medium · Market Microstructure
In the Almgren-Chriss model, a trader's total expected cost is the sum of permanent impact (linear in total order size, independent of speed) and temporary impact (proportional to the square of the trading rate, times the horizon).
If a trader halves the execution horizon from T to T/2 while keeping the total order size fixed, and temporary impact dominates permanent impact, how does total expected cost change?
- Total expected cost approximately doubles, because temporary impact cost is proportional to (trading rate squared) times horizon, and halving the horizon while keeping total shares fixed doubles the trading rate, which quadruples the rate-squared term but only for half as long — netting to double the cost.
- Total expected cost quadruples, because the trading rate doubles and temporary impact scales as the square of the rate, irrespective of the shorter horizon.
- Total expected cost is halved, because finishing in half the time reduces exposure to permanent impact by 50%.
- Total expected cost is unchanged, because permanent impact does not depend on speed and fully offsets the temporary impact change.
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