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Algorithmic Trading 2008, London 07. 04. 2008
Adaptive Arrival Price
Julian Lorenz (ETH Zurich, Switzerland)
Robert Almgren (Adjunct Professor, New York University)
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22008 Julian Lorenz, [email protected]
Outline
Evolution of algorithmic trading
“Classic” arrival price algorithms
Price adaptive algorithmsSingle-update and multi-update strategiesDynamic programmingImproved shortfall statistics and efficient frontiers
ExtensionsBayesian adaptive trading with price trend
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32008 Julian Lorenz, [email protected]
Electronic/Algorithmic Trading
Use computers to execute orders
Agency trading
Executing orders for clients
Investment decision is made
Large and increasing fraction of total order flow
92% of hedge funds and
11% of all trades, 17% by 2007 (Tabb Group)
Expected 50% of traded volume by 2010 (The Economist, ‘07)
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42008 Julian Lorenz, [email protected]
Evolution of Algorithmic Trading
1. VWAP Automatization of routine trading tasks
2. Arrival PriceQuantitative modeling and optimizationMarket impact, volatility, alpha, etc
3. AdaptivityExecution trajectory responds to marketbehavior in a variety of waysHow to do this optimally?
Evol
utio
n
Use computers to execute orders (agency trading)
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52008 Julian Lorenz, [email protected]
VWAPEasy to understand and implement
“Spread trades out over time”
Criticism
For large trades in illiquid securities, VWAP essentially
reflects trade itself and provides little incentive for
low-cost execution
Artificial: does not correspond to any investment goal
VWAP Arrival Price
AdaptivityVWAP
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62008 Julian Lorenz, [email protected]
Arrival Price
“Trader‘s Dilemma“
Control statistical properties of shortfall
Risk-Reward tradeoff
Benchmark: Pre-trade (decision) priceIf you could execute entire order instantly at this price, you would
VWAP Arrival Price
AdaptivityVWAP
Why trade slowly?Reduce market impact
Why trade rapidly?Minimize risk
Anticipated drift
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72008 Julian Lorenz, [email protected]
Arrival Price: Efficient Strategies
Minimal varianceAdmissible Strategies
Efficient Strategies(Efficient Frontier)
VWAP
Rapid trading
E-V Plane
Minimal expected cost
Mean-Variance (Markowitz optimality): Strategies that minimizeE for fixed V,V for fixed E, orE + λ V for risk aversion parameter λ
Equivalentformulations!
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82008 Julian Lorenz, [email protected]
Arrival Price: Almgren/Chriss Trajectories
κ=10
x(t)
Urgencycontrols curvature
[Almgren, Chriss ’00]: Static trajectories specified at t=0
E-V Plane
tT
Xx(t)
tT
x(t)X
x(t) = stock holding at time tT = time horizonX = initial shares
Efficient trajectories given by
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92008 Julian Lorenz, [email protected]
Adaptivity
1. Adapt to varying volume and volatility
trade more when liquidity is present
trade less when volatility risk is lower
2. Adapt to price movement: “scaling”
trade faster or slower when price move is favorable?
Intuitively, adaptivity makes a lot of sense
VWAP Arrival Price
AdaptivityVWAP
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102008 Julian Lorenz, [email protected]
Scaling in response to price motionsCommon wisdom: Depending on asset price process
Mean-reversion ï aggressive in the money (AIM)Momentum ï passive in the money (PIM)Pure random walk ï no response
[Almgren, L.]: AIM improves mean-variance tradeoff, especially
1. Single-Update: [Algorithmic Trading III, Spring 2007]Adapt strategy exactly once during trading
2. Multi-Update: [In preparation]Any desired degree of precision
for nonzero risk aversion (middle of frontier)and large portfolio sizes
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112008 Julian Lorenz, [email protected]
Intuition behind AIM for Arrival PriceIntroduce intertemporal anti-correlation between
investment gains/losses in first part of execution, and
trading costs (market impact) in second part of execution
Caveat: “Cap winners, let losers run” is deadly if real price process
has momentum
If make money in first part, spend parts on higher impact
Higher impact = trade faster (i.e. reduces market risk)
Trade this volatility reduction for expected cost reduction
(mean-variance tradeoff!)
