Multiscale Stochastic Volatility Models Heston 1 Multiscale Stochastic Volatility Models Heston 1.5

  • View
    0

  • Download
    0

Embed Size (px)

Text of Multiscale Stochastic Volatility Models Heston 1 Multiscale Stochastic Volatility Models Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    Multiscale Stochastic Volatility Models

    Heston 1.5

    Jean-Pierre Fouque

    Department of Statistics & Applied Probability University of California Santa Barbara

    Modeling and Managing Financial Risks

    Paris, January 10-13, 2011

    Jean-Pierre Fouque Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    References:

    Multiscale Stochastic Volatility for Equity, Interest-Rate

    and Credit Derivatives

    J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Sølna Cambridge University Press (in press).

    A Fast Mean-Reverting Correction to Heston Stochastic

    Volatility Model

    J.-P. Fouque and M. Lorig SIAM Journal on Financial Mathematics (to appear).

    Jean-Pierre Fouque Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    Outline 1 Multiscale Models

    Motivation for Multiple Time Scales Multiscale Dynamics Singular-Regular Perturbations: Black-Scholes 2.0

    2 Heston Model Dynamics Why We like Heston Problems with Heston

    3 Perturbation around Heston Heston + Fast Factor Option Pricing

    4 Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

    Jean-Pierre Fouque Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    Motivation for Multiple Time Scales Multiscale Dynamics Singular-Regular Perturbations: Black-Scholes 2.0

    Outline 1 Multiscale Models

    Motivation for Multiple Time Scales Multiscale Dynamics Singular-Regular Perturbations: Black-Scholes 2.0

    2 Heston Model Dynamics Why We like Heston Problems with Heston

    3 Perturbation around Heston Heston + Fast Factor Option Pricing

    4 Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

    Jean-Pierre Fouque Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    Motivation for Multiple Time Scales Multiscale Dynamics Singular-Regular Perturbations: Black-Scholes 2.0

    Volatility Not Constant

    1950 1960 1970 1980 1990 2000 −0.1

    −0.08

    −0.06

    −0.04

    −0.02

    0

    0.02

    0.04 Daily Log Returns

    S&P500 Simulated GBM

    Jean-Pierre Fouque Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    Motivation for Multiple Time Scales Multiscale Dynamics Singular-Regular Perturbations: Black-Scholes 2.0

    Skews of Implied Volatilities (SP500 on June 1, 2007)

    0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.05

    0.1

    0.15

    0.2

    0.25

    0.3

    0.35

    0.4

    0.45

    Moneyness=K/X

    Im pl

    ie d

    V ol

    at ili

    ty Implied Volatilities on 1 June 2007

    1 mo 2 mo 3 mo 6 mo 9 mo 12 mo 18 mo

    Jean-Pierre Fouque Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    Motivation for Multiple Time Scales Multiscale Dynamics Singular-Regular Perturbations: Black-Scholes 2.0

    What’s Wrong with Heston?

    Single factor of volatility running on single time scale not sufficient to describe dynamics of the volatilty process.

    Not just Heston . . . Any one-factor stochastic volatility model has trouble fitting implied volatility levels accross all strikes and maturities.

    Emperical evidence suggests existence of several stochastic volatility factors running on different time scales

    −→

    Jean-Pierre Fouque Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    Motivation for Multiple Time Scales Multiscale Dynamics Singular-Regular Perturbations: Black-Scholes 2.0

    Evidence

    Melino-Turnbull 1990, Dacorogna et al. 1997, Andersen-Bollerslev 1997,

    Fouque-Papanicolaou-Sircar 2000,

    Alizdeh-Brandt-Diebold 2001, Engle-Patton 2001, Lebaron 2001,

    Chernov-Gallant-Ghysels-Tauchen 2003,

    Fouque-Papanicolaou-Sircar-Solna 2003, Hillebrand 2003, Gatheral 2006,

    Christoffersen-Jacobs-Ornthanalai-Wang 2008.

