Interval Arithmatic and Automatic Differentiation in Optimization - - PowerPoint PPT Presentation

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary

Interval Arithmatic and Automatic Differentiation

in Optimization and Model Calibration Grzegorz Kozikowski

University of Manchester

Tampere 15th May 2013 HPCFinance

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary

Agenda

1

Aims

2

Calibration of Heston Model

3

Interval Analysis

4

Interval based algorithm for Heston calibration

5

Summary

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary

Aims Global optimization algorithm for non-linear regression. The approach is based on Interval Analysis and Automatic Differentiation using HPC technologies.

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary

Aims Global optimization algorithm for non-linear regression. The approach is based on Interval Analysis and Automatic Differentiation using HPC technologies. Application Calibration of the Heston Model, ill-posed optimization problems.

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary

Aims Global optimization algorithm for non-linear regression. The approach is based on Interval Analysis and Automatic Differentiation using HPC technologies. Application Calibration of the Heston Model, ill-posed optimization problems. Optimization problem Let us f : Rn → R: minimize

x

f(x) subject to: hi(x) ≤ ai, i = 1, . . . , n. gj(x) ≤ bj, j = 1, . . . , m.

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary

Optimization methods: Class of the gradient based optimization algorithms using real numbers:

• no global extremum guaranteed
• sequential structure
• discrete optimization

Heuristic algorithms

• no global extremum guaranteed
• embarrassingly parallel problem
• discrete optimization

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary

Optimization methods: Consider a contour line plot of some function f: from starting points (0, 0), (0, 4), (4, 0), converges to 1 from starting points (4, 4), (2, 2), converges to point 2 Unsolved problem in real numbers!!!

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary

Optimization methods: Global optimization based on Interval Arithmetic/Automatic Differentiation

• global extremum guaranteed
• embarrassingly parallel problem
• intervals

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Heston Model Calibration function

Heston Model

Heston Stochastic Volatility Model The dynamics of the Heston model can be described as follows: dSt = µStdt +

• VtStdZ 1

t

dVt = κ(θ − Vt)dt + σ

• VtdZ 2

t

dZ 1

t dZ 2 t = ρdt

St asset price at time t µ - constant drift of the asset price dt - time increment Vt - variance of the asset price at time t dZt increment of a Brownian motion at time t κ - constant mean reversion factor of the asset price variance θ - long-term level of asset price variance σ - volatility of volatility ρ - correlation between Brownian motions 7 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Heston Model Calibration function

Heston Model - semi-closed form solution

Solution of Heston PDE is: C(S, V , t, ξ) = SP1 − e−rTKP2 where ξ = (V0, κ, θ, σ, ρ, µ) and: Pj(S, V , t, ξ, K) = 1

2 + 1 π

∞

0 Re( e−iφln(K)fj(S,V ,t,ξ) iφ

)dφ fj(S, V , t, ξ) = eC(T−t,φ)+D(T−t,φ)Vt+iφln(St)) C(T − t, φ) = rφir + a

σ2 [(bj − ρσφi + d)τ − 2ln( 1−gedr 1−g )]

D(T − t, φ) = bj−ρσφi+d

σ2

( 1−edr

1−gedr )

g = bj−ρσφi+d

bj−ρσφi−d

d =

• (ρσφi − bj)2 − σ2(2ujφi − φ2)

u1 = 1

2, u2 = − 1 2, a = κθ, b1 = κ + λ − ρσ, b2 = κ + λ,

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Heston Model Calibration function

Calibration

Minimize errors between prices predicted by the Heston model and the market option data (different maturities, strike prices, options): ri(ξ) = Ci(ξ) − C mkt

i

Global optimization - Calibration of the Heston Model The aim is to find the set of input-parameters ξ = (V0, κ, θ, σ, ρ, µ), X = {ξ : ξǫX} that minimize an error function: min

ξ

= G(ξ) =

• Nquotes
• i=1

(Ci(ξ) − C mkt

i

)2

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Heston Model Calibration function

