4th International Summer School
Achievements and Applications of Contemporary
Informatics, Mathematics and Physics
Nationa...
Outline

•   Stochastic Differential Equations

•   Parameter Estimation

•   Various Statistical Models

•   C-MARS

•   ...
Stock Markets
Stochastic Differential Equations



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



       dX t = a ( X t , t )dt + b( X t , t )dWt
                   drift    and      ...
Regression

               X = ( X1 , X 2 ,..., X m ) and output variable Y ;
                                      T
Inpu...
Generalized Additive Models


       (                          )                  ( )
                                   ...
Generalized Additive Models


 •   Given data ( yi , xi ) (i = 1,2,...,N ),



 •   penalized residual sum of squares
    ...
Generalized Additive Models

          min             t
          t , β0 , f
                                            ...
Generalized Additive Models

          min             t
          t , β0 , f
                                            ...
Generalized Additive Models

          min             t
          t , β0 , f
                                            ...
Generalized Additive Models




                              Ind j : = d j ( D j ) ⋅ v j (V j )
MARS




y                                                    y

    • •                   •                              ...
C-MARS



                    N                            M max           2

                  ∑( y − f (x ) ) + ∑ µ     ...
C-MARS

Tikhonov regularization:

                                         2
              PRSS = y −ψ (d ) θ             ...
C-MARS

Tikhonov regularization:

                                         2
              PRSS = y −ψ (d ) θ             ...
C-MARS



         cluster




         cluster




                   robust optimization
Stochastic Differential Equations Revisited



       dX t = a ( X t , t )dt + b( X t , t )dWt
                   drift   ...
Stochastic Differential Equations



       dX t = a ( X t , t )dt + b( X t , t )dWt
                  drift    and    dif...
Stochastic Differential Equations Revisited



       dX t = a ( X t , t )dt + b( X t , t )dWt
                   drift   ...
Stochastic Differential Equations


 Milstein Scheme :




         ˆ         ˆ                               ˆ           ...
Stochastic Differential Equations


 •   step length   h j = t j +1 − t j := ∆ t j
                                       ...
Stochastic Differential Equations


 •   More simple form:

                                X j = G j + H j c j + ( H j ′ ...
Stochastic Differential Equations

                                                                          g
           ...
Stochastic Differential Equations

 • penalized sum of squares PRRS

                                         ∑{          ...
Stochastic Differential Equations


             θ = (α , β , ϕ              )          (                       )         ...
Appendix Stochastic Differential Equations

  b             2         N −1             2

  ∫  f p′′ (U p )  dU p ≅ ∑  ...
Stochastic Differential Equations

                                                  2                                2   ...
Stochastic Differential Equations


                             2
       min          &
                    X − Aθ       ...
Stochastic Differential Equations

min     t
 t ,θ

                    0N      A  t   −X 
                         ...
Stochastic Differential Equations


 (t , θ , χ ,η , κ1 , κ 2 ) is a primal dual optimal solution if and only if



      ...
Stochastic Differential Equations

Ex.:


              dVt = (θtT ( µ − rt ) + rt )Vt  dt − ct dt + θtT σVt dWt ,
     ...
Nonlinear Regression


                                                   2

                           ∑ j ( j )
      ...
Nonlinear Regression

                                                                        β k +1 := β k + qk
 • Gauss-...
Nonlinear Regression


alternative solution



 min    t,
  t,q


 subject to     ( ∇F (β )∇   T
                         ...
Nonlinear Regression


alternative solution



 min    t,
  t,q


 subject to     ( ∇F (β )∇   T
                         ...
Nonlinear Regression


 alternative solution



  min      t,
   t,q


  subject to           ( ∇F (β )∇   T
             ...
Portfolio Optimization

     max utility !   or

     min costs !


              martingale method:




                 ...
Portfolio Optimization

     max utility !   or

     min costs !


              martingale method:

                    ...
Portfolio Optimization

     max utility !   or

     min costs !


              martingale method:




                 ...
Portfolio Optimization

     max utility !   or

     min costs !


              martingale method:




                 ...
References
Aster, A., Borchers, B., and Thurber, C., Parameter Estimation and Inverse Problems, Academic Press, 2004.
Boyd...
References
Nemirovski, A., Modern Convex Optimization, lecture notes, Israel Institute of Technology (2005).
Nesterov, Y.E...
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Prediction of Financial Processes

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AACIMP 2009 Summer School lecture by Gerhard Wilhelm Weber. "Modern Operational Research and Its Mathematical Methods" course.

