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 1 / 30
Aims Calibration of Heston Model Interval Analysis Interval based algorithm for Heston calibration Summary Agenda Aims 1 Calibration of Heston Model 2 Interval Analysis 3 Interval based algorithm for Heston calibration 4 Summary 5 2 / 30
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. 3 / 30
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. 3 / 30
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 : R n → R : minimize f ( x ) x subject to: h i ( x ) ≤ a i , i = 1 , . . . , n . g j ( x ) ≤ b j , j = 1 , . . . , m . 3 / 30
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 4 / 30
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!!! 5 / 30
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 6 / 30
Aims Calibration of Heston Model Heston Model Interval Analysis Calibration function Interval based algorithm for Heston calibration Summary Heston Model Heston Stochastic Volatility Model The dynamics of the Heston model can be described as follows: � V t S t dZ 1 dS t = µ S t dt + t � V t dZ 2 dV t = κ ( θ − V t ) dt + σ t dZ 1 t dZ 2 t = ρ dt S t asset price at time t µ - constant drift of the asset price dt - time increment V t - 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 Heston Model Interval Analysis Calibration function Interval based algorithm for Heston calibration Summary Heston Model - semi-closed form solution Solution of Heston PDE is: C ( S , V , t , ξ ) = SP 1 − e − rT KP 2 where ξ = ( V 0 , κ, θ, σ, ρ, µ ) and: � ∞ 0 Re ( e − i φ ln ( K ) f j ( S , V , t ,ξ ) P j ( S , V , t , ξ, K ) = 1 2 + 1 ) d φ π i φ f j ( S , V , t , ξ ) = e C ( T − t ,φ )+ D ( T − t ,φ ) V t + i φ ln ( S t ) ) σ 2 [( b j − ρσφ i + d ) τ − 2 ln ( 1 − ge dr C ( T − t , φ ) = r φ ir + a 1 − g )] D ( T − t , φ ) = b j − ρσφ i + d ( 1 − e dr 1 − ge dr ) σ 2 g = b j − ρσφ i + d b j − ρσφ i − d ( ρσφ i − b j ) 2 − σ 2 (2 u j φ i − φ 2 ) � d = u 1 = 1 2 , u 2 = − 1 2 , a = κθ , b 1 = κ + λ − ρσ , b 2 = κ + λ , 8 / 30
Aims Calibration of Heston Model Heston Model Interval Analysis Calibration function Interval based algorithm for Heston calibration Summary Calibration Minimize errors between prices predicted by the Heston model and the market option data (different maturities, strike prices, options): r i ( ξ ) = C i ( ξ ) − C mkt i Global optimization - Calibration of the Heston Model The aim is to find the set of input-parameters ξ = ( V 0 , κ, θ, σ, ρ, µ ) , X = { ξ : ξǫ X } that minimize an error function: � N quotes � � � ( C i ( ξ ) − C mkt ) 2 min = G ( ξ ) = � i ξ i =1 9 / 30
Aims Calibration of Heston Model Heston Model Interval Analysis Calibration function Interval based algorithm for Heston calibration Summary Most optimization require the gradient and the Hessian: r i ( ξ ) = C i ( ξ ) − C mkt i the gradient: 2 � N quotes r i ( ξ ) ∗ dr i ( ξ ) i =1 dV 0 2 � N quotes r i ( ξ ) ∗ dr i ( ξ ) i =1 d κ N quotes 2 � N quotes r i ( ξ ) ∗ dr i ( ξ ) � i =1 d θ ∇ G ( ξ ) = 2 r i ( ξ ) ∇ r i ( ξ ) = = [0] 2 � N quotes r i ( ξ ) ∗ dr i ( ξ ) i =1 d σ i =1 2 � N quotes r i ( ξ ) ∗ dr i ( ξ ) i =1 d ρ 2 � N quotes r i ( ξ ) ∗ dr i ( ξ ) i =1 d µ the Hessian: N quotes ∇ 2 G ( ξ ) = 2 � ( r i ( ξ ) ∇ 2 r i ( ξ ) + ∇ r i ( ξ ) ∇ r i ( ξ ) T ) i =1 10 / 30