Most optimization require the gradient and the Hessian: ri(ξ) = Ci(ξ) − C mkt

i

the gradient: ∇G(ξ) = 2

Nquotes

• i=1

ri(ξ)∇ri(ξ) =           2 Nquotes

i=1

ri(ξ) ∗ dri(ξ)

dV0

2 Nquotes

i=1

ri(ξ) ∗ dri(ξ)

dκ

2 Nquotes

i=1

ri(ξ) ∗ dri(ξ)

dθ

2 Nquotes

i=1

ri(ξ) ∗ dri(ξ)

dσ

2 Nquotes

i=1

ri(ξ) ∗ dri(ξ)

dρ

2 Nquotes

i=1

ri(ξ) ∗ dri(ξ)

dµ

          = [0] the Hessian: ∇2G(ξ) = 2

Nquotes

• i=1

(ri(ξ)∇2ri(ξ) + ∇ri(ξ)∇ri(ξ)T)

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Introduction Reduction box techniques

Interval arithmetic

Approach to putting bounds in mathematical computation and thus, developing numerical methods that yield reliable results Each single real-number X is represented as a pair of bounds [xl, xu], where xl < xu Interval operators are inclusion isotonic (Z ⊂ X) => f (Z) ⊂ f (X) Optimization - reduction of an input space

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Introduction Reduction box techniques

Interval arithmetic

Approach to putting bounds in mathematical computation and thus, developing numerical methods that yield reliable results Each single real-number X is represented as a pair of bounds [xl, xu], where xl < xu Interval operators are inclusion isotonic (Z ⊂ X) => f (Z) ⊂ f (X) Optimization - reduction of an input space

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Introduction Reduction box techniques

Interval operators

Each single real-number X is represented as a pair of [x1, x2], where x1 < x2 Elementary arithmetic operations: [x1, x2] + [y1, y2] = [x1 + y1, x2 + y2] [x1, x2] − [y1, y2] = [x1 + y2, x2 + y1] [x1, x2] ∗ [y1, y2] = [min(x1y1, x1y2, x2y1, x2y2), max(x1y1, x1y2, x2y1, x2y2)]

[x1,x2] [y1,y2] = [x1, x2] ∗ [ 1 y1 , 1 y2 ]

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Introduction Reduction box techniques

Features

Intervals deal with: Non-continuous and non-differentiable functions Uncertainty of machine representation of numbers (outward rounding) Discretization errors Analysis of the simulation results based on uncertain input-parameters Numerical problems that are unsolved in real-numbers (global

• ptimization)

Interval results of operations are not always sharp (sharp bounds - NP-hard problem)

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Introduction Reduction box techniques

Reduction box techniques

Box consistency Consider an equation f (x1, ..., xn) = 0 where x1ǫX1, ..., xnǫXn. Replace all the variables except the i-th by intervals: q(xi) = f (X1, ...Xi−1, xi, Xi+1, ...Xn) = 0 Eliminate subsets of Xi not satisfying q(xi) = 0

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Introduction Reduction box techniques

Reduction box techniques

Contractors Property that allows us to set constraints for input parameters taking into account bounds of the result. Consider f (x1, x2) where x1ǫX1 and x2ǫX2 and f (x) = ¯ y

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Introduction Reduction box techniques

Contractors - Forward/Backward propagation algorithm If f (x1, x2) = exp(x1) + sin(x2) and x1ǫ[x1] x2ǫ[x2]. To obtain the exact domains for ¯ y = f (x) we use contractors: forward propagation: backward propagation: [a1] = exp([x1]) [a2] = ([¯ y] − [a3]) ∩ [a2] [a2] = sin([x2]) [a3] = ([¯ y] − [a2]) ∩ [a3] [y] = [a2] + [a3] [x2] = sin−1([a3]) ∩ [x2] [x1] = log([a1]) ∩ [x1]

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Introduction Reduction box techniques

Upper-bound test

Upper-bound test

1 Consider f (x1, x2) where x1ǫX1 and x2ǫX2 2 Suppose f ∗ - global minimum of f(x) 3 Evaluate f at point/interval x0: fu(x0) > f ∗ 4 As fu(x0) > f ∗, reduce the box for which f > fu(x0) by

contracting an inequality: f (x) = [−∞, fu(x0)]

5 As a result we obtain a new reduced box 17 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Introduction Reduction box techniques

Upper-bound test

Upper-bound test

1 Consider f (x1, x2) where x1ǫX1 and x2ǫX2 2 Suppose f ∗ - global minimum of f(x) 3 Evaluate f at point/interval x0: fu(x0) > f ∗ 4 As fu(x0) > f ∗, reduce the box for which f > fu(x0) by

contracting an inequality: f (x) = [−∞, fu(x0)]

