1. No category

A closed-form execution strategy to target
VWAP
IPAM, UCLA, April 2015
´
Alvaro
Cartea and Sebastian Jaimungal
University College London, UK and University of
Toronto, Canada
[email protected]
April 13, 2015
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VWAP
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Executions are delegated to brokers
How can investors measure the performance of brokers?
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Obtaining what the market has borne is sometimes enough,
A popular benchmark is
RT
Su µu du
,
VWAP = 0R T
µu du
0
(1)
where St is the midprice and µt is the volume traded by the entire
market.
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Siblings:
VWAP and other Volume-based Algorithms
3 / 32
Targeting VWAP via POV or POCV
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Brokers may guarantee VWAP and charge a fee
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Account for price impact:
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Incentive is to outperform VWAP
slice parent order over a time window,
do ‘bits’ in a volume-based fashion to target VWAP.
Formulation: set up a performance criteria where the investor/broker
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seeks to execute a large order over a trading horizon T ,
maximises sales proceeds when liquidating shares,
accounts for permanent and temporary impact of trades,
tracks (penalises deviations from) POV or POCV,
ensures that by the end of trading horizon target quantity is achieved.
4 / 32
Volume patterns
log(1+volume)
14
0.1
0.08
12
0.06
10
0.04
8
6
0.02
10:00
12:00
Time
14:00
16:00
0
Figure : ORCL traded volume for orders sent to NASDAQ in all of 2013 using
5 minute buckets – heat-map of the data and 25%, 50% and 75% quantiles.
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Price Impact of Market Orders
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Permanent Impact
We assume a linear relationship between net order-flow and changes in
the midprice, thus for every trading day we perform the regression
∆Sn = b µn + εn ,
(2)
where
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∆Sn = Snτ − S(n−1)τ is the change in the midprice,
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µn is net order-flow defined as the difference between the volume of
buy and sell MOs during the time interval [(n − 1)τ, nτ ], and
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εn is the error term (assumed normal). In the empirical analysis we
choose τ = 5 min.
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-6
x 10
2500
2000
1.5
Frequency
Permanent Impact
2
1
0.5
0
1500
1000
500
50
100
150
200
0
-2
250
Trading Day of 2013
-1
0
1
Net Order-Flow (×105 )
2
5
E[ (µ+ − µ− ) | ∆S ]
1
x 10
0.5
0
-0.5
-1
-0.1
-0.05
0
0.05
0.1
Price Change (∆S)
Figure : Order-Flow and effect on the drift of midprice of INTC.
8 / 32
Temporary Impact
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Assume that temporary price impact is linear in the rate of trading
so the difference between the execution price that the investor
receives and the midprice is k ν,
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To do this we take a snapshot of the LOB each second,
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Determine the price per share for various volumes (by walking
through the LOB),
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Compute the difference between the price per share and the best
quote at that time,
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Perform a linear regression.
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0.03
24.6
Price Impact
0.025
Price
24.55
24.5
24.45
24.4
24.35
0.02
0.015
0.01
0.005
24.3
0
2
4
0
0
6
Volume
1
2
3
Volume
4
x 10
4
5
4
x 10
-5
Price Impact (k)
10
-6
10
-7
10
-8
10
10
12
Time
14
16
Figure : An illustration of how the temporary impact may be estimated from
snapshots of the LOB using INTC on Nov 1, 2013. The first panel is at
11:00am, the second from 11:00am to 11:01am and the third contains the
entire day.
10 / 32
Parameters
Table : Permanent and temporary price impact parameters for Nasdaq stocks,
ADV of MOs, average midprice, σ volatility (hourly) of arithmetic price
changes, mean arrival (hourly) of MOs λ± , and average volume of MOs E[η ± ].
Data are from Nasdaq 2013.
FARO
SMH
NTAP
mean
stdev
mean
stdev
mean
stdev
ADV
midprice
σ
b
k
b/k
23,914
40.55
0.151
1.41E-04
1.86E-04
1.02
14,954
6.71
0.077
9.61E-05
2.56E-04
0.83
233,609
37.90
0.067
5.45E-06
8.49E-07
7.43
148,580
2.44
0.039
4.20E-06
8.22E-07
6.24
1,209,628
38.33
0.078
5.93E-06
3.09E-06
2.04
642,376
3.20
0.045
2.31E-06
1.75E-06
0.77
λ+
E[η + ]
λ−
E[η − ]
16.81
103.56
17.62
104.00
9.45
21.16
10.69
21.79
47.29
377.05
46.37
381.70
28.13
118.05
27.62
126.74
300.52
308.45
293.83
312.81
144.48
53.09
136.13
49.86
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-7
x 10
60
4
Frequency
50
k
3
2
1
0
0
40
30
20
10
0.5
0
0
1
b
-6
x 10
1
2
3
4
5
b/k
Figure : Price Impact INTC using daily observations for 2013.