Aims Calibration of Heston Model Introduction Interval Analysis Reduction box techniques Interval based algorithm for Heston calibration Summary 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 [ x l , x u ], where x l < x u Interval operators are inclusion isotonic ( Z ⊂ X ) = > f ( Z ) ⊂ f ( X ) Optimization - reduction of an input space 11 / 30
Aims Calibration of Heston Model Introduction Interval Analysis Reduction box techniques Interval based algorithm for Heston calibration Summary 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 [ x l , x u ], where x l < x u Interval operators are inclusion isotonic ( Z ⊂ X ) = > f ( Z ) ⊂ f ( X ) Optimization - reduction of an input space 11 / 30
Aims Calibration of Heston Model Introduction Interval Analysis Reduction box techniques Interval based algorithm for Heston calibration Summary Interval operators Each single real-number X is represented as a pair of [ x 1 , x 2 ], where x 1 < x 2 Elementary arithmetic operations: [ x 1 , x 2 ] + [ y 1 , y 2 ] = [ x 1 + y 1 , x 2 + y 2 ] [ x 1 , x 2 ] − [ y 1 , y 2 ] = [ x 1 + y 2 , x 2 + y 1 ] [ x 1 , x 2 ] ∗ [ y 1 , y 2 ] = [ min ( x 1 y 1 , x 1 y 2 , x 2 y 1 , x 2 y 2 ) , max ( x 1 y 1 , x 1 y 2 , x 2 y 1 , x 2 y 2 )] [ x 1 , x 2 ] [ y 1 , y 2 ] = [ x 1 , x 2 ] ∗ [ 1 y 1 , 1 y 2 ] 12 / 30
Aims Calibration of Heston Model Introduction Interval Analysis Reduction box techniques Interval based algorithm for Heston calibration Summary 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 optimization) Interval results of operations are not always sharp (sharp bounds - NP-hard problem) 13 / 30
Aims Calibration of Heston Model Introduction Interval Analysis Reduction box techniques Interval based algorithm for Heston calibration Summary Reduction box techniques Box consistency Consider an equation f ( x 1 , ..., x n ) = 0 where x 1 ǫ X 1 , ..., x n ǫ X n . Replace all the variables except the i-th by intervals: q ( x i ) = f ( X 1 , ... X i − 1 , x i , X i +1 , ... X n ) = 0 Eliminate subsets of X i not satisfying q ( x i ) = 0 14 / 30
Aims Calibration of Heston Model Introduction Interval Analysis Reduction box techniques Interval based algorithm for Heston calibration Summary Reduction box techniques Contractors Property that allows us to set constraints for input parameters taking into account bounds of the result. Consider f ( x 1 , x 2 ) where x 1 ǫ X 1 and x 2 ǫ X 2 and f ( x ) = ¯ y 15 / 30
Aims Calibration of Heston Model Introduction Interval Analysis Reduction box techniques Interval based algorithm for Heston calibration Summary Contractors - Forward/Backward propagation algorithm If f ( x 1 , x 2 ) = exp ( x 1 ) + sin ( x 2 ) and x 1 ǫ [ x 1 ] x 2 ǫ [ x 2 ]. To obtain the exact domains for ¯ y = f ( x ) we use contractors: forward propagation: backward propagation: [ a 1 ] = exp ([ x 1 ]) [ a 2 ] = ([¯ y ] − [ a 3 ]) ∩ [ a 2 ] y ] − [ a 2 ]) ∩ [ a 3 ] [ a 2 ] = sin ([ x 2 ]) [ a 3 ] = ([¯ [ x 2 ] = sin − 1 ([ a 3 ]) ∩ [ x 2 ] [ y ] = [ a 2 ] + [ a 3 ] [ x 1 ] = log ([ a 1 ]) ∩ [ x 1 ] 16 / 30
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