5th International Summer School
Achievements and Applications of Contemporary Informatics,
Mathematics and Physics
Nationa...
Outline

•   Stochastic Differential Equations

•   Parameter Identification

•   Uncertainty , Ellipsoidal Calculus

•   ...
Stock Markets
Stochastic Differential Equations



       dX t    a( X t , t )dt b( X t , t )dWt
                   drift    and      di...
Stochastic Differential Equations


Milstein Scheme :




ˆ        ˆ        ˆ                              ˆ              ...
Example: Technology Emissions-Means Model



      E                        E           E
           ( k 1)               ...
Gene-Environmental and Financial Dynamics




     d Ei(t )    ai (E(t ) , t ) dt        bi (E(t ) , t ) dWi (t )



.   (...
Gene-Environment Networks              Errors and Uncertainty



  Errors uncorrelated      Errors correlated       Fuzzy ...
Gene-Environment Networks                 Errors and Uncertainty


Identify groups (clusters) of jointly acting
genetic an...
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




     ...
Gene-Environment Networks                 Errors and Uncertainty



Clusters and Ellipsoids:
Genetic clusters:           C...
Gene-Environment Networks   Ellipsoidal Calculus




      r   1
Gene-Environment Networks   Ellipsoidal Calculus




      r=1
Gene-Environment Networks                         Ellipsoidal Calculus


The Regression Problem:




                     ...
Gene-Environment Networks                   Ellipsoidal Calculus


Measures for the size of intersection:

• Volume       ...
What is a Bubble?

A situation in which prices for securities, especially stocks, rise
far above their actual value.




 ...
Shape of a Bubble




  Dimensions of the ellipsoid   Intersection of the two bubbles
Our Goals
Modelling Bubbles
 Lin, L., and Sornette, D., Diagnostics of rational expectation financial bubbles
 with stocha...
Homotopy

                              transition between bubbles

                                                      ...
In topology, two continuous functions from one topological space to another are called
homotopic if the first can be "cont...
Homotopy

                              transition between bubbles

                                                      ...
First Bubble Model

                                  m                                               m
           dp     ...
Second Bubble Model

 y(t ) ln p(t ) :
                                       dy x (1                       ( x , t )) dt ...
Jump Process




  t
      Φ ( s) d X ( s)                  The integrator X may have jumps.
 0

X (t )     X (0) I (t ) R...
Clarke’s Subdifferential as “Bubble”

                                                         path nowhere differentiable...
Identifying Stochastic Differential Equations


                      2
                                     2
    min    ...
Identifying Stochastic Differential Equations

min    t
 t,

                          0N        A           t            ...
Identifying Stochastic Differential Equations

                                                                           ...
Identifying Stochastic Differential Equations

    Application                                                  F. Yerlika...
Identifying Stochastic Differential Equations

                                                      A. Özmen, G.-W. Weber...
Portfolio Optimization

    max utility !   or

    min costs !     or

    min risk!


             martingale method:


...
Portfolio Optimization

    max utility !   or

    min costs !     or

    min risk!


             martingale method:

 ...
Portfolio Optimization

    max utility !   or

    min costs !     or

    min risk!


             martingale method:


...
Portfolio Optimization

    max utility !   or

    min costs !     or

    min risk!


             martingale method:


...
Portfolio Optimization

    max utility !   or

    min costs !     or

    min risk!


             martingale method:


...
Prediction of Credit Default


 non-default                             default
 cases                                   c...
Prediction of Credit Default



  Optimization problem:

  Simultaneously obtain the thresholds and parameters a and b
  t...
Eco-Finance Networks




 E (k 1) M s ( k ) E (k ) Cs ( k )
                                     s(k ) : FBQ( E (k 1))
   ...
Eco-Finance Networks




 E (k 1) M s ( k ) E (k ) Cs ( k )


                                 IE (k 1)           IMk IE (...
Eco-Finance Networks




                       IE (k 1)     IMk IE (k )


                          IE (t )   IM IE (t )
...
Eco-Finance Networks



 E (t )          M s (t ) E (t ) Cs (t ) E (t ) Ds (t )
where

         s(t ) : F (Q( E(t )))
    ...
Eco-Finance Networks

                                                                                    2
              ...
Generalized Semi-Infinite Programming


                                         ( ), ( , ) C 0 :



