interval arithmatic and automatic di erentiation euroad workshop... · interval analysis summary...
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AimsCalibration of Heston Model
Interval AnalysisSummary
Interval Arithmatic and Automatic Differentiationin Optimization and Model Calibration
Grzegorz Kozikowski
University of Manchester, HPCFinance
Sophia-Antipolis, 11th June 2013Euro AD Workshop
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AimsCalibration of Heston Model
Interval AnalysisSummary
Experience with AD and HPC:
Bachelor Thesis:
Implementation of Automatic Differentiation library using IntervalAnalysis and OpenCL
Forward/Reverse Mode for the Gradient, the Hessian (CPU and GPU)
PARA 2012 Proceedings LNCS 7782, 489-503, 2013
Master Thesis:
Parallel approach to Monte Carlo simulation for Option PriceSensitivities using the Adjoint and Interval Analysis (CUDA)
the Adjoint for the first and the second-order Greeks (CPU and GPU)
PPAM 2013 Proceedings
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AimsCalibration of Heston Model
Interval AnalysisSummary
Agenda
1 Aims
2 Calibration of Heston Model
3 Interval Analysis
4 Summary
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AimsCalibration of Heston Model
Interval AnalysisSummary
Aims
Global optimization algorithm for non-linear regression.The approach is based on Interval Analysis and AutomaticDifferentiation using HPC technologies.
Application
Calibration of the Heston Model, ill-posed optimization problems.
Optimization problem
Let us f : Rn → R:
minimizex
f (x)
subject to:
hi (x) ≤ ai , i = 1, . . . , n.
gj(x) ≤ bj , j = 1, . . . ,m.
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AimsCalibration of Heston Model
Interval AnalysisSummary
Aims
Global optimization algorithm for non-linear regression.The approach is based on Interval Analysis and AutomaticDifferentiation using HPC technologies.
Application
Calibration of the Heston Model, ill-posed optimization problems.
Optimization problem
Let us f : Rn → R:
minimizex
f (x)
subject to:
hi (x) ≤ ai , i = 1, . . . , n.
gj(x) ≤ bj , j = 1, . . . ,m.
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AimsCalibration of Heston Model
Interval AnalysisSummary
Aims
Global optimization algorithm for non-linear regression.The approach is based on Interval Analysis and AutomaticDifferentiation using HPC technologies.
Application
Calibration of the Heston Model, ill-posed optimization problems.
Optimization problem
Let us f : Rn → R:
minimizex
f (x)
subject to:
hi (x) ≤ ai , i = 1, . . . , n.
gj(x) ≤ bj , j = 1, . . . ,m.3 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
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!!!(gradient based optimization algorithms or heuristics)
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AimsCalibration of Heston Model
Interval AnalysisSummary
Optimization methods:
Global optimization based on IntervalArithmetic/Automatic Differentiation- global extremum guaranteed- embarrassingly parallel problem- intervals
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AimsCalibration of Heston Model
Interval AnalysisSummary
Heston ModelCalibration function
Heston Model
Heston Stochastic Volatility Model
The dynamics of the Heston model can be described as follows:
dSt = µStdt +√
VtStdZ 1t
dVt = κ(θ − Vt)dt + σ√
VtdZ 2t
dZ 1t dZ 2
t = ρdtSt asset price at time tµ - constant drift of the asset pricedt - time incrementVt - variance of the asset price at time tdZt increment of a Brownian motion at time tκ - constant mean reversion factor of the asset price varianceθ - long-term level of asset price varianceσ - volatility of volatilityρ - correlation between Brownian motions
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AimsCalibration of Heston Model
Interval AnalysisSummary
Heston ModelCalibration function
Heston Model - a 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 ) = 12 + 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 + aσ2 [(bj − ρσφi + d)τ − 2ln( 1−gedr
1−g )]
D(T − t, φ) =bj−ρσφi+d
σ2 ( 1−edr1−gedr )
g =bj−ρσφi+dbj−ρσφi−d
d =√
(ρσφi − bj)2 − σ2(2ujφi − φ2)u1 = 1
2 , u2 = −12 , a = κθ, b1 = κ+ λ− ρσ, b2 = κ+ λ,
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AimsCalibration of Heston Model
Interval AnalysisSummary
Heston ModelCalibration function
Calibration
Minimize errors between prices predicted by the Heston model andthe market option data (different maturities, strike prices, options):
ri (ξ) = Ci (ξ)− Cmkti
Global optimization - Calibration of the Heston Model
The aim is to find the set of input-parameters
ξ = (V0, κ, θ, σ, ρ, µ), ξεX
that minimize an error function:
minξεX
G (ξ) = minξεX
√√√√Nquotes∑i=1
(Ci (ξ)− Cmkti )2
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AimsCalibration of Heston Model
Interval AnalysisSummary
Heston ModelCalibration function
Calibration
