Quasi-Monte Carlo Algorithms with Applications in Numerical Analysis and Finance

Transcription

1 Quasi-Monte Carlo Algorithms with Applications in Numerical Analysis and Finance Reinhold F. Kainhofer Rigorosumsvortrag, Institut für Mathematik Graz, 16. Mai 2003

2 Inhalt 1. Quasi-Monte Carlo Methoden 2. Sublineare Dividendenschranken in der Risikotheorie 3. Runge-Kutte QMC Methoden für retardierte Differentialgleichungen (a) Lösungsskizze des Algorithmus (b) RKQMC Lösungsmethoden (Hermite Interpolation, QMC Methoden) (c) Konvergenzbeweis (d) Numerische Beispiele 4. QMC Methoden für singuläre gewichtete Integration (a) Konvergenztheorem und -beweis (b) Hlawka-Mück Konstruktion (c) Numerisches Beispiel (Bewertung von asiatischen Optionen)

3 Quasi-Monte Carlo methods Integrals approximated by a discrete sum over N (quasi-)random points: f(x)dx = 1 [0,1] s N N f(x i ) i=1

4 Quasi-Monte Carlo methods Integrals approximated by a discrete sum over N (quasi-)random points: MC methods: x i random points [0,1] s f(x)dx = 1 N N f(x i ) QMC methods: x i low-discrepancy sequences Low discrepancy sequences: deterministic point sequences {x i } 1 n N [0, 1) s, good uniform distribution i=1

5 Quasi-Monte Carlo methods Integrals approximated by a discrete sum over N (quasi-)random points: MC methods: x i random points [0,1] s f(x)dx = 1 N N f(x i ) QMC methods: x i low-discrepancy sequences Low discrepancy sequences: deterministic point sequences {x i } 1 n N [0, 1) s, good uniform distribution discrepancy of the point set S D N (S) = sup J [0,1] s i=1 A(J, S) N λ s (J )

6 Quasi-Monte Carlo methods Integrals approximated by a discrete sum over N (quasi-)random points: MC methods: x i random points [0,1] s f(x)dx = 1 N N f(x i ) QMC methods: x i low-discrepancy sequences Low discrepancy sequences: deterministic point sequences {x i } 1 n N [0, 1) s, good uniform distribution discrepancy of the point set S D N (S) = sup J [0,1] s i=1 A(J, S) N λ s (J ) Koksma-Hlawka inequality (f of bounded Variation): 1 N N f(x n ) f(u)du [0,1) V ([0, 1)s, f)dn (x 1,..., x N ). s n=1

7 Low-Discrepancy sequences ( (log N) DN s ) (S) O N

8 Low-Discrepancy sequences ( (log N) DN s ) (S) O Halton-sequence in bases (b 1,... b s ): inversion of digit expansion of n in base b i at the comma (t, s) nets in base b (Niederreiter, Sobol, Faure): net-like structure best possible uniform distribution on elementary intervals N

9 Low-Discrepancy sequences ( (log N) DN s ) (S) O Halton-sequence in bases (b 1,... b s ): inversion of digit expansion of n in base b i at the comma (t, s) nets in base b (Niederreiter, Sobol, Faure): net-like structure best possible uniform distribution on elementary intervals N 1 Monte Carlo, 3000 points 1 Halton sequence, 3000 points

10 Low-Discrepancy sequences ( (log N) DN s ) (S) O Halton-sequence in bases (b 1,... b s ): inversion of digit expansion of n in base b i at the comma (t, s) nets in base b (Niederreiter, Sobol, Faure): net-like structure best possible uniform distribution on elementary intervals N 1 Monte Carlo, 3000 points 1 Halton sequence, 3000 points Problem: correlations between elements

11 Risk Model with constant interest force and non-linear dividend barrier reserve R t dividends dividend barrier b claims ~ F(y) premiums t b x ruin time t φ(u, b)... probability of survival W (u, b)... expected value of discounted dividend payments