                     ...
Motivations


    • Present a new method for optimal control
      of Stochastic Hybrid Systems.
    • More flexible than ...
Problem Formulation:




     •                                      standard Brownian motion

     •                     ...
Applications:

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




     •   Systems biolo...
Method: 1st step

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




     •      ...
Method: 2nd and 3rd step

     2.     Rewrite original problem as deterministic PDE optimization program




     3.     S...
References
Aster, A., Borchers, B., Thurber, C., Parameter Estimation and Inverse Problems, Academic Press, 2004.
Boyd, S....
References
Nemirovski, A., Modern Convex Optimization, lecture notes, Israel Institute of Technology (2005).
Nesterov, Y.E...
Appendix   TEM Model
Appendix                                       The mixed-integer problem



   : nxn constant matrix with entries       re...
Appendix                                           Numerical Example
                                                     ...
Appendix                                       Numerical Example
Apply 3rd-order Heun’s time discretization :
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New Mathematical Tools for the Financial Sector

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AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 5.
More info at http://summerschool.ssa.org.ua

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New Mathematical Tools for the Financial Sector

  1. 1. 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 New Mathematical Tools for the Financial Sector Gerhard-Wilhelm Weber Institute of Applied Mathematics Middle East Technical University, Ankara, Turkey Faculty of Economics, Management and Law, University of Siegen, Germany Center for Research on Optimization and Control, University of Aveiro, Portugal Universiti Teknologi Malaysia, Skudai, Malaysia
  2. 2. Outline • Stochastic Differential Equations • Parameter Identification • Uncertainty , Ellipsoidal Calculus • Bubbles • Programming Aspects • Portfolio Optimization • Hybrid Control • Outlook and Conclusion
  3. 3. Stock Markets
  4. 4. Stochastic Differential Equations dX t a( X t , t )dt b( X t , t )dWt drift and diffusion term Ex.: price, wealth, interest rate, volatility processes Wt N (0, t ) (t [0, T ]) Wiener process
  5. 5. Stochastic Differential Equations Milstein Scheme : ˆ ˆ ˆ ˆ 1 ˆ Xj 1 Xj a ( X j , t j )(t j 1 t j ) b( X j , t j )(W j 1 Wj ) (b b)( X j , t j ) (W j 1 W j ) 2 (t j 1 tj) 2 and, based on our finitely many data: Wj ( W j )2 Xj a ( X j , t j ) b( X j , t j ) 1 2(b b)( X j , t j ) 1 . hj hj
  6. 6. Example: Technology Emissions-Means Model E E E ( k 1) (k ) (k ) M (k ) M M M E E E ( k 1) (k ) (k ) (k ) 0 M M M M u(k ) IE( k +1) IM( k ) IE( k )
  7. 7. Gene-Environmental and Financial Dynamics d Ei(t ) ai (E(t ) , t ) dt bi (E(t ) , t ) dWi (t ) . (k ) Wi ( k ) 1 ( Wi ( k ) )2 Ei ai (E ( k ) , t ( k ) ) bi (E ( k ) , t ( k ) ) ( k ) (b'bi )(E ( k ) , t ( k ) ) i (k ) 1 h 2 h IE( k +1) IM( k ) IE( k )
  8. 8. Gene-Environment Networks Errors and Uncertainty Errors uncorrelated Errors correlated Fuzzy values Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics θ2 θ1
  9. 9. Gene-Environment Networks Errors and Uncertainty Identify groups (clusters) of jointly acting genetic and environmental variables disjoint overlapping stable clustering
  10. 10. Gene-Environment Networks Errors and Uncertainty 2) Interaction of Genetic Clusters
  11. 11. Gene-Environment Networks Errors and Uncertainty 3) Interaction of Environmental Clusters
