A Fast Mean-Reverting Correction to Hestons Stochastic Volatility - - PowerPoint PPT Presentation

Heston Model Multiscale Model Numerical Work

A Fast Mean-Reverting Correction to Heston’s Stochastic Volatility Model

Jean-Pierre Fouque1 Matthew J Lorig2

1Department of Statistics & Applied Probability

University of California - Santa Barbara www.physics.ucsb.edu/∼mjlorig/

2Department 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

1

Heston Model Motivation and Dynamics Why We like Heston Problems with Heston

2

Multiscale Model Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

3

Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation and Dynamics Why We like Heston Problems with Heston

Outline

1

Heston Model Motivation and Dynamics Why We like Heston Problems with Heston

2

Multiscale Model Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

3

Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation and Dynamics Why We like Heston Problems with Heston

Volatility Not Constant

1950 1960 1970 1980 1990 2000 −0.1 −0.08 −0.06 −0.04 −0.02 0.02 0.04 Daily Log Returns S&P500 Simulated GBM

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation and Dynamics Why We like Heston Problems with Heston

Heston Under Risk-Neutral Measure

Motivated by notion that volatility not constant dXt = rXtdt + √ ZtXtdW x

t

dZt = 휅(휃 − Zt)dt + 휎 √ Zt dW z

t

d ⟨W x, W z⟩t = 휌dt One-factor stochastic volatility model Square of volatility, Zt, follows CIR process

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation and Dynamics Why We like Heston Problems with Heston

Outline

1

Heston Model Motivation and Dynamics Why We like Heston Problems with Heston

2

Multiscale Model Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

3

Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation and Dynamics Why We like Heston Problems with Heston

Formulas!

Explicit formulas for European options: PH(t, x, z) = e−r휏 1 2휋 ∫ e−ikq ˆ G(휏, k, z)ˆ h(k)dk q(t, x) = r(T − t) + log x, ˆ h(k) = ∫ eikqh(eq)dq, ˆ G(휏, k, z) = eC(휏,k)+zD(휏,k) . . . 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 Multiscale Model Numerical Work Motivation and Dynamics Why We like Heston Problems with Heston

Pretty Pictures!

100 150 K 0.5 1.0 1.5 2.0 T 0.10 0.15 0.20 0.25 Σ Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation and Dynamics Why We like Heston Problems with Heston

Pretty Pictures Explained

Implied volatility 휎Imp(T, K) defined by PBS(휎Imp(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 Multiscale Model Numerical Work Motivation and Dynamics Why We like Heston Problems with Heston

Outline

1

Heston Model Motivation and Dynamics Why We like Heston Problems with Heston

2

Multiscale Model Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

3

Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation and Dynamics Why We like Heston Problems with Heston

Captures Some . . . Not All Features of Smile

Misprices far ITM and OTM European

• ptions [5]

[12]

−0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 Days to Maturity = 583 log(K/x) Implied Volatility Market Data Heston Fit Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation and Dynamics Why We like Heston Problems with Heston

Simultaneous Fit Across Expirations Is Poor

Particular difficulty fitting short expirations [7]

−0.1 −0.05 0.05 0.1 0.15 0.1 0.11 0.12 0.13 0.14 0.15 0.16 Days to Maturity = 65 log(K/x) Implied Volatility Market Data Heston Fit Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

Outline

1

Heston Model Motivation and Dynamics Why We like Heston Problems with Heston

2

Multiscale Model Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

3

Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation for Multiple Time Scales Multiscale Dynamics 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 Multiscale Model Numerical Work Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

Evidence

[1] [2] [3] [4] [6] [8] [9] [10] [11]

