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Pricing Central Tendency in Volatility
Stanislav Khrapov
NES Anniversary, Moscow
December 14, 2012
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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
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Motivation and Contribution The Model Results
Volatility
020406080
100120140
Volatility measures
RVVIX
19971999
20012003
20052007
20092011
−40−20
0204060
Difference
RV-VIX
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Motivation and Contribution The Model Results
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
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Motivation and Contribution The Model Results
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
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Motivation and Contribution The Model Results
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)
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Motivation and Contribution The Model Results
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)
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Motivation and Contribution The Model Results
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)
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Motivation and Contribution The Model Results
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)
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Motivation and Contribution The Model Results
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
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Motivation and Contribution The Model Results
The Model
Risk-neutral:
dpt = rdt + 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
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Motivation and Contribution The Model Results
The Model
Risk-neutral:
dpt = rdt + 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
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Motivation and Contribution The Model Results
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
]
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Motivation and Contribution The Model Results
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
]
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Motivation and Contribution The Model Results
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
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Motivation and Contribution The Model Results
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
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Motivation and Contribution The Model Results
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
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Motivation and Contribution The Model Results
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
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Motivation and Contribution The Model Results
Moment Conditions
First moment:
EPt[(
1− AyhL)× (1− AσhL)× Vt+2h,h
]= Const
EPt[Vt+2h,h − ρ0 − ρ1Vt+h,h − ρ2Vt ,h
]= 0
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Motivation and Contribution The Model Results
Moment Conditions
First moment:
EPt[(
1− AyhL)× (1− AσhL)× Vt+2h,h
]= Const
EPt[Vt+2h,h − ρ0 − ρ1Vt+h,h − ρ2Vt ,h
]= 0
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Motivation and Contribution The Model Results
Moment Conditions
Second moment:
EPt
(1− γ1L) ×(1− γ2L) ×(1− γ3L) ×(1− γ4L) ×(1− γ5L) × V2
t+5h,h
= Const
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Motivation and Contribution The Model Results
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
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Motivation and Contribution The Model Results
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
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Motivation and Contribution The Model Results
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
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Motivation and Contribution The Model Results
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)
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Motivation and Contribution The Model Results
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
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Motivation and Contribution The Model Results
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
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Motivation and Contribution The Model Results
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
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Motivation and Contribution The Model Results
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
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Motivation and Contribution The Model Results
Volatility Premia
5 10 15 20−4
−3
−2
−1
0
1
Forecast horizon, days
Mean p
rem
ium
, var
units
VP
CP
TP
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Motivation and Contribution The Model Results
Conclusion
Joint estimation of volatility modelLong-term mean is changingCorresponding risk has a priceCorresponding premium is large
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Thank you!
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, 2011, The effects of the monetary policy regime shift toinflation targeting on the real interest rate in the UnitedKingdom, Economic Modelling 28, 754–759.
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