Optimal Hedge Tracking Portfolios in a Limit Order Book

Abstract

Derivative hedging under transaction costs has attracted considerable attention over the past three decades. Yet comparatively little effort has been made towards integrating this problem in the context of trading through a limit order book. In this paper, we propose a simple model for a wealth-optimizing option seller, who hedges his position using a combination of limit and market orders, while facing certain constraints as to how far he can deviate from a targeted (Bachelierian) delta strategy. By translating the control problem into a three-dimensional Hamilton–Jacobi–Bellman quasi-variational inequality (HJB QVI) and solving numerically, we are able to deduce optimal limit order quotes alongside the regions surrounding the targeted delta surface in which the option seller must place limit orders vis-à-vis the more aggressive market orders. Our scheme is shown to be monotone, stable, and consistent and thence, modulo a comparison principle, convergent in the viscosity sense.

A Leland model for delta hedging in central risk books

“…6 Optimal market and limit orders for option hedging are computed in Agliardi (2016), Ellersgaard and Tegnér (2017), Cartea et al (2019) by numerically solving the corresponding dynamic programming equations. Agliardi (2016), Ellersgaard and Tegnér (2017) do not consider adverse selection. The model of Cartea et al (2019) is closest to the one we study in the present paper.…”

“…Conversely, no exposure corresponds to posting no new limit orders and canceling all existing ones. 6 Optimal market and limit orders for option hedging are computed in Agliardi (2016), Ellersgaard and Tegnér (2017), Cartea et al (2019) by numerically solving the corresponding dynamic programming equations. Agliardi (2016), Ellersgaard and Tegnér (2017) do not consider adverse selection.…”

“…1919The even more general case of a completely free choice of market and limit orders needs to be studied by numerically solving the corresponding dynamic‐programming equations (Agliardi, 2016; Cartea et al., 2019; Ellersgaard & Tegnér, 2017). In contrast, the approach we present here retains most of the tractability of the standard Leland model by still reducing optimization to a pointwise problem.…”

“…66Optimal market and limit orders for option hedging are computed in Agliardi (2016), Ellersgaard and Tegnér (2017), Cartea et al. (2019) by numerically solving the corresponding dynamic programming equations.…”

“…(2019) by numerically solving the corresponding dynamic programming equations. Agliardi (2016), Ellersgaard and Tegnér (2017) do not consider adverse selection. The model of Cartea et al.…”