New Mathematical Tools for the Financial Sector
description
Transcript of 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