12 Integro-differential equation for W (u, b): (c+i u) W u + 1 α m b m 1 W b (i+λ) W + λ u 0 W (u z,b)df (z)=0 with boundary condition W u u=b = 1. Solution is fixed point of integral operator apply it iteratively or recursively Ag(u, b) = + t 0 ( (c λe (λ+i)t +u)e it c ( g (c + u)e it c z, b m + t ) ) 1/m df (z)dt 0 α ( (b m + α) t 1/m ( g b m + t ) 1/m ( z, b m + t ) ) 1/m df (z)dt 0 α α ( ) t + λe λt e is (c + i u)e is 1 t t mα ( b m + α s ) 1 1/m ds dt, λe (λ+i)t t = Solution is just a high-dimensional integration problem.

13 Numerical solution of delayed differential equations using QMC methods 1. Sketch of the numerical solution 2. The RKQMC solution methods (Hermite Interpolation, QMC methods) 3. Convergence proofs 4. Numerical examples

14 The problem Heavily varying delay differential equations (DDE) or DDE with heavily varying solutions. y (t) = f (t, y(t), y(t τ 1 (t)),..., y(t τ k (t))), for t t 0, k 1, y(t) = φ(t), for t t 0, with

15 The problem Heavily varying delay differential equations (DDE) or DDE with heavily varying solutions. y (t) = f (t, y(t), y(t τ 1 (t)),..., y(t τ k (t))), for t t 0, k 1, y(t) = φ(t), for t t 0, with f(t, y(t), y ret (t))...piecewise smooth in y and y ret, bounded and Borel measurable in t y(t)...solution, d-dimensional real-valued function τ 1 (t),..., τ k (t)...cont. delay functions, bounded from below by τ 0 > 0, satisfy t 1 τ j (t 1 ) t 2 τ j (t 2 ) for t 1 t [ 2 ] φ(t)...initial function, cont. on inf (t τ j(t)), t 0. t 0 t,1 j k

16 Sketch of the numerical solution For heavily oscillating DDE: conventional Runge-Kutta methods unstable. Hermite interpolation for retarded argument ODE

17 Sketch of the numerical solution For heavily oscillating DDE: conventional Runge-Kutta methods unstable. Hermite interpolation for retarded argument ODE use RKQMC methods for ODE: Large Runge-Kutta error for heavily oscillating DE Idea (Stengle, Lécot): integrate over whole step size

18 Sketch of the numerical solution For heavily oscillating DDE: conventional Runge-Kutta methods unstable. Hermite interpolation for retarded argument ODE use RKQMC methods for ODE: Large Runge-Kutta error for heavily oscillating DE Idea (Stengle, Lécot): integrate over whole step size Runge Kutta: Integration over y and t discretized RK(Q)MC: Integration over y discretized, numerical Integration in t (using MC or QMC integration to minimize the error)

19 Hermite interpolation Use hermite interpolation for the retarded arguments:

20 Hermite interpolation Use hermite interpolation for the retarded arguments: z j (t) := y(t τ j (t)) = { φ(t τ j (t)), if t τ j (t) t 0 P q (t τ j (t); (y i ); (y i )) otherwise

21 Hermite interpolation Use hermite interpolation for the retarded arguments: z j (t) := y(t τ j (t)) = DDE transforms to a ODE: { φ(t τ j (t)), if t τ j (t) t 0 P q (t τ j (t); (y i ); (y i )) otherwise y (t) = f(t, y(t), y(t τ 1 (t)),..., y(t τ k (t))) f(t, y(t), z 1 (t),..., z k (t)) =: g(t, y(t)).

22 Hermite interpolation Use hermite interpolation for the retarded arguments: z j (t) := y(t τ j (t)) = DDE transforms to a ODE: { φ(t τ j (t)), if t τ j (t) t 0 P q (t τ j (t); (y i ); (y i )) otherwise y (t) = f(t, y(t), y(t τ 1 (t)),..., y(t τ k (t))) f(t, y(t), z 1 (t),..., z k (t)) =: g(t, y(t)). Solution has to be piecewise r/2-times continuously differentiable in t.