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

Outline

1

Heston Model Motivation and Dynamics Why We like Heston Problems with Heston

2

Multiscale Model Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

3

Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

Multiscale Under Risk-Neutral Measure

dXt = rXtdt + √ Zt f(Yt)XtdW x

t

dYt = Zt 휖 (m − Yt)dt + 휈 √ 2 √ Zt 휖 dW y

t

dZt = 휅(휃 − Zt)dt + 휎 √ Zt dW z

t

d 〈 W i, W j〉

t = 휌ijdt

i, j ∈ {x, y, z} Volatility controlled by product √Zt f(Yt) Yt modeled as OU process running on time-scale 휖/Zt Note: f(y) = 1 ⇒ model reduces to Heston

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

Outline

1

Heston Model Motivation and Dynamics Why We like Heston Problems with Heston

2

Multiscale Model Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

3

Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

Option Pricing PDE

Price of European Option Expressed as Pt = 피 [ e−r(T−t)h(XT)

• Xt, Yt, Zt

] =: P휖(t, Xt, Yt, Zt) Using Feynman-Kac, derive following PDE for P휖 ℒ휖P휖(t, x, y, z) = 0, ℒ휖 = ∂ ∂t + ℒ(X,Y,Z) − r , P휖(T, x, y, z) = h(x)

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

Some Book-Keeping of ℒ휖

ℒ휖 has convenient form ℒ휖 = z 휖 ℒ0 + z √휖ℒ1 + ℒ2, ℒ0 = 휈2 ∂2 ∂y2 + (m − y) ∂ ∂y ℒ1 = 휌yz휎휈 √ 2 ∂2 ∂y∂z + 휌xy휈 √ 2 f(y)x ∂2 ∂x∂y ℒ2 = ∂ ∂t + 1 2f 2(y)zx2 ∂2 ∂x2 + r ( x ∂ ∂x − ⋅ ) + 1 2휎2z ∂2 ∂z2 + 휅(휃 − z) ∂ ∂z + 휌xz휎f(y)zx ∂2 ∂x∂z .

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

Perturbative Solution

PDE has no analytic solution for general f(y) Perform singular perturbation with respect to 휖 P휖 = P0 + √휖P1 + 휖P2 + . . . Turns out P0 and P1 functions of t, x, and z only Find P0(t, x, z) = PH(t, x, z) with effective correlation 휌 → 휌xz ⟨f⟩

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

More Formulas!

P1(t, x, z) = e−r휏 2휋 ∫ e−ikq( 휅휃ˆ f0(휏, k) + zˆ f1(휏, k) ) × ˆ G(휏, k, z)ˆ h(k)dk, ˆ f0(휏, k) = ∫ 휏 ˆ f1(t, k)dt, ˆ f1(휏, k) = ∫ 휏 b(s, k)eA(휏,k,s)ds. b(휏, k) = − ( V1D(휏, k) ( −k2 + ik ) + V2D2(휏, k) (−ik) +V3 ( ik3 + k2) + V4D(휏, k) ( −k2)) . A(휏, k, s) solves ODE in 휏

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Outline

1

Heston Model Motivation and Dynamics Why We like Heston Problems with Heston

2

Multiscale Model Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

3

Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Group Parameters

P1 is linear function of four constants V1 = 휌yz휎휈 √ 2 〈 휙′〉 , V2 = 휌xz휌yz휎2휈 √ 2 〈 휓′〉 , V3 = 휌xy휈 √ 2 〈 f휙′〉 , V4 = 휌xy휌xz휎휈 √ 2 〈 f휓′〉 . 휓(y) and 휙(y) solve Poisson equations Each Vi has unique effect on implied volatility

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Effect of V1 and V2 on Implied Volatility

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.12 0.14 0.16 0.18 0.2 0.22 0.24 log(K/x) σimp V1 Effect −0.02 −0.01 0.005 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.12 0.14 0.16 0.18 0.2 0.22 log(K/x) σimp V2 Effect −0.02 −0.01 0.005

Vi = 0 corresponds to Heston

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Effect of V3 and V4 on Implied Volatility