5 As a result we obtain a new reduced box 17 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Introduction Reduction box techniques

Upper-bound test

Upper-bound test

1 Consider f (x1, x2) where x1ǫX1 and x2ǫX2 2 Suppose f ∗ - global minimum of f(x) 3 Evaluate f at point/interval x0: fu(x0) > f ∗ 4 As fu(x0) > f ∗, reduce the box for which f > fu(x0) by

contracting an inequality: f (x) = [−∞, fu(x0)]

5 As a result we obtain a new reduced box 17 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Introduction Reduction box techniques

Upper-bound test

Upper-bound test

1 Consider f (x1, x2) where x1ǫX1 and x2ǫX2 2 Suppose f ∗ - global minimum of f(x) 3 Evaluate f at point/interval x0: fu(x0) > f ∗ 4 As fu(x0) > f ∗, reduce the box for which f > fu(x0) by

contracting an inequality: f (x) = [−∞, fu(x0)]

5 As a result we obtain a new reduced box

x0 - the middle of box of the point generated by some optimization methods for real numbers.

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Introduction Reduction box techniques

Monotonicity/Concavity tests

Monotonicity test Let ∇(X) the interval gradient of a box X. If 0 ǫ∇(X), then eliminate X Concavity test Let H(X) the interval Hessian of a box X. If some Hii(X) < 0, then eliminate X Disjoint boxes If X1 ∩ X2 = ∅ and fupper(X1) < flower(X2) then eliminate X2 (inclusion isotonic property)

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Introduction Reduction box techniques

Taylor series - the gradient Consider a Taylor expansion of f around x (the middle of the box): f (y)ǫf (x) + n

i=1(yi − xi)gi.

Eliminate all y that: f (y)ǫf (x) + n

i=1(yi − xi)gi > ¯

f Taylor series - the Hessian Consider a Taylor expansion of f around x (the middle of the box): f (y)ǫf (x) + (y − x)Tg(x) + 1

2H(x)(y − x).

Eliminate all y that: f (y)ǫf (x) + (y − x)Tg(x) + 1

2H(x)(y − x) > ¯

f

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Introduction Reduction box techniques

Newton Method

Newton Method - equation f(X)=0 From the mean value theorem: f (x) − f (x⋆) = (x − x⋆)f ′(ξ) and (x⋆ < ξ < x) If x⋆ is a zero of f then: x⋆ = x − f (x)

F ′(ξ)

If f ′(ξ)ǫf ′(X) and x = m(X) then: N(Xn) = m(Xn) − f (m(Xn))

F ′(Xn) ,

Xn+1 = Xn ∩ N(Xn)

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

1 Begin with a box X in which the global minimum is sought 2 Contract ξ to the conditionCi(ξ) = C mkt

i

± Error

3 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 4 If ξ = ∅ then evaluate G(ξ) otherwise go to Stage 2 5 Contract ξ to the condition Ci(ξ) = C mkt

i

± Error

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

1 Begin with a box X in which the global minimum is sought 2 Contract ξ to the conditionCi(ξ) = C mkt

i

± Error

3 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 4 If ξ = ∅ then evaluate G(ξ) otherwise go to Stage 2 5 Contract ξ to the condition Ci(ξ) = C mkt

i

± Error

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

1 Begin with a box X in which the global minimum is sought 2 Contract ξ to the conditionCi(ξ) = C mkt

i

± Error

3 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 4 If ξ = ∅ then evaluate G(ξ) otherwise go to Stage 2 5 Contract ξ to the condition Ci(ξ) = C mkt

i

± Error

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

1 Begin with a box X in which the global minimum is sought 2 Contract ξ to the conditionCi(ξ) = C mkt

i

± Error

3 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 4 If ξ = ∅ then evaluate G(ξ) otherwise go to Stage 2 5 Contract ξ to the condition Ci(ξ) = C mkt

i

± Error

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

1 Begin with a box X in which the global minimum is sought 2 Contract ξ to the conditionCi(ξ) = C mkt

i

± Error

3 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 4 If ξ = ∅ then evaluate G(ξ) otherwise go to Stage 2 5 Contract ξ to the condition Ci(ξ) = C mkt

i

± Error

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

1 Begin with a box X in which the global minimum is sought 2 Contract ξ to the conditionCi(ξ) = C mkt

i

± Error

3 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 4 If ξ = ∅ then evaluate G(ξ) otherwise go to Stage 2 5 Contract ξ to the condition Ci(ξ) = C mkt

i

± Error

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

7 Find minimum fsomemin of G(ξ) by using heuristic/gradient

methods

8 If interval contraction was not successful, fsomemin is global

minimum

9 Contract ξ to G(ξ) = [0, fsomemin] 10 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 11 Contract ξ to the condition: ∇G(ξ) = 0 12 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 22 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