23 Hermite interpolation Use hermite interpolation for the retarded arguments: z j (t) := y(t τ j (t)) = DDE transforms to a ODE: { φ(t τ j (t)), if t τ j (t) t 0 P q (t τ j (t); (y i ); (y i )) otherwise y (t) = f(t, y(t), y(t τ 1 (t)),..., y(t τ k (t))) f(t, y(t), z 1 (t),..., z k (t)) =: g(t, y(t)). Solution has to be piecewise r/2-times continuously differentiable in t. Resulting ODE has to fulfill requirements for RKQMC methods (Borelmeasurable in t, continously differentiable in y(t)).

24 Runge Kutta QMC methods for ODE G. Stengle, Ch. Lécot, I. Coulibaly, A. Koudiraty y (t) = f(t, y(t)), 0 < t < T, y(0) = y 0 f smooth in y, bounded and Borel measurable in t.

25 Runge Kutta QMC methods for ODE G. Stengle, Ch. Lécot, I. Coulibaly, A. Koudiraty y (t) = f(t, y(t)), 0 < t < T, y(0) = y 0 f smooth in y, bounded and Borel measurable in t. f is Taylor-expanded only in y integral equation in t.

26 Runge Kutta QMC methods for ODE G. Stengle, Ch. Lécot, I. Coulibaly, A. Koudiraty y (t) = f(t, y(t)), 0 < t < T, y(0) = y 0 f smooth in y, bounded and Borel measurable in t. f is Taylor-expanded only in y integral equation in t. y n+1 = y n + h n s!n 0 j<n G s ( t j,n ; y) G s ( t j,n ; y)... differential increment function of scheme

27 Runge Kutta QMC methods for ODE G. Stengle, Ch. Lécot, I. Coulibaly, A. Koudiraty y (t) = f(t, y(t)), 0 < t < T, y(0) = y 0 f smooth in y, bounded and Borel measurable in t. f is Taylor-expanded only in y integral equation in t. y n+1 = y n + h n s!n 0 j<n G s ( t j,n ; y) G s ( t j,n ; y)... differential increment function of scheme G 1 (u; y) = f (u, y) G 2 (ū; y) = f (ū 1, y) + 1 β f (ū 2, y)) + 1 α f (ū 2, y + αh n f (ū 1, y))) L 2 G 3 (ū; y) = a 1 f(ū 1, y) + a 2,l f ( ū 2, y + b 2,l h n f(ū 1, y) ) + l=1 l=1 L 3 ( + a 3,l ū 3, y + b (1) 3,l h nf (ū 1, y) + b (2) 3,l h nf ( ū 2, y n + c 3,l h n (ū 1, y n ) ))

28 RKQMC for Volterra functional equations y (t) = f(t, y(t), z(t)) t t 0 z(t) = (F y)(t) y(s) = φ(s) s t 0

29 RKQMC for Volterra functional equations y (t) = f(t, y(t), z(t)) t t 0 z(t) = (F y)(t) y(s) = φ(s) s t 0 (F y)(t) contains dependence on retarded values. F and f are Lipschitz continuous in all arguments except t.

30 RKQMC for Volterra functional equations y (t) = f(t, y(t), z(t)) t t 0 z(t) = (F y)(t) y(s) = φ(s) s t 0 (F y)(t) contains dependence on retarded values. F and f are Lipschitz continuous in all arguments except t. RKQMC method (n 0): y n+1 = y n + h n z(t) = ( F y j )(t) N i=1 G s (t n,i ; y n, z(t))

31 Convergence: RKQMC for DDE Theorem[K., 2002] If (i) RKQMC method converges for ordinary differential equations with order p (ii) the increment function G s of the method and F are Lipschitz (iii) the interpolation fulfills a Lipschitz condition (iv) Hermite interpolation is used with order r (v) the initial error e 0 vanishes then the method converges.