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.12 0.14 0.16 0.18 0.2 0.22 0.24 log(K/x) σimp V3 Effect −0.02 −0.01 0.005 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.1 0.15 0.2 0.25 log(K/x) σimp V4 Effect −0.02 −0.01 0.005

Vi = 0 corresponds to Heston

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Outline

1

Heston Model Motivation and Dynamics Why We like Heston Problems with Heston

2

Multiscale Model Motivation for Multiple Time Scales Multiscale Dynamics Option Pricing

3

Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Captures More Features of Smile

Better fit for far ITM and OTM European

• ptions

583 days to maturity

−0.15 −0.1 −0.05 0.05 0.1 0.15 0.2 0.1 0.11 0.12 0.13 0.14 0.15 0.16 log(K/x) Implied Volatility Market Data Heston Fit Multiscale Fit Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Simultaneous Fit Across Expirations Is Improved

Vast im- provement for short expirations 65 days to maturity

−0.1 −0.05 0.05 0.1 0.15 0.1 0.11 0.12 0.13 0.14 0.15 0.16 log(K/x) Implied Volatility Market Data Heston Fit Multiscale Fit Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Heston Model Multiscale Model Numerical Work Multiscale Implied Volatility Surface Multiscale Fit to Data

Summary

Heston model provides easy way to calculate option prices in stochastic volatility setting, but fails to capture some features of implied volatility surface Multiscale model offers improved fit to implied volatility surface while maintaining convenience of option pricing formulas

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Appendix For Further Reading

For Further Reading I

Sassan Alizadeh, Michael W. Brandt, and Francis X. Diebold. Range-Based Estimation of Stochastic Volatility Models. SSRN eLibrary, 2001. Torben G. Andersen and Tim Bollerslev. Intraday periodicity and volatility persistence in financial markets. Journal of Empirical Finance, 4(2-3):115–158, June 1997. Mikhail Chernov, A. Ronald Gallant, Eric Ghysels, and George Tauchen. Alternative models for stock price dynamics. Journal of Econometrics, 116(1-2):225–257, 2003.

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Appendix For Further Reading

For Further Reading II

Robert F. Engle and Andrew J. Patton. What good is a volatility model? 2008. Gabriele Fiorentini, Angel Leon, and Gonzalo Rubio. Estimation and empirical performance of heston’s stochastic volatility model: the case of a thinly traded market. Journal of Empirical Finance, 9(2):225–255, March 2002. Jean-Pierre Fouque, George Papanicolaou, Ronnie Sircar, and Knut Solna. Short time-scale in sp500 volatility. The Journal of Computational Finance, 6(4), 2003.

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Appendix For Further Reading

For Further Reading III

Jim Gatheral. The Volatility Surface: a Practitioner’s Guide. John Wiley and Sons, Inc., 2006. Eric Hillebrand. Overlaying time scales and persistence estimation in garch(1,1) models. Econometrics 0301003, EconWPA, January 2003. Blake D. Lebaron. Stochastic Volatility as a Simple Generator of Financial Power-Laws and Long Memory. SSRN eLibrary, 2001.

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Appendix For Further Reading

For Further Reading IV

Angelo Melino and Stuart M. Turnbull. Pricing foreign currency options with stochastic volatility. Journal of Econometrics, 45(1-2):239–265, 1990. Ulrich A. Muller, Michel M. Dacorogna, Rakhal D. Dave, Richard B. Olsen, Olivier V. Pictet, and Jacob E. von Weizsacker. Volatilities of different time resolutions – analyzing the dynamics of market components. Journal of Empirical Finance, 4(2-3):213–239, June 1997.

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0

Appendix For Further Reading

For Further Reading V

J.E. Zhang and Jinghong Shu. Pricing standard & poor’s 500 index options with heston’s model. Computational Intelligence for Financial Engineering, 2003.

• Proceedings. 2003 IEEE International Conference on,

pages 85–92, March 2003.

Jean-Pierre Fouque, Matthew J Lorig Heston 2.0