Heston Model Multiscale Model Numerical Work A Fast Mean-Reverting Correction to Heston’s Stochastic Volatility Model Jean-Pierre Fouque 1 Matthew J Lorig 2 1 Department of Statistics & Applied Probability University of California - Santa Barbara www.physics.ucsb.edu/ ∼ mjlorig/ 2 Department of Physics University of California - Santa Barbara www.pstat.ucsb.edu/faculty/fouque/ WCMF, 2009 Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Multiscale Model Numerical Work Outline Heston Model 1 Motivation and Dynamics Why We like Heston Problems with Heston Multiscale Model 2 Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing Numerical Work 3 Multiscale Implied Volatility Surface Multiscale Fit to Data Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation and Dynamics Multiscale Model Why We like Heston Numerical Work Problems with Heston Outline Heston Model 1 Motivation and Dynamics Why We like Heston Problems with Heston Multiscale Model 2 Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing Numerical Work 3 Multiscale Implied Volatility Surface Multiscale Fit to Data Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation and Dynamics Multiscale Model Why We like Heston Numerical Work Problems with Heston Volatility Not Constant Daily Log Returns 0.04 S&P500 Simulated GBM 0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1 1950 1960 1970 1980 1990 2000 Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation and Dynamics Multiscale Model Why We like Heston Numerical Work Problems with Heston Heston Under Risk-Neutral Measure Motivated by notion that volatility not constant √ Z t X t dW x dX t = rX t dt + t √ Z t dW z dZ t = 휅 ( 휃 − Z t ) dt + 휎 t d ⟨ W x , W z ⟩ t = 휌 dt One-factor stochastic volatility model Square of volatility, Z t , follows CIR process Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation and Dynamics Multiscale Model Why We like Heston Numerical Work Problems with Heston Outline Heston Model 1 Motivation and Dynamics Why We like Heston Problems with Heston Multiscale Model 2 Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing Numerical Work 3 Multiscale Implied Volatility Surface Multiscale Fit to Data Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation and Dynamics Multiscale Model Why We like Heston Numerical Work Problems with Heston Formulas! Explicit formulas for European options: ∫ P H ( t , x , z ) = e − r 휏 1 e − ikq ˆ G ( 휏, k , z ) ˆ h ( k ) dk 2 휋 q ( t , x ) = r ( T − t ) + log x , ∫ e ikq h ( e q ) dq , h ( k ) = ˆ G ( 휏, k , z ) = e C ( 휏, k )+ zD ( 휏, k ) ˆ . . . C ( 휏, k ) and D ( 휏, k ) solve ODE’s in 휏 = T − t . Note: audience tunes out if you put too many equations on a slide Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation and Dynamics Multiscale Model Why We like Heston Numerical Work Problems with Heston Pretty Pictures! 2.0 T 1.5 1.0 0.5 0.25 0.20 Σ 0.15 0.10 100 K 150 Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation and Dynamics Multiscale Model Why We like Heston Numerical Work Problems with Heston Pretty Pictures Explained Implied volatility 휎 Imp ( T , K ) defined by P BS ( 휎 Imp ( T , K )) = P ( T , K ) P is price of option with strike K and expiration T Heston captures well-documented features of implied volatility surface: smile and skew Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation and Dynamics Multiscale Model Why We like Heston Numerical Work Problems with Heston Outline Heston Model 1 Motivation and Dynamics Why We like Heston Problems with Heston Multiscale Model 2 Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing Numerical Work 3 Multiscale Implied Volatility Surface Multiscale Fit to Data Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation and Dynamics Multiscale Model Why We like Heston Numerical Work Problems with Heston Captures Some . . . Not All Features of Smile Days to Maturity = 583 0.19 Market Data Heston Fit 0.18 Misprices 0.17 far ITM Implied Volatility 0.16 and OTM European 0.15 options [5] 0.14 [12] 0.13 0.12 0.11 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 log(K/x) Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation and Dynamics Multiscale Model Why We like