Balancing Small Transaction Costs with Loss of Optimal Allocation in Dynamic Stock Trading Strategies

Trading With Small Price Impact

“…If θ 0 t = (t, S t ) is a delta-hedge in a complete Markovian setting then this is the "Cash-Gamma," i.e., the second derivative of the option price with respect to the underlying, multiplied by the squared value of the latter. 5 This bound is derived by noticing that the deviations from the frictionless target are approximately uniform in this case(Janeček and Shreve 2004;Rogers 2004;Goodman and Ostrov 2010; Muhle-Karbe 2013, 2015;Kallsen and Li 2013), so that the corresponding average squared deviation equals one-third of the halfwidth of the no-trade region determined inMartin (2012);Soner and Touzi (2013); Muhle-Karbe (2013, 2015);Kallsen and Li (2013).6 To see this, note that the approximate probability density of the deviation is a "hat function" in this case, so that the corresponding average squared deviation is given by one-sixth of the halfwidth of the no-trade region determined byKorn (1998);Altarovici et al (2015).7 This correspondence remains true with several risky assets, where optimal liquidation has been studied bySchied et al (2010);Schöneborn (2011).…”

Long Horizons, High Risk Aversion, and Endogenous Spreads

“…More broadly, our results are relevant for the literature on exponential utility with transaction costs, both in the context of portfolio choice (Mokkhavesa and Atkinson ; Liu ; Goodman and Ostrov ) and of option pricing (Davis, Panas, and Zariphopoulou ; Whalley and Wilmott ; Barles and Soner ). In contrast to these papers, we remove consumption and random endowment from our model, focusing instead on long‐horizon asymptotics for tractability.…”

Option Pricing and Hedging With Small Transaction Costs

“…As first pointed out by Rogers () (also compare Goodman and Ostrov ), the utility loss due to transaction costs is composed of two parts. On the one hand, there is the displacement loss due to following the strategy instead of the frictionless maximizer .…”

“…To further simplify the formulas for both parts of the utility loss, replace-at the leading order O(ε 2/3 )-the terms ϕ 2 by their expectation 1 3 ( ϕ + ) 2 under the uniform distribution on [ ϕ − , ϕ + ] (compare Rogers 2004;Goodman and Ostrov 2010), which is justified below.…”