  12. 12. Gene-Environment Networks Errors and Uncertainty 3) Interaction of Genetic & Environmental Clusters Determine the degree of connectivity
  13. 13. 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)
  14. 14. Gene-Environment Networks Ellipsoidal Calculus r 1
  15. 15. Gene-Environment Networks Ellipsoidal Calculus r=1
  16. 16. Gene-Environment Networks Ellipsoidal Calculus The Regression Problem: measurement Maximize (overlap of ellipsoids) T R R ˆ X r( ) X r( ) ˆ Er( ) Er( ) 1 r 1 r 1 prediction
  17. 17. 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 E r , r r semidefinite programming interior point methods
  18. 18. What is a Bubble? A situation in which prices for securities, especially stocks, rise far above their actual value. When does it burst? When investors realize how far prices have risen from actual values, the bubble bursts.
  19. 19. Shape of a Bubble Dimensions of the ellipsoid Intersection of the two bubbles
  20. 20. Our Goals Modelling Bubbles 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. Developing a method to contract a bubble to one point or shrink them, e.g., as soon as possible.
  21. 21. Homotopy 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.
  22. 22. 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.
  23. 23. Homotopy 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.
  24. 24. First Bubble Model m m dp p (1 ( p, t )) dt p dW ( p, t ) 0 dp p m dt p K (tc t ) ( m 1) 1 (m 1) K ( ) tc p0 (m 1) ( p, t ) ~ (t ) 1 m tc 2 [ p(t ) ] m 1 2 d~ tc ~ dt ( tc ) dW
  25. 25. Second Bubble Model y(t ) ln p(t ) : dy x (1 ( x , t )) dt ( ) x dW dx x m (1 ( x , t )) dt xmdW ( x , t ) 0: dy x dt ( ) x dW showing that x (t ) dt E [dy] m ( x, t ) (x , t) 0: d2y dy y (t ) A B ( Tc t ) 1 dt 2 dt 1/ dp 1 1 (m 1) Tc ( ) B ( ) A p (Tc ) dt t t0 1 2 ~ (t ) m 1 ( x, t ) ~ (t ) 1 m tc 2 [ x (t ) ] m 1 ( x, t ) tc [ x (t ) ] 2 2
  26. 26. Jump Process t Φ ( s) d X ( s) The integrator X may have jumps. 0 X (t ) X (0) I (t ) R(t ) J (t ) X ( 0 ) is a nonrandom initial condition. t I (t ) Γ ( s) dW ( s) Ito integral of an adapted process (s) . 0 t R(t ) ( s) ds Riemann integral for some adapted process (s) . 0 J (t ) lim J ( s ) Adapted right-continuous pure Jump process, J (0) 0 . s t
  27. 27. Clarke’s Subdifferential as “Bubble” path nowhere differentiable, we discretize -2 -1 0 1 This constitutes a “homotopy of bubbles”. fC (t ) co ξ | lim f (tk ), tk t (k ), tk D f (k IN) k
  28. 28. Identifying Stochastic Differential Equations 2 2 min X A μ L 2 Tikhonov regularization 2 min t, Conic quadratic programming t, subject to A X t, 2 L 2 M Interior Point Methods
  29. 29. Identifying Stochastic Differential Equations min t t, 0N A t X subject to : , 1 0T m 0 primal problem 06( N 1) L t 06( N 1) : , 0 0T m M LN 1 , L6( N 1) 1 LN 1 : x ( x1 , x2 ,..., xN 1 )T R N 1 | xN+1 x12 2 2 x2 ... xN max ( X T , 0) 1 0T N 1) , 6( M 2 0T N 1 0T N 6( 1) 0 1 subject to 1 2 , dual problem AT 0m LT 0m 0m 1 LN 1 , 2 L6 ( N 1) 1
  30. 30. Identifying Stochastic Differential Equations A. Özmen, G.-W. Weber, I.Batmaz Important new class of (Generalized) Partial Linear Models: E Y X ,T G XT T , e.g., GPLM (X ,T ) = LM (X ) + MARS (T ) y c-(x, )=[ (x )] c+(x, )=[ (x )] x X * * 2 L* * 2 CMARS 2 2
  31. 31. Identifying Stochastic Differential Equations Application F. Yerlikaya Özkurt, G.-W. Weber, P. Taylan 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.
  32. 32. Identifying Stochastic Differential Equations A. Özmen, G.-W. Weber, I.Batmaz Robust CMARS: confidence interval (T j ) ... ...................... . outlier ... outlier semi-length of confidence interval RCMARS