Heston Numerical Work Problems with Heston Simultaneous Fit Across Expirations Is Poor Days to Maturity = 65 0.16 Market Data Heston Fit 0.15 Particular 0.14 difficulty Implied Volatility fitting short 0.13 expirations [7] 0.12 0.11 0.1 −0.1 −0.05 0 0.05 0.1 0.15 log(K/x) Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation for Multiple Time Scales Multiscale Model Multiscale Dynamics Numerical Work Option Pricing Outline Heston Model 1 Motivation and Dynamics Why We like Heston Problems with Heston Multiscale Model 2 Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing Numerical Work 3 Multiscale Implied Volatility Surface Multiscale Fit to Data Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation for Multiple Time Scales Multiscale Model Multiscale Dynamics Numerical Work Option Pricing What’s Wrong with Heston? Single factor of volatility running on single time scale not sufficient to describe dynamics of the volatility process. Not just Heston . . . Any one-factor stochastic volatility model has trouble fitting implied volatility levels across all strikes and maturities [7] Empirical evidence suggests existence of several stochastic volatility factors running on different time scales Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation for Multiple Time Scales Multiscale Model Multiscale Dynamics Numerical Work Option Pricing Evidence [1] [2] [3] [4] [6] [8] [9] [10] [11] Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation for Multiple Time Scales Multiscale Model Multiscale Dynamics Numerical Work Option Pricing Outline Heston Model 1 Motivation and Dynamics Why We like Heston Problems with Heston Multiscale Model 2 Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing Numerical Work 3 Multiscale Implied Volatility Surface Multiscale Fit to Data Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation for Multiple Time Scales Multiscale Model Multiscale Dynamics Numerical Work Option Pricing Multiscale Under Risk-Neutral Measure √ Z t f ( Y t ) X t dW x dX t = rX t dt + t √ dY t = Z t Z t √ 휖 dW y 휖 ( m − Y t ) dt + 휈 2 t √ Z t dW z dZ t = 휅 ( 휃 − Z t ) dt + 휎 t 〈 W i , W j 〉 d t = 휌 ij dt i , j ∈ { x , y , z } Volatility controlled by product √ Z t f ( Y t ) Y t modeled as OU process running on time-scale 휖/ Z t Note: f ( y ) = 1 ⇒ model reduces to Heston Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation for Multiple Time Scales Multiscale Model Multiscale Dynamics Numerical Work Option Pricing Outline Heston Model 1 Motivation and Dynamics Why We like Heston Problems with Heston Multiscale Model 2 Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing Numerical Work 3 Multiscale Implied Volatility Surface Multiscale Fit to Data Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation for Multiple Time Scales Multiscale Model Multiscale Dynamics Numerical Work Option Pricing Option Pricing PDE Price of European Option Expressed as � [ ] � e − r ( T − t ) h ( X T ) P t = 피 � X t , Y t , Z t =: P 휖 ( t , X t , Y t , Z t ) Using Feynman-Kac, derive following PDE for P 휖 ℒ 휖 P 휖 ( t , x , y , z ) = 0 , ℒ 휖 = ∂ ∂ t + ℒ ( X , Y , Z ) − r , P 휖 ( T , x , y , z ) = h ( x ) Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation for Multiple Time Scales Multiscale Model Multiscale Dynamics Numerical Work Option Pricing Some Book-Keeping of ℒ 휖 ℒ 휖 has convenient form ℒ 휖 = z 휖 ℒ 0 + z √ 휖 ℒ 1 + ℒ 2 , ℒ 0 = 휈 2 ∂ 2 ∂ y 2 + ( m − y ) ∂ ∂ y √ √ ∂ 2 ∂ 2 2 f ( y ) x ℒ 1 = 휌 yz 휎휈 2 ∂ y ∂ z + 휌 xy 휈 ∂ x ∂ y ( ) 2 f 2 ( y ) zx 2 ∂ 2 ℒ 2 = ∂ ∂ t + 1 x ∂ ∂ x 2 + r ∂ x − ⋅ 2 휎 2 z ∂ 2 ∂ 2 ∂ z 2 + 휅 ( 휃 − z ) ∂ + 1 ∂ z + 휌 xz 휎 f ( y ) zx ∂ x ∂ z . Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
Heston Model Motivation for Multiple Time Scales Multiscale Model Multiscale Dynamics Numerical Work Option Pricing Perturbative Solution PDE has no analytic solution for general f ( y ) Perform singular perturbation with respect to 휖 P 휖 = P 0 + √ 휖 P 1 + 휖 P 2 + . . . Turns out P 0 and P 1 functions of t , x , and z only Find P 0 ( t , x , z ) = P H ( t , x , z ) with effective correlation 휌 → 휌 xz ⟨ f ⟩ Jean-Pierre Fouque, Matthew J Lorig Heston 2.0
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