12 / 32
Targeting VWAP via POV
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The Model
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The investor searches for an optimal speed to liquidate N shares
over a trading horizon T .
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Inventory
dQtν = −νt dt ,
Q0ν = N ,
(3)
where νt is the speed of liquidation.
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The rest of the market trades at speed µ+ for buy side and µ− for
sell side.
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And the idea is to target, at every instant in time, a percentage
ρ ∈ [0, 1] of the overall traded volume.
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The overall traded volume, at every instant in time, is
−
µ+
t + µt + νt .
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The Model, price impact
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Execution price
Sˆtν = Stν − k νt ,
(4)
k > 0 is the temporary impact parameter.
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The midprice satisfies
−
dStν = b µ+
t − (νt + µt ) dt + dMt ,
S0ν = S ,
(5)
where b ≥ 0 is the permanent impact parameter, and
M = {Mt }0≤t≤T is a martingale (independent of all other
processes).
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Cash
dXtν = Sˆtν νt dt ,
X0v = X0 .
(6)
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The Model (cont)
Performance criteria is
Z
ν
ν
ν
ν
H (t, x, S, µ, q) = Et,x,S,µ,q Xτ ν +Qτ ν (Sτ ν −α Qτ ν )− ϕ˜
ν
τν
(νu −
2
χνu )
du
,
t
(7)
where µ = {µ+ , µ− },
−
χνt := ρ˜ × µ+
t + µt + νt ,
(8)
τ ν = T ∧ inf{t : Qtν = 0}, and her value function
H(t, x, S, µ, q) = sup H ν (t, x, S, µ, q) ,
(9)
ν∈A
where A is the set of admissible strategies consisting of F-predictable
RT
processes such that 0 |νu | du < +∞, P-a.s.
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DPE
The value function satisfies
0 = ∂t + LM + Lµ H
n
o
+ sup (S − kν) ν ∂x H − ν ∂q H + b ((µ+ − µ− ) − ν) ∂S H − ϕ (ν − ρ µ)2 ,
ν
(10)
for q > 0, with terminal condition
H(t, x, S, µ, q) = x + q(S − αq) ,
(11)
where µ = µ+ + µ− , Lµ represents the infinitesimal generator of µ, LM
the infinitesimal generator of M, and we have introduced the re-scaled
parameters
ϕ = ϕ(1
˜ − ρ˜)2
and
ρ = ρ˜/(1 − ρ˜) .
17 / 32
Solving the DPE
The DPE (10) admits the solution
H(t, x, S, µ, q) = x + q S + h0 (t, µ) + h1 (t, µ) q + h2 (t) q 2 ,
q>0
where
1
T −t
+
k + ϕ α − 12 b
h2 (t) = −
ϕρ
h1 (t, µ) =
(T − t) + ζ
Z
h0 (t, µ) =
"
T
Et,µ
t
−
1
b,
2
(12a)
T
−
Et,µ µ+
ds
s + µs
t
b
+
(T − t) + ζ
Z
!−1
(12b)
T
Z
−
((T − s) + ζ) Et,µ µ+
s − µs ds ,
t
#
− 2
(h1 (t, µs ) − 2ϕρ(µ+
s + µs ))
2 +
− 2
− ϕρ (µs + µs ) ds ,
4(k + ϕ)
(12c)
and the constant
ζ=
k +ϕ
.
α − 12 b
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Optimal liquidation speed (Target POV!)
νt∗ =
∗
1
Qν
(T − t) + ζ t
(13a)
(
ϕ
+
ρ
k +ϕ
(µ+
t
+
µ−
t )
1
−
(T − t) + ζ
Z
)
T
E[
µ+
s
+
µ−
s
| Ftµ
] ds
t
(13b)
−
b
k +ϕ
Z
t
T
(T − s) + ζ +
E µs − µ−
| Ftµ ds ,
s
(T − t) + ζ
∀t < τ ν
∗
(13c)
where Ftµ denotes the natural filtration generated by µ, and νt∗ = 0 for
t ≥ τ , is the admissible optimal control we seek.
19 / 32
Optimal liquidation speed (targeting POV)
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The first component is TWAP-like.