Minimize errors between prices predicted by the Heston model andthe market option data (different maturities, strike prices, options):
ri (ξ) = Ci (ξ)− Cmkti
Global optimization (the minmax form)
The aim is to find the set of input-parameters
ξ = (V0, κ, θ, σ, ρ, µ), ξεX
that:minξεX
maxiε1,...,Nquotes
|Ci (ξ)− Cmkti |
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AimsCalibration of Heston Model
Interval AnalysisSummary
Heston ModelCalibration function
Most optimization require the gradient and the Hessian:
ri (ξ) = Ci (ξ)− Cmkti
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 ) = [...]NxN
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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Interval arithmetic
Approach to putting bounds in mathematical computation andthus, 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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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Interval arithmetic
Approach to putting bounds in mathematical computation andthus, 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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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
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, 1y2
]
Conclusion: Even if interval operations are more computationally expensive, they take infinite spaces into account
(non differentiable functions). Many numerical algorithms using intervals may efficiently solve the mathematical
problems that are unsolved in real number arithmetics.
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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Reduction box techniques
Box consistency
Consider an equation f (x1, ..., xn) = 0 where x1εX1, ..., xnεXn.
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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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Reduction box techniques
Contractors
Property that allows us to set constraints for input parameterstaking into account bounds of the result.Consider f (x1, x2) where x1εX1 and x2εX2 and f (x) = y
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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Contractors - Forward/Backward propagation algorithm
If f (x) = exp(x) + sin(x) and xε[x ] .To obtain the exact domains for y = f (x) we use contractors:
forward propagation: backward propagation:[a1] = exp([x ]) [a2] = ([y ]− [a3]) ∩ [a2][a2] = sin([x ]) [a3] = ([y ]− [a2]) ∩ [a3]
[y ] = [a2] + [a3] [x ] = sin−1([a3]) ∩ [x ][x ] = log([a1]) ∩ [x ]
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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Forward/Backward algorithm + Automatic Differentiation
If f ′(x) = exp(x) + cos(x) and xε[x ] .To obtain the exact domains for 0 = f ′(x) we use contractors:
forward propagation: backward propagation:[a1] = exp([x ]) [a2] = (0− [a3]) ∩ [a2][a2] = cos([x ]) [a3] = (0− [a2]) ∩ [a3]
[y ] = [a2] + [a3] [x ] = cos−1([a3]) ∩ [x ][x ] = log([a1]) ∩ [x ]
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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Monotonicity/Concavity tests
Monotonicity test
Let ∇(X ) the interval gradient of a box X.If 0 /∈ ∇(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
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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Taylor’s expansions
Taylor series - the gradient
Consider a Taylor expansion of f around x (the middle of the box):
f (y)εf (x) +n∑
i=1
(yi − xi )gi (1)
Eliminate all y that:
f (y)εf (x) +n∑
i=1
(yi − xi )gi > f (2)
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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Taylor’s expansions
Taylor series - the Hessian
Consider a Taylor expansion of f around x (the middle of the box):
f (y)εf (x) + (y − x)Tg(x) +1
2H(x)(y − x) (3)
Eliminate all y that:
f (y)εf (x) + (y − x)Tg(x) +1
2H(x)(y − x) > f (4)
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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Newton Method - equation f(X)=0
From the mean value theorem:f (xu)− f (xl) = (xu − xl)f ′(ξ) and (xl < ξ < xu)
If xl is a zero of f then: xl = xu − f (xu)f ′(ξ)
If f ′(ξ)εf ′([xl , xu]) and m = mid([xl , xu]) then:
N([xl , xu]) = m − f (m)F ′([xl ,xu ]) ,
[xl , xu] = [xl , xu] ∩ N([xl , xu])20 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Newton Method - equation f(X)=0
From the mean value theorem:f (xu)− f (xl) = (xu − xl)f ′(ξ) and (xl < ξ < xu)
If xl is a zero of f then: xl = xu − f (xu)f ′(ξ)
This method can be extended to multi-dimensional case
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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
1 Begin with a box X in which the global minimum is sought
2 Use bid-ask spread to contract ξ to the conditionCi (ξ) = [Cmkt
i − bid ,Cmkti + ask]
3 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
