1© 2016 The MathWorks, Inc.

Solving Large Optimization Problems in Finance:

How MATLAB Can Help You

Jorge Paloschi

Consulting Services

MathWorks Spain

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Optimization

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Optimization in Financial Applications with MATLAB

Financial Optimization

Optimization Methods

Customized Optimization Models

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Financial Optimization

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Financial Applications and Optimization

Asset

Management

Trading

Risk

Management

Portfolio Optimization

Machine Learning

Regression

Maximum Likelihood Estimation

Distribution Fitting

Curve Fitting

ALM

Econometrics

Pricing &

Valuation

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MATLAB – The Financial Development Platform

Financial

Statistics/Machine Learning Optimization

Financial Instruments Econometrics

MATLAB

Parallel Computing MATLAB Parallel Server

Files

Databases

Datafeeds

Access

Reporting

Applications

Production

Share

Data Analysis and Visualization

Financial Modeling

Application Development

Research and Quantify

Datafeed

Database

Spreadsheet Link EX

MATLAB

Compiler

SDK

MATLAB

Compiler

Rep

ort G

en

era

tor

Pro

du

ctio

n S

erv

er

Trading Toolbox

Deep Learning Risk management

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– Mean-Variance Portfolio Optimization

– Conditional Value-at-Risk Portfolio Optimization

– Mean-Absolute Deviation Portfolios

– Time Series Regression Models

– Conditional Mean Variance Models

– Multivariate Models

– Linear/ Nonlinear Regression

– Probability disribution fitting

– Machine Learning, e.g., SVM, NN,...

– Nonlinear Regression, Convolutional Neural Networks

Financial Optimization in MATLAB

Financial

Statistics

Econometrics

Neural Network

Optimization

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Optimization Methods

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Optimization Problem

Objective Function

uxl

eqbxeqA

bAx

)(min xfx

Typically a linear or nonlinear function

Linear constraints inequalities equalities bounds

Nonlinear constraints inequalities equalities

Decision variables (can be discrete or integer)

0)(

0)(

xeq

c

xc

Subject to Constraints

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How to solve an optimization problem ?

What do you know about your optimization problem ?

information

accuracy

information

runtime

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Variables & Constraints

Variables

Continuous Discrete

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

100

0 5 10 15 200

10

20

30

40

50

60

70

80

90

100

Integer Programming

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Constraints

Linear Nonlinear

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0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-2

-1

0

1

2

-2

-1

0

1

2

-8

-6

-4

-2

0

2

4

6

8

10

Objective Function

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

Least-squares/curve fitting

Quadratic f(x) = xTAxLinear f(x) = ATx

General f(x)

f(c)=S[g(xk;c) – yk]2

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Numerical Optimization

x

xxfxxfxf

2

)()()(

function [f,df] = objective(x)f = ... % function value

df = ... % gradient vector

Fewer function evaluations

More accurate

Whenever possible, provide gradient/hessian information!

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Derivative-Free Optimization

?f fmincon

Repeatedly sample

several points

Direct Search

Genetic Algorithm

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Approaches in MATLAB

Local Optimization – Finds local minima/maxima

Uses supplied gradients or estimates them

Applicable for large scale problems with smooth objective function

– Faster/fewer function evaluations

Global Optimization– No gradient information required

– Solve problems with non-smooth, discontinuous objective function

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Solvers

Variables

Continuous

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

100

0 5 10 15 200

10

20

30

40

50

60

70

80

90

100

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Constraints

Linear

NonlinearDiscrete

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-2

-1

0

1

2

-2

-1

0

1

2

-8

-6

-4

-2

0

2

4

6

8

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

Least-squares

Quadratic f(x) = xTAx

Linear f(x) = ATx

General f(x)

f(c)=S[g(xk;c) – yk]2

Objective function Solver

linprog

quadprog

intlinprog

lsqlinlsqnonlin

fminuncfmincon

Optimization Toolbox

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Solvers

Variables

Continuous

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

100

0 5 10 15 200

10

20

30

40

50

60

70

80

90

100

Constraints

Discrete

Nonlinearf(x)

Objective function Solver

GlobalSearch,

MultiStart

Non-smooth patternsearch,

ga,

simulannealbnd

ga

Global Optimization Toolbox

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How to Improve Performance?