32 Convergence: RKQMC for DDE Theorem[K., 2002] If (i) RKQMC method converges for ordinary differential equations with order p (ii) the increment function G s of the method and F are Lipschitz (iii) the interpolation fulfills a Lipschitz condition (iv) Hermite interpolation is used with order r (v) the initial error e 0 vanishes then the method converges. If at least ii and iii hold, the error is bounded by e j+1 e 0 e Lt j + (et jl 1) L ( E ODE + L1 L 2 E interpol ). G r

33 Special case: one retarded argument Choose the RKQMC method: G s Lipschitz (L 2 ) in 2 nd and 3 rd argument, bounded variation (in the sense of Hardy and Krause). c 1, c 2, c 3 such that for a p > 0 loc. trunc. error ɛ n c 1 (h n )h p n RK error δ n c 2 (h n ) e n QMC error d n c 3 (h n )D N (S) interpolation order p, fulfils a certain Lipschitz condition

34 Special case: one retarded argument Choose the RKQMC method: G s Lipschitz (L 2 ) in 2 nd and 3 rd argument, bounded variation (in the sense of Hardy and Krause). c 1, c 2, c 3 such that for a p > 0 loc. trunc. error ɛ n c 1 (h n )h p n RK error δ n c 2 (h n ) e n QMC error d n c 3 (h n )D N (S) interpolation order p, fulfils a certain Lipschitz condition Then the error e n = y n y(t n ) of the method is bounded from above by: e n e 0 e t n(c 2 + L 2 s! ) e t n(c 2 + L2 s! ) 1 + c 2 + L 2 s! { c 3 DN(X) + L } 2 s! MHq + c 1 H p

35 Numerical examples y (t) = 3y(t 1) sin(λt) + 2y(t 1.5) cos(λt), t 0 y(t) = 1, t 0,

36 Numerical examples y (t) = 3y(t 1) sin(λt) + 2y(t 1.5) cos(λt), t 0 y(t) = 1, t 0, Log 2 S meth,λ Log 2 Λ Butcher Heun Runge RKQMC1, N 1000 RKQMC2, N 1000 RKQMC3, N 1000 RKQMC3, N 100 RKQMC3, N 10 Fig: Error of the RK and RKQMC methods, h n = for RK, h n = 0.01 for RKQMC

37 Numerical examples y (t) = 3y(t 1) sin(λt) + 2y(t 1.5) cos(λt), t 0 y(t) = 1, t 0, Log 2 S meth,λ Log 2 Λ Butcher Heun Runge RKQMC1, N 1000 RKQMC2, N 1000 RKQMC3, N 1000 RKQMC3, N 100 RKQMC3, N 10 Fig: Error of the RK and RKQMC methods, h n = for RK, h n = 0.01 for RKQMC slowly varying (small λ): conventional RK better rapidly varying (high λ): RKQMC outperform higher order RK

38 Advantage of RKQMC for unstable DDE y (t) = π λ 2 ( ( y t 2 3 ) ( y t 2 1 )), t 0 2λ 2λ y(t) = sin(λtπ), t < 0,

39 Advantage of RKQMC for unstable DDE y (t) = π λ 2 ( ( y t 2 3 ) ( y t 2 1 )), t 0 2λ 2λ y(t) = sin(λtπ), t < 0, Exact Solution: y(t) = sin(λtπ) No discontinuities in any derivative k-th derivative only bounded by λ k exploding error

40 Advantage of RKQMC for unstable DDE y (t) = π λ 2 ( ( y t 2 3 ) ( y t 2 1 )), t 0 2λ 2λ y(t) = sin(λtπ), t < 0, Exact Solution: y(t) = sin(λtπ) No discontinuities in any derivative k-th derivative only bounded by λ k exploding error Log 2 S Λ, meth Butcher Heun Runge RKQMC1, N 1000 RKQMC2, N 1000 RKQMC3, N Log 2 Λ RKQMC schemes can delay the instability of the solution for heavily oscillating delay differential equations.

