14. Execution strategies
14.1 Introduction
Opportunistic algorithms: what and when to buy and sell
Execution strategies (algorithmic trading): where and how to buy and sell
Minimizing losses related to trading process. Commissions, fees, taxes are
fixed. What else? Order slicing into child orders to minimize market impact
(X = Σxk) and placement…
Where to trade: liquidity aggregators; smart routing; co-location.
Implementation shortfall (Perold (1988))
IS = Σ xkpk - p0Σxk + (pN – p0)(X - Σxk) + C (fixed cost)
execution cost
opportunity cost
<--------------->
<----------------->
Benchmark-driven and cost-driven algorithms
1
14. Execution strategies
14.2 Benchmark-based schedules
14.2.1 Time-weighed average price (TWAP)
Child orders are spread uniformly – but randomized in size and time
to prevent informational leak (risk of exposure).
Periodic benchmarks adjust the size.
14.2.2 Volume-weighed average price (VWAP)
Pronounced intra-day trading patterns. N trades during time T
(νk is volume at period k)
N
VWAP =
N
v p / v
k 1
k
k
k 1
k
N
v
From historical market data: uk = vk /
i 1
i
The size of child order k for the order of size X: xk = Xuk
2
14. Execution strategies
14.2 Benchmark-based schedules (continued)
14.2.3 Percentage of volume (POV)
Percentage of the total trading volume γ = xk /(Xk + xk)
xk= γXk /(1 - γ)
14.2.4 Participation weighed price (PWP)
VWAP and POV combo. If participation rate is chosen γ and order
volume is N, PWP is VWAP calculated over N/γ units traded after
order N was submitted.
3
14. Execution strategies
14.3 Cost-driven schedules
14.3.1 Risk-neutral framework
Volatility risk is neglected. Implementation shortfall is minimized.
Bertsimas & Lo (1988):
N
min E[  x k p k ] with constraint
k 1
N
x
k 1
k
X
Price follows the random walk with permanent market impact
pk = pk-1 + θxk + εk
Volume to be bought: wk = wk-1 - xk, w1 = X, wN+1 = 0
Bellman equation: Vk (pk-1, wk) = Min E{pkxk + Vk+1(pk, wk+1)}
x1* = … = x *N - result of permanent impact.
Obizhaeva & Wang (2005) , Gatheral (2009)
4
14. Execution strategies
14.3 Cost-driven schedules
14.3.2 Risk-averse framework
Almgren & Chriss (2000)
Sell X units within time T; N periods τ = T/N; tk = k*τ;
n = {n0, …, nN}, - ni number of units sold during ti-1 < t ≤ ti;
x = {x0, …, xN}; xk is the remaining number of units at time tk to be
sold; x0 = X; xN = n0 = 0.
i k
xk =X -  ni =
i 1
i N
n
i  k 1
i
Price S follows the arithmetic random walk with no drift. Market impact
has a permanent part g(nk/τ) that lasts entire trading time T and the
temporary part τh(nk/τ) that affects price only during one interval τ.
5
14. Execution strategies
14.3 Cost-driven schedules
14.3.2 Risk-averse framework (continued)
Sk = Sk-1 + στ1/2 dξ1 – τg(nk/τ)
Ŝk = Sk-1 + στ1/2 dξ1 – τh(nk/τ)
N
N
k 1
k 1
N
IS = XS0 - n k Sˆ k =   x k ( 1 / 2 d k  g (nk /  ))   nk h(nk /  )
N
E(x) =
x
k 1
k 1
N
k
g (nk /  ))   nk h(nk /  )
k 1
N
2
V(x) =    x k
2
k 1
min U = E(x) +λV(x) ; λ - risk aversion
6
14. Execution strategies
14.3 Cost-driven schedules
14.3.2 Risk-averse framework (continued 2)
Linear model: g(nk/τ) = γ nk/τ; h(nk/τ) = ε sgn(nk) + ηnk/τ; ε is fees
~ 2 τ2xk
δU/δxk => xk-1 - 2xk + xk+1 = 
~ 2 =  2 / ~
nk =
~     / 2
2 X sinh(  / 2)
cosh(  (T  t k 1 / 2 ))
cosh( T )
2(cosh(κτ) – 1) = ~ 2 τ2
When τ approaches zero, ~   and ~ 2   2 - exponential decay.
7
14. Execution strategies
14.3 Cost-driven schedules
14.3.2 Risk-averse framework (continued 3)
Huberman & Stahl (2005), Almgren & Lorenz (2007), Jondeau et al (2008),
Shied & Schöneborn (2009): time-dependent volatility and liquidity.
Power-law decay of market impact (Bouchaud et al (2004) in equity
markets, Schmidt (2010a) in FX): F(ni, tk - ti);
sk = -F(nk, 0) - initial market impact
k
Sk = S0 +
 [
i 1
1/ 2
d i  F (ni , (k  i ) )]
Note: kth child order is executed not at Sk but at Svwap k;
for a sell order Sk < Svwap k ≤ Sk-1; ck(nk) = Sk-1 - Svwap k
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14. Execution strategies
14.3 Cost-driven schedules
14.3.2 Risk-averse framework (continued 4)
ck(nk) = αnk
sk(nk) = βnk
F(nk, t) = γsk(nk)/(t-tk)m, t > tk
The case with one lagged impact:
xk-1 - 2xk + xk+1 = κ02τ2xk
κ02 = λσ2/ (α + βγτ-m)
9
14. Execution strategies
14.3 Cost-driven schedules
70
tau = 720s
60
lambda=1.0E-7
lambda=5.0E-7
Order size, MM EUR
50
40
30
20
10
0
12
24
36
Time, min
48
60
10
14. Execution strategies
14.4 The taker’s dilemma
Taker’s order or maker’s order: fast execution or saving the spread?
Distance from best price; for offers, D = P – BO; for bids, D = BB – P.
Loss function for one order:
L1 (V , D,  )  aV   T (V , D )  D 
Loss function of N = nV:
n


