Superhedging in illiquid markets

arXiv:0807.2962v1 [q-fin.PR] 18 Jul 2008 Superhedging in illiquid markets Teemu Pennanen∗ November 5, 2018 Abstract We study contingent claims in a discrete-time market model where trading costs are given by convex functions and portfolios are constrained by convex sets. In addition to classical frictionless markets and markets with transaction costs or bid-ask spreads, our framework covers markets with nonlinear illiquidity effects for large instantaneous trades. We de- rive dual characterizations of superhedging conditions for contingent claim processes in a market without a cash account. The characterizations are given in terms of stochastic discount factors that correspond to martin- gale densities in a market with a cash account. The dual representations are valid under a topological condition and a weak consistency condition reminiscent of the “law of one price”, both of which are implied by the no arbitrage condition in the case of classical perfectly liquid market models. We give alternative sufficient conditions that apply to market models with nonlinear cost functions and portfolio constraints. Key words Illiquidity, portfolio constraints, claim processes, superhedging, deflators, convex duality 1 Introduction The notion of arbitrage is often given a central role when studying pricing and hedging of contingent claims in financial markets. In classical perfectly liquid market models, there are two good reasons for this. First, a violation of the no arbitrage condition leads to an unnatural situation where one can find self- financing trading strategies that generate infinite proceeds out of zero initial investment. Second, as discovered by Schachermayer [35], the no arbitrage condition implies the closedness of the set of claims that can be superhedged with zero cost. The closedness yields dual characterizations of superhedging conditions in terms of e.g. martingale measures and state price deflators. In illiquid markets, however, things are different. A violation of the no arbitrage condition does not necessarily mean that one can generate infinite proceeds by simple scaling of arbitrage strategies. Indeed, illiquidity effects may ∗Department of Mathematics and Systems Analysis, Helsinki University of Technology, P.O.Box 1100, FI-02015 TKK, Finland, teemu.pennanen@tkk.fi 1 come into play when trades get larger; see C¸etin and Rogers [4] or Pennanen [24, 25]. On the other hand, even in the case of linear models, the no arbitrage condition is not necessary for closedness of the set of claims hedgeable with zero cost. There may exist other economically meaningful conditions that yield the closedness and corresponding dual characterizations of superhedging conditions. This paper studies superhedging in a nonlinear discrete time model from [24, 25] where trading costs are given by convex cost functions and portfolios may be constrained by convex constraints. The model generalizes many better- known models such as the classical linear model, the transaction cost model of Jouini and Kallal [15], the sublinear model of Kaval and Molchanov [18], the illiquidity model of C¸etin and Rogers [4] as well as the linear models with portfolio constraints of Pham and Touzi [27], Napp [22], Evstigneev, Sch¨urger and Taksar [10] and Rokhlin [34]. Our model covers nonlinear illiquidity ef- fects associated with instantaneous trades (market orders) but it assumes that agents have no market power in the sense that trades do not affect the costs of subsequent trades. This is analogous to the models of C¸etin, Jarrow and Protter [3], C¸etin, Soner and Touzi [5] and Rogers and Singh [33], the last one of which gives economic motivation for the assumption. We avoid long term price impacts because they interfere with convexity which is essential in many aspects of pricing and hedging. Convexity becomes an important issue also in numerical calculations; see e.g. Edirisinghe, Naik and Uppal [9]. Unlike in most of the above papers, we do not assume the existence of a cash account a priori. This is important when studying contingent claim processes which may give pay-outs not only at maturity but throughout the whole life time of the claim. Such claim processes are common in practice where wealth cannot be transferred quite freely in time. Accordingly, much as in Jaschke and K¨uchler [14], we study pricing in terms of premium processes instead of a single premium (price) in the beginning. This is quite natural since much of trading in practice consists of exchanging sequences of cash flows. The usual pricing and hedging problems are obtained as special cases when the premium process is null after the initial date and claims have pay-outs only at maturity. There is a simple and quite natural condition under which these more general pricing problems are well-defined and nontrivial in convex, possibly nonlinear market models. The condition is reminiscent of the “law of one price” (or the “no good deal of the second kind” in [14]) which is weake

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