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Prediction of Financial Processes

  1. 1. 4th International Summer School Achievements and Applications of Contemporary Informatics, Mathematics and Physics National University of Technology of the Ukraine Kiev, Ukraine, August 5-16, 2009 Prediction of Financial Processes Parameter Estimation in Stochastic Differential Equations by Continuous Optimization Gerhard- Gerhard-Wilhelm Weber * Vefa Gafarova, Nüket Erbil, Cem Ali Gökçen, Azer Kerimov 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 Pakize Taylan Dept. Mathematics, Dicle University, Diyarbakır, Turkey
  2. 2. Outline • Stochastic Differential Equations • Parameter Estimation • Various Statistical Models • C-MARS • Accuracy vs. Stability • Tikhonov Regularization • Conic Quadratic Programming • Nonlinear Regression • Portfolio Optimization • 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 Wt N (0, t ) (t ∈ [0, T ]) Wiener process
  5. 5. 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
  6. 6. Regression X = ( X1 , X 2 ,..., X m ) and output variable Y ; T Input vector linear regression : m Y = E (Y X 1 ,..., X m ) + ε = β0 + ∑ X j β j + ε , j =1 β = ( β 0 , β1 ,..., β m ) which minimizes T 2 ( ) N RSS ( β ) := ∑ yi − x β T i i =1 ˆ = ( X T X )−1 X T y , β ( ) −1 Cov( β) = X T X ˆ σ2
  7. 7. Generalized Additive Models ( ) ( ) m E Yi xi1 , xi 2 ,..., xi m = β0 + ∑ f j x i j j =1 f j are estimated by a smoothing on a single coordinate. Standard convention : ( ) E f j ( xij ) = 0 . • Backfitting algorithm (Gauss-Seidel) ri j = yi − β 0 − ∑ f k ( xik ) , ˆ k≠ j it “cycles” and iterates.
  8. 8. Generalized Additive Models • Given data ( yi , xi ) (i = 1,2,...,N ), • penalized residual sum of squares 2 N  m  m b PRSS (β 0 , f1 ,..., f m ) : = ∑  yi − β 0 − ∑ f j ( xij )  + ∑ µ j ∫  2  f j'' (t j )  dt j  i =1  j =1  j =1 a µ j ≥ 0. • New estimation methods for additive model with CQP :
  9. 9. Generalized Additive Models min t t , β0 , f 2 N  m  subject to ∑ i=1  yi − β0 − ∑ f j ( xij )  ≤ t 2 , t ≥ 0, j =1  2 ∫  f j (t j )  dt j ≤ M j (j = 1, 2,..., m), ''   dj splines: f j ( x) = ∑ θl j hl j ( x). l =1 By discretizing, we get min t t , β0 , f W ( β 0 , θ ) 2 ≤ t 2 , t ≥ 0, 2 subject to 2 V j ( β0 ,θ ) ≤ M j (j = 1,..., m). 2
  10. 10. Generalized Additive Models min t t , β0 , f 2 N  m  subject to ∑ i=1  yi − β0 − ∑ f j ( xij )  ≤ t 2 , t ≥ 0, j =1  2 ∫  f j (t j )  dt j ≤ M j (j = 1, 2,..., m), ''   dj splines: f j ( x) = ∑ θl j hl j ( x). l =1 By discretizing, we get min t t , β0 , f W ( β 0 , θ ) 2 ≤ t 2 , t ≥ 0, 2 subject to 2 V j ( β0 ,θ ) ≤ M j (j = 1,..., m). 2