    Jean-Pierre Fouque Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    Motivation for Multiple Time Scales Multiscale Dynamics Singular-Regular Perturbations: Black-Scholes 2.0

    Outline 1 Multiscale Models

    Motivation for Multiple Time Scales Multiscale Dynamics Singular-Regular Perturbations: Black-Scholes 2.0

    2 Heston Model Dynamics Why We like Heston Problems with Heston

    3 Perturbation around Heston Heston + Fast Factor Option Pricing

    4 Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

    Jean-Pierre Fouque Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    Motivation for Multiple Time Scales Multiscale Dynamics Singular-Regular Perturbations: Black-Scholes 2.0

    Multiscale SV Under Risk-Neutral

    dXt = rXt dt + f (Yt ,Zt)Xt dW (0)⋆ t ,

    dYt =

    ( 1

    ε α(Yt) −

    1√ ε β(Yt)Λ1(Yt ,Zt)

    ) dt +

    1√ ε β(Yt) dW

    (1)⋆ t ,

    dZt = ( δc(Zt) −

    √ δ g(Zt)Λ2(Yt ,Zt)

    ) dt +

    √ δ g(Zt) dW

    (2)⋆ t .

    Time scales: ε≪ 1 ≪ 1/δ Fast factor: Y Slow factor: Z Option pricing:

    Pε,δ(t,Xt ,Yt ,Zt) = IE ⋆ {

    e−r(T−t)h(XT ) | Xt ,Yt ,Zt } ,

    ∂Pε,δ

    ∂t + L(X ,Y ,Z)Pε,δ − rPε,δ = 0.

    Jean-Pierre Fouque Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    Motivation for Multiple Time Scales Multiscale Dynamics Singular-Regular Perturbations: Black-Scholes 2.0

    Outline 1 Multiscale Models

    Motivation for Multiple Time Scales Multiscale Dynamics Singular-Regular Perturbations: Black-Scholes 2.0

    2 Heston Model Dynamics Why We like Heston Problems with Heston

    3 Perturbation around Heston Heston + Fast Factor Option Pricing

    4 Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

    Jean-Pierre Fouque Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    Motivation for Multiple Time Scales Multiscale Dynamics Singular-Regular Perturbations: Black-Scholes 2.0

    Expand

    Pε,δ = ∑

    i≥0

    j≥0

    ( √ ε)i (

    √ δ)j Pi ,j ,

    P0 = PBS computed at volatility σ(z):

    σ̄2(z) = 〈f 2(·, z)〉 = ∫

    f 2(y , z)ΦY (dy).

    √ εP1,0 = V

    ε 2 x

    2 ∂ 2PBS ∂x2

    + V ε3 x ∂

    ∂x

    ( x2 ∂2PBS ∂x2

    ) ,

    √ δ P0,1 = V

    δ 0

    ∂PBS ∂σ

    + V δ1 ∂

    ∂x

    ( ∂PBS ∂σ

    ) .

    In terms of implied volatility:

    I ≈ ( b⋆ + τbδ

    ) +

    ( aε + τaδ

    ) log(K/x) τ

    , τ = T − t.

    Jean-Pierre Fouque Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    Motivation for Multiple Time Scales Multiscale Dynamics Singular-Regular Perturbations: Black-Scholes 2.0

    Time Scales

    The previous asymptotics is a perturbation around Black-Scholes leading to Black-Scholes 2.0 where 2.0 = 1+ 2x(0.5)

    Why not keeping one volatility factor on a time scale of order one? This would be a perturbation around a one-factor stochastic volatility model, but which one?

    Heston of course!

    Jean-Pierre Fouque Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    Dynamics Why We like Heston Problems with Heston

    Outline 1 Multiscale Models

    Motivation for Multiple Time Scales Multiscale Dynamics Singular-Regular Perturbations: Black-Scholes 2.0

    2 Heston Model Dynamics Why We like Heston Problems with Heston

    3 Perturbation around Heston Heston + Fast Factor Option Pricing

    4 Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

    Jean-Pierre Fouque Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    Dynamics Why We like Heston Problems with Heston

    Heston Under Risk-Neutral Measure

    dXt = rXtdt + √

    Zt XtdW x t

    dZt = κ(θ − Zt)dt + σ √

    Zt dW z t

    d 〈W x ,W z〉t = ρdt

    One-factor stochastic volatility model

    Square of volatility, Zt , follows CIR process

    Jean-Pierre Fouque Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    Dynamics Why We like Heston Problems with Heston

    Outline 1 Multiscale Models

    Motivation for Multiple Time Scales Multiscale Dynamics Singular-Regular Perturbations: Black-Scholes 2.0

    2 Heston Model Dynamics Why We like Heston Problems with Heston

    3 Perturbation around Heston Heston + Fast Factor Option Pricing

    4 Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

    Jean-Pierre Fouque Heston 1.5

  • Multiscale Models Heston Model

    Perturbation around Heston Numerical Work

    Dynamics Why We