Robert Almgren
Quantitative Brokers and NYU
Feb 2009
Quantitative Challenges
in Algorithmic Execution
quantitativebrokers
Sell-side (broker) viewpoint
2
"Flow" business
no pricing computations
use for standardized exchange-traded products
Agency trading: execute large transactions
profit from commissions
Optimize each order individually
no knowledge of overall program optimize whatever metrics you set
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Algo trading business
Three legs of quantitative framework
1. optimal trading strategies 2. post-trade reporting
3. pre-trade cost estimation
Client uses these to improve investment results
Very highly developed in equities
all brokers provide rich suite of
transaction cost tools and execution algorithms In development for other asset classes
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Algorithmic trading system
4 Algorithmic Trading System Orders and execution parameters
Fill reports and execution quality analysis
Client
hedge fund mutual fundMarkets
Fill reports Child ordersMarket Data
Broker
Time slicingand order submission using quantitative analysis
Trades and quotes News feeds etc. "Buy 50,000 XYZ, moderate urgency, max price $55, complete by 14:00"
"You are done on 23,000 XYZ, at 15c better than strike.
Today's average so far on 17 orders is 13 bp better than VWAP"
To Cincinnati exchange: "Limit buy order 100 XYZ @ 53.22" equities options futures foreign exchange etc.
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Post-trade report
Design to reflect criteria for good trade
Mean and standard deviation of results Relative to arrival price, VWAP, close
Subdivide by all possible parameters
strategy, urgency, buy/sell, primary exchange, duration, order size, index membership, etc.
Execution optimizes to these benchmarks
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Post-trade cost reporting
Benchmarks: strike, VWAP, close
Sample mean
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Quantitative pre-trade cost modeling
7
Trade size as % of daily volume Execution price relative to pre-trade price, as fraction of daily vol Regression model Individual algo executions
Active area of current research:
• temporary/permanent impact
• linear/nonlinear models
• time rate of decay of impact
• non-Gaussian residuals How much slippage will it
cost me to execute this trade? Calibrate from historical trades.
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Algo trading ranking
8
Institutional Investor's Alpha, Sept 2008,
Top Equity Trading Firms
B of A #1 2008 up from #4 2007
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Outline
1. Arrival price framework
trading fast vs trading slow
2. Adaptive vs non-adaptive strategies
non-adaptivity in the base case
price limits, short-term signals, views
3. Mean-variance adaptivity
subtle aspects of time-dependent MV Julian Lorenz PhD thesis
4. Other asset classes
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1. Arrival Price Framework
Agency trading is about access to liquidity
dispersed in "space": smart order routing
tactics: technology and microstructure
dispersed in time: time slicing and trajectories
strategy: optimization and differential equations
How to determine optimal trajectory
how fast should you trade?
what is the exact optimal profile?
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What is a good trade
"Implementation shortfall" (Perold 1988)
Actual trade price vs "ideal"
Depends on benchmark
most important: decision price or "arrival price" other possibilities: VWAP, close, etc
Trade to minimize discrepancy to benchmark
average price should be close or better
result should be predictable (low variance)
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How fast should you trade?
Why trade slowly?
Market impact: wait for counterparties to appear
Why trade fast?
Arrival price answer: reduce volatility exposure Other answers
short-term alpha, control info leak, dynamic views
These answers determine optimal strategy
Grinold & Kahn 1996, Almgren & Chriss 2000
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Arrival price solution
13 Time Shares remaining to execute Order entry time Imposed end time High urgency (immediate) E large, V small Low urgency (VWAP) E small, V large
Mean variance min E+λV
Urgency (risk-aversion) λ set by client. Or proxy by target percentage of volume
Mean variance popular instead of utility function because
• simple, graphical
• indifferent to total wealth
E V VWAP Fast Variance of cost Expected cost
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2. Adaptive vs. non-adaptive
Arrival price trajectory is determined at start
compute optimal trade list (calculus of variations)
Information is revealed as execution proceeds
standard filtration: price moves are only new info changes in parameters, changes in preferences
Should you use new info to change trade list?
