that are sufficient to superhedge a given claim process. We also extend the fundamental theorem of asset pricing for general conical models.Key words. superhedging, physical delivery, illiquidity, transaction costs, convex duality AMS subject classifications. 91B25, 52A07, 46A20DOI. 10.1137/0907541821. Introduction. This paper studies superhedging of contingent claims with physical delivery in markets with temporary illiquidity effects. Our market model is a generalization of the currency market model of Kabanov [12]. In Kabanov's model, price dynamics and transaction costs are modeled implicitly by solvency cones, i.e., sets of portfolios which can be transformed into the zero portfolio by self-financing transactions at a given time and state. An essential difference between Kabanov's model and more traditional models of mathematical finance (including, e.g., the transaction cost models of Jouini and Kallal [11], Cvitanić and Karatzas [6], and Kaval and Molchanov [18]) is that Kabanov's model focuses on contingent claims with physical delivery, i.e., claims whose payouts are given in terms of portfolios of assets instead of a single reference asset like cash. Accordingly, notions of arbitrage as well as the corresponding dual variables are defined in terms of vector-valued processes; see, e.g., [15,31].Astic and Touzi [2] extended Kabanov's model by allowing general convex solvency regions in the case of finite probability spaces. Nonconical solvency regions allow for the modeling of temporary illiquidity effects where marginal trading costs may depend on the magnitude of a trade as, e.g., in Ç etin, Jarrow, and Protter [3], Ç etin and Rogers [4], Ç etin, Soner, and Touzi [5], Rogers and Singh [27], or Pennanen [19,20]. These models cover nonlinear illiquidity effects, but they assume that agents have no market power in the sense that their trades do not affect the costs of subsequent trades; see [27] for further motivation of this assumption.

Arbitrage and deflators in illiquid markets

“…When moving to nonconical models, such as the one studied here, the situation changes. This is the topic of the follow-up paper [30]; see also [31], where the approach is further extended to a nonconical version of the currency market model of [18].…”

Risk Arbitrage and Hedging to Acceptability

“…satisfies K 0 t Ă K 8 t , see [23]. Therefore, k t`α x P K t for all k t P K t , x P K 0 t , and all α P R. Furthermore, pSNAq trivially implies A t,T X L 0 pK t , F t q " t0u for all t.…”

A note on super-hedging for investor–producers

“…This definition implies that arbitrages might exist, but they are limited for elements of X 0 0 (T ) and even not possible for the recession cone. Pennanen and Penner [16] proved that the set X 0 0 (T ) is closed in probability under this condition. Hence, it is Fatou-closed.…”