  33. 33. Portfolio Optimization max utility ! or min costs ! or min risk! martingale method: Optimization Problem Representation Problem or stochastic control
  34. 34. Portfolio Optimization max utility ! or min costs ! or min risk! martingale method: Parameter Estimation Optimization Problem Representation Problem or stochastic control
  35. 35. Portfolio Optimization max utility ! or min costs ! or min risk! martingale method: Optimization Problem Representation Problem Parameter Estimation or stochastic control
  36. 36. Portfolio Optimization max utility ! or min costs ! or min risk! martingale method: Optimization Problem Representation Problem Parameter Estimation or stochastic control
  37. 37. Portfolio Optimization max utility ! or min costs ! or min risk! martingale method: Optimization Problem Representation Problem Parameter Estimation or stochastic control
  38. 38. Prediction of Credit Default non-default default cases cases cut-off value ROC curve c score value TPF, sensitivity c = cut-off value K. Yildirak, E. Kürüm, E., G.-W. Weber FPF, 1-specificity
  39. 39. Prediction of Credit Default Optimization problem: 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
  40. 40. Eco-Finance Networks E (k 1) M s ( k ) E (k ) Cs ( k ) s(k ) : FBQ( E (k 1)) 1 if Ei (k ) i Qi ( E (k )) : 0 else θ2,2 θ2,1 θ1,1 θ1,2
  41. 41. Eco-Finance Networks E (k 1) M s ( k ) E (k ) Cs ( k ) IE (k 1) IMk IE (k ) IE (t ) IM IE (t ) E (t ) M s (t ) E (t ) Cs (t ) E (t ) Ds (t )
  42. 42. Eco-Finance Networks IE (k 1) IMk IE (k ) IE (t ) IM IE (t ) modules
  43. 43. Eco-Finance Networks E (t ) M s (t ) E (t ) Cs (t ) E (t ) Ds (t ) where s(t ) : F (Q( E(t ))) Q( E(t )) (Q1 ( E(t )),..., Qn ( E(t ))) 0 for Ei (t ) i ,1 1 for i ,1 Ei (t ) i,2 Qi ( E (t )) : ... parameter estimation: di for i ,di Ei (t ) (i) estimation of thresholds (ii) calculation of matrices and vectors describing the system between thresholds
  44. 44. Eco-Finance Networks 2 l 1 min M E C E D E 0 (mij ), (ci ), (di ) subject to n pij ( mij , y ) j ( y) (j 1, ..., n ) i 1 n qi (ci , y ) ( y) ( 1, ..., m ) ( y Y (C , D )) i 1 n i (di , y ) ( y) i 1 mii i ,min (i 1,..., n) & overall box constraints
  45. 45. Generalized Semi-Infinite Programming ( ), ( , ) C 0 : ( ) (, ) homeom. asymptotic () : structurally stable effect IR n global local global
  46. 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. Control of Stochastic Hybrid Systems, Robin Raffard Chess Review, Nov. 21, 2005 4-5 546
  47. 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 Control of Stochastic Hybrid Systems, Robin Raffard Chess Review, Nov. 21, 2005 5 47
  48. 48. Applications: • Engineering: Maintain dynamical system in safe domain for maximum time. • Systems biology: Parameter identification. • Finance: Optimal portfolio selection. Control of Stochastic Hybrid Systems, Robin Raffard Chess Review, Nov. 21, 2005 5 48
  49. 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 Control of Stochastic Hybrid Systems, Robin Raffard Chess Review, Nov. 21, 2005 5 49
  50. 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 … Control of Stochastic Hybrid Systems, Robin Raffard Chess Review, Nov. 21, 2005 5 50
  51. 51. References 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).
  52. 52. 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)).
  53. 53. Appendix TEM Model
  54. 54. 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.
  55. 55. Appendix Numerical Example Ö. Defterli, A. Fügenschuh, G.-W. Weber 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
  56. 56. Appendix Numerical Example Apply 3rd-order Heun’s time discretization :
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