41 Time-corrected error QMC integration is more expensive than 1 evaluation (Runge-Kutta)

42 Time-corrected error QMC integration is more expensive than 1 evaluation (Runge-Kutta) Compare a time-corrected error

43 Time-corrected error QMC integration is more expensive than 1 evaluation (Runge-Kutta) Compare a time-corrected error Log 2 S Λ, meth T Log 2 Λ Butcher Heun Runge RKQMC1, N 1000 RKQMC2, N 1000 RKQMC3, N 1000 Fig: Time-corrected error of the RK and RKQMC methods Result: RKQMC methods loose some advantage, but still better than Runge-Kutta for heavily oscillating DDE.

44 Non-uniform QMC integration of singular integrands 1. Convergence theorem 2. Sketch of proof 3. Hlawka-Mück construction for h-distributed low-discrepancy sequences 4. Numerical example (pricing of Asian options)

45 H-Diskrepancy H-discrepancy of a sequence ω = (y 1, y 2,... ): D N,H (ω) = sup 1 J L N A N(J, ω) H(J), with L... support [ of H, J... intervals a, ] b, A N... number of elements of ω lying in J.

46 H-Diskrepancy H-discrepancy of a sequence ω = (y 1, y 2,... ): D N,H (ω) = sup 1 J L N A N(J, ω) H(J), with L... support [ of H, J... intervals a, ] b, A N... number of elements of ω lying in J. Koksma-Hlawka inequality (arbitrary distribution) Let f a function of bounded variation (in the sense of Hardy and Krause) on L and ω = (y 1, y 2,... ) a sequence on L. Then f(x)dh(x) 1 N f(y n ) L N V (f)d N,H(ω). n=1

47 Convergence of the multidimensional QMC estimator

48 Convergence of the multidimensional QMC estimator Theorem[Hartinger, K., Tichy, 2003] Let f(x) be a function on L = [a, b] with singularities only at the left boundary of the definition interval (i.e. f ± only if x (j) a j for at least one j),

49 Convergence of the multidimensional QMC estimator Theorem[Hartinger, K., Tichy, 2003] Let f(x) be a function on L = [a, b] with singularities only at the left boundary of the definition interval (i.e. f ± only if x (j) a j for at least one j), and let furthermore c N,j = min 1 n N y(j) n a j < c j c N,j.

50 Convergence of the multidimensional QMC estimator Theorem[Hartinger, K., Tichy, 2003] Let f(x) be a function on L = [a, b] with singularities only at the left boundary of the definition interval (i.e. f ± only if x (j) a j for at least one j), and let furthermore c N,j = min 1 n N y(j) n a j < c j c N,j. If the improper integral exists, and if D N,H (ω) V [c,b] (f) = o(1), then the QMC estimator converges to the value of the improper integral: lim N 1 N N f(y n ) = n=1 [a,b] f(x) dh(x).

51 Hlawka-Mück Transformation (1 dimension) ỹ k = 1 N N 1 + x k H(x r ) = 1 N r=1 N χ [0,xk ](H(x r )) r=1

52 Hlawka-Mück Transformation (1 dimension) ỹ k = 1 N N 1 + x k H(x r ) = 1 N r=1 N χ [0,xk ](H(x r )) r=1 D N,H ( ω) (1 + 2sM)D N (ω)

53 0 might appear among the ỹ k adapt the construction as follows (needed for singular integrands): We define the sequence ω for i = 1,..., N as: { ỹ ȳ k = k wenn ỹ k 1, N 1 wenn ỹ N k = 0.

54 0 might appear among the ỹ k adapt the construction as follows (needed for singular integrands): We define the sequence ω for i = 1,..., N as: { ỹ ȳ k = k wenn ỹ k 1, N 1 wenn ỹ N k = 0. The discrepancy of this sequence can be bounded as D N,H ( ω) (1 + 4M) s D N (ω).

