Continuous-time duality for superreplication with transient price impact

Abstract

We establish a superreplication duality in a continuous-time financial model as in (Bank and Voß (2018)) where an investor’s trades adversely affect bid- and ask-prices for a risky asset and where market resilience drives the resulting spread back towards zero at an exponential rate. Similar to the literature on models with a constant spread (cf., e.g., Math. Finance 6 (1996) 133–165; Ann. Appl. Probab. 20 (2010) 1341–1358; Ann. Appl. Probab. 27 (2017) 1414–1451), our dual description of superreplication prices involves the construction of suitable absolutely continuous measures with martingales close to the unaffected reference price. A novel feature in our duality is a liquidity weighted $L^{2}$-norm that enters as a measurement of this closeness and that accounts for strategy dependent spreads. As applications, we establish optimality of buy-and-hold strategies for the superreplication of call options and we prove a verification theorem for utility maximizing investment strategies.