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122008 Julian Lorenz, [email protected]
1. Single-Update
Follow Almgren-Chriss trajectory with urgency κ0until T*
At T* switch to different urgency κ1,…,κndepending on performance up to T*
[Almgren, L. 2007]
If price up at T*,trade faster in remainder
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132008 Julian Lorenz, [email protected]
Single-Update (cont.)Switch to κi if based on accumulated gain/loss at T*
C0,C1,…,Cn cost on first and second part
Explicit expressions for E[Ci] and Var[Ci]
Explicit expression for E and Var of composite strategy Numerical optimization of criterion E +λ Var over κ0,…,κn
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142008 Julian Lorenz, [email protected]
Single-Update: Numerical Results
Almgren/Chrissfrontier
family of improved frontiers
Family of frontiersparametrized bymarket power
= measure of trade‘sability to move themarket compared tostock volatility
Larger relative improvementfor large portfolios
(i.e. )
AC static frontierscoincide with
Almgren/Chrissdeterministic strategy
Single Update
Sample cost PDFs:
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152008 Julian Lorenz, [email protected]
2. Multiple Updates: Dynamic Programming
Define value function
and optimal strategies for k-1 periods
Optimal Markovianone-step control
and optimal strategies for k periods
[Almgren, L. 2008]:
Single-update does NOT generalize to multiple updates
“E + λ Var“ not amenable to dynamic programming Squared expectation in Var[X] = E[X²] - E[X]²
Markov property for mean-variance efficient strategies
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162008 Julian Lorenz, [email protected]
In current period sell shares at
Dynamic Programming (cont.)We want to determine
Situation: Sell program (buy program analogously)k periods and x shares leftlimit for expected cost is ccurrent stock price Snext price innovation is σξ ~ N(0,σ2)
Construct optimal strategy for k periods
Use an efficient strategy for remaining shares in remaining k-1 periods
Priceimpactfunction
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172008 Julian Lorenz, [email protected]
Dynamic Programming (cont.)
Strategy π defined by and control function z(ξ)
Optimization of by means of and
andExpressions for cost of strategy π conditional on ξ
Use the laws of total expectation and variance
Specify by its expected cost z(ξ)
Note: must be deterministic, but when we begin , outcomeof is known, i.e. we may choose depending on ξ
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182008 Julian Lorenz, [email protected]
Dynamic Programming (cont.)Value function recursion:
where
Control variablenew stock holding
(i.e. sell y in this period)
Control functiontargeted cost as function of next price change ξ
(For linear price impact)
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192008 Julian Lorenz, [email protected]
Solving the Dynamic ProgramSeries of one-period optimization problems Each step: Determine optimal control
#shares to sell in next periodTarget expected cost for remainder as function z(ξ) of next period stock price change
No closed form solution numerical approximation Nice property: Convex constrained problems
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202008 Julian Lorenz, [email protected]
Behavior of Adaptive Strategy
Formally:
Aggressive in the Money (AIM) Control function z(ξ) is monotone increasing
Spend windfall gains on increased impact coststo reduce total variance
Example: sell program
“If price goes up, sell faster“
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212008 Julian Lorenz, [email protected]
100
101
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
E/Elin
V/Vlin
VWAP
Immediate Sale
−4 −2 0 2 4 6 8 10 12C/E
lin
Efficient FrontiersSimilar results asfor single-update
Sample cost PDFs:
Larger relative improvement for large portfoliosMarket power μ
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222008 Julian Lorenz, [email protected]
Single Update vs. Multi-Update
Full improvement by multi-update
Significant improvement even by single-update
Multi-update naturally more computational intensive
Single-update offers good value for low computation
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232008 Julian Lorenz, [email protected]
Other CriteriaInstead of mean-variance tradeoff: Utility functions
Exponential utility (CARA):Power law utility:etc.
Optimal strategies are AIM or PIM depending on utility[Schied, Schöneborn ’08]
Advantage of mean-variance optimization
Clear and intuitive meaning
Corresponds to how trade results are reported in practice
Independent of client‘s wealth
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242008 Julian Lorenz, [email protected]
Extensions
Non-linear impact functions and cost models
Multiple securities (program trading)
Other asset price processes
Price momentum (drift)
Mean-Reversion
Non-Gaussian returns
etc.
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252008 Julian Lorenz, [email protected]
Trading with Price Trend
Market ModelStock price: Brownian motion with unknown drift α
Prior estimate for drift, update this belief using price observations Temporary and permanent market impact
”Daily Trading Cycle”: institutional traders make decisions overnight and trade across the following day
Price momentum, if large net positions being traded
with
[Almgren, L. ’06]: “Bayesian Adaptive Trading”, Journal of Trading
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262008 Julian Lorenz, [email protected]
Optimal risk-neutral strategy (buy program)x(t) = trading trajectory (shares remaining) in continuous time
T = time horizonη = linear market
impact
v(t) = instantaneous trading rate
[Almgren, L. 2007]: Optimal dynamic strategy given by instantaneous trade rate
: Best estimate of drift using prior and S(t)Trade rate of re-computed static trajectory with current best drift estimate
Locally optimal myopic strategy = Global optimal solution
Highly dynamic
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272008 Julian Lorenz, [email protected]
Trading with Price Trend: ExamplesBuy-program with significant upwards drift prior
Prior driftestimate
Shar
es r
emai
ning
to b
uy
Stock price path
Posterior drift estimate
Time
Slow down trading, if cost lower than expected
(PIM)
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282008 Julian Lorenz, [email protected]
ConclusionAdaptivity is the new frontier of algorithmic trading
Single-update and multi-update arrival price algorithms
Pure random walk, no momentum or reversion
Straightforward mean-variance criterion
Dynamic programming approach
Strategies are “aggressive-in-the-money“
Substantially better than static Almgren/Chriss trajectories
Improved efficient frontiers
Relative improvement bigger for large portfolios
New market power parameter μ