Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

Optimal solutions for optimization in practice 57

Uw w w

Cw w

ww Qww

T

B

I

B

T

B

(,,) ( )

()

()()

γκ

γ

γ

α

κ

κ













 

1

1

1

2

subject to

Awd

This form includes a general transaction cost function C , where:

α  vector of expected returns, derived from an alpha model;

w  vector of portfolio weights;

w B  vector of benchmark portfolio weights;

w 1  vector of initial portfolio weights;

Q  covariance matrix, derived from a risk model;

γ  “ risk tolerance ” (takes values from 0 to 1);

κ  “ cost tolerance ” (takes values from 0 to 1);

A  matrix of constraints;

d  vector of limits.

The transpose of any vector is denoted by T.

As a special case of this, we can consider a piecewise linear form of the trans-

action cost function, which is defined by a set of buy-cost coefficients b i and a

set of sell-cost coefficients s i :

Uw w w

bwws

T

B

iIi

(,,) ( )

((, )

γκ

γ

γ

α

κ

κ













1

1

0

max min

piecewise

∑ ((, )

()()

0

1

2

ww

ww Qww

I

B

T

B



 

This form is the most commonly used to rebalance a portfolio when the

trading costs include both commission-based and market impact terms.

3.3.2 The BITA optimizer—functional summary

The mean – variance optimizer from BITA Risk delivers a solution with substan-
tial scope and functionality; a summary is given below.
● High-performance quadratic optimization, using the Active Set method
● Optimization from cash or portfolio rebalancing
● Linear and piecewise linear cost penalties
● General linear constraints, e.g., asset weight, sector weight, factor weight, asset revision
weight, portfolio return, portfolio beta, all of which can be set as hard or soft constraints

LAwUiijj

j

∑ i