European Finance Association Meeting Bergen
When Uncertainty Blows in the Orchard: Comovement & Equilibrium Volatility Risk Premia Andrea Buraschi
Fabio Trojani
Andrea Vedolin
Imperial College London
University of Lugano
University of Lugano
21st August 2009
c (2009) Buraschi, Trojani, Vedolin – 1 ⃝
⊳ Motivation RV & IV Literature DiB Dispersion Model Empirical Analysis Conclusion Appendix
Motivation
c (2009) Buraschi, Trojani, Vedolin – 2 ⃝
Index & Individual RV and IV Realized and Implied Volatility of Individual Options 0.75
Volatility
realized volatility implied volatility 0.5
0.25
0
01/01/00
01/01/05 Time
0.6
Realized and Implied Volatility of S&P 100 realized volatility implied volatility
Volatility
0.5 0.4 0.3 0.2 0.1 0
01/01/00
01/01/05 Time
There are two noteworthy points here: ① The average implied volatility of the constituents is 32.7% and the average realized volatility is 31.8% which yields a volatility risk premium of 0.9%. ② The average implied volatility of the index is 19.2% and the realized volatility amounts to 16.7%, which yields a volatility risk premium of 2.5%. c (2009) Buraschi, Trojani, Vedolin – 3 ⃝
Index & Individual RV and IV Realized and Implied Volatility of Individual Options 0.75
Volatility
realized volatility implied volatility 0.5
0.25
0
01/01/00
01/01/05 Time
0.6
Realized and Implied Volatility of S&P 100 realized volatility implied volatility
Volatility
0.5 0.4 0.3 0.2 0.1 0
01/01/00
01/01/05 Time
There are two noteworthy points here: ① The average implied volatility of the constituents is 32.7% and the average realized volatility is 31.8% which yields a volatility risk premium of 0.9%. ② The average implied volatility of the index is 19.2% and the realized volatility amounts to 16.7%, which yields a volatility risk premium of 2.5%. c (2009) Buraschi, Trojani, Vedolin – 3 ⃝
Index & Individual RV and IV Realized and Implied Volatility of Individual Options 0.75
Volatility
realized volatility implied volatility 0.5
0.25
0
01/01/00
01/01/05 Time
0.6
Realized and Implied Volatility of S&P 100 realized volatility implied volatility
Volatility
0.5 0.4 0.3 0.2 0.1 0
01/01/00
01/01/05 Time
There are two noteworthy points here: ① The average implied volatility of the constituents is 32.7% and the average realized volatility is 31.8% which yields a volatility risk premium of 0.9%. ② The average implied volatility of the index is 19.2% and the realized volatility amounts to 16.7%, which yields a volatility risk premium of 2.5%. c (2009) Buraschi, Trojani, Vedolin – 3 ⃝
In lieu of Literature Survey Motivation RV & IV Literature DiB Dispersion
⊳
Model Empirical Analysis Conclusion Appendix
Two theories exist so far to explain the difference in index and individual volatility risk premia: ① Market frictions: Option market demand and supply drive premia and demand is different in index and single-stock markets.
Bollen and Whaley (2004): Empirical evidence that net
buying pressure is present in index option markets, especially OTM puts. Changes in IV in stock options is smaller and concentrated on calls.
Gˆarleanu, Pedersen, and Poteshman (2009): Equilibrium
option prices are a function of demand pressure. End users have a net long position in S&P 500 index options, in particular OTM puts and a net short position in single-stock options.
② Risk-based: Driessen, Maenhout, and Vilkov (2009) argue that priced correlation risk explains the differential pricing of index and individual options. They remain, however, agnostic about how correlation risk premia emerge. c (2009) Buraschi, Trojani, Vedolin – 4 ⃝
In lieu of Literature Survey Motivation RV & IV Literature DiB Dispersion
⊳
Model Empirical Analysis Conclusion Appendix
Two theories exist so far to explain the difference in index and individual volatility risk premia: ① Market frictions: Option market demand and supply drive premia and demand is different in index and single-stock markets.
Bollen and Whaley (2004): Empirical evidence that net
buying pressure is present in index option markets, especially OTM puts. Changes in IV in stock options is smaller and concentrated on calls.
Gˆarleanu, Pedersen, and Poteshman (2009): Equilibrium
option prices are a function of demand pressure. End users have a net long position in S&P 500 index options, in particular OTM puts and a net short position in single-stock options.
② Risk-based: Driessen, Maenhout, and Vilkov (2009) argue that priced correlation risk explains the differential pricing of index and individual options. They remain, however, agnostic about how correlation risk premia emerge. c (2009) Buraschi, Trojani, Vedolin – 4 ⃝
In lieu of Literature Survey Motivation RV & IV Literature DiB Dispersion
⊳
Model Empirical Analysis Conclusion Appendix
Two theories exist so far to explain the difference in index and individual volatility risk premia: ① Market frictions: Option market demand and supply drive premia and demand is different in index and single-stock markets.
Bollen and Whaley (2004): Empirical evidence that net
buying pressure is present in index option markets, especially OTM puts. Changes in IV in stock options is smaller and concentrated on calls.
Gˆarleanu, Pedersen, and Poteshman (2009): Equilibrium
option prices are a function of demand pressure. End users have a net long position in S&P 500 index options, in particular OTM puts and a net short position in single-stock options.
② Risk-based: Driessen, Maenhout, and Vilkov (2009) argue that priced correlation risk explains the differential pricing of index and individual options. They remain, however, agnostic about how correlation risk premia emerge. c (2009) Buraschi, Trojani, Vedolin – 4 ⃝
Volatility & Correlation Risk Premia Regression ... in the time series Index Volatility Risk Premium & Common Uncertainty
Common DiB
0.1
9/11
LTCM 0.03
0.05
0.02
0
0.01
−0.05
0
−0.1
−0.01
−0.02
Volatility Risk Premium
0.04
−0.15
1998
1999
2000
2001
2002
2003
2004
2005
2006
−0.2
□
Running a simple regression from the volatility risk premium on the common DiB, we find ...
□
a beta of 0.486 with a t-stat of 6.60 and ...
□
an adjusted 𝑅2 of 0.23 c (2009) Buraschi, Trojani, Vedolin – 5 ⃝
Volatility & Correlation Risk Premia Regression ... in the time series
... and in the cross section Industry wide Volatility Risk Premium & Uncertainty
Index Volatility Risk Premium & Common Uncertainty
Common DiB
0.03
0.06
0.05
0.05
0.02
0
0.01
−0.05
0
−0.1
−0.01
−0.02
□
−0.15
1998
1999
2000
2001
2002
2003
2004
2005
2006
−0.2
Running a simple regression from the volatility risk premium on the common DiB, we find ...
□
a beta of 0.486 with a t-stat of 6.60 and ...