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The second component consists of targeting instantaneous total
order-flow plus a weighted average of future expected total
order-flow. Although the POV target is ρ µt , the strategy targets a
ϕρ
≤ ρ, where equality is achieved if the costs
lower amount since k+ϕ
of missing the target are ϕ → ∞ and k remains finite, or there is no
temporary impact k ↓ 0.
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The last component acts to correct her trading based on her
expectations of the net order-flow from that point in time until the
end of the trading horizon. When there is a current surplus of buy
trades she slows down her trading rate to allow the midprice to
appreciate before liquidating the rest of her order.
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From POV to VWAP
Investor must liquidate all shares. Do this by letting α → ∞
lim νt∗
α→∞
=
∗
1
Qtν
T −t
ϕ
+
k +ϕ
b
−
k +ϕ
1
ρ µt −
T −t
RT
t
Z
!
T
E[ µ+
s
+
µ−
s
| Ftµ
] ds
t
µ
−
(T − s) E [µ+
s − µs | Ft ] ds
,
T −t
and follow this by taking the limit ϕ → ∞, resulting in
lim
lim νt∗
ϕ→∞ α→∞
=
∗
1
−
µ+
(15)
Qtν + ρ
t + µt
T −t
!
Z T
+
µ
1
− −
E µs + µs
Ft ds . (16)
T −t t
21 / 32
Targeting VWAP via POCV
22 / 32
Accumulated Volume
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Accumulated volume V of orders, excluding the agent’s, is
Z t
−
Vt =
µ+
du .
u + µu
0
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The investor’s performance criteria is now modified to
"
H ν (t, x, S, µ, V , q)
=
Et,x,S,µ,V ,q Xτνν + Qτνν (Sτνν − αQτνν )
Z
τν
−ϕ
˜
(N − Quν ) − ρ˜ (Vu + (N − Quν ))
2
#
du
t
ν
and τ = T ∧ inf{t :
I
Qtν
= 0}.
The investor’s value function is
H(t, x, S, µ, V , q) = sup H ν (t, x, S, µ, V , q) .
(17)
ν∈A
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DPE
The value function should satisfy the DPE
2
0 = ∂t + LM + Lµ,V H − ϕ (N − q) − ρ V
(18)
+ sup b ((µ+ − µ− ) − ν) ∂S H + (S − k ν) ν ∂x H − ν ∂q H ,
ν
for q > 0, subject to H(T , x, S, y , q) = x + q (S − αq) and
ϕ = ϕ(1
˜ − ρ˜)2
and
ρ = ρ˜/(1 − ρ˜) .
Here, Lµ,V denotes the infinitesimal generator of the joint process
(µt , Vt )0≤t≤T .
24 / 32
The DPE (18) admits the solution
H(t, x, S, µ, V , q) = x + q S + h0 (t, µ, V ) + h1 (t, µ, V ) q + h2 (t) q 2 ,
q > 0 (19)
where
h2 (t) = −
p
kϕ
γ e ξ (T −t) + e ξ (T −t)
−
γ e ξ(T −t) − e −ξ (T −t)
1
2
b,
(20a)
T
Z
`(u, t) N − ρ Et,µ,V [Vu ] du
h1 (t, µ, V ) = 2 ϕ
t
T
Z
+b
(20b)
−
`(u, t) Et,µ,V (µ+
u − µu ) du ,
t
Z
h0 (t, µ, V ) =
T
h
Et,µ
t
1
4k
the function
`(u, t) :=
h1 (u, µu , Vu )
ξ=
ϕ
,
k
− ϕ N − ρ Vu
γ e ξ (T −u) − e −ξ (T −u)
,
γ e ξ (T −t) − e −ξ (T −t)
and the constants
r
2
and
γ=
α − 12 b +
α − 12 b −
√
√
kϕ
kϕ
2 i
du ,
(20c)
(20d)
.
25 / 32
Optimal control for POCV
Moreover, the trading speed, for q > 0, is given by
∗
1+γ
Qν
γ e ξ(T −t) − e −ξ (T −t) t
Z T
n
h io
∗
2
−ξ
`(u, t)
N − Qtν − ρ E Vu Ftµ,V
du
νt∗ = ξ
(21a)
(21b)
t
−
b
2k
Z
t
T
h
i
− µ,V
`(u, t) E (µ+
−
µ
)
F
du ,
t
u
u
∗
∀ t < τν .
