We solve  the optimal trade execution problem in the Almgren and Chirss framework with the Value-at-risk / Expected Shortfall based criterion of Gatheral and Schied when the underlying unaffected stock price follows a displaced diffusion model. The displaced diffusion model can conveniently model at the same time situations where either an arithmetic Brownian motion (ABM) or a geometric Browinan motion (GBM) type dynamics may prevail, thus serving as a bridge between the ABM and GBM frameworks. We introduce alternative risk criteria and we notice that the optimal trade executionstrategy little depends on the specific risk criterion we adopt. In most situations the solution is close to the simple Volume Weighted Average Price (VWAP) solution regardless of the specific diffusion dynamics or risk criterion that is chosen, especially on realistic trading horizons of a few days or hours. This suggests that more general dynamics need to be considered, and possibly more extreme risk criteria, in order to find a relevant impact on the optimal strategy
We solve a version of the optimal trade execution problem when the mid asset price follows a displaced diffusion . Optimal strategies under various risk criteria, namely value-at-risk, expected shortfall and a version of the cost variance measure are derived and compared. It is well known that displaced diffusions exhibit dynamics which are in-between arithmetic Brownian motions (ABM) or geometric Brownian motions (GBM) depending of the choice of the shift parameter. The model presented in the paper attempts to provide a bridge between the approach of Almgren and Chris (ABM) and Gatheral and Schied (GBM) to optimal execution. We also study the dependence of the optimal solution on the choice of the risk aversion criterion. Optimal solutions across criteria and asset dynamics are comparable although the difference are not negligible for high levels of risk aversion and low market impact assets.
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a version of
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in the Almgren and Chirss framework with the Value-at-risk / Expected Shortfall based criterion of Gatheral and Schied when the underlying unaffected stock
when the mid asset
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model. The displaced diffusion model can conveniently model at the same time situations where either an arithmetic Brownian motion
. Optimal strategies under various risk criteria, namely value-at-risk, expected shortfall and a version of the cost variance measure are derived and compared. It is well known that displaced diffusions exhibit dynamics which are in-between arithmetic Brownian motions
R
a geometric Browinan motion
geometric Brownian motions
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type dynamics may prevail, thus serving as
depending of the choice of the shift parameter. The model presented in the paper attempts to provide
R
ABM and GBM frameworks. We introduce alternative risk criteria and we notice that the optimal trade executionstrategy little depends on the specific risk criterion we adopt. In most situations the solution is close to the simple Volume Weighted Average Price (VWAP) solution regardless of the specific diffusion dynamics or risk criterion that is chosen, especially on realistic trading horizons of a few days or hours. This suggests that more general dynamics need to be considered, and possibly more extreme risk criteria, in order to find a relevant impact on the optimal strategy
approach of Almgren and Chris (ABM) and Gatheral and Schied (GBM) to optimal execution. We also study the dependence of the optimal solution on the choice of the risk aversion criterion. Optimal solutions across criteria and asset dynamics are comparable although the difference are not negligible for high levels of risk aversion and low market impact assets.