  11. 11. Generalized Additive Models min t t , β0 , f 2 N  m  subject to ∑ i=1  yi − β0 − ∑ f j ( xij )  ≤ t 2 , t ≥ 0, j =1  2 ∫  f j (t j )  dt j ≤ M j (j = 1, 2,..., m), ''   dj splines: f j ( x) = ∑ θl j hl j ( x). l =1 By discretizing, we get min t t , β0 , f W ( β 0 , θ ) 2 ≤ t 2 , t ≥ 0, 2 subject to 2 V j ( β0 ,θ ) ≤ M j (j = 1,..., m). 2
  12. 12. Generalized Additive Models Ind j : = d j ( D j ) ⋅ v j (V j )
  13. 13. MARS y y • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • •• c-(x,τ)=[−(x−τ)]+ c+(x,τ)=[+(x−τ)]+ c-(x,τ)=[−(x−τ)]+ c+(x,egressionx−τ)]+ rτ)=[+( w ith τ x τ x
  14. 14. C-MARS N M max 2 ∑( y − f (x ) ) + ∑ µ ∑ ∑ 2 θ  Drα, sψ m (t m )  d t m ∫  2 PRSS := i i m 2 m  i =1 m =1 α =1 r <s α = (α1 ,α 2 ) r , s∈V ( m ) Tradeoff between both accuracy and complexity. { V (m) := κ m | j = 1, 2,..., K m j } ( ) Drα, sψ m (t m ) := ∂αψ m ∂α1 trm ∂α 2 tsm (t m ) t m := (tm1 , tm2 ,..., tm K )T m α = (α1 , α 2 ) α := α1 + α 2 , where α1 , α 2 ∈{0,1}
  15. 15. C-MARS Tikhonov regularization: 2 PRSS = y −ψ (d ) θ + µ Lθ 2 2 2 Lθ 2 Conic quadratic programming: y − ψ (d ) θ 2 min t, t ,θ subject to ψ (d ) θ − y 2 ≤ t , Lθ 2 ≤ M
  16. 16. C-MARS Tikhonov regularization: 2 PRSS = y −ψ (d ) θ + µ Lθ 2 2 2 Lθ 2 Conic quadratic programming: y − ψ (d ) θ 2 min t, t ,θ subject to ψ (d ) θ − y 2 ≤ t , Lθ 2 ≤ M
  17. 17. C-MARS cluster cluster robust optimization
  18. 18. Stochastic Differential Equations Revisited 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
  19. 19. Stochastic Differential Equations dX t = a ( X t , t )dt + b( X t , t )dWt drift and diffusion term Ex.: bioinformatics, biotechnology (fermentation, population dynamics) Universiti Teknologi Malaysia Wt N (0, t ) (t ∈ [0, T ]) Wiener process
  20. 20. Stochastic Differential Equations Revisited 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
  21. 21. Stochastic Differential Equations Milstein Scheme : ˆ ˆ ˆ 1 2 ˆ ( X j +1 = X j + a ( X j , t j )(t j +1 − t j ) + b( X j , t j )(W j +1 − W j ) + (b′b)( X j , t j ) (W j +1 − W j ) 2 − (t j +1 − t j ) ˆ ) and, based on our finitely many data: & ∆W j  ( ∆W j ) 2  X j = a ( X j , t j ) + b( X j , t j ) + 1 2 (b ′b)( X j , t j )  − 1 . hj  hj   
  22. 22. Stochastic Differential Equations • step length h j = t j +1 − t j := ∆ t j  X j +1 − X j  , if j = 1, 2,..., N − 1 &  hj X j :=   X N − X N −1 , if j = N  hN  • Wt N (0, t ), ∆W j (independent), Var( ∆W j ) = ∆ t j • ∆W j = Z j ∆ t j , Zj N (0,1) ( ) & Zj 1 X j = a ( X j , t j ) + b( X j , t j ) + (b′b)( X j , t j ) Z j2 − 1 hj 2