4 If ξ = ∅ then go to Stage 2
5 Use bid-ask spread to contract ξ to the conditionCi (ξ) = [Cmkt
i − bid ,Cmkti + ask]
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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
1 Begin with a box X in which the global minimum is sought
2 Use bid-ask spread to contract ξ to the conditionCi (ξ) = [Cmkt
i − bid ,Cmkti + ask]
3 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
4 If ξ = ∅ then go to Stage 2
5 Use bid-ask spread to contract ξ to the conditionCi (ξ) = [Cmkt
i − bid ,Cmkti + ask]
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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
1 Begin with a box X in which the global minimum is sought
2 Use bid-ask spread to contract ξ to the conditionCi (ξ) = [Cmkt
i − bid ,Cmkti + ask]
3 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
4 If ξ = ∅ then go to Stage 2
5 Use bid-ask spread to contract ξ to the conditionCi (ξ) = [Cmkt
i − bid ,Cmkti + ask]
23 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
1 Begin with a box X in which the global minimum is sought
2 Use bid-ask spread to contract ξ to the conditionCi (ξ) = [Cmkt
i − bid ,Cmkti + ask]
3 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
4 If ξ = ∅ then go to Stage 2
5 Use bid-ask spread to contract ξ to the conditionCi (ξ) = [Cmkt
i − bid ,Cmkti + ask]
23 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
1 Begin with a box X in which the global minimum is sought
2 Use bid-ask spread to contract ξ to the conditionCi (ξ) = [Cmkt
i − bid ,Cmkti + ask]
3 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
4 If ξ = ∅ then go to Stage 2
5 Use bid-ask spread to contract ξ to the conditionCi (ξ) = [Cmkt
i − bid ,Cmkti + ask]
23 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
1 Begin with a box X in which the global minimum is sought
2 Use bid-ask spread to contract ξ to the conditionCi (ξ) = [Cmkt
i − bid ,Cmkti + ask]
3 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
4 If ξ = ∅ then go to Stage 2
5 Use bid-ask spread to contract ξ to the conditionCi (ξ) = [Cmkt
i − bid ,Cmkti + ask]
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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
7 Find minimum fsomemin of G (ξ) by using heuristic/gradientmethods
8 If interval contraction was not successful, fsomemin is globalminimum
9 Contract each |Ci (ξ)− Cmkti | = [0, fsomemin]
10 Evaluate the set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
11 Contract ∇G (ξ) = 0 or∑Nquotes
i=1 ∇ri (ξ) = 0
12 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
7 Find minimum fsomemin of G (ξ) by using heuristic/gradientmethods
8 If interval contraction was not successful, fsomemin is globalminimum
9 Contract each |Ci (ξ)− Cmkti | = [0, fsomemin]
10 Evaluate the set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
11 Contract ∇G (ξ) = 0 or∑Nquotes
i=1 ∇ri (ξ) = 0
12 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
7 Find minimum fsomemin of G (ξ) by using heuristic/gradientmethods
8 If interval contraction was not successful, fsomemin is globalminimum
9 Contract each |Ci (ξ)− Cmkti | = [0, fsomemin]
10 Evaluate the set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
11 Contract ∇G (ξ) = 0 or∑Nquotes
i=1 ∇ri (ξ) = 0
12 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
24 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
7 Find minimum fsomemin of G (ξ) by using heuristic/gradientmethods
8 If interval contraction was not successful, fsomemin is globalminimum
9 Contract each |Ci (ξ)− Cmkti | = [0, fsomemin]
10 Evaluate the set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
11 Contract ∇G (ξ) = 0 or∑Nquotes
i=1 ∇ri (ξ) = 0
12 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
24 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
7 Find minimum fsomemin of G (ξ) by using heuristic/gradientmethods
8 If interval contraction was not successful, fsomemin is globalminimum
9 Contract each |Ci (ξ)− Cmkti | = [0, fsomemin]
10 Evaluate the set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
11 Contract ∇G (ξ) = 0 or∑Nquotes
i=1 ∇ri (ξ) = 0
12 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
24 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
7 Find minimum fsomemin of G (ξ) by using heuristic/gradientmethods
8 If interval contraction was not successful, fsomemin is globalminimum
9 Contract each |Ci (ξ)− Cmkti | = [0, fsomemin]
10 Evaluate the set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
11 Contract ∇G (ξ) = 0 or∑Nquotes
i=1 ∇ri (ξ) = 0
12 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
24 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
13 Use a multidimensional interval Newton method for eachsub-box to solve (∇G (ξ) = 0)
14 Contract ξ to the condition Hii (ξ) < 0 or use Taylorexpansions
15 Eliminate sub-boxes whose lower bound > upper bound of thebox for which G(X) is the lowest