55 Sample problem: Valuing Asian options arithmetic mean until expiration time Pay-Off (discrete Asian option, call) ( 1 n P (S T ) = S ti K n i=1 (S t ) t 0... price process, K... strike price S t = e X t with Levy process (X t ) t 0 Increments h i = X i X i 1 with distribution H (e.g. NIG, Variance-Gamma, Hyperbolic,...) ) +

56 NIG distribution Use the NIG distribution for the increments h i H Q. Advantage: closed under convolution dimension reduction, sample only weekly instead of daily Valuation Using fundamental theorem (Schachermayer): [( ) + ] C t0 := e r(t n t 0 ) E Q 1 n S ti K n r... constant interest rate Q... equivalent martingale measure (Esscher measure) i=1

57 NIG distribution Use the NIG distribution for the increments h i H Q. Advantage: closed under convolution dimension reduction, sample only weekly instead of daily Valuation Using fundamental theorem (Schachermayer): [( ) + ] C t0 := e r(t n t 0 ) E Q 1 n S ti K n r... constant interest rate Q... equivalent martingale measure (Esscher measure) i=1 Straigforward simulation (crude Monte Carlo): sample N price pathes and take the mean

58 Quasi-Monte Carlo schemes Problem: QMC numbers i.i.d. NIG(α, β, δ, µ)

59 Quasi-Monte Carlo schemes Problem: QMC numbers i.i.d. NIG(α, β, δ, µ) 2 Solutions: 1. Hlawka-Mück method for direct creation of (x n ) 1 n N i.i.d. NIG direct QMC calculation of the expectation value

60 Quasi-Monte Carlo schemes Problem: QMC numbers i.i.d. NIG(α, β, δ, µ) 2 Solutions: 1. Hlawka-Mück method for direct creation of (x n ) 1 n N i.i.d. NIG direct QMC calculation of the expectation value 2. Transformation of the integral using a suitable density (Ratio of uniforms, Hat ) variance reduction (if done right) Transformation Using a distribution K( x) = u: P ( x)f H( x)d x Q = P (K 1 (u)) f Q H(K 1 (u)) R n [0,1] n k(k 1 (u)) du Problem: Usual transformation F (x) = u leads to unbounded variation

61 Discrepancy for Hlawka-Mück The discrepancy of (y k ) 1 k N can be bounded by D N ((y k ), ρ) (2 + 6sM(ρ))D N ((y k )) with M(ρ) = sup ρ.

62 Discrepancy for Hlawka-Mück The discrepancy of (y k ) 1 k N can be bounded by with M(ρ) = sup ρ. QMC estimator D N ((y k ), ρ) (2 + 6sM(ρ))D N ((y k )) 1. Creation of low-discrepancy points with density f Q H (K 1 (x)) k(k 1 (x)) (Transformation of the integral R n to [0, 1] n using double-exponential distribution K(x))

63 Discrepancy for Hlawka-Mück The discrepancy of (y k ) 1 k N can be bounded by with M(ρ) = sup ρ. QMC estimator D N ((y k ), ρ) (2 + 6sM(ρ))D N ((y k )) 1. Creation of low-discrepancy points with density f Q H (K 1 (x)) k(k 1 (x)) (Transformation of the integral R n to [0, 1] n using double-exponential distribution K(x)) 2. Transformation [0, 1] n to R n of the sequence using doubleexponential distribution K 1 (x). Estimator similar to crude Monte Carlo

64 Numerical results (4 dimensions) Log 2 err dimensions N MC Import. samp. Ratio of unif. Hlawka Mück HM IS double exp. ROU and Hlawka-Mück are considerably better than Monte Carlo and control variate

65 Dimension 12 Log 2 err 12 dimensions N MC Import. samp. Ratio of unif. Hlawka Mück HM IS double exp. ROU looses performance Hlawka-Mück gives best results