  23. 23. Stochastic Differential Equations • More simple form: X j = G j + H j c j + ( H j ′ H j )d j , & where G j := a( X j , t j ) , H j := b( X j , t j ), c j := Z j hj , ( d j :=1 2 Z j2 − 1 . ) • Our problem: ∑( ) N 2 min X j − (G j + H j c j + ( H j′ H j )d j ) & y 2 j =1 y is a vector which comprises a subset of all the parameters.
  24. 24. Stochastic Differential Equations g 2 2 dp G j = a( X j , t j ) = α 0 + ∑ f p (U j , p ) = α 0 + ∑∑ α lp B p (U j , p ) l p =1 p =1 l =1 2 2 d rh H j c j = b( X j , t j )c j = β 0 + ∑ g r (U j ,r ) = β 0 + ∑∑ β rm Crm (U j ,r ) r =1 r =1 m =1 2 2 d sf Fj d j = b′b( X j , t j )d j = ϕ0 + ∑ hs (U j , s ) = ϕ0 + ∑∑ ϕ sn Dsn (U j , s ) s =1 s =1 n =1 where U j = (U j ,1 , U j ,2 ) := ( X j , t j ) ; • k th order base spline Bη ,k : a polynomial of degree k − 1, with knots, say x η , 1, xη ≤ x < xη +1 Bη ,1 ( x) =  0, otherwise x − xη xη + k − x Bη ,k ( x) = Bη ,k −1 ( x) + Bη +1,k −1 ( x) xη + k −1 − xη xη + k − xη +1
  25. 25. Stochastic Differential Equations • penalized sum of squares PRRS ∑{ Xj ( j j j) } N 2 & − G + H c + F d 2 + λ  f ′′(U )  2 dU PRSS (θ , f , g , h) : = j =1 j j ∑ p∫ p p  pp =1 2 2 + ∑ µr ∫ [ gr (U r )] dU r +∑ϕs ∫ [ hs′′(U s )] dU s ′′ 2 2 r =1 s =1 bκ • λ p , µ r , ϕ s ≥ 0 (smoothing parameters), ∫ = ∫ (κ = p, r , s ) aκ • large values of λ p , µ r , ϕ s yield smoother curves, smaller ones allow more fluctuation ∑{ X j − ( G j + H j c j + Fj d j ) } N 2 & = j =1 2 N  &  2 dp h 2 dr g 2 ds f  ∑  X j −  α 0 + ∑∑ α p Bp (U j , p ) + β0 + ∑∑1 βr Cr (U j ,r ) + ϕ0 + ∑∑ ϕs Ds (U j ,s )  j =1   l l m m n n    p =1 l =1 r =1 m = s =1 n =1 
  26. 26. Stochastic Differential Equations θ = (α , β , ϕ ) ( ) ( ) T T g T , α = α0 ,α ,α α p = α , α ,..., α ( p = 1, 2), T T T T T 1 2 dp 1 2 , p p p β = ( β0 , β , β ) ( ) T T T T 1 2 , β r = β , β ,..., β 1 r 2 r d rh r (r = 1, 2), ( ϕ = (ϕ0 , ϕ1T , ϕ 2 ) , ϕ s = ϕ s , ϕ s2 ,..., ϕ sd ) T T ( s = 1, 2). f T 1 s ∑{ } ( ) N T • Then, & X j − Ajθ 2 & − Aθ 2 . = X A = A1T , A2 ,..., AN T T ( ) j =1 2 T & & & & X = X 1 , X 2 ,..., X N • Furthermore, b 2 N −1 2 ∫  f p′′ (U p ) dU p ≅ a   ∑  f p′′ (U jp )  (U j +1, p − U jp ) j =1   2  dp l l  g N −1 = ∑  ∑ α p B p′′ (U jP )u j  . j =1  l =1   
