hard · Market Microstructure market-impact

Holding everything else fixed, how do these two changes affect the optimal trajectory's urgency parameter κ=√(λσ^2/eta) and the resulting expected cost-versus-variance trade-off?

1. Both raising λ and raising σ increase κ and accelerate liquidation, but only λ reflects a preference change while higher σ raises both the optimal expected impact cost and the residual timing variance the trader is fleeing.
2. Both raising λ and raising σ increase κ and accelerate liquidation identically, since the mean-variance objective depends only on the product λσ^2 and the two inputs are interchangeable in every observable outcome
3. Raising λ increases κ and front-loads the trade, but raising σ actually decreases κ, since higher volatility widens the efficient frontier and rewards patience instead, pushing the two inputs in opposite directions entirely
4. Raising σ increases κ and front-loads the trade, but raising λ leaves κ completely unchanged, since risk aversion merely rescales the trader's objective function and cannot alter the underlying Euler-Lagrange solution's curvature in any way

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