New Mathematical Tools for the Financial Sector

Gerhard-Wilhelm Weber

Institute of Applied Mathematics Middle East Technical University, Ankara, Turkey

New Mathematical Tools

for the Financial Sector

Faculty of Economics, Management and Law, University of Siegen, GermanyCenter for Research on Optimization and Control, University of Aveiro, Portugal

Universiti Teknologi Malaysia, Skudai, Malaysia

5th International Summer School

Achievements and Applications of Contemporary Informatics,

Mathematics and Physics

National University of Technology of the Ukraine

Kiev, Ukraine, August 3-15, 2010

• Stochastic Differential Equations

• Parameter Identification

• Uncertainty , Ellipsoidal Calculus

• Bubbles

• Programming Aspects

• Portfolio Optimization

• Hybrid Control

• Outlook and Conclusion

Outline

Stock Markets

drift and diffusion term

Wiener process

( , ) ( , )t t t tdX a X t dt b X t dW

(0, ) ( [0, ])tW N t t T

Stochastic Differential Equations

Ex.: price, wealth, interest rate, volatility

processes

Milstein Scheme :

and, based on our finitely many data:

2

1 1 1 1 1

1ˆ ˆ ˆ ˆ ˆ( , )( ) ( , )( ) ( )( , ) ( ) ( )2

j j j j j j j j j j j j j j j jX X a X t t t b X t W W b b X t W W t t

2( )( , ) ( , ) 1 2( )( , ) 1 .

j j

j j j j j j j

j j

W WX a X t b X t b b X t

h h

Stochastic Differential Equations

( 1) ( ) ( )

( )

k k k

kME E E

M M M

( +1) ( ) ( ) IE IM IEk k k

( 1) ( ) ( )

( )

( )

0k k k

k

kM

u

E E E

M M M

Example: Technology Emissions-Means Model

( +1) ( ) ( ) IE IM IEk k k

( ) ( ) ( ) 2( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

( )1E (E , ) (E , ) ( )(E , ) 1

2

k k kk k k k k ki i

i i i i ik k

W Wa t b t b'b t

h h

( ) ( ) ( ) ( )E (E , ) (E , )t t t t

i i i id a t dt b t dW

Gene-Environmental and Financial Dynamics

.

Gene-Environment Networks Errors and Uncertainty

Errors uncorrelated Errors correlated Fuzzy values

Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics

θ1

θ2

Identify groups (clusters) of jointly acting

genetic and environmental variables

stable clustering

disjoint

overlapping

Gene-Environment Networks Errors and Uncertainty

2) Interaction of Genetic Clusters

Gene-Environment Networks Errors and Uncertainty

3) Interaction of Environmental Clusters

Gene-Environment Networks Errors and Uncertainty

3) Interaction of Genetic & Environmental Clusters

Determine the degree of connectivity

Gene-Environment Networks Errors and Uncertainty

Clusters and Ellipsoids:

Genetic clusters: C1,C2,…,CR

Environmental clusters: D1,D2,…,DS

Genetic ellipsoids: X1,X2,…,XR Xi = E (μi,Σi)

Environmental ellipsoids: E1,E2,…,ES, Ej = E (ρj,Πj)

Gene-Environment Networks Errors and Uncertainty

r 1

Gene-Environment Networks Ellipsoidal Calculus

r=1

Gene-Environment Networks Ellipsoidal Calculus

The Regression Problem:

Maximize (overlap of ellipsoids)

T R

r

R

r

rrrr EEXX1 1 1

)()()()( ˆˆ

measurement

prediction

Gene-Environment Networks Ellipsoidal Calculus

Measures for the size of intersection:

• Volume → ellipsoid matrix determinant

• Sum of squares of semiaxes → trace of configuration matrix

• Length of largest semiaxes → eigenvalues of configuration matrix

semidefinite programming

interior point methods

rr ,E

r

Gene-Environment Networks Ellipsoidal Calculus

What is a Bubble?

A situation in which prices for securities, especially stocks, rise

far above their actual value.

When investors realize how far prices have risen from actual values,

the bubble bursts.

When does it burst?

Shape of a Bubble

Dimensions of the ellipsoid Intersection of the two bubbles

Our Goals

Modelling Bubbles

Developing a method to contract a bubble to one point or shrink them,

e.g., as soon as possible.

Lin, L., and Sornette, D., Diagnostics of rational expectation financial bubbles

with stochastic mean-reverting termination times, Cornell University Library,

2009.

Abreu, D., and Brunnermeier, M.K., Bubbles and crashes, Econometrica 71(1),

174-203, 2003.

Brunnermeier, M.K., Asset Pricing Under Asymmetric Information, Oxford

University Press, 2001.

Binswanger, M., Stock Market Speculative Bubbles and Economic Growth,

Edward Elgar Publishing Limited, 1999.

