Option Pricing and Hedging With Small Transaction Costs

Abstract

An investor with constant absolute risk aversion trades a risky asset with general Itô-dynamics, in the presence of small proportional transaction costs. In this setting, we formally derive a leading-order optimal trading policy and the associated welfare, expressed in terms of the local dynamics of the frictionless optimizer. By applying these results in the presence of a random endowment, we obtain asymptotic formulas for utility indifference prices and hedging strategies in the presence of small transaction costs.

Trading with small nonlinear price impact

“…, and it converges to zero between these boundaries. This corresponds to the instantaneous reflection off these trading boundaries that is asymptotically optimal for small proportional transaction costs [12,42,43,49,60].…”

“…The asymptotically optimal trading rate is in turn given by (3.6); the corresponding performance loss is given by the formula from Theorem 3.3. For exponential utility, the certainty equivalent loss obtained by disregarding the frictionless Lagrange multiplierŷ corresponds to the adjustment of the utility-indifference price of Hodges and Neuberger [36]; compare [11,42,61].…”

“…When these quantities are time dependent and random, they need to be averaged both across time and states. As for other frictions [42,43,50], averaging across states is performed under the frictionless marginal pricing measureQ-the small friction is priced like a marginal path-dependent option. The comparative statics of the certainty equivalent loss (1.2) also are consistent with their counterparts for proportional [2,42], quadratic [50], or fixed transaction costs [7]; the elasticity of price impact p only governs the powers to which the inputs are raised, and determines the universal constant c p .…”

“…The connection between asymptotics of utility maximization problems with small transaction costs and ergodic control problems with "frozen coefficients" has first been established using PDE techniques by Soner and Touzi [60]. For proportional costs, non-Markovian extensions of these results have been obtained formally by [42,43] and proved rigorously by [1,2,34]. A closely related strand of research studies "pathwise" criteria, where the goal is to trade off the error of tracking an exogenous target strategy against the trading costs incurred by the hedge.…”

Utility Indifference Option Pricing Model with a Non-Constant Risk-Aversion under Transaction Costs and Its Numerical Approximation

“…In Kallsen and Muhle-Karbe (2015), asymptotic formulas for utility indifference prices and hedging strategies in the presence of small transaction costs were obtained. Perrakis and Lefoll (2000) derived optimal perfect hedging portfolios in the presence of transaction costs.…”