Home
/
Today's Focus
Adverse Selection and Inventory Models
Hard
·
28 min read
Market Microstructure
Market Making
Adverse Selection
Stochastic Control
Avellaneda-Stoikov
1
Article
2
Notebook
3
Quiz
Quick Quiz
1.
In Glosten-Milgrom, a market buy raises the dealer's belief
μ
t
=
Pr
[
V
=
V
H
]
\mu_t=\Pr[V=V_H]
μ
t
=
Pr
[
V
=
V
H
]
. By what mechanism does this cause a permanent price impact?
The buy reduces available ask liquidity, widening the spread and raising the mid
The buy is evidence an informed trader may have acted, raising the dealer's value estimate, so the mid is revised up
The buy order mechanically increases demand, pushing the price up
The dealer raises the ask to avoid further adverse selection from the same trader
2.
Avellaneda-Stoikov reservation price
r
=
S
−
q
γ
σ
2
(
T
−
t
)
r=S-q\gamma\sigma^2(T-t)
r
=
S
−
q
γ
σ
2
(
T
−
t
)
. With
q
=
−
20
q=-20
q
=
−
20
,
γ
=
0.01
\gamma=0.01
γ
=
0.01
,
σ
=
0.25
\sigma=0.25
σ
=
0.25
,
T
−
t
=
0.5
T-t=0.5
T
−
t
=
0.5
, the reservation price relative to mid is:
r
=
S
+
0.025
r=S+0.025
r
=
S
+
0.025
(using
σ
=
0.25
\sigma=0.25
σ
=
0.25
as the variance instead of
σ
2
=
0.0625
\sigma^2=0.0625
σ
2
=
0.0625
)
r
=
S
+
0.00625
r=S+0.00625
r
=
S
+
0.00625
; the maker skews quotes above mid to buy back the short
r
=
S
−
0.00625
r=S-0.00625
r
=
S
−
0.00625
; the maker skews below mid to sell
r
=
S
r=S
r
=
S
; a short position has no effect on the reservation price
3.
The Avellaneda-Stoikov half-spread is
1
γ
ln
(
1
+
γ
/
k
)
+
1
2
γ
σ
2
(
T
−
t
)
\frac{1}{\gamma}\ln(1+\gamma/k)+\frac12\gamma\sigma^2(T-t)
γ
1
ln
(
1
+
γ
/
k
)
+
2
1
γ
σ
2
(
T
−
t
)
. What does each term represent and what happens to the spread as
t
→
T
t\to T
t
→
T
?
Both terms grow as
t
→
T
t\to T
t
→
T
, so the maker widens spreads near the close
The first is adverse selection and grows; the second is inventory risk and shrinks; total grows
Both terms vanish as
t
→
T
t\to T
t
→
T
, so the maker quotes a zero spread at the close
A time-invariant floor plus inventory risk that vanishes as
t
→
T
t\to T
t
→
T
4.
In the Avellaneda-Stoikov model, with zero inventory (
q
=
0
q=0
q
=
0
) the optimal bid and ask are symmetric about the mid-price
S
t
S_t
S
t
.
True
False
5.
Glosten-Milgrom predicts the spread narrows as trading proceeds. What is the intuition, and does it match equity-market evidence?
GM predicts a constant spread, independent of the number of trades
More trades cut uncertainty
μ
(
1
−
μ
)
\mu(1-\mu)
μ
(
1
−
μ
)
, narrowing the spread (as observed)
More trades lower order-arrival rates, and lower rates narrow the spread
More trades raise the dealer's inventory, widening the spread — contradicting GM
6.
Order arrival intensity is
λ
(
δ
)
=
A
e
−
k
δ
\lambda(\delta)=Ae^{-k\delta}
λ
(
δ
)
=
A
e
−
k
δ
. Ignoring inventory, the half-spread that maximises expected income per unit time from one side is:
δ
∗
=
1
/
k
\delta^*=1/k
δ
∗
=
1/
k
, from maximising
δ
A
e
−
k
δ
\delta\,A e^{-k\delta}
δ
A
e
−
k
δ
δ
∗
=
k
\delta^*=k
δ
∗
=
k
— the spread equals the decay parameter
δ
∗
=
A
/
k
\delta^*=A/k
δ
∗
=
A
/
k
— proportional to the baseline arrival rate
δ
∗
=
0
\delta^*=0
δ
∗
=
0
— quote at the mid to maximise fill rate
Submit
←
Notebook