optimal adaptive execution of portfolio transactions
DESCRIPTION
Optimal Adaptive Execution of Portfolio Transactions. Julian Lorenz Joint work with Robert Almgren (Banc of America Securities, NY). Execution of Portfolio Transactions. Sell 100,000 Microsoft shares today!. Broker/Trader. Fund Manager. Problem: Market impact. - PowerPoint PPT PresentationTRANSCRIPT
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15. 05. 2007
Optimal Adaptive Execution of Portfolio Transactions
Julian Lorenz
Joint work with Robert Almgren (Banc of America Securities, NY)
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22007 Julian Lorenz, [email protected]
Execution of Portfolio Transactions
Fund Manager Broker/Trader
Sell 100,000 Microsoft shares
today!
How to optimize the trade schedule over the day?
Problem: Market impactTrading Large Volumes Moves the Price
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32007 Julian Lorenz, [email protected]
Market Model
Stock price follows random walk
Execution strategy:
sell shares between t0 and t1
Sell program
t1 and t2 …
s.t.
for initial position of X shares
Pure sell program:
,
Discrete times
= shares hold at time
i.e.
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42007 Julian Lorenz, [email protected]
Benchmark: Pre-Trade Book Value
Cost C() = Pre-Trade Book Value – Capture of Trade
C() is independent of S0
Market Impact and Cost of a Strategy
Linear Temporary Market Impact
Selling xk-1 – xk shares in [tk-1, tk] at discount to Sk-1
with
x
x
X=x0=100
N=10
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52007 Julian Lorenz, [email protected]
Trader‘s Dilemma
Random variable!
Optimal trade schedules seek risk-reward balance
Obviously by immediate liquidation
No risk, but high market impact cost
Minimal RiskŒ
tT
x(t)X
Linear strategy
Minimal Expected Cost
But: High exposure to price volatility
High risk
tT
x(t)X
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62007 Julian Lorenz, [email protected]
Efficient Strategies
Minimal varianceŒ Admissible Strategies
Efficient Strategies
Linear Strategy
ImmediateSale
E-V Plane
Minimal expected cost
Risk-Reward Tradeoff: Mean-Variance
Œ
Variance as risk measure
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72007 Julian Lorenz, [email protected]
Almgren/Chriss Deterministic Trading (1/2)R. Almgren, N. Chriss: "Optimal execution of portfolio transactions", Journal of Risk (2000).
Deterministic trading strategy
functions of decision variables (x1,…,xN)
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82007 Julian Lorenz, [email protected]
Almgren/Chriss Deterministic Trading (2/2)
DeterministicTrajectories
for some
Dynamic strategies:xi = xi(1,…,i-1)
Almgren/Chriss Trajectories:xi deterministic
Dynamic strategies improve (w.r.t. mean-variance) !We show:
C() normally distributed
Straightforward QP
E-V Plane
tT
X x(t)
tT
x(t)XT=1, =10
x(t)
Urgency controls curvature
By dynamic programming
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92007 Julian Lorenz, [email protected]
Definitions
Adapted trading strategy: xi may depend on 1…,i-1
Efficient trading strategies
„no other admissible strategy offers lower variance for same level of expected cost“
i.e.
adapted strategies for X shares in N periods with expected cost
Admissible trading strategies for expected cost
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112007 Julian Lorenz, [email protected]
Dynamic Programming (1/4)
i.e. minimal variance to sell x shares in k periods with
Define value function
and optimal strategies for k-1 periods
Optimal Markovian one-step control
+ and optimal strategies for k periods
For type “ “ DP is straightforward.
Here: in value function & terminal constraint … ?…ultimately interested in
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122007 Julian Lorenz, [email protected]
Dynamic Programming (2/4)
We want to determine
Situation: k periods and x shares left Limit for expected cost is c Current stock price S Next price innovation is ~ N(0,2)
Construct optimal strategy for k periods
In current period sell shares atŒ
Use efficient strategy for remaining k-1 periods
Specify by its expected cost z()
Note: must be deterministic, but when we begin , outcomeof is known, i.e. we may choose depending on
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132007 Julian Lorenz, [email protected]
Dynamic Programming (3/4)
Strategy defined by control and control function z()
Conditional on :
Using the laws of total expectation and variance
One-step optimization of and by means of and
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142007 Julian Lorenz, [email protected]
Dynamic Programming (4/4)
Theorem:
where
Control variablenew stock holding
(i.e. sell x – x’ in this period)
Control functiontargeted cost as function of next price change
Solve recursively!
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152007 Julian Lorenz, [email protected]
Solving the Dynamic Program
Difficulty for numerical treatment:
No closed-form solution
Need to determine a control function
Approximation: is piecewise constant
Theorem:
In each step, the optimization problem is a convex constrained problem in {x‘, z1, … , zk}.
Nice convexity property
For fixed determine
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162007 Julian Lorenz, [email protected]
Behavior of Adaptive Strategy
Theorem:
„Aggressive in the Money“
At all times, the control function z() is monotone increasing
Recall:
High expected cost = sell quickly (low variance)
z() specifies expected cost for remainder as a function of the next price change
Low expected cost = sell slowly (high variance)
If price goes up ( > 0), sell faster in remainder
Spend part of windfall gains on increased impact coststo reduce total variance
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172007 Julian Lorenz, [email protected]
Numerical Example
Respond only to up/down
Discretize state space of
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182007 Julian Lorenz, [email protected]
Sample Trajectories of Adaptive Strategy
Aggressive in the money …
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202007 Julian Lorenz, [email protected]
Family of New Efficient Frontiers
Family of frontiersparametrized by size of trade X
Almgren/Chriss deterministic
strategy
Adaptivestrategies
Sample cost PDFs:
Distribution plots obtained by Monte Carlo simulation
Almgren/Chriss frontier
Improved frontiers
Œ
‹
‹
Œ
Larger improvement for large portfolios
(i.e. )
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222007 Julian Lorenz, [email protected]
Extensions
Non-linear impact functions
Multiple securities („basket trading“)
Dynamic Programming approach also applicable for other mean-variance problems, e.g. multiperiod portfolio optimization