1. Risk-Averse Adaptive Execution of Portfolio Transactions
Julian Lorenz
Institute of Theoretical Computer Science, ETH Zurich
This is joint work with R. Almgren (Bank of America Securities, on leave from University of Toronto)

2. Execution of Financial Decisions
Portfolio optimization tells a great deal about investments that optimally balance risk and expected returns
Markowitz, CAPM, …
But how to implement them?
How to sell out of a large or illiquid portfolio position
within a given time horizon?

3. Price Appreciation, Market Impact, Timing Risk
We have to deal with …
Market risk
Market impact
Price appreciation,
Timing risk
Market impact
trade fast
trade slowly
We want to balance market risk and market impact.

4. Benchmark: Arrival Price
Goal: Find optimized execution strategy
Benchmark: Implementation shortfall
= „value of position at time of decision-making“
- „capture of trade“
This benchmark is also known as „Arrival Price“ (i.e. price prevailing at decision-making).
Average price achieved
Arrival price
Other common benchmark: Market VWAP

5. Discrete Trading Model
Trading is possible at N discrete times
No interest on cash position
A trading strategy is given by (xi)i=0..N+1 where
xk = #units hold at t=k (i.e. we sell nk=xk-xk+1 at price Sk)
Boundary conditions: x0 = X and xN+1 = 0
Price dynamics:
Exogenous: Arithmetic Random Walk
Sk = Sk-1 + (k+), k=1..N
with k ~ N(0,1) i.i.d
Endogenous: Market Impact
Permanent
Temporary

6. Permanent vs. Temporary Market Impact
Simplified model of market impact:

7. Permanent vs. Temporary Market Impact
Permanent market impact
with k ~ N(0,1) i.i.d
Temporary market impact
Simplest case: Linear impact functions
(„Quadratic cost model“)

8. Shortfall of a Trading Strategy
In fact, permanent impact is fairly easy tractable. Hence, we
will focus on temporary impact.
The capture of a trading strategy (xi)i=1..N is
with nk=xk-1-xk.
Assuming linear impact, the implementation shortfall is

9. Mean-Variance Optimization
In the spirit of Markowitz‘ portfolio optimization, we want to optimize
The Lagrangian for this problem is
¸ 0 is the Lagrange multiplier or can be seen as a
measure of risk aversion by itself.

10. Efficient Trading Frontier
Similar to portfolio optimization, this leads to an efficient frontier of trading strategies:

11. Bibliography This is the model as first proposed in
R. Almgren, N. Chriss: "Optimal execution of portfolio transactions", Journal of Risk 3, 2000.
It was extended in a series of publications, e.g.
Konishi, Makimoto: “Optimal slice of a block trade”, Journal of Risk, 2001.
Almgren, Chriss: "Bidding principles“, Risk, 2003.
Almgren: "Optimal execution with nonlinear impact functions and
trading-enhanced risk", Applied Mathematical Finance, 2003.
Huberman, Stanzl: “Optimal Liquidity Trading”, Review of Finance, 2005.

12. Response of Finance Industry
The model has been remarkably influential in the finance industry:
Neil Chriss and Robert Almgren pioneered much of the early research in the field... [The] efficient trading frontier will truly revolutionize financial decision-making for years to come.
Robert Kissell and Morton Glantz, „Optimal Trading Strategies“, 2003
Almgren's paper, […] coauthored with Neil Chriss, head of quantitative strategies for giant hedge fund SAC Capital Advisors, helped lay the groundwork for the arrival-price algorithms currently being developed on Wall Street.
Justin Shack, „The orders of battle", Institutional Investor, 2004

13. Optimal Static Trading Strategies (I)
Almgren/Chriss brought up arguments, why in this arithmetic Brownian motion setting together with mean-variance utility, an optimal trading strategy would not depend on the stock price process.
They therefore considered the model, where xk are static
variables.
Then

14. Optimal Static Trading Strategies (II)
Then
becomes a straightforward
convex minimization problem in x1,…,xN with solution
But is xk really path-independent?

15. Binomial Model (I) Consider the following arithmetic binomial model:
sell x2+
(S0+2, x2+)
(S0+, x1) 2
sell (x1 - x2+)
(S0, x2+)
sell (X-x1)
sell x2+ 1
sell x2-
(S0, X)
sell (x1 - x2-)
(S0, x2-) 2
(S0 - , x1)
(S0 – 2, x2-)
sell x2-
Then we have the shortfall

16. Binomial Model (II) A trading strategy is defined by (x1,x2+,x2-)
For the variance we have to deal with path dependent stock
holdings x2 and with covariances, e.g
One calculates (with and )
The path-independent solution forces  = 0 with optimum )
For  < 0, first order decrease in variance ( ) and only second-order increase in expectation. )
Path-independent solution is non-optimal.

17. Binomial Model (III) Intuition? Suppose price moves up:
Less than anticipated cost (investment gain)
Sell faster and allow to burn off some of the profit
Increase in cost anticorrelated with investment gain
How to compute optimal path-dependent strategies?
In fact, „Optimal Execution“ can be seen as a multiperiod portfolio optimization problem with quadratic transaction costs and the additional constraint that at the end we are only allowed to hold cash.

18. Continuous Time Continuous-time formulation:
Strategy v(t) must be adapted to the filtration of B.
s.t.
We would like to use dynamic programming, but variance doesn‘t directly fit into „expected utility“ framework.

19. Mean-Variance and Expected Utility
Theorem:
Corollary:

20. Dynamic Programming (I)
Hence, mean-variance optimization is essentially equivalent
to minimizing expectation of the utility function
Value function at t in state (x,y,s)
Terminal utility function {
Force complete liquidation
There is only terminal utility, no „consumption“ process.

21. Dynamic Programming (II)
The HJB-Equation for the process
leads to
with the optimal trade rate .
With =T-t we get the final PDE that is to be solved for >0:

22. Further Research Directions
Summary:
We showed that the path-independent trading strategies given by Almgren/Chriss can be further improved.
Using the dynamic programming paradigm, we derived a PDE which characterizes optimal adaptive strategies.
Ongoing work:
Find explicit analytic solutions for strategies
Multiple-security portfolios (with correlations), „basket trading“
Nonlinear impact functions
Other stochastic processes for security e.g. geometric Brownian motion …