Utility‐based pricing and hedging of contingent claims in Almgren‐Chriss model with temporary price impact

Abstract

In this paper, we construct the utility-based optimal hedging strategy for a European-type option in the Almgren-Chriss model with temporary price impact. The main mathematical challenge of this work stems from the degeneracy of the second order terms and the quadratic growth of the first-order terms in the associated Hamilton-Jacobi-Bellman equation, which makes it difficult to establish sufficient regularity of the value function needed to construct the optimal strategy in a feedback form. By combining the analytic and probabilistic tools for describing the value function and the optimal strategy, we establish the feedback representation of the latter. We use this representation to derive an explicit asymptotic expansion of the utility indifference price of the option, which allows us to quantify the price impact in options' market via the price impact coefficient in the underlying market.

A two-player portfolio tracking game

“…1. Similar to Carlin et al [8] and Schied and Zhang [19] the agent's individual optimization problems in ( 6) and ( 7) are intertwined through common aggregated temporary and permanent price impact affecting their performance functionals J 1 and J 2 in (4) and ( 5) (in contrast to, e.g, Huang et al [16], Casgrain and Jaimungal [10,11] or Ekren and Nadtochiy [14] where agents only interact through permanent or temporary price impact, respectively). One can think of both players as strategic agents who compete for liquidity while concurrently trading in a single illiquid risky asset to meet their tracking objectives for the purpose of, e.g., hedging fluctuations of random endowments.…”

“…t and X1 t . The optimal signal processes ξ1 in (14) and ξ2 in (15) are convex combinations of weighted averages of expected future target positions of the processes ξ 1 , ξ 2 and the expected terminal positions Ξ 1 T , Ξ 2 T , where the weights w 1 t , w 2 t , w 3 t , w 4 t systematically shift toward the desired individual terminal state as t ↑ T (Lemma 3.4 2.) implies that lim t↑T ξi t = Ξ i T P-a.s. for both players i = 1, 2).…”

“…Moreover, the weights w 3 t and w 4 t dictate the actual trading direction with respect to the other agent's tracking target. Indeed, observe that if w 3 t predominates w 4 t in (14), the first player's optimal signal ξ1 directs to also trade in parallel in the same direction as the second player, that is, in the direction of the expected future average positions of ξ 2 . In contrast, if w 4 t outweighs w 3 t , then the optimal signal imposes to trade in the opposite direction of the second player's target strategy, i.e., toward the expected weighted averages of −ξ 2 .…”

“…Our case studies in Section 4 illustrate how these two market types, plastic and elastic, correlate accordingly with the sizes of the weights w 3 and w 4 ; we refer to the graphical illustration in Figure 6 below (note, however, that the dependence of the functions w 3 , w 4 on γ and λ, as well as their interrelations with σ and T are nontrivial). In this regard, depending on the illiquidity parameters the optimal signal processes ξ1 in (14) and ξ2 in (15) account for different types of regimes. It turns out that this leads to qualitative different behavioral patterns in the Nash equilibrium of Theorem 3.1 where both predation and cooperation between the agents can occur in a coexisting manner (see Section 4 below).…”

“…As a consequence, both agents will choose their dynamic trading strategies from a suitable set of adapted stochastic processes rather than opting for static strategies from a set of deterministic functions as in the papers cited above (except for the numerical study in Carmona and Yang [9]). Other recent work on price impact games with Almgren-Chriss type price impact include, e.g., Huang et al [16], Casgrain and Jaimungal [10,11] where agents pursue optimal liquidation or trading and interact through common aggregated permanent price impact; Ekren and Nadtochiy [14] describe an equilibrium between competing market makers with constant absolute risk aversion who are intertwined through jointly incurred temporary price impact while hedging fractions of the same European option in a Markovian framework. Price impact games of liquidating agents in a market model with transient price impact are analyzed, e.g., in Schied and Zhang [20], Schied et al [21], and Strehle [24].…”