Affine Option Pricing Model in Discrete Time Eric Renault 1 Stanislav - - PowerPoint PPT Presentation

Affine Option Pricing Model in Discrete Time

Eric Renault1 Stanislav Khrapov2

1Brown University

Providence, RI

2New Economic School

Moscow, Russia

Third International Moscow Finance Conference November 8, 2013

Introduction

Introduction The Model Option Pricing Estimation Conclusion

Continuous vs Discrete

Continuous time affine models with stochastic volatility: Cox, Ingersoll, and Ross (1985, Econometrica) Heston (1993, RFS) Duffie, Pan, and Singleton (2000, Econometrica)

Introduction The Model Option Pricing Estimation Conclusion

Continuous vs Discrete

Popular because of computational convenience ... ... with historical probability measure (P): Robustness to temporal aggregation: Meddahi and Renault (2004, JoE) Robustness to cross-sectional aggregation (portfolio)

Introduction The Model Option Pricing Estimation Conclusion

Continuous vs Discrete

Popular because of computational convenience ... ... with historical probability measure (P): Robustness to temporal aggregation: Meddahi and Renault (2004, JoE) Robustness to cross-sectional aggregation (portfolio) ... with risk neutral probability measure (Q): Structure preserving change of measure (affine structure preserved) Analytical tractability of computing derivative prices (inverse Fourier transform)

Introduction The Model Option Pricing Estimation Conclusion

Discrete Time Extension

Volatility model: Darolles, Gourieroux, and Jasiak (2006, JTSA) Gourieroux and Jasiak (2006, JoF) Option pricing model with conditional skewness: Feunou and Tedongap (2012, JBES)

Introduction The Model Option Pricing Estimation Conclusion

Discrete Time Extension

Advantages of discrete time: Computational/Statistical tractability More flexibility for higher order moments

Introduction The Model Option Pricing Estimation Conclusion

Discrete Time Extension

Advantages of discrete time: Computational/Statistical tractability More flexibility for higher order moments Challenges of discrete time: Accommodating leverage effect Keeping the advantage of structure preserving change of measure historical/risk-neutral

Affine Stochastic Volatility Model

Introduction The Model Option Pricing Estimation Conclusion

The Model

CAR Volatility: Darolles, Gourieroux, and Jasiak (2006, JTSA) E

• exp
• −uσ 2

t+1

• It
• = exp
• −a(u)σ 2

t − b(u)

Introduction The Model Option Pricing Estimation Conclusion

The Model

CAR Volatility: Darolles, Gourieroux, and Jasiak (2006, JTSA) E

• exp
• −uσ 2

t+1

• It
• = exp
• −a(u)σ 2

t − b(u)

• Log Excess Return

E

• exp{−vrt+1}
• It ∪σ 2

t+1

• = exp
• −α (v)σ 2

t+1 −β (v)σ 2 t −γ (v)

Introduction The Model Option Pricing Estimation Conclusion

The Model

Joint Return and Volatility E

• exp
• −uσ 2

t+1 − vrt+1

• It
• = exp
• −l (u,v)σ 2

t − g (u,v)

Introduction The Model Option Pricing Estimation Conclusion

The Model

Joint Return and Volatility E

• exp
• −uσ 2

t+1 − vrt+1

• It
• = exp
• −l (u,v)σ 2

t − g (u,v)

• l (u,v)

=

a[u +α (v)]+β (v) g (u,v)

=

b[u +α (v)]+γ (v)

Introduction The Model Option Pricing Estimation Conclusion

The Model

Joint Return and Volatility E

• exp
• −uσ 2

t+1 − vrt+1

• It
• = exp
• −l (u,v)σ 2

t − g (u,v)

• l (u,v)

=

a[u +α (v)]+β (v) g (u,v)

=

b[u +α (v)]+γ (v)

α (v) = 0 ⇐ ⇒ leverage!

Introduction The Model Option Pricing Estimation Conclusion

Risk-Neutral Distribution

The affine structure is kept from P to Q when Pricing Kernel = Exponential Affine

Introduction The Model Option Pricing Estimation Conclusion

Risk-Neutral Distribution

The affine structure is kept from P to Q when Pricing Kernel = Exponential Affine Stochastic Discount Factor (SDF): Mt,t+1 (θ) = exp(−rf,t)exp

• m0 (θ)+ m1 (θ)σ 2

t −θ1σ 2 t+1 −θ2rt+1

• risk prices θ1 ≤ 0 and θ2 ≥ 0

m0 (θ), m1 (θ): bonds and stocks are priced correctly

Introduction The Model Option Pricing Estimation Conclusion

Risk-Neutral Distribution

Risk-neutral pricing: EQ H

• rt+1,σ 2

t+1

• It
• = exp(rf,t)E
• Mt,t+1 (θ)H
• rt+1,σ 2

t+1

• It
• for any function H

Introduction The Model Option Pricing Estimation Conclusion

Risk-Neutral Distribution

Risk-neutral pricing: EQ H

• rt+1,σ 2

t+1

• It
• = exp(rf,t)E
• Mt,t+1 (θ)H
• rt+1,σ 2

t+1

• It
• for any function H

Risk-neutral distribution: EQ exp

• −uσ 2

t+1 − vrt+1

• It
• = exp
• −l∗ (u,v)σ 2

t − g∗ (u,v)

• with

l∗ (u,v)

=

l (θ1 + u,θ2 + v)− l (θ1,θ2) g∗ (u,v)

=

g (θ1 + u,θ2 + v)− g (θ1,θ2)

Introduction The Model Option Pricing Estimation Conclusion

Affine Moments

Volatility moments: E

• σ 2

t+1

• It
• =

a′ (0)σ 2

t + b′ (0)

V

• σ 2

t+1

• It
• =

−a′′ (0)σ 2

t − b′′ (0)

Introduction The Model Option Pricing Estimation Conclusion

Affine Moments

Volatility moments: E

• σ 2

t+1

• It
• =

a′ (0)σ 2

t + b′ (0)

V

• σ 2

t+1

• It
• =

−a′′ (0)σ 2

t − b′′ (0)

Return expectation: E [rt+1|Iσ

t ] = α′ (0)σ 2 t+1 +β ′ (0)σ 2 t +γ′ (0)

Introduction The Model Option Pricing Estimation Conclusion

Affine Moments

Volatility moments: E

• σ 2

t+1

• It
• =

a′ (0)σ 2

t + b′ (0)

V

• σ 2

t+1

• It
• =

−a′′ (0)σ 2

t − b′′ (0)

Return expectation: E [rt+1|Iσ

t ] = α′ (0)σ 2 t+1 +β ′ (0)σ 2 t +γ′ (0)

Leverage effect:

φ ≈ Corr

• rt+1,σ 2

t+1

• It
• = α′ (0)
• V
• σ 2

t+1

• It
• V [rt+1|It]

1/2 α (v) = 0 ⇐ ⇒ leverage!

Option Pricing

Introduction The Model Option Pricing Estimation Conclusion

Generalized Black-Scholes

Assume α (v),β (v),γ (v) are quadratic, then rt+1|Iσ

t ∼ N (E [rt+1|Iσ t ],V [rt+1|Iσ t ])

Introduction The Model Option Pricing Estimation Conclusion

Generalized Black-Scholes

Assume α (v),β (v),γ (v) are quadratic, then rt+1|Iσ

t ∼ N (E [rt+1|Iσ t ],V [rt+1|Iσ t ])

Option price: Ct (xt,φ) = EQ

t

• BS
• Stξt,t+1 (φ),
• 1−φ 2

σ 2

t+1,K

• where xt = log(K/St) is the moneyness and

logξt,t+1 (φ) = EQ [rt+1|Iσ

t ]+ 1

2V Q [rt+1|Iσ

t ]

is price distortion

Introduction The Model Option Pricing Estimation Conclusion

Leverage and Volatility Smirk

Two effects of φ: Price distortion Stξt,t+1 (φ) Volatility

• 1−φ 2

σ 2

t+1

Introduction The Model Option Pricing Estimation Conclusion

Leverage and Volatility Smirk

Two effects of φ: Price distortion Stξt,t+1 (φ) Volatility

• 1−φ 2

σ 2

t+1

Around φ = 0 the first order effect is through volatility: Ct (xt,φ) ≈ Ct (xt,0)+ kφ · CovQ

σ 2

t+1,Φ(d)

• It
• with

d = 1 2σt+1 − xt

σt+1

Introduction The Model Option Pricing Estimation Conclusion

Leverage and Volatility Smirk

Two effects of φ: Price distortion Stξt,t+1 (φ) Volatility

• 1−φ 2

σ 2

t+1

Around φ = 0 the first order effect is through volatility: Ct (xt,φ) ≈ Ct (xt,0)+ kφ · CovQ

σ 2

t+1,Φ(d)

• It
• with

d = 1 2σt+1 − xt

σt+1

Cov() more positive out of the money

= ⇒ the smile is pushed down on the out of the money side

Introduction The Model Option Pricing Estimation Conclusion

Leverage and Volatility Smirk

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 Log-moneyness, log(K/S) 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 Implied vol φ

0.0

• 0.1
• 0.2
• 0.3
• 0.4
• 0.5

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 Log-moneyness, log(K/S) 0.20 0.22 0.24 0.26 0.28 0.30 0.32 Implied vol T

10 20 30 40 50 60

Estimation

Introduction The Model Option Pricing Estimation Conclusion

Maximum Likelihood

Joint likelihood f

• rt+1,σ 2

t+1

• σ 2

t ;c,ρ,δ,φ,θ2

• =f
• rt+1
• σ 2

t+1,σ 2 t ;φ,θ2

• × f
• σ 2

t+1

• σ 2

t ;c,ρ,δ

Introduction The Model Option Pricing Estimation Conclusion

Maximum Likelihood

Joint likelihood f

• rt+1,σ 2

t+1

• σ 2

t ;c,ρ,δ,φ,θ2

• =f
• rt+1
• σ 2

t+1,σ 2 t ;φ,θ2

• × f
• σ 2

t+1

• σ 2

t ;c,ρ,δ

• where

f

• rt+1
• σ 2

t+1,σ 2 t ;φ,θ2

• ∼

Normal f

• σ 2

t+1

• σ 2

t ;c,ρ,δ

• ∼

nc − Gamma

σ 2

t+1 is ARG(1) from Gourieroux and Jasiak (2006, JoF)

Introduction The Model Option Pricing Estimation Conclusion

Spectral GMM

Singleton (2001, JoE), Chacko and Viceira (2003, JoE) Moment functions: gt (u,θ) = Zt ·

• exp
• −uσ 2

t+1

• − exp
• −a(u)σ 2

t − b(u)

• exp{−urt+1}− exp
• −α (u)σ 2

t+1 −β (u)σ 2 t −γ (u)

Introduction The Model Option Pricing Estimation Conclusion

Spectral GMM

Singleton (2001, JoE), Chacko and Viceira (2003, JoE) Moment functions: gt (u,θ) = Zt ·

• exp
• −uσ 2

t+1

• − exp
• −a(u)σ 2

t − b(u)

• exp{−urt+1}− exp
• −α (u)σ 2

t+1 −β (u)σ 2 t −γ (u)

• Moments to match:

E

• Re{gt (u,θ)}

Im{gt (u,θ)}

• = 0

Introduction The Model Option Pricing Estimation Conclusion

Model Fit

−0.003 −0.002 −0.001 0.000 0.001 0.002 0.003 0.004 V −0.003 −0.002 −0.001 0.000 0.001 0.002 0.003 0.004 EV −0.10 −0.08 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 0.08 R −0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 ER 1997 1998 1999 2000 2001 2002 2003 2004 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016

V EV

1997 1998 1999 2000 2001 2002 2003 2004 −0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15

R ER

Introduction The Model Option Pricing Estimation Conclusion

Parameter Estimates

MLE GMM

ˆ θ

t

ˆ θ

t c 2.4e-5 [29.9] 6.7e-6 [4.5]

ρ

0.66 [39.9] 0.91 [28.5]

δ

1.45 [29.5] 1.18 [6.4]

φ

• 0.21

[-14.3]

• 0.22

[-10.2]

θ2

1.57 [0.7] 1.90 [0.9]

Introduction The Model Option Pricing Estimation Conclusion

Vol Risk Price Calibration

ˆ θ1 = argmin

θ1

RMSEIV (θ1) =

• 1

N

N

∑

j=1

• IV Market

j

− IV Model

j

(θ1) 2

−2500 −2000 −1500 −1000 −500 theta1 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 IVRMSE −1700 −1680 −1660 −1640 −1620 −1600 theta1 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016 0.0018 IVRMSE +4.134

Introduction The Model Option Pricing Estimation Conclusion

Vol Risk Price Calibration

ˆ θ1 = argmin

θ1

RMSEIV (θ1) =

• 1

N

N

∑

j=1

• IV Market

j

− IV Model

j

(θ1) 2

−2500 −2000 −1500 −1000 −500 theta1 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 IVRMSE −1700 −1680 −1660 −1640 −1620 −1600 theta1 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016 0.0018 IVRMSE +4.134

Introduction The Model Option Pricing Estimation Conclusion

Conclusion

We have been able to build a discrete time version of Heston’s model

Introduction The Model Option Pricing Estimation Conclusion

Conclusion

We have been able to build a discrete time version of Heston’s model Advantages of discrete time:

Easier theoretical derivations (impact of leverage on volatility smile, etc...) Easier for statistical inference More flexibility for higher order moments

Introduction The Model Option Pricing Estimation Conclusion

Conclusion

We have been able to build a discrete time version of Heston’s model Advantages of discrete time:

Easier theoretical derivations (impact of leverage on volatility smile, etc...) Easier for statistical inference More flexibility for higher order moments

Work in progress: take advantage of this flexibility for empirical fit better than standard Heston:

Two volatility factors (slow and fast mean reverting) Mixture component in return rt+1 given Iσ

t for more kurtosis

(gamma mixture to keep the affine structure) reminiscent of jumps in continuous time

Thank you!

Chacko, George, and Luis M Viceira, 2003, Spectral GMM estimation

• f continuous-time processes, Journal of Econometrics 116,

259–292. Cox, John C, Jonathan E Ingersoll, and Stephen A Ross, 1985, A Theory of the Term Structure of Interest Rates, Econometrica 53, 385–407. Darolles, Serge, Christian Gourieroux, and Joann Jasiak, 2006, Structural Laplace Transform and Compound Autoregressive Models, Journal of Time Series Analysis 27, 477–503. Duffie, Darrell, Jun Pan, and Kenneth J Singleton, 2000, Transform Analysis and Asset Pricing for Affine Jump-Diffusions, Econometrica 68, 1343–1376. Feunou, Bruno, and Romeo Tedongap, 2012, A Stochastic Volatility Model With Conditional Skewness, Journal of Business and Economic Statistics 30, 576–591. Gourieroux, Christian, and Joann Jasiak, 2006, Autoregressive gamma processes, Journal of Forecasting 25, 129–152. Heston, Steven L, 1993, A Closed-Form Solution for Options With Stochastic Volatility With Applications to Bond and Currency Options, Review of Financial Studies 6, 327–343. Meddahi, Nour, and Eric Renault, 2004, Temporal aggregation of