7 Find minimum fsomemin of G(ξ) by using heuristic/gradient

methods

8 If interval contraction was not successful, fsomemin is global

minimum

9 Contract ξ to G(ξ) = [0, fsomemin] 10 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 11 Contract ξ to the condition: ∇G(ξ) = 0 12 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 22 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

7 Find minimum fsomemin of G(ξ) by using heuristic/gradient

methods

8 If interval contraction was not successful, fsomemin is global

minimum

9 Contract ξ to G(ξ) = [0, fsomemin] 10 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 11 Contract ξ to the condition: ∇G(ξ) = 0 12 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 22 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

7 Find minimum fsomemin of G(ξ) by using heuristic/gradient

methods

8 If interval contraction was not successful, fsomemin is global

minimum

9 Contract ξ to G(ξ) = [0, fsomemin] 10 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 11 Contract ξ to the condition: ∇G(ξ) = 0 12 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 22 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

7 Find minimum fsomemin of G(ξ) by using heuristic/gradient

methods

8 If interval contraction was not successful, fsomemin is global

minimum

9 Contract ξ to G(ξ) = [0, fsomemin] 10 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 11 Contract ξ to the condition: ∇G(ξ) = 0 12 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 22 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

7 Find minimum fsomemin of G(ξ) by using heuristic/gradient

methods

8 If interval contraction was not successful, fsomemin is global

minimum

9 Contract ξ to G(ξ) = [0, fsomemin] 10 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 11 Contract ξ to the condition: ∇G(ξ) = 0 12 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 22 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

13 Use multidimensional interval Newton method for each

sub-box to solve (∇G(ξ) = 0)

14 Contract ξ to the condition Hii(ξ) < 0 or use Taylor

expansions

15 Eliminate sub-boxes whose lower bound > upper bound of the

box for which G(X) is the lowest

16 Choose the best sub-boxes 23 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

13 Use multidimensional interval Newton method for each

sub-box to solve (∇G(ξ) = 0)

14 Contract ξ to the condition Hii(ξ) < 0 or use Taylor

expansions

15 Eliminate sub-boxes whose lower bound > upper bound of the

box for which G(X) is the lowest

16 Choose the best sub-boxes 23 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

13 Use multidimensional interval Newton method for each

sub-box to solve (∇G(ξ) = 0)

14 Contract ξ to the condition Hii(ξ) < 0 or use Taylor

expansions

15 Eliminate sub-boxes whose lower bound > upper bound of the

box for which G(X) is the lowest

16 Choose the best sub-boxes 23 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

13 Use multidimensional interval Newton method for each

sub-box to solve (∇G(ξ) = 0)

14 Contract ξ to the condition Hii(ξ) < 0 or use Taylor

expansions

15 Eliminate sub-boxes whose lower bound > upper bound of the

box for which G(X) is the lowest

16 Choose the best sub-boxes 23 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

16 Choose the best sub-boxes

Interval results of arithmetic operations are not always sharp (f (xǫX) ⊂ f (X), but not always f (xǫX) ⊆ f (X)) !!! Conclusion: The sub-box whose error function is the lowest known value is not always the best. Glower(ξb1) < Glower(ξb2) < .... < Glower(ξbm) Answer: Try to contract the difference between the best pair of the sub-boxes. If successful, eliminate all the other.

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

16 Choose the best sub-boxes

Interval results of arithmetic operations are not always sharp (f (xǫX) ⊂ f (X), but not always f (xǫX) ⊆ f (X)) !!! Conclusion: The sub-box whose error function is the lowest known value is not always the best. Glower(ξb1) < Glower(ξb2) < .... < Glower(ξbm) Answer: Try to contract the difference between the best pair of the sub-boxes. If successful, eliminate all the other.