  27. 27. Appendix Stochastic Differential Equations b 2 N −1 2 ∫  f p′′ (U p )  dU p ≅ ∑  B j ′′u jα p  = AP α p 2 p B ( p = 1, 2) a   j =1   2 ( ) T Ap := B1p′′T u1 , B2p′′T u2 ,..., BN −1′′T u N −1 B p u j := U j +1, p − U j , p ( j = 1, 2,..., N − 1). b N −1 2 [ gr′′(U r )] dU r ≅ ∑ C rj ′′v j β r  = ArC β r 2 ∫ (r = 1, 2) 2 a j =1   2 ( ) T ArC := C1r′′T v1 , C2 ′′T v2 ,..., CN −1′′T vN −1 r r v j := U j +1,r − U j ,r ( j = 1, 2,..., N − 1). b 2 N −1 2  h ′′ (U )  dU ≅  D s′′ w ϕ  = A Dϕ ∫ s s  s ∑ j j s 2 s s ( s = 1, 2) 2 a j =1 ( ) T A := D ′′ w1 , D2 ′′T w2 ,..., DN −1′′T wN −1 s D s 1 s T s w j := U j +1, s − U j , s ( j = 1, 2,..., N − 1).
  28. 28. Stochastic Differential Equations 2 2 2 & − Aθ 2 + λ A Bα 2 + µ AC β ∑ p p p ∑ r r r + ∑ ϕ s AsDϕ s 2 2 PRSS (θ , f , g , h) = X 2 2 2 2 p =1 r =1 s =1 Let us assume that λ p = µr = ϕ s =: µ = δ : 2 • 2 & PRSS (θ , f , g , h) ≈ X − Aθ + δ 2 Lθ 2 , 2 2 where L is a 6( N − 1) × m matrix: 0 A1B 0 0 0 0 0 0 0    0 0 A2B 0 0 0 0 0 0  0  θ = (α T , β T , ϕ T ) . T 0 0 0 A1C 0 0 0 0 L :=  , 0 0 0 0 0 A2C 0 0 0  0 0 0 0 0 0 0 A1D 0    0 0 0 0 0 0 0 0 A2D 
  29. 29. Stochastic Differential Equations 2 min & X − Aθ + µ Lθ 2 θ 2 2 Tikhonov regularization min t, t ,θ subject to & Aθ − X ≤ t, 2 Lθ 2 ≤ M Conic quadratic programming
  30. 30. Stochastic Differential Equations min t t ,θ  0N A  t   −X  & subject to χ :=  T    + , 0m    θ  0   1 primal problem  06( N −1) L  t   06( N −1)  η :=    +  ,  0 0T   θ   M  m   χ ∈ LN +1 , η ∈ L6( N −1)+1 { LN + 1 := x = ( x1 , x2 ,..., xN )T ∈ R N +1 | xN+1 ≥ x12 + x2 + ... + xN 2 2 } & ( max ( X T , 0) κ 1 + 0T N −1) , − M κ 2 6( )  0T 1  0T N −1) 0 1  κ1 +  T κ2 =  , N 6( subject to  T  dual problem A 0m   L 0m   0m  κ 1 ∈ LN +1 , κ 2 ∈ L6 ( N −1)+1
  31. 31. Stochastic Differential Equations (t , θ , χ ,η , κ1 , κ 2 ) is a primal dual optimal solution if and only if  0N A  t   −X & χ :=  T    + , 0m   θ   0    1  06( N −1) L  t   06( N −1)  η :=    +    0 0T   θ   M  m    0T 1  0T N −1) 0 1  κ1 +  T κ2 =   N 6(  T A 0m   L 0m   0m  κ 1T χ = 0, κ 2 η = 0 T κ 1 ∈ LN +1 , κ 2 ∈ L6( N −1)+1 χ ∈ LN +1 , η ∈ L6( N −1)+1.
  32. 32. Stochastic Differential Equations Ex.: dVt = (θtT ( µ − rt ) + rt )Vt  dt − ct dt + θtT σVt dWt ,   drt = α ⋅ ( R − rt ) dt + σ t ⋅ rt τ ⋅ dWt , dX t = µ ( t , X t , Zt ) dt + σ ( t , X t , Zt ) dWt . nonlinear regression
  33. 33. Nonlinear Regression 2 ∑ j ( j ) N min f ( β ) =  d − g x ,β   j =1  N =: ∑ f j2 ( β ) j =1 F ( β ) := ( f1 ( β ),..., f N ( β ) ) T min f ( β ) = F T ( β ) F ( β )
  34. 34. Nonlinear Regression β k +1 := β k + qk • Gauss-Newton method : ∇F ( β )∇T F ( β )q = −∇F ( β ) F ( β ) • Levenberg-Marquardt method : λ ≥0 ( ) ∇F ( β )∇T F (β ) + λ I p q = −∇F ( β ) F ( β )