16 Choose the best sub-boxes
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AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
13 Use a multidimensional interval Newton method for eachsub-box to solve (∇G (ξ) = 0)
14 Contract ξ to the condition Hii (ξ) < 0 or use Taylorexpansions
15 Eliminate sub-boxes whose lower bound > upper bound of thebox for which G(X) is the lowest
16 Choose the best sub-boxes
25 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
13 Use a multidimensional interval Newton method for eachsub-box to solve (∇G (ξ) = 0)
14 Contract ξ to the condition Hii (ξ) < 0 or use Taylorexpansions
15 Eliminate sub-boxes whose lower bound > upper bound of thebox for which G(X) is the lowest
16 Choose the best sub-boxes
25 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
13 Use a multidimensional interval Newton method for eachsub-box to solve (∇G (ξ) = 0)
14 Contract ξ to the condition Hii (ξ) < 0 or use Taylorexpansions
15 Eliminate sub-boxes whose lower bound > upper bound of thebox for which G(X) is the lowest
16 Choose the best sub-boxes
25 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
16 Choose the best sub-boxes
Interval results of arithmetic operations are not always sharp(f (xεX ) ⊂ f (X ), but not always f (xεX ) ⊆ f (X )) !!!Conclusion:The sub-box whose error function is the lowest known value is notalways the best.
Glower (ξb1) < Glower (ξb2) < .... < Glower (ξbm)
Answer:Try to contract the difference between the best pair of thesub-boxes. If successful, eliminate all the other. 26 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
16 Choose the best sub-boxes
Interval results of arithmetic operations are not always sharp(f (xεX ) ⊂ f (X ), but not always f (xεX ) ⊆ f (X )) !!!Conclusion:The sub-box whose error function is the lowest known value is notalways the best.
Glower (ξb1) < Glower (ξb2) < .... < Glower (ξbm)
Answer:Try to contract the difference between the best pair of thesub-boxes. If successful, eliminate all the other. 26 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
16 Choose the best sub-boxes
Interval results of arithmetic operations are not always sharp(f (xεX ) ⊂ f (X ), but not always f (xεX ) ⊆ f (X )) !!!Conclusion:The sub-box whose error function is the lowest known value is notalways the best.
Glower (ξb1) < Glower (ξb2) < .... < Glower (ξbm)
Answer:Try to contract the difference between the best pair of thesub-boxes. If successful, eliminate all the other. 26 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
13 Use multidimensional Newton method for each sub-box tosolve (∇G (ξ) = 0)
14 Contract ξ to the condition Hii (ξ) < 0 or use Taylorexpansions
15 Eliminate sub-boxes whose lower bound > upper bound of thebox 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 upperbound of ε is less than error tolerance (for each dimension)
27 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
13 Use multidimensional Newton method for each sub-box tosolve (∇G (ξ) = 0)
14 Contract ξ to the condition Hii (ξ) < 0 or use Taylorexpansions
15 Eliminate sub-boxes whose lower bound > upper bound of thebox 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 upperbound of ε is less than error tolerance (for each dimension)
27 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach
13 Use multidimensional Newton method for each sub-box tosolve (∇G (ξ) = 0)
14 Contract ξ to the condition Hii (ξ) < 0 or use Taylorexpansions
15 Eliminate sub-boxes whose lower bound > upper bound of thebox 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 upperbound of ε is less than error tolerance (for each dimension)
27 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
IntroductionReduction box techniquesOverviewStage 1Stage 2Stage 3
Heston Calibration - Interval Approach1 Begin with a box X in which the global minimum is sought
2 Use bid-ask spread to contract ξ to the condition Ci (ξ) = [Cmkti − bid, Cmkt
i + ask]
3 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
4 If ξ = ∅ then go to Stage 2
5 Use bid-ask spread to contract ξ to the condition Ci (ξ) = [Cmkti − bid, Cmkt
i + ask]
6 Find minimum fsomemin of G(ξ) by using heuristic/gradient methods
7 If interval contraction was not successful, fsomemin is global minimum
8 Contract each |Ci (ξ)− Cmkti | = [0, fsomemin ]
9 Evaluate the set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
10 Contract ∇G(ξ) = 0 or∑Nquotes
i=1 ∇ri (ξ) = 0
11 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
12 Use multidimensional Newton method for each sub-box to solve (∇G(ξ) = 0)
13 Contract ξ to the condition Hii (ξ) < 0 or use Taylor expansions
14 Eliminate sub-boxes whose lower bound > upper bound of the box for which G(X) is the lowest
15 Choose the best sub-boxes
16 Evaluate the joint set: ξ = ξ1 ∩ ξ2 ∩ ... ∩ ξNquotes
17 Repeat 6-17 until the difference between lower and upper bound of ε is less than error tolerance (for eachdimension)