Garber, P.M., and Flood, R.P., Speculative Bubbles Speculative Attacks, and

Policy Switching, The MIT Press, 1997.

Homotopy

In topology, two continuous functions from one topological space to another are called

homotopic if the first can be "continuously deformed" into the second one.

Such a deformation is called a homotopy between the two functions.

Formally, a homotopy between two continuous functions f and g from a topological space X

to a topological space Y is defined to be a continuous function

H : X × [0,1] → Y such that, if x ∈ X, then H(x,0) = f(x) and H(x,1) = g(x).

If we think of the second parameter of H as time, then H describes a

continuous deformation of f into g:

at time 0 we have the function f and

at time 1 we have the function g.

transition between bubbles

?

concept of homotopy

In topology, two continuous functions from one topological space to another are called

homotopic if the first can be "continuously deformed" into the second one.

Such a deformation is called a homotopy between the two functions.

Formally, a homotopy between two continuous functions f and g from a topological space X

to a topological space Y is defined to be a continuous function

H : X × [0,1] → Y such that, if x ∈ X, then H(x,0) = f(x) and H(x,1) = g(x).

If we think of the second parameter of H as time, then H describes a

continuous deformation of f into g:

at time 0 we have the function f and

at time 1 we have the function g.

Homotopy

In topology, two continuous functions from one topological space to another are called

homotopic if the first can be "continuously deformed" into the second one.

Such a deformation is called a homotopy between the two functions.

Formally, a homotopy between two continuous functions f and g from a topological space X

to a topological space Y is defined to be a continuous function

H : X × [0,1] → Y such that, if x ∈ X, then H(x,0) = f(x) and H(x,1) = g(x).

If we think of the second parameter of H as time, then H describes a

continuous deformation of f into g:

at time 0 we have the function f and

at time 1 we have the function g.

transition between bubbles

?

concept of homotopy

First Bubble Model

dWpdttppdp mm )),(1(

0),( tp dtpdp m

)( ttKp c

)1(1 m )(K )1()1(

0 mptm

c

12 ])([2

1)(

~),( m

c tpmtttp

dWdtttd cc )(~~

Second Bubble Model

showing that

( ) ln ( ) :y t p t

dWxdttxxdy )()),(1(

dWxdttxxdx mm )),(1(

( , ) 0 :x t dWxdtxdy )( ][)( dyEdttx

( , ) ( , ) 0 :x t x tm

dt

dy

dt

yd2

21)()( tTBAty c

)1(1 m

/1

0

)(tt

cdt

dpT )(

1

1B )( cTpA

12 ])([2

1)(

~),( m

c txmtttx12

])([2

)(~

),( m

c txtttx

0

( ) ( )

t

d XΦ s s

0( ) ( ) ( ) ( ) ( )X t X I t R t J t

The integrator X may have jumps.

0( ) ( ) ( )

tI t Γ s dW s )(s

t

dsstR0

)()( )(s

0)0(J( ) lim ( )s t

J t J s

)0(X is a nonrandom initial condition.

Ito integral of an adapted process .

Riemann integral for some adapted process .

Adapted right-continuous pure Jump process, .

Jump Process

Clarke’s Subdifferential as “Bubble”

path nowhere differentiable,

we discretize

-2 -1 0 1

This constitutes a “homotopy of bubbles”.

N( ) co | lim ( ), ( ), ( )C k k k fk

f t ξ f t t t k t D k I

22

22

min X A Lμ Tikhonov regularization

,

2

2

subject to

min ,

,

tt

A X t

L M

Identifying Stochastic Differential Equations

Conic quadratic programming

Interior Point Methods

,

6( 1)6( 1)

1 6( 1) 1

min

subject to : ,1 0

: ,0

,

0

0

00

0

t

N

T

m

NN

T

m

N N

t

t

t

M

L L

A X

L

1 1 2 2 2

1 2 1 1 1 2: ( , ,..., ) | ...N T N

N N+ NL x x x x x x xx R

6( 1) 1

1 6( 1) 2

6( 1)

1 2

1

1 2

max ( ,0) ,

11 0subject to ,

,

0

0 0

00 0

N

T T

N

T T

N N

T Tmm m

N

M

L L

X

A Ldual problem

primal problem

Identifying Stochastic Differential Equations

Identifying Stochastic Differential Equations

Important new class of (Generalized) Partial Linear Models:

, ,

( , ) GPLM ( ) ( )LM MARS= +

TE Y X T G X T

X T X T

e.g.,

2 2* * * *

2 2X L

x

y

+( , )=[ ( )]c x x( , )=[ ( )]-c x x

CMARS

A. Özmen, G.-W. Weber, I.Batmaz

Application

Evaluation of the models based on performance values:

• CMARS performs better than Tikhonov regularization with respect to all the measures for both data sets.