□
an adjusted 𝑅2 of 0.23
2000 - 2004 βˆ = 1.480 t-Stat = 3.92 ¯ 2 = 0.54 R
1996 - 2000 βˆ = 0.730 t-Stat = 3.86 ¯ 2 = 0.53 R
Energy
Volatility Risk Premium
9/11
LTCM
0.1
Volatility Risk Premium
0.04
Energy
Media
0.04
0.03
0.02
0.01
0 0
Materials
Automotive
Banking Consumer Durables Consumer Durables Financial Services Materials Automotive Materials Financial Services Insurance Banking Energy Telecommunications Consumer Non-Durables Consumer Non-Durables Consumer Durables Computer Hardware Computer Hardware Computer Hardware Telecommunications Telecommunications Media Media Health Services Health ServicesServices Health Food Food Tobacco Beverage Food Beverage Tobacco Beverage Tobacco
Banking Automotive
Financial Services 2004 - 2008 βˆ = 0.611 t-Stat = 2.78 ¯ 2 = 0.37 R
Insurance
Insurance
0.1
0.2
0.3 0.4 Uncertainty-DiB
0.5
0.6
0.7
□
Doing the same in the cross section, we find ...
□
highly significant beta coefficients notwithstanding the ...
□
different periods.
c (2009) Buraschi, Trojani, Vedolin – 5 ⃝
Dispersion Trading: Exposure to Correlation Motivation RV & IV Literature DiB Dispersion
⊳
□
So called dispersion trades take advantage of the difference between index and individual volatility risk premia.
□
Using option data on the index and the constituent stocks, we find that ...
Model Empirical Analysis Conclusion
Date
Description
Loss
% DiB
% S&P 500 Corr
August 1998
LTCM & Russian Default
-122%
+33%
+106%
February 2001
DotCom Bubble
-111%
+10%
+46%
September 2001
Terrorists’ Attacks
-141%
+25%
+40%
Appendix
□
Strong negative returns are related to increases in the common DiB component and realized correlation.
Z We ask: Can we give a structural explanation to these returns? c (2009) Buraschi, Trojani, Vedolin – 6 ⃝
Motivation
⊳ Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC Empirical Analysis
Model
Conclusion Appendix
c (2009) Buraschi, Trojani, Vedolin – 7 ⃝
Economy’s State Variables Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
⊳
Empirical Analysis Conclusion Appendix
There are two firms in the economy which produce their perishable good. Dividend dynamics are as follows: 𝑑 log 𝐷𝑖 (𝑡) = 𝑑𝜇𝐷𝑖 (𝑡) =
𝜇𝐷𝑖 (𝑡)𝑑𝑡 + 𝜎𝐷𝑖 𝑑𝑊𝐷𝑖 (𝑡), (𝑎0𝐷𝑖 + 𝑎1𝐷𝑖 𝜇𝐷𝑖 (𝑡)) 𝑑𝑡 + 𝜎𝜇𝐷𝑖 𝑑𝑊𝜇𝐷𝑖 (𝑡),
and a signal 𝑧(𝑡), with dynamics: 𝑑𝑧(𝑡) 𝑑𝜇𝑧 (𝑡)
= (𝛼𝐷1 𝜇𝐷1 (𝑡) + 𝛼𝐷2 𝜇𝐷2 (𝑡) + 𝛽𝜇𝑧 (𝑡)) 𝑑𝑡 + 𝜎𝑧 𝑑𝑊𝑧 (𝑡), = (𝑎0𝑧 + 𝑎1𝑧 𝜇𝑧 (𝑡)) 𝑑𝑡 + 𝜎𝜇𝑧 𝑑𝑊𝜇𝑧 (𝑡). ↷
We interpret the signal as an economy-wide business cycle indicator. Dividends and the signal are observable, but their expected growth rates are unobservable and have to be estimated given the information. The following quantities are of key interest: ① The precision of the signal growth rate and ... ② The informativeness of the signal through 𝛼𝐷𝑖 , 𝛽. c (2009) Buraschi, Trojani, Vedolin – 8 ⃝
Economy’s State Variables Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
⊳
Empirical Analysis Conclusion Appendix
There are two firms in the economy which produce their perishable good. Dividend dynamics are as follows: 𝑑 log 𝐷𝑖 (𝑡) = 𝑑𝜇𝐷𝑖 (𝑡) =
𝜇𝐷𝑖 (𝑡)𝑑𝑡 + 𝜎𝐷𝑖 𝑑𝑊𝐷𝑖 (𝑡), (𝑎0𝐷𝑖 + 𝑎1𝐷𝑖 𝜇𝐷𝑖 (𝑡)) 𝑑𝑡 + 𝜎𝜇𝐷𝑖 𝑑𝑊𝜇𝐷𝑖 (𝑡),
and a signal 𝑧(𝑡), with dynamics: 𝑑𝑧(𝑡) 𝑑𝜇𝑧 (𝑡)
= (𝛼𝐷1 𝜇𝐷1 (𝑡) + 𝛼𝐷2 𝜇𝐷2 (𝑡) + 𝛽𝜇𝑧 (𝑡)) 𝑑𝑡 + 𝜎𝑧 𝑑𝑊𝑧 (𝑡), = (𝑎0𝑧 + 𝑎1𝑧 𝜇𝑧 (𝑡)) 𝑑𝑡 + 𝜎𝜇𝑧 𝑑𝑊𝜇𝑧 (𝑡). ↷
We interpret the signal as an economy-wide business cycle indicator. Dividends and the signal are observable, but their expected growth rates are unobservable and have to be estimated given the information. The following quantities are of key interest: ① The precision of the signal growth rate and ... ② The informativeness of the signal through 𝛼𝐷𝑖 , 𝛽. c (2009) Buraschi, Trojani, Vedolin – 8 ⃝
Economy’s State Variables Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
⊳
Empirical Analysis Conclusion Appendix
There are two firms in the economy which produce their perishable good. Dividend dynamics are as follows: 𝑑 log 𝐷𝑖 (𝑡) = 𝑑𝜇𝐷𝑖 (𝑡) =
𝜇𝐷𝑖 (𝑡)𝑑𝑡 + 𝜎𝐷𝑖 𝑑𝑊𝐷𝑖 (𝑡), (𝑎0𝐷𝑖 + 𝑎1𝐷𝑖 𝜇𝐷𝑖 (𝑡)) 𝑑𝑡 + 𝜎𝜇𝐷𝑖 𝑑𝑊𝜇𝐷𝑖 (𝑡),
and a signal 𝑧(𝑡), with dynamics: 𝑑𝑧(𝑡) 𝑑𝜇𝑧 (𝑡)
= (𝛼𝐷1 𝜇𝐷1 (𝑡) + 𝛼𝐷2 𝜇𝐷2 (𝑡) + 𝛽𝜇𝑧 (𝑡)) 𝑑𝑡 + 𝜎𝑧 𝑑𝑊𝑧 (𝑡), = (𝑎0𝑧 + 𝑎1𝑧 𝜇𝑧 (𝑡)) 𝑑𝑡 + 𝜎𝜇𝑧 𝑑𝑊𝜇𝑧 (𝑡). ↷
We interpret the signal as an economy-wide business cycle indicator. Dividends and the signal are observable, but their expected growth rates are unobservable and have to be estimated given the information. The following quantities are of key interest: ① The precision of the signal growth rate and ... ② The informativeness of the signal through 𝛼𝐷𝑖 , 𝛽. c (2009) Buraschi, Trojani, Vedolin – 8 ⃝
Subjective Expected Growth Rates Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
⊳
Empirical Analysis Conclusion Appendix
The subjective expected growth rate of cash flows and signals is ( ) 𝑛 𝑛 𝑌 𝑚 (𝑡) := 𝐸 (𝜇𝐷1 (𝑡), 𝜇𝐷2 (𝑡), 𝜇𝑧 (𝑡)) ∣ℱ𝑡 . Let 𝑌 (𝑡) = (log 𝐷1 (𝑡), log 𝐷2 (𝑡), 𝑧(𝑡)). The beliefs dynamics of agent 𝑛 have the functional form (Kalman-Bucy filter): 𝑑𝑚𝑛 (𝑡)
=
(𝑎0 + 𝑎1 𝑚𝑛 (𝑡))𝑑𝑡 + 𝛾 𝑛 (𝑡)𝐴′ 𝐵 −1 𝑑𝑊𝑌𝑛 (𝑡),
𝑑𝛾 𝑛 (𝑡)/𝑑𝑡
=
𝑎1 𝛾 𝑛 (𝑡) + 𝛾 𝑛 (𝑡)𝑎′1 + 𝑏𝑛 𝑏𝑛′ − 𝛾 𝑛 (𝑡)𝐴′ (𝐵𝐵 ′ )−1 𝐴𝛾 𝑛 (𝑡),
𝑛 𝑛 with initial conditions 𝑚𝑛 (0) = 𝑚𝑛 0 and 𝛾 (0) = 𝛾0 , where 𝑑𝑊𝑌𝑛 (𝑡) := 𝐵 −1 (𝑑𝑌 (𝑡) − 𝐴𝑚𝑛 (𝑡)𝑑𝑡) is the innovation process induced by investor’s 𝑛 belief and filtration .
Subjective beliefs, 𝑚𝑛 (𝑡), depend on two main drivers: ( 𝐴 ) 𝐵 ① The average subjective uncertainty, 𝜎 ¯𝜇𝑧 = 0.5 𝜎𝜇𝑧 + 𝜎𝜇𝑧 , in the 𝑛 𝑛 economy through 𝛾 (𝑡) and 𝑏 . Z A higher 𝜎 ¯𝜇𝑧 implies a higher estimation risk and therefore a more volatile updating process for estimation. ② The informativeness of the signal through 𝐴. Z For 𝛽 = 0, the signal produces unbiased estimates of the linear combination of firms’ growth rates. For 𝛽 ∕= 0, the signal is biased by another unobservable, orthogonal variable. c (2009) Buraschi, Trojani, Vedolin – 9 ⃝
Subjective Expected Growth Rates Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
⊳
Empirical Analysis Conclusion Appendix
The subjective expected growth rate of cash flows and signals is ( ) 𝑛 𝑛 𝑌 𝑚 (𝑡) := 𝐸 (𝜇𝐷1 (𝑡), 𝜇𝐷2 (𝑡), 𝜇𝑧 (𝑡)) ∣ℱ𝑡 . Let 𝑌 (𝑡) = (log 𝐷1 (𝑡), log 𝐷2 (𝑡), 𝑧(𝑡)). The beliefs dynamics of agent 𝑛 have the functional form (Kalman-Bucy filter): 𝑑𝑚𝑛 (𝑡)
=
(𝑎0 + 𝑎1 𝑚𝑛 (𝑡))𝑑𝑡 + 𝛾 𝑛 (𝑡)𝐴′ 𝐵 −1 𝑑𝑊𝑌𝑛 (𝑡),
𝑑𝛾 𝑛 (𝑡)/𝑑𝑡
=
𝑎1 𝛾 𝑛 (𝑡) + 𝛾 𝑛 (𝑡)𝑎′1 + 𝑏𝑛 𝑏𝑛′ − 𝛾 𝑛 (𝑡)𝐴′ (𝐵𝐵 ′ )−1 𝐴𝛾 𝑛 (𝑡),
𝑛 𝑛 with initial conditions 𝑚𝑛 (0) = 𝑚𝑛 0 and 𝛾 (0) = 𝛾0 , where 𝑑𝑊𝑌𝑛 (𝑡) := 𝐵 −1 (𝑑𝑌 (𝑡) − 𝐴𝑚𝑛 (𝑡)𝑑𝑡) is the innovation process induced by investor’s 𝑛 belief and filtration .
Subjective beliefs, 𝑚𝑛 (𝑡), depend on two main drivers: ( 𝐴 ) 𝐵 ① The average subjective uncertainty, 𝜎 ¯𝜇𝑧 = 0.5 𝜎𝜇𝑧 + 𝜎𝜇𝑧 , in the 𝑛 𝑛 economy through 𝛾 (𝑡) and 𝑏 . Z A higher 𝜎 ¯𝜇𝑧 implies a higher estimation risk and therefore a more volatile updating process for estimation. ② The informativeness of the signal through 𝐴. Z For 𝛽 = 0, the signal produces unbiased estimates of the linear combination of firms’ growth rates. For 𝛽 ∕= 0, the signal is biased by another unobservable, orthogonal variable. c (2009) Buraschi, Trojani, Vedolin – 9 ⃝
Subjective Expected Growth Rates Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
⊳
Empirical Analysis Conclusion Appendix
The subjective expected growth rate of cash flows and signals is ( ) 𝑛 𝑛 𝑌 𝑚 (𝑡) := 𝐸 (𝜇𝐷1 (𝑡), 𝜇𝐷2 (𝑡), 𝜇𝑧 (𝑡)) ∣ℱ𝑡 . Let 𝑌 (𝑡) = (log 𝐷1 (𝑡), log 𝐷2 (𝑡), 𝑧(𝑡)). The beliefs dynamics of agent 𝑛 have the functional form (Kalman-Bucy filter): 𝑑𝑚𝑛 (𝑡)
=
(𝑎0 + 𝑎1 𝑚𝑛 (𝑡))𝑑𝑡 + 𝛾 𝑛 (𝑡)𝐴′ 𝐵 −1 𝑑𝑊𝑌𝑛 (𝑡),
𝑑𝛾 𝑛 (𝑡)/𝑑𝑡
=
𝑎1 𝛾 𝑛 (𝑡) + 𝛾 𝑛 (𝑡)𝑎′1 + 𝑏𝑛 𝑏𝑛′ − 𝛾 𝑛 (𝑡)𝐴′ (𝐵𝐵 ′ )−1 𝐴𝛾 𝑛 (𝑡),
𝑛 𝑛 with initial conditions 𝑚𝑛 (0) = 𝑚𝑛 0 and 𝛾 (0) = 𝛾0 , where 𝑑𝑊𝑌𝑛 (𝑡) := 𝐵 −1 (𝑑𝑌 (𝑡) − 𝐴𝑚𝑛 (𝑡)𝑑𝑡) is the innovation process induced by investor’s 𝑛 belief and filtration .
Subjective beliefs, 𝑚𝑛 (𝑡), depend on two main drivers: ( 𝐴 ) 𝐵 ① The average subjective uncertainty, 𝜎 ¯𝜇𝑧 = 0.5 𝜎𝜇𝑧 + 𝜎𝜇𝑧 , in the 𝑛 𝑛 economy through 𝛾 (𝑡) and 𝑏 . Z A higher 𝜎 ¯𝜇𝑧 implies a higher estimation risk and therefore a more volatile updating process for estimation. ② The informativeness of the signal through 𝐴. Z For 𝛽 = 0, the signal produces unbiased estimates of the linear combination of firms’ growth rates. For 𝛽 ∕= 0, the signal is biased by another unobservable, orthogonal variable. c (2009) Buraschi, Trojani, Vedolin – 9 ⃝
Investors’ Disagreement Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
⊳
Empirical Analysis Conclusion Appendix
The process ) ⎛ ( 𝐴 ⎞ 𝐵 Ψ𝐷1 (𝑡) 𝐷1 (𝑡) − 𝑚𝐷1 (𝑡)) /𝜎𝐷1 (𝑚𝐴 𝐵 ⎠, (𝑡) − 𝑚 Ψ(𝑡) := ⎝ Ψ𝐷2 (𝑡) ⎠ = ⎝ 𝑚 𝐷 𝐷2 (𝑡)) /𝜎𝐷2 ( 2𝐴 Ψ𝑧 (𝑡) 𝑚𝑧 (𝑡) − 𝑚𝐵 𝑧 (𝑡) /𝜎𝑧 ⎛
⎞
(1)
is the disagreement process in the economy. The dynamics of the disagreement becomes a function of economic uncertainty and the informativeness of the signal: ⎛ ⎞ 𝑑Ψ(𝑡)
⎜ = 𝐵 −1 ⎝𝑎1 𝐵 +
𝛾 𝐵 (𝑡) | {z }
Economic uncertainty
⎟ 𝐴′ 𝐵 −1 ⎠ Ψ(𝑡)𝑑𝑡
( 𝐴 ) ′ −1 𝐵 −1 +𝐵 𝛾 (𝑡) − 𝛾 (𝑡) 𝐴 𝐵 𝑑𝑊𝑌𝐴 (𝑡). {z } | Diff. in Uncertainty
c (2009) Buraschi, Trojani, Vedolin – 10 ⃝
Why A Common Signal? Consumer Durables
Consumer Services 0.2
0.6 Russian Default LTCM
0.4
9/11/2001
0.15 0.1
0.2
0.05 1998
2000
2002
2004
2006
2008
1998
2000
2002
2004
2006
2008
2000
2002
2004
2006
2008
2000
2002
2004
2006
2008
Retailing
Media 0.25 0.15
0.2
0.1
0.15 0.1
0.05 1998
2000
2002
2004
Transportation
2006
2008
1998
0.5
0.2
0.4
0.15
Technology, Hardware & Equipment
0.3 0.1
0.2 0.1
0.05 1998
2000
2002
2004
2006
2008
1998
□ The steady-state solution of the Riccati equation 1 reveals that disagreement processes
are only correlated if the business cycle indicator is present. □ Empirical evidence points towards a strong common counter-cyclical component in
the cross-section of disagreement proxies! □ Simple firm-by-firm regressions from the common DiB on the firm specific DiB yields an
average beta of 0.64 with a t-stat of 13.86 and an adjusted 𝑅2 of 0.50. c (2009) Buraschi, Trojani, Vedolin – 11 ⃝
Uncertainty & Correlation in DiB Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
⊳
Empirical Analysis
We ask how we can generate co-movement given that fundamentals are very weakly linked. The correlation among firm-specific uncertainty depends crucially on two different quantities: ① The informativeness of the signal about the firms’ dividend growth rates, i.e. the size of the weights used for updating in the signal (𝛼𝐷𝑖 , 𝑖 = 1, 2 and 𝛽). ↶
Conclusion Appendix
② The amount of subjective economic uncertainty . In particular, we study the impact of ... – ... the average subjective ( ) uncertainty, defined as
𝜎 ¯𝜇𝑧 ≡ 0.5 𝜎𝜇𝐴𝑧 + 𝜎𝜇𝐵𝑧 and ...
– ... the difference in agents’ subjective uncertainty:
Δ𝜎𝜇𝑧 ≡ 𝜎𝜇𝐴𝑧 − 𝜎𝜇𝐵𝑧 .
c (2009) Buraschi, Trojani, Vedolin – 12 ⃝
Uncertainty & Correlation in DiB Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
⊳
Empirical Analysis
We ask how we can generate co-movement given that fundamentals are very weakly linked. The correlation among firm-specific uncertainty depends crucially on two different quantities: ① The informativeness of the signal about the firms’ dividend growth rates, i.e. the size of the weights used for updating in the signal (𝛼𝐷𝑖 , 𝑖 = 1, 2 and 𝛽). ↶
Conclusion Appendix
② The amount of subjective economic uncertainty . In particular, we study the impact of ... – ... the average subjective ( ) uncertainty, defined as
𝜎 ¯𝜇𝑧 ≡ 0.5 𝜎𝜇𝐴𝑧 + 𝜎𝜇𝐵𝑧 and ...
– ... the difference in agents’ subjective uncertainty:
Δ𝜎𝜇𝑧 ≡ 𝜎𝜇𝐴𝑧 − 𝜎𝜇𝐵𝑧 .
c (2009) Buraschi, Trojani, Vedolin – 12 ⃝
Uncertainty & Correlation in DiB Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
⊳
Empirical Analysis
We ask how we can generate co-movement given that fundamentals are very weakly linked. The correlation among firm-specific uncertainty depends crucially on two different quantities: ① The informativeness of the signal about the firms’ dividend growth rates, i.e. the size of the weights used for updating in the signal (𝛼𝐷𝑖 , 𝑖 = 1, 2 and 𝛽). ↶
Conclusion Appendix
② The amount of subjective economic uncertainty . In particular, we study the impact of ... – ... the average subjective ( ) uncertainty, defined as
𝜎 ¯𝜇𝑧 ≡ 0.5 𝜎𝜇𝐴𝑧 + 𝜎𝜇𝐵𝑧 and ...
– ... the difference in agents’ subjective uncertainty:
Δ𝜎𝜇𝑧 ≡ 𝜎𝜇𝐴𝑧 − 𝜎𝜇𝐵𝑧 .
c (2009) Buraschi, Trojani, Vedolin – 12 ⃝
Uncertainty Correlation cont’d Instantaneous Correlation between ΨD1 and Ψz
Instantaneous Correlation between ΨD1 and ΨD2
0.8
σ ¯µz = 0.10
ρ(dΨD1 dΨz )
0.35 0.3
ρ(dΨD1 dΨD2 )
σ ¯µz = 0.10
0.7
0.25 0.2 0.15
0.6 0.5 0.4 0.3 0.2
0.1 0.5
0.05 0.4
σ ¯µz = 0.02
0 0.1
0.3
0.2 0.3
0.1 0.5
σ ¯µz = 0.02 0.4
0.5 0.3
0.2 0.4
αD2
0.5
0.1
αD1
0.4 0.3
0.2
αD1
0.2 0.1
0.1
αD2
□
As 𝛼𝐷𝑖 approaches 0, the signal becomes non-informative about the expected dividend growth rate of 𝐷𝑖 (𝑡). ⇒ The correlation between the idiosyncratic disagreement processes is abysmally small.
□
Higher economic uncertainty implies hightened correlation. c (2009) Buraschi, Trojani, Vedolin – 13 ⃝
Conditional DiB Correlation: Common Component Conditional Correlation between DiBs
Motivation
Cond. Correl Financial & Consumer Services
Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
⊳
Empirical Analysis Conclusion Appendix
□
9/11/2001
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.4
□
1
−0.2
1999
1999.5
2000
2000.5
2001
2001.5
2002
Cond. Correl Consumer Services & Transportation
1
−0.4
9/11/2001: The conditional correlation between the sector specific DiBs increases on average sixfold. This increase is common to all sectors.
c (2009) Buraschi, Trojani, Vedolin – 14 ⃝
Conditional DiB Correlation: Idiosyncratic Component Motivation
1.2
Cond. Correl Financial & Consumer Services
Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
⊳
Empirical Analysis Conclusion Appendix
Credit Crisis 1
□
0.6
9/11/2001
0.8
0.4
0.6
0.2
0.4
0
0.2
−0.2
0
−0.4
−0.2
−0.6
−0.4
□
0.8
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Cond. Correl Consumer Services & Transportation
Conditional Correlation between DiBs
−0.8
Credit Crisis: The origin of this crisis lies in the financial sector. After the interventions in the banking sector, uncertainty in this sector vanishes while the repercussions on the real sector still remain ambiguous. c (2009) Buraschi, Trojani, Vedolin – 15 ⃝
Financial Markets and Equilibrium Preferences: Two groups of investors with life-time utility: Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
⊳
Empirical Analysis Conclusion Appendix
𝑉 𝑛 = sup
𝑛 𝑐𝑛 𝐷1 ,𝑐𝐷2
𝐸𝑛
(∫
0
∞
𝑒−𝛿𝑡
(
𝑐𝑛𝐷1 (𝑡)1−𝛾 1−𝛾
+
𝑛 1−𝛾 ) 𝑐𝐷2 (𝑡)
1−𝛾
) 𝑑𝑡 ℱ0𝑌 ,
(2) where 𝑐𝑛𝐷𝑖 (𝑡) is the consumption of agent 𝑛 = 𝐴, 𝐵 at time 𝑡 of good 𝑖, 𝛾 > 0 is the RRA, and 𝛿 ≥ 0 is the time preference rate. Financial market: An incomplete market, completed by the firm’s capital structure: □ □
A risk-free bond and a European stock option written on each stock of firm 𝑖 (in zero net supply). A stock for each firm 𝑖 (in positive supply).
Definition 1 An equilibrium is a price process for financial assets and individual consumption/portfolio policies such that: 1. Given equilibrium prices, all agents in the economy solve the optimization problem (2), subject to their budget constraint. 2. Good and financial markets clear. c (2009) Buraschi, Trojani, Vedolin – 16 ⃝
Equilibrium Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
Standard computations yield the equilibrium quantities: The investors’ state price densities are: 𝐴
⊳
Empirical Analysis
=
𝜉 𝐵 (𝑡)
=
The weighting process 𝜆(𝑡) = 𝑦𝐴 𝜉 𝐴 (𝑡)/(𝑦𝐵 𝜉 𝐵 (𝑡)) follows the dynamics:
Conclusion Appendix
𝜉 (𝑡)
( )𝛾 𝑒−𝛿𝑡 −𝛾 1/𝛾 𝐷1 (𝑡) 1 + 𝜆(𝑡) , 𝑦1 ( )𝛾 𝑒−𝛿𝑡 −𝛾 1/𝛾 𝐷1 (𝑡) 1 + 𝜆(𝑡) 𝜆(𝑡)−1 . 𝑦2
𝑑𝜆(𝑡) =− 𝜆(𝑡)
( 2 ∑
𝐴 Ψ𝐷𝑖 (𝑡)𝑑𝑊𝐷 (𝑡) + 𝑖
𝑖=1
(
2 ∑
𝛼𝐷𝑖 Ψ𝐷𝑖 (𝑡)
𝑖=1
)
)
𝜎𝐷𝑖 + 𝛽Ψ𝑧 (𝑡) 𝑑𝑊𝑧𝐴 (𝑡) . 𝜎𝑧
And the relative price of the second good is: 𝑟𝑝(𝑡) =
(
𝐷2 (𝑡) 𝐷1 (𝑡)
)−𝛾
.
Separability of the utility functions yields such a simple form!
c (2009) Buraschi, Trojani, Vedolin – 17 ⃝
Endogenous Stock Return Correlation Stock Return Correlation, αD1 = αD2 = 0.1
Stock Return Correlation, αD1 = αD2 = 0.45
Stock Return Correlation
σ ¯µz = 0.1 0.5 0.4 0.3 0.2 0.1 σ ¯µz = 0.02 0
0.3 0.225
0
0.15
0.075 0.15
ΨD1
0.075
0.225 0.3
0
Stock Return Correlation
0.7
0.6
σ ¯µz = 0.1
0.6 0.5 0.4 0.3 0.2 0.1
σ ¯µz = 0.02 0.3
0
0.225
0 0.075
Ψz
0.15 0.15
ΨD1
0.075
0.225 0.3
0
Ψz
Stock returns correlate because of a ... ① A market-clearing effect (see Cochrane et al., 2008). For symmetric economies this effect tends to be small. ② A risk-sharing effect. A higher Ψ(𝑡) implies a higher demand of the pessimist for protection from the optimist. ③ A DiB-comovement effect. c (2009) Buraschi, Trojani, Vedolin – 18 ⃝
Index and Individual Stock VolRP Index and Single Stock VolRP, αD1 = αD2 = 0.1 4
Option 1 (¯ σµz = 0.01) Index (¯ σµz = 0.01) Option 1 (¯ σµz = 0.10) Index (¯ σµz = 0.10)
6
Volatility Risk Premium, %
Volatility Risk Premium, %
7
Option 1 (¯ σµz = 0.01) Index (¯ σµz = 0.01) Option 1 (¯ σµz = 0.10) Index (¯ σµz = 0.10)
3.5
3
2.5
2
1.5
0.5
0 −1
Index and Single Stock VolRP, αD1 = αD2 = 0.45
5
4
3
2
1
−0.5
0
Moneyness
0.5
1
0 −1
−0.5
0
0.5
1
Moneyness
□
The smile of the index is steeper than for individual options due to the more negative skewness.
□
The wedge between the index and individual volatility risk premia is driven by a correlation risk premium.
□
Higher economic uncertainty increases this wedge. c (2009) Buraschi, Trojani, Vedolin – 19 ⃝
Risk Premia in a Simulated Economy Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
Goal: We want to design a trading strategy that takes advantage of the correlation risk premium. We do this ...
□
... by carrying out so called generalized dispersion trades with ATM straddles. This means we are short the index options while being long the individual constituents options.
□
To identify the impact of DiB, we sort the constituents by DiB. We do not trade the whole book of individual stocks due to spread risk. If our model is correct, then ...
□
the stocks with the lowest DiB, should carry the smallest volatility risk premia and hence be the cheapest options.
□
... and ex post returns from this strategy should still have a significant exposure to the Common DiB.
⊳
Empirical Analysis Conclusion Appendix
c (2009) Buraschi, Trojani, Vedolin – 20 ⃝
Risk Premia in a Simulated Economy Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
Goal: We want to design a trading strategy that takes advantage of the correlation risk premium. We do this ...
□
... by carrying out so called generalized dispersion trades with ATM straddles. This means we are short the index options while being long the individual constituents options.
□
To identify the impact of DiB, we sort the constituents by DiB. We do not trade the whole book of individual stocks due to spread risk. If our model is correct, then ...
□
the stocks with the lowest DiB, should carry the smallest volatility risk premia and hence be the cheapest options.
□
... and ex post returns from this strategy should still have a significant exposure to the Common DiB.
⊳
Empirical Analysis Conclusion Appendix
c (2009) Buraschi, Trojani, Vedolin – 20 ⃝
Risk Premia in a Simulated Economy Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
Goal: We want to design a trading strategy that takes advantage of the correlation risk premium. We do this ...
□
... by carrying out so called generalized dispersion trades with ATM straddles. This means we are short the index options while being long the individual constituents options.
□
To identify the impact of DiB, we sort the constituents by DiB. We do not trade the whole book of individual stocks due to spread risk. If our model is correct, then ...
□
the stocks with the lowest DiB, should carry the smallest volatility risk premia and hence be the cheapest options.
□
... and ex post returns from this strategy should still have a significant exposure to the Common DiB.
⊳
Empirical Analysis Conclusion Appendix
c (2009) Buraschi, Trojani, Vedolin – 20 ⃝
Risk Premia in a Simulated Economy Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
Goal: We want to design a trading strategy that takes advantage of the correlation risk premium. We do this ...
□
... by carrying out so called generalized dispersion trades with ATM straddles. This means we are short the index options while being long the individual constituents options.
□
To identify the impact of DiB, we sort the constituents by DiB. We do not trade the whole book of individual stocks due to spread risk. If our model is correct, then ...
□
the stocks with the lowest DiB, should carry the smallest volatility risk premia and hence be the cheapest options.
□
... and ex post returns from this strategy should still have a significant exposure to the Common DiB.
⊳
Empirical Analysis Conclusion Appendix
c (2009) Buraschi, Trojani, Vedolin – 20 ⃝
Simulated Option Trading Strategies Motivation Model Model 𝑚(𝑡) Ψ(𝑡) Common DiB Definition Equilibrium Return Corr IV MC
Return Standard Deviation Skewness Kurtosis Annualized Sharpe Ratio CAPM Alpha t-Stat CAPM Beta t-Stat Common DiB Beta t-Stat
⊳
Empirical Analysis
DiB Sorted Strategy
Index
Short Index Put
0.127 0.231 -3.127 7.532 1.867 0.091★★ 2.26 0.172 0.89 1.293★★★ 2.82
0.010 0.038 -7.423 5.819 0.683
0.117 0.653 -8.239 8.477 0.607 0.070★ 1.71 2.182★ 1.83 2.193★★ 2.28
Conclusion Appendix
□
Annualized Sharpe ratios are above 1.
□
Dispersion trades involve smaller third and fourth moments than the index strategies.
□
The CAPM Beta is not statistically significant, indicating that market risk is hedged away.
□
The exposure to the Common DiB is priced. c (2009) Buraschi, Trojani, Vedolin – 21 ⃝
Motivation Model
⊳ Empirical Analysis Data Reg Cheapest Trading Conclusion Appendix
Empirical Analysis
c (2009) Buraschi, Trojani, Vedolin – 22 ⃝
Data Motivation Model Empirical Analysis Data Reg Cheapest Trading
⊳
Conclusion Appendix
Options Data: From the Optionsmetrics IVY DB we take S&P 100 index option information and on all constituents. We use the mid point of the bid ask spread as the option’s price. Stock Returns Data: From CRSP we take the stock price information for the individual stocks. From Optionmetrics we get the index price information. Realized volatility is calculated over a 21-day window. Difference in Beliefs Proxy: Individual Firms: From the I/B/E/S database we use analysts forecasts of earnings per share and compute for each firm the mean absolute difference scaled by an indicator of earnings uncertainty. Index: Using the cross-section of all constituents on the S&P 100, we build a market-wide DiB index using dynamic factor analysis. Other: Announcement dates, skewness measures, Fama and French and Carhart factor, macro factors, earnings uncertainty. c (2009) Buraschi, Trojani, Vedolin – 23 ⃝
Empirical Evidence VolRP & CorrRP Individual:
Index:
Correlation:
7 ∑
𝑅𝑉𝑖,𝑡 − 𝐼𝑉𝑖,𝑡 = 𝛽0 + 𝛽1 𝐷𝐼𝐵𝑖,𝑡 + 𝛽2 𝐷𝐼𝐵 𝑡 + 𝛽𝑗 Control(𝑗)𝑖,𝑡 + 𝛾𝑘 Control𝑡 + 𝜖𝑖,𝑡 , {z } | 𝑗=3 𝑘=1 Volatility Risk Premium 7 ∑ 𝑅𝑉𝑡 − 𝐼𝑉𝑡 = 𝛽0 + 𝛽1 𝐷𝐼𝐵𝑡 + 𝛽𝑘 Control𝑡 + 𝜖𝑡 | {z } 𝑘=2 Index Volatility Risk Premium 3 ∑ 𝐶𝑂𝑅𝑅𝑃𝑡 = 𝛽0 + 𝛽1 𝐷𝐼𝐵𝑡 + 𝛽𝑘 Control𝑡 + 𝜖𝑡 , 𝑘=2
Individual Constant DiB Common DiB Market Vola Macro Factor Liquidity
−0.001★★★ (-8.21) −0.037★★★ (-22.18) −0.012★★ (-2.37) −0.028★★ (-2.45) 0.031★★ (2.25) 0.002★ (1.92)
CAPM Beta Skewness Adj. 𝑅2
2 ∑
0.012★ (1.71) 0.26
−0.002★★ (-2.38) −0.028★★★ (-8.93) −0.018★★ (-2.39) 0.019★ (1.93) 0.001★ (1.88) −0.021★ (-1.72) 0.010 (1.50) 0.30
Index
Correlation
−0.001★★★ (-7.89)
−0.002★★★ (-2.71)
−0.028★★★ (-5.28) −0.097★★★ (-3.01) 0.010★★ (2.29)
−0.098★★ (-2.12) −0.041★★ (-2.35) 0.018 (1.49)
0.014★★ (1.89) 0.22
0.08
c (2009) Buraschi, Trojani, Vedolin – 24 ⃝
Enhanced Dispersion Trades Motivation Model Empirical Analysis Data Reg Cheapest Trading
⊳
Conclusion Appendix
To determine the cheapest options in our data sample, each month, we sort according to the following risk factors : ① Beta: The investor calculates monthly CAPM betas and chooses the options of the firms in the decile with the lowest estimated beta exposure. ② Liquidity: This investor builds a monthly measure of liquidity, proxied by the monthly average of the daily trading volume divided by the number of shares outstanding. She then chooses the options in the highest liquidity decile. ③ DiB: This investor is long the options of stocks in the lowest DiB-sorted decile. As a “benchmark” strategy we apply a sorting according to the volatility risk premium.
c (2009) Buraschi, Trojani, Vedolin – 25 ⃝
Enhanced Dispersion Trades Motivation Model Empirical Analysis Data Reg Cheapest Trading
⊳
Conclusion Appendix
To determine the cheapest options in our data sample, each month, we sort according to the following risk factors : ① Beta: The investor calculates monthly CAPM betas and chooses the options of the firms in the decile with the lowest estimated beta exposure. ② Liquidity: This investor builds a monthly measure of liquidity, proxied by the monthly average of the daily trading volume divided by the number of shares outstanding. She then chooses the options in the highest liquidity decile. ③ DiB: This investor is long the options of stocks in the lowest DiB-sorted decile. As a “benchmark” strategy we apply a sorting according to the volatility risk premium.
c (2009) Buraschi, Trojani, Vedolin – 25 ⃝
Enhanced Dispersion Trades Motivation Model Empirical Analysis Data Reg Cheapest Trading
⊳
Conclusion Appendix
To determine the cheapest options in our data sample, each month, we sort according to the following risk factors : ① Beta: The investor calculates monthly CAPM betas and chooses the options of the firms in the decile with the lowest estimated beta exposure. ② Liquidity: This investor builds a monthly measure of liquidity, proxied by the monthly average of the daily trading volume divided by the number of shares outstanding. She then chooses the options in the highest liquidity decile. ③ DiB: This investor is long the options of stocks in the lowest DiB-sorted decile. As a “benchmark” strategy we apply a sorting according to the volatility risk premium.
c (2009) Buraschi, Trojani, Vedolin – 25 ⃝
Enhanced Dispersion Trades Motivation Model Empirical Analysis Data Reg Cheapest Trading
⊳
Conclusion Appendix
To determine the cheapest options in our data sample, each month, we sort according to the following risk factors : ① Beta: The investor calculates monthly CAPM betas and chooses the options of the firms in the decile with the lowest estimated beta exposure. ② Liquidity: This investor builds a monthly measure of liquidity, proxied by the monthly average of the daily trading volume divided by the number of shares outstanding. She then chooses the options in the highest liquidity decile. ③ DiB: This investor is long the options of stocks in the lowest DiB-sorted decile. As a “benchmark” strategy we apply a sorting according to the volatility risk premium.
c (2009) Buraschi, Trojani, Vedolin – 25 ⃝
Enhanced Dispersion Trades Motivation Model Empirical Analysis Data Reg Cheapest Trading
⊳
Conclusion Appendix
To determine the cheapest options in our data sample, each month, we sort according to the following risk factors : ① Beta: The investor calculates monthly CAPM betas and chooses the options of the firms in the decile with the lowest estimated beta exposure. ② Liquidity: This investor builds a monthly measure of liquidity, proxied by the monthly average of the daily trading volume divided by the number of shares outstanding. She then chooses the options in the highest liquidity decile. ③ DiB: This investor is long the options of stocks in the lowest DiB-sorted decile. As a “benchmark” strategy we apply a sorting according to the volatility risk premium.
c (2009) Buraschi, Trojani, Vedolin – 25 ⃝
Correlation Trading Strategies
Mean StDev Skewness Kurtosis Sharpe Ratio CAPM Alpha t-Stat CAPM Beta t-Stat Common DiB Beta t-Stat Liq Beta t-Stat
DiB sorted Strategy
Beta sorted Strategy
Liquidity sorted Strategy
Na¨ıve Strategy
0.1654 0.3795 -0.1226 0.0593 1.5097 0.1347★★★ 2.67 0.1604 0.16 0.4778★★ 1.97 0.6864 1.07
0.0370 0.4765 -0.1852 0.1583 0.2695 0.0308 0.77 -0.2167 -0.22 0.2159 1.44 0.6499 1.01
0.0226 0.4091 -0.6906 1.1180 0.1915 0.0196 0.47 0.6027 0.73 0.3334★★ 1.99 0.1151 0.21
0.0952 0.4170 -0.0779 0.8430 0.7908 0.0363 0.84 -0.2379 -0.28 2.5191★★★ 2.57 0.0922 0.16
S&P 100
-0.0043 0.0135 -1.6909 7.6652 -1.1110
Short Index Put
0.1753 0.5498 -0.5595 0.4942 1.1046 0.1253★ 1.93 14.4033★★★ 3.36 0.4432★ 1.78 0.5978 0.81
□
Annualized Sharpe ratios are above one and the strategy out-performs a selling all put index strategy.
□
The CAPM Beta is not statistically significant which indicates that we successfully hedged away the market risk.
□
However, the exposure to Common DiB is economically and statistically significant.
c (2009) Buraschi, Trojani, Vedolin – 26 ⃝
Correlation Trading Strategies
Mean StDev Skewness Kurtosis Sharpe Ratio CAPM Alpha t-Stat CAPM Beta t-Stat Common DiB Beta t-Stat Liq Beta t-Stat
DiB sorted Strategy
Beta sorted Strategy
Liquidity sorted Strategy
Na¨ıve Strategy
0.1654 0.3795 -0.1226 0.0593 1.5097 0.1347★★★ 2.67 0.1604 0.16 0.4778★★ 1.97 0.6864 1.07
0.0370 0.4765 -0.1852 0.1583 0.2695 0.0308 0.77 -0.2167 -0.22 0.2159 1.44 0.6499 1.01
0.0226 0.4091 -0.6906 1.1180 0.1915 0.0196 0.47 0.6027 0.73 0.3334★★ 1.99 0.1151 0.21
0.0952 0.4170 -0.0779 0.8430 0.7908 0.0363 0.84 -0.2379 -0.28 2.5191★★★ 2.57 0.0922 0.16
S&P 100
-0.0043 0.0135 -1.6909 7.6652 -1.1110
Short Index Put
0.1753 0.5498 -0.5595 0.4942 1.1046 0.1253★ 1.93 14.4033★★★ 3.36 0.4432★ 1.78 0.5978 0.81
□
Annualized Sharpe ratios are above one and the strategy out-performs a selling all put index strategy.
□
The CAPM Beta is not statistically significant which indicates that we successfully hedged away the market risk.
□
However, the exposure to Common DiB is economically and statistically significant.
c (2009) Buraschi, Trojani, Vedolin – 26 ⃝
Correlation Trading Strategies
Mean StDev Skewness Kurtosis Sharpe Ratio CAPM Alpha t-Stat CAPM Beta t-Stat Common DiB Beta t-Stat Liq Beta t-Stat
DiB sorted Strategy
Beta sorted Strategy
Liquidity sorted Strategy
Na¨ıve Strategy
0.1654 0.3795 -0.1226 0.0593 1.5097 0.1347★★★ 2.67 0.1604 0.16 0.4778★★ 1.97 0.6864 1.07
0.0370 0.4765 -0.1852 0.1583 0.2695 0.0308 0.77 -0.2167 -0.22 0.2159 1.44 0.6499 1.01
0.0226 0.4091 -0.6906 1.1180 0.1915 0.0196 0.47 0.6027 0.73 0.3334★★ 1.99 0.1151 0.21
0.0952 0.4170 -0.0779 0.8430 0.7908 0.0363 0.84 -0.2379 -0.28 2.5191★★★ 2.57 0.0922 0.16
S&P 100
-0.0043 0.0135 -1.6909 7.6652 -1.1110
Short Index Put
0.1753 0.5498 -0.5595 0.4942 1.1046 0.1253★ 1.93 14.4033★★★ 3.36 0.4432★ 1.78 0.5978 0.81
□
Annualized Sharpe ratios are above one and the strategy out-performs a selling all put index strategy.
□
The CAPM Beta is not statistically significant which indicates that we successfully hedged away the market risk.
□
However, the exposure to Common DiB is economically and statistically significant.
c (2009) Buraschi, Trojani, Vedolin – 26 ⃝
Correlation Trading Strategies
Mean StDev Skewness Kurtosis Sharpe Ratio CAPM Alpha t-Stat CAPM Beta t-Stat Common DiB Beta t-Stat Liq Beta t-Stat
DiB sorted Strategy
Beta sorted Strategy
Liquidity sorted Strategy
Na¨ıve Strategy
0.1654 0.3795 -0.1226 0.0593 1.5097 0.1347★★★ 2.67 0.1604 0.16 0.4778★★ 1.97 0.6864 1.07
0.0370 0.4765 -0.1852 0.1583 0.2695 0.0308 0.77 -0.2167 -0.22 0.2159 1.44 0.6499 1.01
0.0226 0.4091 -0.6906 1.1180 0.1915 0.0196 0.47 0.6027 0.73 0.3334★★ 1.99 0.1151 0.21
0.0952 0.4170 -0.0779 0.8430 0.7908 0.0363 0.84 -0.2379 -0.28 2.5191★★★ 2.57 0.0922 0.16
S&P 100
-0.0043 0.0135 -1.6909 7.6652 -1.1110
Short Index Put
0.1753 0.5498 -0.5595 0.4942 1.1046 0.1253★ 1.93 14.4033★★★ 3.36 0.4432★ 1.78 0.5978 0.81
□
Annualized Sharpe ratios are above one and the strategy out-performs a selling all put index strategy.
□
The CAPM Beta is not statistically significant which indicates that we successfully hedged away the market risk.
□
However, the exposure to Common DiB is economically and statistically significant.
c (2009) Buraschi, Trojani, Vedolin – 26 ⃝
Correlation Trading Strategies
Mean StDev Skewness Kurtosis Sharpe Ratio CAPM Alpha t-Stat CAPM Beta t-Stat Common DiB Beta t-Stat Liq Beta t-Stat
DiB sorted Strategy
Beta sorted Strategy
Liquidity sorted Strategy
Na¨ıve Strategy
0.1654 0.3795 -0.1226 0.0593 1.5097 0.1347★★★ 2.67 0.1604 0.16 0.4778★★ 1.97 0.6864 1.07
0.0370 0.4765 -0.1852 0.1583 0.2695 0.0308 0.77 -0.2167 -0.22 0.2159 1.44 0.6499 1.01
0.0226 0.4091 -0.6906 1.1180 0.1915 0.0196 0.47 0.6027 0.73 0.3334★★ 1.99 0.1151 0.21
0.0952 0.4170 -0.0779 0.8430 0.7908 0.0363 0.84 -0.2379 -0.28 2.5191★★★ 2.57 0.0922 0.16
S&P 100
-0.0043 0.0135 -1.6909 7.6652 -1.1110
Short Index Put
0.1753 0.5498 -0.5595 0.4942 1.1046 0.1253★ 1.93 14.4033★★★ 3.36 0.4432★ 1.78 0.5978 0.81
□
Annualized Sharpe ratios are above one and the strategy out-performs a selling all put index strategy.
□
The CAPM Beta is not statistically significant which indicates that we successfully hedged away the market risk.
□
However, the exposure to Common DiB is economically and statistically significant.
c (2009) Buraschi, Trojani, Vedolin – 26 ⃝
Motivation Model Empirical Analysis
⊳ Conclusion Conclusion Thank You Appendix
Conclusion
c (2009) Buraschi, Trojani, Vedolin – 27 ⃝
Conclusion Motivation Model Empirical Analysis Conclusion Conclusion Thank You
⊳
Appendix
Z Disagreement is a key priced state variable that drives the second moments of stock returns, as well as the volatility risk premia of individual and index options. Z Stochastic correlation arises endogenously in our economy due to diverging optimal consumer policies. Z The wedge between the index and constituents volatility risk premia is driven by a correlation risk premium. This premium depends positively on disagreement, the amount of economic uncertainty, and the precision of the business-cycle indicator. Z Using a comprehensive data set, we find empirical evidence of the above hypotheses. Z A trading strategy exploiting the difference in the volatility risk premium of index and individual options yields economically significant returns even in the presence of transaction costs.
c (2009) Buraschi, Trojani, Vedolin – 28 ⃝
Thank You! Motivation Model Empirical Analysis Conclusion Conclusion Thank You
⊳
Appendix
Thank you very much for coming!
c (2009) Buraschi, Trojani, Vedolin – 29 ⃝
Motivation Model Empirical Analysis Conclusion
⊳ Appendix Appendix
Appendix
c (2009) Buraschi, Trojani, Vedolin – 30 ⃝
Appendix: Dispersion Trading Motivation
Consider a delta-hedged portfolio which is long the stock option and short the index. The P&L of individual stock is: ⎡
Model Empirical Analysis
⎢ ⎢ P&L = 𝜃 ⎢ ⎣
Conclusion Appendix Appendix
⊳
⎤
(
)2 𝑑𝑆 √ 𝑆𝜎 𝑑𝑡 | {z }
Standardized move of underlying stock 𝑆
⎥ ⎥ −1⎥ , ⎦
where 𝜃 is defined as the options sensitivity with respect to a change in the time to maturity. The P&L of the index is then simply a weighted sum of the individual stock P&Ls: P&LIndex
𝑛 ∑ ∑ 𝜔 𝑖 𝜔 𝑗 𝜎𝑖 𝜎𝑗 ) 𝜔𝑖2 𝜎𝑖2 ( 2 = 𝜃𝐼 𝑛 𝑖 − 1 + 𝜃𝐼 (𝑛𝑖 𝑛𝑗 − 𝛿𝑖𝑗 ) . 2 2 𝜎 𝜎 𝐼𝑛𝑑𝑒𝑥 𝑖=1 𝐼𝑛𝑑𝑒𝑥 𝑖∕=𝑗
Hence, the dispersion trade itself has the following P&L: P&LDispTrade =
𝑛 ∑ 𝑖=1
↶
𝜃𝑖
(
𝑛2𝑖
)
− 1 + 𝜃𝐼
(
𝑛2𝐼
)
−1 .
c (2009) Buraschi, Trojani, Vedolin – 31 ⃝
