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

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,

Timing riskMarket impact

trade fast trade slowly

We want to balance market risk and market impact.

Market risk Market impact

We have to deal with …

Benchmark: Arrival Price

Benchmark: Implementation shortfall

Other common benchmark: Market VWAP

= „value of position at time of decision-making“

- „capture of trade“

Goal: Find optimized execution strategy

This benchmark is also known as „Arrival Price“ (i.e. price prevailing at decision-making).

Arrival price Average price achieved

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 i.i.d

Endogenous: Market Impact

- Permanent

- Temporary

Permanent vs. Temporary Market Impact

Simplified model of market impact:

Permanent vs. Temporary Market Impact

• Permanent market impact

with k i.i.d

• Temporary market impact

with k i.i.d

(„Quadratic cost model“)

Simplest case: Linear impact functions

Shortfall of a Trading Strategy

The capture of a trading strategy (xi)i=1..N is

with nk=xk-1-xk.

Assuming linear impact, the implementation shortfall is

In fact, permanent impact is fairly easy tractable. Hence, we

will focus on temporary impact.

Mean-Variance Optimization

¸ 0 is the Lagrange multiplier or can be seen as a measure of risk aversion by itself.

In the spirit of Markowitz‘ portfolio optimization, we want to optimize

The Lagrangian for this problem is

Efficient Trading Frontier

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

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.

Response of 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

The model has been remarkably influential in the finance industry:

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

Optimal Static Trading Strategies (II)

Then

convex minimization problem in x1,…,xN with solution

becomes a straightforward

But is xk really path-independent?

Binomial Model (I)

Consider the following arithmetic binomial model:

(S0, X)

sell (X-x1)

(S0 - , x1)

sell (x1 - x2

+)

sell (x1 - x2

-)

(S0+, x1)

(S0 – 2, x2-)

(S0, x2-)

(S0+2, x2+)

(S0, x2+)

sell x2+

sell x2+

sell x2-

sell x2-

Then we have the shortfall

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)

Intuition?

Suppose price moves up:

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.

• Less than anticipated cost (investment gain)

• Sell faster and allow to burn off some of the profit

• Increase in cost anticorrelated with investment gain

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.

Mean-Variance and Expected Utility

Theorem:

Corollary:

Dynamic Programming (I)

Hence, mean-variance optimization is essentially equivalentto 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)

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

• 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• …

Ongoing work:

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.