Derivatives calculations– Gradients

– Hessian matrix

Default approach– Approximation: Finite differences perturbations ( VERY EXPENSIVE! )

Alternative approaches– Cheaper approximations

LBFGS

Supply your own approximation

Analytical calculation

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Hessian Analytical Calculation

Not easy, in general, to calculate analytical Hessian

Can MATLAB help?

– YES! Symbolic MathToolbox

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Sample Problem

Problem: Volatility surface estimation for the pricing of

call options. Calibration using market options.

Objective: g(x ) = ||f(x)||2

– fj(x) = C(xj,Kj,Tj) – Cj

K strikes

T maturities

C closing prices

BSM model (Black Scholes and Merton)

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Numerical results ( problem size 209)

Gradients Function evaluations Time (secs)

Numerical 100,056 1351

Numerical

(using parallel)

100,056 642

Analytical 52 3.41

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MATLAB Code

%Define symbolic residuals

x = evalin(symengine,['n:=',num2str(n),';[x[j] $ j = 1..n]']);

f = evalin(symengine,['[f[j] $ j = 1..n]']);

for jj=1:n

d1 = (log(S(jj)/K(jj)) +

(r+x(jj)^2/2)*T(jj))/(x(jj)*sqrt(T(jj)));

d2 = d1 - x(jj)*sqrt(T(jj));

val = S(jj)*0.5*(1+erf(d1/sqrt(2))) - ...

K(jj)*exp(-r*T(jj)) * 0.5*(1+erf(d2/sqrt(2)));

f(jj) = (val-CP(jj));

end

obj = f*f.';

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MATLAB Code

%Define objective gradient and hessian

grad = diff(obj,x);

hess = jacobian(grad,x);

%Transform symbolics into MATLAB function handles

symGrad = matlabFunction(grad);

symHess = matlabFunction(hess);

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Example of Use

References

– Paloschi J and Krishnamurthy S – Improving MATLAB performance when solving

financial optimization problems – Wilmott Magazine – May 2011

– https://uk.mathworks.com/matlabcentral/fileexchange/33597-improving-matlab-

performance-when-solving-financial-optimization-problems

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Customized Optimization Models

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Supported Portfolio Optimization Models

Mean-Variance Portfolio Optimization

Conditional Value-at-Risk Portfolio Optimization

Mean-Absolute Deviation Portfolio Optimization

Financial Toolbox

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MATLAB – The Financial Development Platform

Financial

Statistics/Machine Learning Optimization

Financial Instruments Econometrics

MATLAB

Parallel Computing MATLAB Parallel Server

Files

Databases

Datafeeds

Access

Reporting

Applications

Production

Share

Data Analysis and Visualization

Financial Modeling

Application Development

Research and Quantify

Datafeed

Database

Spreadsheet Link EX

MATLAB

Compiler

SDK

MATLAB

Compiler

Rep

ort G

en

era

tor

Pro

du

ctio

n S

erv

er

Trading Toolbox

Deep Learning Risk management

28

Customized Portfolio Optimization -

Deployment

Compile your MATLAB

optimization model for your

dedicated platform

Make it available for your

enterprise environment

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Customer User Stories Involving Optimization in Finance

https://uk.mathworks.com/company/user_stories/search.html?q=&fq=produ

ct:OP%20marketing-industry:financial-services

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Summary

Optimization for financial applications is built within MATLAB

toolboxes covering many standard applications

A large variety of optimization algorithms available in MATLAB®

Optimization Toolbox™ and Global Optimization Toolbox™

Customized optimization models made easy by

– quick modeling (Math to MATLAB)

– advanced optimization process diagnostics

– rapid deployment

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Thank you !