15. 05. 2007 optimal adaptive execution of portfolio transactions julian lorenz joint work with...
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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)
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
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.
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
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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Efficient Strategies
Minimal varianceΠAdmissible 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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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)
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
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
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
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
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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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!
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
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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Numerical Example
Respond only to up/down
Discretize state space of
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Sample Trajectories of Adaptive Strategy
Aggressive in the money …
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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. )
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
