[Paper Review] Hedging with Transient Price Impact

Peter Bank, Mete Soner+1 more|arXiv (Cornell University)|Oct 12, 2015

Stochastic processes and financial applications17 references3 citations

TL;DR

This paper solves the optimal hedging problem in a Bachelier model with transient price impact by framing it as a cost-minimal tracking of a predictable frictionless hedging strategy. The key result is that optimal trades target a weighted average of future expected target positions, generalizing Garleanu and Pedersen’s insight beyond Markovian diffusion settings using convex analysis instead of dynamic programming.

ABSTRACT

We consider the problem of hedging a European contingent claim in a Bachelier model with transient price impact as proposed by Almgren and Chriss. Following the approach of Rogers and Singh and Naujokat and Westray, the hedging problem can be regarded as a cost optimal tracking problem of the frictionless hedging strategy. We solve this problem explicitly for general predictable target hedging strategies. It turns out that, rather than towards the current target position, the optimal policy trades towards a weighted average of expected future target positions. This generalizes an observation of Garleanu and Pedersen from their homogenous Markovian optimal investment problem to a general hedging problem. Our findings complement a number of previous studies in the literature on optimal strategies in illiquid markets where the frictionless strategy is confined to diffusions. The consideration of general predictable reference strategies is made possible by the use of a convex analysis approach instead of the more common dynamic programming methods.

Motivation & Objective

• To address optimal hedging of European options in illiquid markets with transient price impact.
• To extend prior work limited to diffusion-based frictionless strategies to general predictable target hedging strategies.
• To develop a solution method that avoids dynamic programming and instead uses convex analysis for broader applicability.

Proposed method

• Formulate the hedging problem as a cost-optimal tracking of a predictable frictionless hedging strategy under transient price impact.
• Apply convex analysis techniques to derive the optimal trading policy, avoiding the limitations of dynamic programming.
• Characterize the optimal strategy as a function of expected future target positions, weighted by a kernel reflecting market impact decay.
• Use the Almgren-Chriss framework for transient price impact to model market impact dynamics.
• Derive explicit solutions for the optimal trading trajectory under general predictable target strategies.
• Generalize the observation that optimal trades do not follow the current target but anticipate future positions, using a weighted average of expectations.

Experimental results

Research questions

• RQ1 How can optimal hedging be achieved in a Bachelier model with transient price impact when the frictionless strategy is a general predictable process?
• RQ2 What is the optimal trade-off between tracking error and transaction costs in the presence of transient impact?
• RQ3 Why does the optimal strategy deviate from the current target position and instead target a weighted average of future expected positions?
• RQ4 How does the use of convex analysis enable solutions beyond the scope of dynamic programming in illiquid market models?
• RQ5 To what extent does the optimal strategy generalize Garleanu and Pedersen’s result on weighted averages of future positions to non-Markovian, predictable strategies?

Key findings

• The optimal hedging strategy does not trade toward the current target position but toward a weighted average of expected future target positions.
• The weights in the average are determined by the decay profile of transient price impact, reflecting the market's memory of past trades.
• The solution is explicitly derived for any predictable target hedging strategy, not restricted to diffusion processes.
• The convex analysis approach provides a more general framework than dynamic programming, enabling broader applicability to non-Markovian settings.
• The result generalizes Garleanu and Pedersen’s observation on weighted averaging of future positions to a wider class of hedging problems.
• The framework allows for explicit computation of optimal trading trajectories under transient impact, offering practical implementation in illiquid markets.