• On the other hand, GLM with CMARS (GPLM) performs better than both Tikhonov regularization and CMARS with respect to all the measures for both data sets.

Identifying Stochastic Differential Equations

F. Yerlikaya Özkurt, G.-W. Weber, P. Taylan

Robust CMARS:

RCMARSsemi-length of confidence interval

.. ..outlier outlier

confidence interval

. ......( )jT

... . .. ... .. .... .. . . . ..

Identifying Stochastic Differential Equations

A. Özmen, G.-W. Weber, I.Batmaz

max utility ! or

min costs ! or

min risk!

martingale method:

Optimization Problem

Representation Problem

or stochastic control

Portfolio Optimization

max utility ! or

min costs ! or

min risk!

martingale method:

Optimization Problem

Representation Problem

or stochastic control

Parameter Estimation

Portfolio Optimization

max utility ! or

min costs ! or

min risk!

martingale method:

Optimization Problem

Representation Problem

or stochastic control

Parameter Estimation

Portfolio Optimization

max utility ! or

min costs ! or

min risk!

martingale method:

Optimization Problem

Representation Problem

or stochastic control

Parameter Estimation

Portfolio Optimization

max utility ! or

min costs ! or

min risk!

martingale method:

Optimization Problem

Representation Problem

or stochastic control

Parameter Estimation

Portfolio Optimization

non-default

casesdefault

cases

score value

TP

F, sensitiv

ity

FPF, 1-specificity

ROC curve

cut-off value

c

c = cut-off value

Prediction of Credit Default

K. Yildirak, E. Kürüm, E., G.-W. Weber

Simultaneously obtain the thresholds and parameters a and b

that maximize AUC,

while balancing the size of the classes (regularization),

and guaranteeing a good accuracy

discretization of integral

nonlinear regression problem

Optimization problem:

Prediction of Credit Default

( ) ( )( 1) M ( )s k s kE k E k C

( ) : ( ( 1))

1 if ( )( ( )) :

0 else

B

i i

i

s k F Q E k

E kQ E k

Eco-Finance Networks

θ1,1 θ1,2

θ2,1

θ2,2

( ) ( )( 1) M ( )s k s kE k E k C

( 1) IM ( )kIE k IE k

( ) ( )IE t IM IE t

( ) ( ) ( )( ) ( ) ( )s t s t s tE t M E t C E t D

Eco-Finance Networks

modules

( 1) IM ( )kIE k IE k

( ) ( )IE t IM IE t

Eco-Finance Networks

)))(((:)( tEQFts

1( ( )) ( ( ( )),..., ( ( )))nQ E t Q E t Q E t

1,)( ii tE

2,1, )( iii tE

)(, tEidi i

where

0 for

1 for( ( )) :

...

for

i

i

Q E t

dparameter estimation:

(i) estimation of thresholds

(ii) calculation of matrices and

vectors describing the system

between thresholds

( ) ( ) ( )( ) ( ) ( )s t s t s tE t M E t C E t D

Eco-Finance Networks

min

( ), ( ), ( )ij i im c d

1

1

1

,min

( 1, ..., )

( 1, ..., )

( , ) ( )

( , ) ( )

( , ) ( )

&

n

ij ij j

i

n

i i

i

n

i i

i

ii i

j n

m

p m y y

q c y y

d y y

m

overall box constraints

( ( , ))y Y C D

( 1,..., )i n

subject to

21

0

l

M E C E D E

Eco-Finance Networks

: structurally stable

global local global

)(

nIR

asymptotic

effect

)(

homeom.

:),(),( 0C

),(

Generalized Semi-Infinite Programming

Chess Review, Nov. 21, 2005Control of Stochastic Hybrid Systems, Robin Raffard 46

Motivations

• Present a new method for optimal control

of Stochastic Hybrid Systems.

• More flexible than Hamilton-Jacobi,

because handles more problem formulations.

• In implementation, up to dimension 4-5

in the continuous state.

4-55

Chess Review, Nov. 21, 2005Control of Stochastic Hybrid Systems, Robin Raffard 47

Problem Formulation:

• standard Brownian motion

• continuous state

Solves an SDE whose jumps are governed by the discrete state.

• discrete state

Continuous time Markov chain.

• control

5

Chess Review, Nov. 21, 2005Control of Stochastic Hybrid Systems, Robin Raffard 48

Applications:

• Systems biology: Parameter identification.

• Finance: Optimal portfolio selection.

• Engineering: Maintain dynamical system in safe domain for maximum time.