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AimsCalibration of Heston Model
Interval AnalysisSummary
QuestionsBibliography
Summary
The Gradient and the Hessian are evaluated by AutomaticDifferentiation routines
If some intermediate result has an infinite bound, useextended interval analysis and contractors
During interval contraction or Gauss-Newton method, theinitial box might be divided into many sub-boxes
All the sub-boxes can be processed in parallel (GPU, multicoreCPUs)
Application of the Pareto frontier algorithm for intervalsand additional contraction
29 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
QuestionsBibliography
Summary
The Gradient and the Hessian are evaluated by AutomaticDifferentiation routines
If some intermediate result has an infinite bound, useextended interval analysis and contractors
During interval contraction or Gauss-Newton method, theinitial box might be divided into many sub-boxes
All the sub-boxes can be processed in parallel (GPU, multicoreCPUs)
Application of the Pareto frontier algorithm for intervalsand additional contraction
29 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
QuestionsBibliography
Summary
The Gradient and the Hessian are evaluated by AutomaticDifferentiation routines
If some intermediate result has an infinite bound, useextended interval analysis and contractors
During interval contraction or Gauss-Newton method, theinitial box might be divided into many sub-boxes
All the sub-boxes can be processed in parallel (GPU, multicoreCPUs)
Application of the Pareto frontier algorithm for intervalsand additional contraction
29 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
QuestionsBibliography
Summary
The Gradient and the Hessian are evaluated by AutomaticDifferentiation routines
If some intermediate result has an infinite bound, useextended interval analysis and contractors
During interval contraction or Gauss-Newton method, theinitial box might be divided into many sub-boxes
All the sub-boxes can be processed in parallel (GPU, multicoreCPUs)
Application of the Pareto frontier algorithm for intervalsand additional contraction
29 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
QuestionsBibliography
Summary
The Gradient and the Hessian are evaluated by AutomaticDifferentiation routines
If some intermediate result has an infinite bound, useextended interval analysis and contractors
During interval contraction or Gauss-Newton method, theinitial box might be divided into many sub-boxes
All the sub-boxes can be processed in parallel (GPU, multicoreCPUs)
Application of the Pareto frontier algorithm for intervalsand additional contraction
29 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
QuestionsBibliography
Questions
Questions
30 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
QuestionsBibliography
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
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AimsCalibration of Heston Model
Interval AnalysisSummary
QuestionsBibliography
Thank You
Thank You very muchfor Your attention
32 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
QuestionsBibliography
Features
Intervals deal with:
Non-continuous and non-differentiable functions
Uncertainty of machine representation of numbers (outwardrounding)
Discretization errors
Analysis of the simulation results based on uncertaininput-parameters
Numerical problems that are unsolved in real-numbers (globaloptimization)
Interval results of operations are not always sharp(sharp bounds - NP-hard problem)
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AimsCalibration of Heston Model
Interval AnalysisSummary
QuestionsBibliography
Upper-bound test
Upper-bound test
1 Consider f (x1, x2) where x1εX1 and x2εX2
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) bycontracting 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 optimizationmethods for real numbers.
32 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
QuestionsBibliography
Upper-bound test
Upper-bound test
1 Consider f (x1, x2) where x1εX1 and x2εX2
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) bycontracting 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 optimizationmethods for real numbers.
32 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
QuestionsBibliography
Upper-bound test
Upper-bound test
1 Consider f (x1, x2) where x1εX1 and x2εX2
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) bycontracting 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 optimizationmethods for real numbers.
32 / 32
AimsCalibration of Heston Model
Interval AnalysisSummary
QuestionsBibliography
Upper-bound test
Upper-bound test
1 Consider f (x1, x2) where x1εX1 and x2εX2
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) bycontracting 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 optimizationmethods for real numbers.
32 / 32