  35. 35. Nonlinear Regression alternative solution min t, t,q subject to ( ∇F (β )∇ T ) F ( β ) + λ I p q − ( −∇F ( β ) F ( β ) ) 2 ≤ t , t ≥ 0, || Lq || 2 ≤ M conic quadratic programming
  36. 36. Nonlinear Regression alternative solution min t, t,q subject to ( ∇F (β )∇ T ) F ( β ) + λ I p q − ( −∇F ( β ) F ( β ) ) 2 ≤ t , t ≥ 0, || Lq || 2 ≤ M conic quadratic programming interior point methods
  37. 37. Nonlinear Regression alternative solution min t, t,q subject to ( ∇F (β )∇ T ) F ( β ) + λ I p q − ( −∇F ( β ) F ( β ) ) 2 ≤ t , t ≥ 0, || Lq || 2 ≤ M  1  min Q(q) := f ( β ) + qT ∇F ( β ) F ( β ) + qT ∇F ( β )∇T F ( β )q  q 2  subject to q 2 ≤∆  trust region
  38. 38. Portfolio Optimization max utility ! or min costs ! martingale method: Optimization Problem Representation Problem or stochastic control
  39. 39. Portfolio Optimization max utility ! or min costs ! martingale method: Parameter Estimation Optimization Problem Representation Problem or stochastic control
  40. 40. Portfolio Optimization max utility ! or min costs ! martingale method: Optimization Problem Representation Problem Parameter Estimation or stochastic control
  41. 41. Portfolio Optimization max utility ! or min costs ! martingale method: Optimization Problem Representation Problem Parameter Estimation or stochastic control
  42. 42. References Aster, A., Borchers, B., and Thurber, C., Parameter Estimation and Inverse Problems, Academic Press, 2004. Boyd, S., and Vandenberghe, L., Convex Optimization, Cambridge University Press, 2004. Buja, A., Hastie, T., and 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., and Stuetzle, W., Projection pursuit regression, J. Amer. Statist Assoc. 76 (1981) 817-823. Hastie, T., and Tibshirani, R., Generalized additive models, Statist. Science 1, 3 (1986) 297-310. Hastie, T., and Tibshirani, R., Generalized additive models: some applications, J. Amer. Statist. Assoc. 82, 398 (1987) 371-386. Hastie, T., Tibshirani, R., and Friedman, J.H., The Element of Statistical Learning, Springer, 2001. Hastie, T.J., and Tibshirani, R.J., Generalized Additive Models, New York, Chapman and Hall, 1990. Kloeden, P.E, Platen, E., and Schurz, H., Numerical Solution of SDE Through Computer Experiments, Springer Verlag, New York, 1994. Korn, R., and Korn, E., Options Pricing and Portfolio Optimization: Modern Methods of Financial Mathematics, Oxford University Press, 2001. Nash, G., and Sofer, A., Linear and Nonlinear Programming, McGraw-Hill, New York, 1996. Nemirovski, A., Lectures on modern convex optimization, Israel Institute of Technology (2002).
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