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

16 Choose the best sub-boxes

Interval results of arithmetic operations are not always sharp (f (xǫX) ⊂ f (X), but not always f (xǫX) ⊆ f (X)) !!! Conclusion: The sub-box whose error function is the lowest known value is not always the best. Glower(ξb1) < Glower(ξb2) < .... < Glower(ξbm) Answer: Try to contract the difference between the best pair of the sub-boxes. If successful, eliminate all the other.

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

13 Use multidimensional Newton method for each sub-box to

solve (∇G(ξ) = 0)

14 Contract ξ to the condition Hii(ξ) < 0 or use Taylor

expansions

15 Eliminate sub-boxes whose lower bound > upper bound of the

box for which G(X) is the lowest

16 Choose the best sub-boxes 17 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 18 Repeat Stage 2 until the difference between lower and upper

bound of ǫ is less than error tolerance (for each dimension)

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

13 Use multidimensional Newton method for each sub-box to

solve (∇G(ξ) = 0)

14 Contract ξ to the condition Hii(ξ) < 0 or use Taylor

expansions

15 Eliminate sub-boxes whose lower bound > upper bound of the

box for which G(X) is the lowest

16 Choose the best sub-boxes 17 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 18 Repeat Stage 2 until the difference between lower and upper

bound of ǫ is less than error tolerance (for each dimension)

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Heston Calibration - Interval Approach

13 Use multidimensional Newton method for each sub-box to

solve (∇G(ξ) = 0)

14 Contract ξ to the condition Hii(ξ) < 0 or use Taylor

expansions

15 Eliminate sub-boxes whose lower bound > upper bound of the

box for which G(X) is the lowest

16 Choose the best sub-boxes 17 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes 18 Repeat Stage 2 until the difference between lower and upper

bound of ǫ is less than error tolerance (for each dimension)

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Comments

The Gradient and the Hessian are evaluated by Automatic Differentiation routines If some intermediate result has an infinite bound, use extended interval analysis and contractors During interval contraction or Gauss-Newton method, the initial box might be divided into many sub-boxes All the sub-boxes can be processed in parallel (GPU, multicore CPUs)

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Comments

The Gradient and the Hessian are evaluated by Automatic Differentiation routines If some intermediate result has an infinite bound, use extended interval analysis and contractors During interval contraction or Gauss-Newton method, the initial box might be divided into many sub-boxes All the sub-boxes can be processed in parallel (GPU, multicore CPUs)

26 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Comments

The Gradient and the Hessian are evaluated by Automatic Differentiation routines If some intermediate result has an infinite bound, use extended interval analysis and contractors During interval contraction or Gauss-Newton method, the initial box might be divided into many sub-boxes All the sub-boxes can be processed in parallel (GPU, multicore CPUs)

26 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Stage 1 Stage 2 Stage 3 Comments

Comments

The Gradient and the Hessian are evaluated by Automatic Differentiation routines If some intermediate result has an infinite bound, use extended interval analysis and contractors During interval contraction or Gauss-Newton method, the initial box might be divided into many sub-boxes All the sub-boxes can be processed in parallel (GPU, multicore CPUs)

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Questions Bibliography

Conclusion

Conclusion Heston calibration requires an implementation of extended interval analysis for complex numbers Parallel architectures are suitable for global optimization using Interval Analysis Interval analysis deals with numerical problems that are unsolved in real-numbers (global optimization)

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Questions Bibliography

Questions Questions

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Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Questions Bibliography

Bibliography

Hansen Eldon, Global Optimization using Interval Arithmetics New York Basel, 2006 Grzegorz Kozikowski, Evaluation of Option Price Sensitivities based on the Automatic Differentiation Methods using CUDA, Warsaw University of Technology 2013, Master Thesis Grzegorz Kozikowski, Library for Automatic Differentiation using OpenCL, Warsaw University of Technology 2011, Bachelor Thesis Grzegorz Kozikowski and Bartlomiej Jacek Kubica, Interval Arithmetic and Automatic Differentiation using OpenCL technology, Springer 2013, LNCS 7738 Max E. Jerell, Automatic Differentiation and Interval Arithmetic for Estimation of Disequilibrium Models, 1997 Luenberger G. David, Ye Yinyu: Linear and nonlinear programming, Springer 2008 Bazara S. Mokhatar, Nonlinear programming, Theory and algorithms, Willey Interscience 29 / 30

Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Questions Bibliography

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