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Arrival price answer: No
New price information does not change trade list
15 Shares remaining Price Reevaluate trajectory at intermediate time
Price up or price down does not change optimal strategy for remaining time with same market parameters
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Arrival price time-stability
New information does not change list
Reevaluate trajectory partway through
trading gains or losses are sunk costs
risk-reward tradeoff for tail is same as original
Depends on
constant risk aversion (mean-variance) constant market parameters
arithmetic Brownian motion
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Percentage of volume parameter
17
Specify urgency by initial trade rate (slope of tangent at t=0)
"Start at 25% of expected mkt vlm"
Reevaluate with same parameter gives different trajectory
Time
Shares remaining to execute
This is a "false" reason for adapting, since it is not optimal in any sense Continuous-time adapting:
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Other reasons to adapt
Asset price approaches limit price
18
Price not worth buying above this levelLimit price: customer says it is
As price gets close to limit, do you
•
Speed up because may not be able to complete?•
Slow down because asset is getting less valued? Answer depends on value of unexecuted shares.quantitativebrokers
Customer belief on process
19
Price
Example: Price is up at mid-program
Momentum belief: Price will continue up
• Accelerate buy programs
• Slow down sell programs
Reversion belief: Price will revert down
• Slow down buy programs
• Accelerate sell programs
Almgren/Lorenz 2006 : Trade decisions
determined overnight cause momentum effects Behavioral finance loss aversion:
"Sell winners too early and ride losers too long" (Shefrin/Statman 1985)
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Liquidity and volatility fluctuations
20
Liquidity or market impact:
estimate from microstructure model
Instantaneous volatility:
estimate from HF modeling
Arrival price trajectory depends on
balance of market impact cost and volatility risk. Should adapt dynamically to variations
Discrete time: Nitin Walia, Princeton senior thesis 2006 (Cheridito) Continuous time: interesting nonlinear PDE
Interesting research problem
σ(t)
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PDE for random liquidity
Liquidity and volatility log-OU processes
Value function u solves
= instantaneous liquidity,
= time to expiration
21
Robert Almgren February 10, 2009 7
To summarize, the PDE (4) is to be solved for R(τ, ξ) on the domain τ > 0 and −∞ < ξ < ∞, with the singular initial condition
R(τ, ξ) ∼ expτ(ξ), τ → 0.
This condition corresponds to the balance Rτ ∼ −e−ξR2; these two terms are of size O!τ−2" and all other terms are of size O!τ−1".
To trace the change of variables in detail, write
R(τ, ξ, ζ) = S! τ, ξ, βLζ − βVξ ", χ = βLζ − βVξ so that Rτ = Sτ Rξξ = Sξξ − 2βVSξχ + βV2Sχχ Rξ = Sξ − βVSχ Rξζ = βLSξχ − βLβVSχχ Rζ = βLSχ Rζζ = βL2Sχχ and ξ Rξ + ζ Rζ = ξ Sξ β2 L Rξξ + 2βLβV Rξζ + βV2 Rζζ = βL2 Sξξ.
We thus wind up with the PDE for S(τ, ξ, χ)
Sτ + α ξ Sξ = exp # 2 βL ! χ + βVξ" $ − e−ξS2 + 12 αβL2Sξξ
in which χ appears only as a parameter (no χ-derivatives). We solve this equation separately for each value of χ, and since we are interested only in the case χ = 0 this reduces to (4).
It is convenient to make the change of variables
R(τ, ξ) = eξ u(τ, ξ)
for then
uτ + α!ξ −β2"uξ = e(γ−1)ξ − u2 − α
%
ξ − 12β2& u + 12 αβ2uξξ (5) with initial data
u(τ, ξ) ∼ τ1 , τ → 0.
Then the dimensional rate of selling is simply
v = ¯κ x u(τ, ξ).
Robert Almgren February 10, 2009 7
To summarize, the PDE (4) is to be solved for R(τ, ξ) on the domain τ > 0 and −∞ < ξ < ∞, with the singular initial condition
R(τ, ξ) ∼ expτ(ξ), τ → 0.
This condition corresponds to the balance Rτ ∼ −e−ξR2; these two terms are of size O!τ−2" and all other terms are of size O!τ−1".
To trace the change of variables in detail, write
R(τ, ξ, ζ) = S! τ, ξ, βLζ − βVξ ", χ = βLζ − βVξ so that Rτ = Sτ Rξξ = Sξξ − 2βVSξχ + βV2Sχχ Rξ = Sξ − βVSχ Rξζ = βLSξχ − βLβVSχχ Rζ = βLSχ Rζζ = βL2Sχχ and ξ Rξ + ζ Rζ = ξ Sξ β2 L Rξξ + 2βLβV Rξζ + βV2 Rζζ = βL2 Sξξ.
We thus wind up with the PDE for S(τ, ξ, χ)
Sτ + α ξ Sξ = exp # 2 βL ! χ + βVξ" $ − e−ξS2 + 12 αβL2Sξξ
in which χ appears only as a parameter (no χ-derivatives). We solve this equation separately for each value of χ, and since we are interested only in the case χ = 0 this reduces to (4).
It is convenient to make the change of variables
R(τ, ξ) = eξ u(τ, ξ) for then uτ + α!ξ −β2"uξ = e(γ−1)ξ − u2 − α % ξ − 12β2& u + 1 2 αβ 2u ξξ (5)
with initial data
u(τ, ξ) ∼ τ1 , τ → 0.
Then the dimensional rate of selling is simply
v = ¯κ x u(τ, ξ).
