Pricing Central Tendency in Volatility

Pricing Central Tendency in Volatility

Stanislav Khrapov

NES Anniversary, Moscow

December 14, 2012

Motivation and Contribution The Model Results

Market Returns

400600800

1000120014001600

S&P500 index

SPX

19971999

20012003

20052007

20092011

−10−5

05

1015

S&P500 log returns

logR


Volatility

020406080

100120140

Volatility measures

RVVIX

19971999

20012003

20052007

20092011

−40−20

0204060

Difference

RV-VIX


Persistence

0 10 20 30 40 50 60 70 80 90Lags, days

−0.2

0.0

0.2

0.4

0.6

0.8

1.0Autocorrelation function

VIXRVlogR


Thick Tails

Min Max Mean Std Skewness Kurtosis

logR -9.47 10.96 0.01 1.32 -0.25 7.98

VIX 9.89 80.86 21.69 8.83 2.09 7.40

RV 2.38 118.75 13.37 8.61 3.41 20.12


Contribution

Two-component volatility (central tendency)Engle and Lee (1996), Andersen and Lund (1997),Balduzzi, Das, and Foresi (1998), Reschreiter (2010, 2011)

Both volatility risks are pricedAdrian and Rosenberg (2008), Todorov (2010)

Explicit expressions for innovations, moments, etcBollerslev and Zhou (2002), Eraker (2009), Todorov (2010)

Joint estimation under P and QChernov and Ghysels (2000), Garcia, Lewis, Pastorello,and Renault (2011), Bollerslev, Gibson, and Zhou (2011)


Contribution






Contribution






Contribution






The Model

Historical:

dpt = (r + µπ)dt + σtdW rt

dσ2t = κσ

(yt − σ2

t

)dt + ησσtdW σ

t

dyt = κy (µ− yt)dt + ηy√

ytdW yt

κσ = κσ − λσησ, κy = κy − λyηy

pt - log priceσ2

t - stochastic volatilityyt - central tendency


The Model

Risk-neutral:

dpt = rdt + σtdW rt

dσ2t = κσ

(yt − σ2

t

)dt + ησσtdW σ

t


ytdW yt


pt - log priceσ2



The Model

Risk-neutral:

dpt = rdt + σtdW rt

dσ2t = κσ

(yt − σ2

t

)dt + ησσtdW σ

t


ytdW yt


pt - log priceσ2



Data

Objective measure:

RVt ,1 ≡n∑

j=1

r2t+ j−1

n ,t+ jn

a.s.−→ Vt ,1

Risk-neutral measure:

VIXt ,22 = EQt[Vt ,22

]


Data

Objective measure:

RVt ,1 ≡n∑

j=1

r2t+ j−1

n ,t+ jn

a.s.−→ Vt ,1

Risk-neutral measure:

VIXt ,22 = EQt[Vt ,22

]


Discretization

[σ2

t+h, yt+h]′ - VAR(1)-type

Vt ,h ≡ 1h

´ t+ht σ2

s ds, Yt ,h ≡ 1h

´ t+ht ysds

[Vt ,h,Yt ,h

]′ - VARMA(1,1)-type

Vt ,h - ARMA(2,2)-type


Discretization

[σ2


Vt ,h ≡ 1h

´ t+ht σ2

s ds, Yt ,h ≡ 1h

´ t+ht ysds

[Vt ,h,Yt ,h




Discretization

[σ2


Vt ,h ≡ 1h

´ t+ht σ2

s ds, Yt ,h ≡ 1h

´ t+ht ysds

[Vt ,h,Yt ,h




Discretization

[σ2


Vt ,h ≡ 1h

´ t+ht σ2

s ds, Yt ,h ≡ 1h

´ t+ht ysds

[Vt ,h,Yt ,h




Moment Conditions

First moment:

EPt[(

1− AyhL)× (1− AσhL)× Vt+2h,h

]= Const

EPt[Vt+2h,h − ρ0 − ρ1Vt+h,h − ρ2Vt ,h

]= 0


Moment Conditions

First moment:

EPt[(

1− AyhL)× (1− AσhL)× Vt+2h,h

]= Const

EPt[Vt+2h,h − ρ0 − ρ1Vt+h,h − ρ2Vt ,h

]= 0


Moment Conditions

Second moment:

EPt

(1− γ1L) ×(1− γ2L) ×(1− γ3L) ×(1− γ4L) ×(1− γ5L) × V2

t+5h,h

= Const


Premia

Volatility premium

VPt ,H = EPt[Vt ,H

]− EQ

t[Vt ,H

]

