MATH 5440: Week 4 Assignement
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MATH 5440: Week 4 Assignement
Due Date: February 17, 2023 at 10am
Exercise 1 Example impact computations
This exercise provides impact curve examples for two impact models.
Assume that (0, 1) represents a full trading day.
Closed-form examples I
Consider the original OW model
dIt = −βItdt + λdQt
with β, λ > 0.
1. Assume the trader sends in a Time Weighted Average Price (TWAP) order for the day. Hence, one has Qt = Qt for a constant Q for all t ∈ (0, 1). Furthermore, assume that I0 = 0. Compute and plot It .
2. Assume the unperturbed price S is a martingale. Compute and plot the expected fundamental P&L Y and the mark-to-market P&L X for t ∈ (0, 1).
3. Assume the impact state halves by the start of the next day I1+ = 
I1− .
The trader does not trade on the second day. Compute and plot I, X, and Y for t ∈ (1, 2).
Numerical examples II
Consider the impact model
λ
vt
with vt the intraday market activity captured by the volatility of market trades.
Assume the intraday volume profile v follows the deterministic curve vt = e4 · (t−0.7)2 .
1. Plot the function vt to visualize the intraday volume profile.
2. Numerically solve questions 1-3 for this model.
Exercise 2 Globally concave AFS model
This exercise establishes an order-based impact formula from a globally concave AFS model. Consider a globally concave impact model
It = sign(Jt)|Jt|c
with local dynamics
dJt = −βJtdt + λdQt
for β, λ > 0 and c ∈ (0, 1].
Consider an order of size Q > 0 traded over [0, T]. Let the unperturbed price S be a martingale, J0 = 0, and Q0 = 0.
1. Map the objective function in impact space.
2. Derive the optimal execution strategy. What is IT as a function of Q?
3. Consider a TWAP execution. What is IT as a function of Q?
2023-02-17