L( N  nV ) (V , D,  )  aV    kT (V , D )  nD 


k 1
How to define T(V, D)?
The random walk model (Lo et al (2002))
The limit-buy order with limit price Pl will be executed within the given time
interval if and only if Pmin is less than or equal to Pl => first passage time.
Cancellations from empirical data described with the gamma distribution.
11
14. Execution strategies
14.4 The taker’s dilemma (continued)
Censoring limit-order execution time (Eisler et al (2009)).
Partial fill is neglected...
Time to fill a limit order: TTF; Order lifetime before cancellation: LT
Kaplan-Meyer estimator: execution and cancellation are independent.
Then
PTTF (t ) 

PFPT (t ) PLT ( t )
P
FPT
( ) PLT (  )d
0
Empirical FPT and TTF have power-law decay
Gaps in empirical distributions...
12
14. Execution strategies
14.4 The taker’s dilemma – Simulations
Input:
• Best price - observable
• Volume at best price - observable
• Deal depletion rate – from simulations
• Total depletion rate (deals + cancellations) – from simulations
Output:
• Expected loss
• Expected execution time
13
14. Execution strategies
14.4 The taker’s dilemma - Simulations
New Order
Simulate total rate
Read price & volume
Up
Same
Down
Re-submit
Orders
Yes
Execute at
deal rate
No
Order on
top?
Yes
Order on
top?
No
Execute at
deal rate
Deplete at
total rate
Order on top
Record
Partial Fill
No
Fully
Filled?
Yes
14
Record Execution Time
14. Execution strategies
14.4 The taker’s dilemma: Resampling (Loss in pips)
Size,
MM
SR
RE MCMC1 MCMC2 BB1
BB3
BB5 BB10
1
2
5
10
20
0.65
0.74
1.01
1.35
1.89
0.62
0.73
1.00
1.34
1.89
0.61
0.73
1.04
1.47
2.16
0.66
0.76
1.06
1.56
2.04
0.69
0.81
1.12
1.54
2.20
0.69
0.81
1.13
1.53
2.16
0.51
0.67
1.05
1.56
2.33
0.72
0.82
1.10
1.48
2.07
RE yields a loss somewhat smaller than SR.
MCMC1 and MCMC2 yield practically the same results higher than SR.
The bootstrap results be rather sensitive to the block size.
15
14. Execution strategies
14.5 The taker’s dilemma: Convergence of simulations
2
1.8
1.6
Loss, pips
1.4
1.2
1
0.8
0.6
1MM
5MM
10MM
20MM
0.4
0.2
0
1
2
3
4
5
6
7
8
Number of runs, thousands
9
10
11
16
14. Execution strategies
14.6 The taker’s dilemma: Loss in pips
Week ending
Order size, MM
Taker
Automatch
Maker
Week ending
Order size, MM
Taker
Automatch
Maker
1
1.71
1.71
0.82
1
1.58
1.58
0.70
2/6/2009
2
5 10
1.82 2.10 2.46
1.73 1.86 2.18
1.00 1.44 1.99
2/20/2009
2
5 10
1.68 1.94 2.27
1.59 1.69 1.92
0.83 1.17 1.61
20
1
3.06 1.66
2.84 1.67
2.80 0.77
20
1
2.80 1.56
2.44 1.57
2.26 0.67
2/13/2009
2
5 10
1.77 2.01 2.33
1.68 1.80 2.09
0.94 1.35 1.86
2/27/2009
2
5 10
1.66 1.90 2.22
1.58 1.66 1.87
0.80 1.13 1.57
20
2.87
2.71
2.62
20
2.74
2.38
2.22
17
14. Execution strategies
14.4 The taker’s dilemma: Expected execution time (sec)
Automatch
Maker
1
0.0
5.9
Automatch
Maker
1
0.0
6.6
2/6/2009
2
5
1.2 11.2
11.6 29.0
2/20/2009
2
5
1.3 12.0
12.6 31.0
10 20
36.3 90.9
57.1 111.4
1
0.0
6.1
10 20
38.9 98.6
61.1 119.9
1
0.0
5.2
2/13/2009
2
5
1.2 11.3
12.0 29.8
2/27/2009
2
5
0.9 8.7
10.1 25.2
10 20
37.4 95.7
59.0 116.1
10 20
30.2 79.1
50.4 99.2
18