MATH 5440: Week 7 Assignement
Due Date: March 10, 2023 at 10am
Exercise 1 Trading two consecutive orders
This exercise reconciles two points of view on consecutive trades:
(a) One can treat the second order as the first order’s continuation.
(b) One can treat the second order as a separate trade from an external source.
One shows that the two points of view lead to the same optimal trading strategy. Consider a generalized OW model with parameters β > 0, λ = eγ satisfying the no price-manipulation condition 2β + γ′ > 0. For a given trading process Q, the price impact is the solution to the SDE
dIt = −βtItdt + λtdQt. Consider a deterministic alpha signal αt.
1. In the continuation approach, Q0 = Q ̃ and I0 = I ̃ are non-zero and capture the first order’s effect. Derive the optimal impact state It0.
2. In the separate approach, Q0 = 0 and I0 = 0. The external impact I ̄ captures the first order’s effect. The first order’s external impact satisfies
∀t>0; dI ̄=−βI ̄dt. ttt
Derive the optimal impact state It1. 3. Reconcile the two points of view.
Exercise 2 Absence of price manipulation strategies
This exercise finds a lower bound on the decay parameter βt of a general- ized OW model based on a liquidity profile vt to rule out price manipulation
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strategies. Assume the intraday volume profile v follows the deterministic
vt = e4·(t−0.7)2 .
Local square root model under the calendar clock I
The impact model considered is dIt=−βItdt+√λ dQt.
1. Establish the lower bound on β that guarantees no price manipula-
2. For β = 0.01, find a pair of trades that lead to a price manipulation paradox.
Local log model under the volume clock II
The impact model considered is
dIt = −βvtItdt + vλ dQt.
1. Establishalowerboundonβthatguaranteesnopricemanipulation.
2. For β = 0.01, find a pair of trades that lead to a price manipulation
Exercise 3 Optimal execution for large orders
This exercise solves an optimal execution problem under a globally con- cave AFS price impact model. The AFS model is particularly relevant when submitting sizable orders. In that regime, the instantaneous liquidity condi- tions are of second order, and price impact’s concavity drives trading costs. The interval [0, T ] represents a single trading day. Consider
It = h(Jt) with h(x) = sign(x)|x|c and local dynamics
dJt = −βJtdt + λdQt
for c ∈ (0, 1]. The corresponding self-financing equation is
dY =QdS −h(J ̄)dQ −λh′(J ̄)d[Q,Q]. ttttt2tt
Assume that I0 = 0.
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1. Under this model, solve the optimal execution problem for the tar- get impact state I∗. Assume given a deterministic alpha αt and an overnight alpha α ̃.
2. What is the relationship between the order size QT and the implied overnight alpha α ̃ when there is no trading alpha?
3. Assume that I0 > 0 from a previous order. Solve the idealized optimal execution problem for the target impact state I∗ without trading alpha. How does the order size QT change given I0?
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