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Almgren-Chriss Optimal Execution
Hard
·
25 min read
Market Microstructure
Optimal Execution
Market Impact
Stochastic Control
Almgren-Chriss
1
Article
2
Notebook
3
Quiz
Quick Quiz
1.
The shortfall variance for a trajectory
x
(
t
)
x(t)
x
(
t
)
is
V
a
r
[
I
S
]
=
σ
2
∫
0
T
x
(
t
)
2
d
t
\mathrm{Var}[\mathrm{IS}]=\sigma^2\int_0^T x(t)^2\,dt
Var
[
IS
]
=
σ
2
∫
0
T
x
(
t
)
2
d
t
. Why does it depend on remaining inventory
x
(
t
)
x(t)
x
(
t
)
, not on the trading rate
v
(
t
)
v(t)
v
(
t
)
?
It depends on
v
(
t
)
v(t)
v
(
t
)
: trading faster raises price uncertainty
The
x
(
t
)
2
x(t)^2
x
(
t
)
2
term comes from the quadratic utility, not from price risk
Mid-price diffusion while the position is open: a larger
x
(
t
)
x(t)
x
(
t
)
means more exposure to
σ
d
W
\sigma\,dW
σ
d
W
It comes from permanent impact, which is proportional to total shares sold
2.
The optimal trajectory
x
∗
(
t
)
=
X
sinh
(
κ
(
T
−
t
)
)
/
sinh
(
κ
T
)
x^*(t)=X\,\sinh(\kappa(T-t))/\sinh(\kappa T)
x
∗
(
t
)
=
X
sinh
(
κ
(
T
−
t
))
/
sinh
(
κ
T
)
is front-loaded relative to TWAP. Why?
It captures price appreciation by selling before temporary impact reverts
Early on,
x
(
t
)
x(t)
x
(
t
)
is large, so the variance penalty
λ
σ
2
x
(
t
)
2
\lambda\sigma^2 x(t)^2
λ
σ
2
x
(
t
)
2
is large; selling faster reduces it sooner
The
sinh
\sinh
sinh
form is an approximation to exponential decay; front-loading is a linearisation artefact
Front-loading lowers temporary impact by selling slowly early when the book is thin
3.
At
λ
=
0
\lambda=0
λ
=
0
, minimising
η
∫
0
T
v
(
t
)
2
d
t
\eta\int_0^T v(t)^2\,dt
η
∫
0
T
v
(
t
)
2
d
t
subject to
∫
0
T
v
d
t
=
X
\int_0^T v\,dt=X
∫
0
T
v
d
t
=
X
gives TWAP (
v
=
X
/
T
v=X/T
v
=
X
/
T
). Which inequality demonstrates this?
Grönwall's inequality applied to the inventory ODE
Jensen's inequality in the form
E
[
v
]
2
≥
E
[
v
2
]
\mathbb{E}[v]^2\ge\mathbb{E}[v^2]
E
[
v
]
2
≥
E
[
v
2
]
Cauchy-Schwarz:
X
2
≤
T
∫
v
2
d
t
X^2\le T\int v^2\,dt
X
2
≤
T
∫
v
2
d
t
, equality iff
v
v
v
constant
The triangle inequality applied to the
L
2
L^2
L
2
norm of
v
v
v
4.
Increasing the risk-aversion
λ
\lambda
λ
raises expected implementation shortfall while lowering its variance, so the risk-neutral trader (
λ
=
0
\lambda=0
λ
=
0
) achieves the lowest expected cost.
True
False
5.
The efficient frontier is the set of (expected IS, variance) pairs from optimal strategies over all
λ
≥
0
\lambda\ge 0
λ
≥
0
. Which strategy is guaranteed to lie ON the frontier?
A non-monotone schedule that buys shares back mid-execution
TWAP — the
λ
=
0
\lambda=0
λ
=
0
(risk-neutral) minimum-cost endpoint
Sell half the position immediately, then TWAP the remainder
VWAP — trading in proportion to historical volume
6.
Why is permanent impact a sunk cost that does not affect the optimal Almgren-Chriss schedule?
Permanent impact is zero in Almgren-Chriss; only temporary impact is modelled
The total permanent-impact cost depends only on
X
X
X
and
γ
\gamma
γ
, not on the schedule
{
n
k
}
\{n_k\}
{
n
k
}
Permanent impact reverses after the trade, so it nets to zero
It affects future periods, so subsequent trades in the schedule hedge it away
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