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

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risk averse adaptive execution of portfolio transactions

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)

execution of financial decisions
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?

price appreciation market impact timing risk
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.

benchmark arrival price
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

discrete trading model
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
permanent vs temporary market impact
Permanent vs. Temporary Market Impact

Simplified model of market impact:

permanent vs temporary market impact1
Permanent vs. Temporary Market Impact
  • Permanent market impact

with k ~ N(0,1) i.i.d

  • Temporary market impact

with k ~ N(0,1) i.i.d

Simplest case: Linear impact functions

(„Quadratic cost model“)

shortfall of a trading strategy
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

mean variance optimization
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.

efficient trading frontier
Efficient Trading Frontier

Similar to portfolio optimization, this leads to an efficient frontier of trading strategies:

  • 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.

response of finance industry
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

optimal static trading strategies i
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



optimal static trading strategies ii
Optimal Static Trading Strategies (II)


becomes a straightforward

convex minimization problem in x1,…,xN with solution

But is xk really path-independent?

binomial model i
Binomial Model (I)

Consider the following arithmetic binomial model:

sell x2+

(S0+2, x2+)

(S0+, x1)


sell (x1 - x2+)

(S0, x2+)

sell (X-x1)

sell x2+


sell x2-

(S0, X)

sell (x1 - x2-)

(S0, x2-)


(S0 - , x1)

(S0 – 2, x2-)

sell x2-

Then we have the shortfall

binomial model ii
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.


binomial model iii
Binomial Model (III)


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.

continuous time
Continuous Time

Continuous-time formulation:

Strategy v(t) must be adapted to the filtration of B.


We would like to use dynamic programming, but variance doesn‘t directly fit into „expected utility“ framework.

dynamic programming i
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.

dynamic programming ii
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:

further research directions
Further Research Directions


  • 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
  • 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: