pricing central tendency in volatility
Post on 20-Jan-2015
296 Views
Preview:
DESCRIPTION
TRANSCRIPT
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
Motivation and Contribution The Model Results
Volatility
020406080
100120140
Volatility measures
RVVIX
19971999
20012003
20052007
20092011
−40−20
0204060
Difference
RV-VIX
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
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
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)
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)
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)
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)
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
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
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
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
]
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
]
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
Motivation and Contribution The Model Results
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.
top related