Central tendency premium

CPt ,H = EPt[Yt ,H

]− EQ

t[Yt ,H

]Transient premium

TPt ,H = VPt ,H − CPt ,H


Premia

Volatility premium

VPt ,H = EPt[Vt ,H

]− EQ

t[Vt ,H

]Central tendency premium

CPt ,H = EPt[Yt ,H

]− EQ

t[Yt ,H

]

Transient premium



Premia

Volatility premium

VPt ,H = EPt[Vt ,H

]− EQ

t[Vt ,H

]Central tendency premium

CPt ,H = EPt[Yt ,H

]− EQ

t[Yt ,H

]Transient premium



Parameter estimates

µ 0.0046 (0.0005)

κσ 0.8989 (0.0057) κy 0.0178 (0.0038)

ησ 0.1041 (0.0225) ηy 0.0073 (0.0033)

λσ 0.2013 (0.0786) λy 1.0929 (0.4835)


Parameter estimates

µ 0.0046 (0.0005)

κσ 0.8989 (0.0057) κy 0.0178 (0.0038)

ησ 0.1041 (0.0225) ηy 0.0073 (0.0033)

λσ 0.2013 (0.0786) λy 1.0929 (0.4835)

µ - unconditional mean


Parameter estimates

µ 0.0046 (0.0005)

κσ 0.8989 (0.0057) κy 0.0178 (0.0038)

ησ 0.1041 (0.0225) ηy 0.0073 (0.0033)

λσ 0.2013 (0.0786) λy 1.0929 (0.4835)

κ - speed of mean reversion


Parameter estimates

µ 0.0046 (0.0005)

κσ 0.8989 (0.0057) κy 0.0178 (0.0038)

ησ 0.1041 (0.0225) ηy 0.0073 (0.0033)

λσ 0.2013 (0.0786) λy 1.0929 (0.4835)

η - instantaneous SD


Parameter estimates

µ 0.0046 (0.0005)

κσ 0.8989 (0.0057) κy 0.0178 (0.0038)

ησ 0.1041 (0.0225) ηy 0.0073 (0.0033)

λσ 0.2013 (0.0786) λy 1.0929 (0.4835)

λ - price of a shock


Volatility Premia

5 10 15 20−4

−3

−2

−1

0

1

Forecast horizon, days

Mean p

rem

ium

, var

units

VP

CP

TP


Conclusion

Joint estimation of volatility modelLong-term mean is changingCorresponding risk has a priceCorresponding premium is large

Thank you!

Adrian, Tobias, and Joshua Rosenberg, 2008, Stock Returnsand Volatility: Pricing the Short-Run and Long-RunComponents of Market Risk, Journal of Finance 63,2997–3030.

Andersen, Torben G, and Jesper Lund, 1997, StochasticVolatility and Mean Drift in the Short Rate Diffusion: Sourcesof Steepness, Level and Curvature in the Yield Curve, .

Balduzzi, Pierluigi, Sanjiv Ranjan Das, and Silverio Foresi,1998, The Central Tendency: A Second Factor in BondYields, Review of Economics and Statistics 80, 62–72.

Bollerslev, Tim, Michael Gibson, and Hao Zhou, 2011, Dynamicestimation of volatility risk premia and investor risk aversionfrom option-implied and realized volatilities, Journal ofEconometrics 160, 235–245.

Bollerslev, Tim, and Hao Zhou, 2002, Estimating stochasticvolatility diffusion using conditional moments of integratedvolatility, Journal of Econometrics 109, 33 – 65.

Chernov, Mikhail, and Eric Ghysels, 2000, A study towards aunified approach to the joint estimation of objective and riskneutral measures for the purpose of options valuation,Journal of Financial Economics 56, 407–458.

Engle, Robert F, and Gary G J Lee, 1996, Estimating DiffusionModels of Stochastic Volatility, in Peter E Rossi, ed.:Modeling Stock Market Volatility: Bridging The Gap toContinuous Time . pp. 333–355 (Academic Press).

Eraker, Bjorn, 2009, The Volatility Premium, .Garcia, Rene, Marc-Andre Lewis, Sergio Pastorello, and Eric

Renault, 2011, Estimation of objective and risk-neutraldistributions based on moments of integrated volatility,Journal of Econometrics 160, 22–32.

Reschreiter, Andreas, 2010, Inflation and the Mean-RevertingLevel of the Short Rate, The Manchester School 78, 76–91.

, 2011, The effects of the monetary policy regime shift toinflation targeting on the real interest rate in the UnitedKingdom, Economic Modelling 28, 754–759.

Todorov, Viktor, 2010, Variance Risk-Premium Dynamics: TheRole of Jumps, Review of Financial Studies 23, 345–383.

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Pricing Central Tendency in Volatility