ξ τ
quantitativebrokers
Implementation in practice
22
Liquidity indicator for buy and sell Trade price and
bid/offer
Arrival price trajectory modified by liquidity
quantitativebrokers
Gamma hedging
23 Cumulative shares purchased Price Target quantity X(T) = Δ0+Γ (S(T)-S0) S(t) T Cannot go backwards (agency trading) Variant of "departure price:"benchmark is close price
Interesting research problem
Problem: charaterize distribution of final over-traded quantity
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3. Mean-variance adaptivity
Broker should optimize to post-trade benchmark
Shows historical sample mean and variance
Optimal arrival price trajectories adapt to price
not related to momentum or mean reversion
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3 kinds of adaptivity
1. No adaptivity
Strategy required to be fixed in time
2. Dynamic programming
Adapt freely to minimize E+λV
at each instant, using price observed to date
Classic arrival price: 1 & 2 give same solution
Unlike option pricing! hedge depends on price
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Dr. Strangelove Strategy
3. Rule of adaptivity is fixed in advance
Minimize E+λV measured at initial time
Not allowed to modify rule at intermediate times even if risk preferences are different at that time
This corresponds to ex-post mean-var report
Optimal solutions adapt dynamically to price
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Effect of trading gains/losses
27
Price
Buy order: price up = trading loss Compensate by slowing rest of program
to reduce market impact costs
Buy order: price down = trading gain Accelerate rest of program,
spending the gain on market impact costs Introduce anti-correlation between
• trading gain/loss in first part of program
• market impact costs in remainder
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Single-update strategy
28 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time t Shares remaining x(t) −1 0 1 100 101 102 Normalized cost C 0 Urgency κ Nonadaptivestrategy price move to that timeChange once, based on
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Dynamic programming
Standard dynamic programming:
control variable is number of shares to trade
Modified dynamic programming
extra variable is return target for remainder
Solve for entire efficient frontier over time
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Dynamic programming for frontier
30
Efficient frontier at tj-1 for x shares
c z- z0 z+ !j "#$ !j %#$ ... ... E[C]
Efficient frontier at tj for x’ shares
Var[C] Var[C]
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Continuous-time update
31 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time t Shares remaining x(t) 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 Time t S(t)−S 0 Almgren/Lorenz 2008quantitativebrokers
Improved efficient frontier
32 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 κ = 8 7.1 4.9 6.0 VWAP µ=0 0.05 0.1 0.15 0.2 µ=0 0.05 0.1 0.15 0.2 V/V lin E/E lin 0 0.2 0.4 0.6 0.8 κ=7.1 κ=4.9 κ=6.0 Nondimensional cost C μ = "market power"
ability of portfolio to push market relative to volatility
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Related to short-term investment performance
Cvitanic, Lazrak, Wang 2008:
problems with ex-post Sharpe ratio measurement
"Problem" disappears with exponential utility
Schied & Schöneborn 2008:
static pre-computed solutions are optimal
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4. Algo trading outside equities
Same factors drive other markets:
clients are becoming sophisticated
clients want transaction and execution research markets are becoming more electronic
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Development in other assets
35
Electronic Milestone 3rd Generation Algorithms
(Adaptive)
2nd Generation Algorithms
(Arrival Price, Departure Price)
Smart Order Routings 1st Generation Algorithms
(TVOL, TWAP, TVOL)
Advanced Order Types Limit Orders RFQ
Evolutionary Cycle:
Equities 32 years Instinet 1976 Futures 16 years SFE 1992 FX 15 years EBS 1993
Fixed Income 9 years TradeWeb 1998
EQUITY
FUTURES FX
quantitativebrokers
In order of maturity
1. Equities
poster child for agency algo execution
2. Equity-linked products: futures and options
equity players are expanding there
3. Foreign exchange
electronic trading is widespread, algos coming
4. Fixed income
most traditional, furthest from algo development product complexity combines with market
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ASSET MANAGEMENT GROUP
Best Execution
Guidelines
for
Fixed-Income
Securities
WHITE PAPER J A N U A R Y 2 0 0 8“... It is clear that the duty to seek best execution imposed on an asset manager is the same regardless of
whether the manager is undertaking equity or fixed income transactions. The characteristics of the
fixed-income markets present a manager with practical difficulties, though, in assessing and documenting
fixed-income best execution not faced when undertaking equity
quantitativebrokers
Larry Tabb, Advanced Trading, April 2008
“The opportunity may be right to open up the fixed-income markets to alternative execution ... the
industry may need to migrate from a traditional OTC market without a central venue to a more traditional exchange model in which there are not only liquidity providers making two-sided markets but a vibrant
agency model as well. In this model, investors work with brokers on a commission-basis and trade on
behalf of their clients in a more transparent and open community.”
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