(21c)
And the optimal inventory path
∗
Qtν = `(t, 0) N
Z tZ T
n h i
N − ρ E Vs Fuµ,V
+
+
`(u, s) ϕ
k
0
u
b
E
2k
h
io
− µ,V
ds du .
(µ+
s − µ s ) Fu
(22)
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Interpreting liquidation speed
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(21a) is an Almgren-Chriss like strategy which approaches TWAP
near maturity.
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(21b) corrects for the weighted average of the difference
between
ν∗
what the investor hash liquidated
so
far
N
−
Q
and
her
future
t
i
µ,V
targeted volume ρ E Vu Ft
.
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(21c) contributes a weighted average of future net order-flow and
accounts for the permanent impact on the midprice. If future net
order-flow is expected to be buy heavy, then investor slows down
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Finally, as t → T the second and third terms become negligible and
the investor ignores order-flow entirely and instead focuses on
completing her trades.
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Targeting VWAP
Let α → ∞, so that the investor ensures that she completely liquidates
(T −u))
her position by the terminal time, γ → 1, therefore `(u, t) → sinh(ξ
sinh(ξ (T −t))
and so we have
∗
lim νt∗
α→∞
=
−
ξ
ξ
Qtν
sinh (ξ (T − t))
2
T
Z
t
−
b
2k
Z
t
io
h ∗
sinh (ξ (T − u)) n
du
N − Qtν − ρ E Vu Ftµ,V
sinh (ξ (T − t))
T
i
sinh (ξ (T − u)) h +
µ,V
E (µu − µ−
du .
u ) Ft
sinh (ξ (T − t))
28 / 32
Strategy Performance
Assume that other market participants trades follow
+
+ +
+
dµ+
t = −κ µt dt + η1+N + dNt ,
(23a)
−
− −
−
dµ−
t = −κ µt dt + η1+N − dNt ,
(23b)
t−
t−
where
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κ± ≥ 0 are the mean-reversion rates,
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Nt+ and Nt− are independent homogeneous Poisson processes with
intensities λ+ and λ− , respectively,
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{η1± , η2± , . . . } are non-negative i.i.d. random variables with
distribution function F , with finite first moment, independent from
all processes.
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In addition, we require κ± > λ± E[η1± ] to ensure that µ± remain
bounded P-a.s..
29 / 32
VWAP as target
Compute VWAP as
RT
VWAP =
−
Suν (µ+
u + µu + νu ) du
,
RT +
µu + µ−
u + νu du
0
0
(24a)
while the execution price is computed as
Xν
Exec. Price = T =
N
RT
0
Sˆuν νu du
.
N
(24b)
30 / 32
POV-VWAP performance
Table : The statistics of the execution price, VWAP, and relative error
(computed as (Exec. Price − VWAP)/VWAP for each simulation) and reported
in basis points (i.e., ×104 ).
quantile
FARO
mean
stdev
5%
25%
50%
75%
95%
%t : νt∗ < 0
SMH
NTAP
VWAP
Rel.Error
VWAP
Rel.Error
VWAP
Rel.Error
$ 40.54
$0.11
$40.35
$40.46
$40.54
$40.61
$40.72
8.9
16.9
-4.2
-0.4
2.5
12.0
42.6
$ 37.90
$0.04
$37.83
$37.87
$37.90
$37.92
$37.96
2.98
6.10
-1.03
-0.16
0.53
3.60
15.29
$ 38.30
$0.06
$38.20
$38.26
$38.30
$38.34
$38.40
0.19
0.87
-0.78
-0.20
0.03
0.36
1.66
27.3%
18.4%
0.6%
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POCV-VWAP performance
Table : The statistics of the execution price, VWAP, and relative error
(computed as (Exec. Price − VWAP)/VWAP for each simulation) and reported
in basis points (i.e., ×104 ). For POCV we set ϕ = 105 × k
quantile
FARO
mean
stdev
5%
25%
50%
75%
95%
%t : νt∗ < 0
SMH
NTAP
VWAP
Rel.Error
VWAP
Rel.Error
VWAP
Rel.Error
$40.54
$0.11
$40.35
$40.46
$40.54
$40.61
$40.73
5.0
14.2
-13.6
-3.0
1.9
11.0
31.9
$37.89
$0.04
$37.83
$37.87
$37.89
$37.92
$37.96
2.75
6.02
-5.43
-0.78
1.79
5.51
13.76
$38.30
$0.06
$38.20
$38.26
$38.30
$38.34
$38.40
1.88
2.35
-1.65
0.28
1.72
3.27
6.00
0.83%
0.013%
0.55%
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