5

Chess Review, Nov. 21, 2005Control of Stochastic Hybrid Systems, Robin Raffard 49

Method: 1st step

1. Derive a PDE satisfied by the objective function in terms of the generator:

• Example 1:

If

then

• Example 2:

If

then

5

Chess Review, Nov. 21, 2005Control of Stochastic Hybrid Systems, Robin Raffard 50

Method: 2nd and 3rd step

2. Rewrite original problem as deterministic PDE optimization program

3. Solve PDE optimization program using adjoint method

Simple and robust …

5

Aster, A., Borchers, B., Thurber, C., Parameter Estimation and Inverse Problems, Academic Press, 2004.

Boyd, S., Vandenberghe, L., Convex Optimization, Cambridge University Press, 2004.

Buja, A., Hastie, T., Tibshirani, R., Linear smoothers and additive models, The Ann. Stat. 17, 2 (1989) 453-510.

Fox, J., Nonparametric regression, Appendix to an R and S-Plus Companion to Applied Regression,

Sage Publications, 2002.

Friedman, J.H., Multivariate adaptive regression splines, Annals of Statistics 19, 1 (1991) 1-141.

Friedman, J.H., Stuetzle, W., Projection pursuit regression, J. Amer. Statist Assoc. 76 (1981) 817-823.

Hastie, T., Tibshirani, R., Generalized additive models, Statist. Science 1, 3 (1986) 297-310.

Hastie, T., Tibshirani, R., Generalized additive models: some applications, J. Amer. Statist. Assoc.

82, 398 (1987) 371-386.

Hastie, T., Tibshirani, R., Friedman, J.H., The Element of Statistical Learning, Springer, 2001.

Hastie, T.J., Tibshirani, R.J., Generalized Additive Models, New York, Chapman and Hall, 1990.

Kloeden, P.E, Platen, E., Schurz, H., Numerical Solution of SDE Through Computer Experiments,

Springer Verlag, New York, 1994.

Korn, R., Korn, E., Options Pricing and Portfolio Optimization: Modern Methods of Financial Mathematics,

Oxford University Press, 2001.

Nash, G., Sofer, A., Linear and Nonlinear Programming, McGraw-Hill, New York, 1996.

Nemirovski, A., Lectures on modern convex optimization, Israel Institute of Technology (2002).

References

Nemirovski, A., Modern Convex Optimization, lecture notes, Israel Institute of Technology (2005).

Nesterov, Y.E , Nemirovskii, A.S., Interior Point Methods in Convex Programming, SIAM, 1993.

Önalan, Ö., Martingale measures for NIG Lévy processes with applications to mathematical finance,

presentation in: Advanced Mathematical Methods for Finance, Side, Antalya, Turkey, April 26-29, 2006.

Taylan, P., Weber G.-W., Kropat, E., Approximation of stochastic differential equations by additive

models using splines and conic programming, International Journal of Computing Anticipatory Systems 21

(2008) 341-352.

Taylan, P., Weber, G.-W., Beck, A., New approaches to regression by generalized additive models

and continuous optimization for modern applications in finance, science and techology, in the special issue

in honour of Prof. Dr. Alexander Rubinov, of Optimization 56, 5-6 (2007) 1-24.

Taylan, P., Weber, G.-W., Yerlikaya, F., A new approach to multivariate adaptive regression spline

by using Tikhonov regularization and continuous optimization, to appear in TOP, Selected Papers at the

Occasion of 20th EURO Mini Conference (Neringa, Lithuania, May 20-23, 2008) 317- 322.

Seydel, R., Tools for Computational Finance, Springer, Universitext, 2004.

Stone, C.J., Additive regression and other nonparametric models, Annals of Statistics 13, 2 (1985) 689-705.

Weber, G.-W., Taylan, P., Akteke-Öztürk, B., and Uğur, Ö., Mathematical and data mining contributions

dynamics and optimization of gene-environment networks, in the special issue Organization in Matter

from Quarks to Proteins of Electronic Journal of Theoretical Physics.

Weber, G.-W., Taylan, P., Yıldırak, K., Görgülü, Z.K., Financial regression and organization, to appear

in the Special Issue on Optimization in Finance, of DCDIS-B (Dynamics of Continuous, Discrete and

Impulsive Systems (Series B)).

References

TEM Model Appendix

The mixed-integer problem

: nxn constant matrix with entries representing the effect

which the expression level of gene has on the change of expression of gene

Genetic regulation network

mixed-integer nonlinear optimization problem (MINLP):

subject to

: constant vector representing the lower bounds for the decrease of the transcript concentration.

in order to bound the indegree of each node, introduce

binary variables :

is a given parameter.

Appendix

Numerical Example

Data

Gebert et al. (2004a)

Apply 3rd-order Heun method

Take

using the modeling language Zimpl 3.0, we solve

by SCIP 1.2 as a branch-and-cutframework,

together with SOPLEX 1.4.1 as our LP-solver

AppendixÖ. Defterli, A. Fügenschuh, G.-W. Weber

Apply 3rd-order Heun’s time discretization :

Appendix Numerical Example