Title: A New Class of Asset Pricing Models with Lvy processes: Theory and Applications
1A New Class of Asset Pricing Models with Lévy
processes Theory and Applications
www.ornthanalai.com
FMA Dallas Meeting, Sept 9th 2008
22/17
Another Black Monday for Wall Street ?
Major indexes suffered their biggest percentage
declines since the 1987 crash as the government's
bailout plan was defeated Monday -Business
Week, September 29th 2008
32/17
Another Black Monday for Wall Street ?
Major indexes suffered their biggest percentage
declines since the 1987 crash as the government's
bailout plan was defeated Monday -Business
Week, September 29th 2008
43/17
The Issues in Modeling Asset prices
Stylized facts of asset returns
- Presence of jumps in returns
- Return volatility is time-varying
- Leverage effect
- Volatility also jumps ?
53/17
The Issues in Modeling Asset prices
Stylized facts of asset returns
- Presence of jumps in returns
- Return volatility is time-varying
- Leverage effect
- Volatility also jumps ?
Derivative valuation
- Asset return dynamics are different under the
risk-neutral probability - How to economically link the risk-neutral and
realized return dynamics together?
64/17
Objectives of this paper
- Develop a new class of discrete-time asset
pricing models - Capturing the key stylized facts of asset
returns - Simple (analytical) valuation of various
derivatives - Ease of implementation
- Conduct empirical studies on index returns and
options data - Specification analysis of different jump
dynamics - Estimation of the risk premia? What is the
premium for bearing the market jump risk?
74/17
Objectives of this paper
- Develop a new class of discrete-time asset
pricing models - Capturing the key stylized facts of asset
returns - Simple (analytical) valuation of various
derivatives - Ease of implementation
- Conduct empirical studies on index returns and
options data - Specification analysis of different jump
dynamics - Estimation of the risk premia
- What is the premium for bearing the market jump
risk?
85/17
Construction of the Models
95/17
Construction of the Models
105/17
Construction of the Models
116/17
Application of Lévy GARCH models
- Joint MLE of SP 500 index Options and Returns
- This method helps with the identification of the
pricing kernel that underlies the two markets - I use 10 years of time-series cross-sectional
data - The most extensive joint estimation in the
literature - Pan (2002) and Eraker (2004) also apply joint
estimation method on smaller data sets - I obtain precise estimates of the long-run
market risk premia - The most important issue in empirical option
pricing research
127/17
Empirical findings from Joint Estimation
- Data favor small jumps that arrive at high
frequency - Models that use small frequent jumps
(Infinite-activity) outperform those that use
large and rare jumps (Poisson) in fitting returns
as well as pricing options -
- Jump risk is economically important
- You need jump risk to reconcile the difference
between investors belief of the returns and the
actual realized returns - Models without the jump risk factor cannot
jointly fit the returns and options data with an
economically reasonable level of the equity
premium
137/17
Empirical findings from Joint Estimation
- Data favor small jumps that arrive at high
frequency - Models that use small frequent jumps
(Infinite-activity) outperform those that use
large and rare jumps (Poisson) in fitting returns
as well as pricing options -
- Jump risk is economically important
- Models without the jump risk factor cannot
jointly fit the returns and options data with an
economically reasonable level of the equity
premium - You need jump risk to reconcile investors belief
of the stock returns with their actual realized
returns
148/17
Methodological Contributions
- I provide a general solution for the pricing
transform of asset return dynamics - The setup allows for a wide variety of asset
return specifications - Closed-form valuation of various derivatives
- I develop the risk-neutralization framework for
this class of models - The models allow for time-varying equity risk
premium - The equity return risk premium is tractable and
has an affine structure - The framework can be applied to nonaffine GARCH
dynamics - Derivatives can be priced using through Monte
Carlo simulation - Paramount importance
158/17
Methodological Contributions
- I provide a general solution for the pricing
transform of asset return dynamics - The setup allows for a wide variety of asset
return specifications - Closed-form valuation of various derivatives
- I develop the risk-neutralization framework for
this class of models - The models allow for time-varying equity risk
premium - The equity return risk premium is tractable and
has an affine structure - The framework can be applied to nonaffine GARCH
dynamics - Derivatives can be priced using through Monte
Carlo simulation - Paramount importance
168/17
Methodological Contributions
- I provide a general solution for the pricing
transform of asset return dynamics - The setup allows for a wide variety of asset
return specifications - Closed-form valuation of various derivatives
- I develop the risk-neutralization framework for
this class of models - The models allow for time-varying equity risk
premium - The equity return risk premium is tractable and
has an affine structure - The framework can be applied to nonaffine GARCH
dynamics - Derivatives can be priced using through Monte
Carlo simulation
179/17
Setup of the Empirics Return Dynamic
- I study two-factor return models that consist of
a Normal and a Jump component - The return dynamic
- The conditional equity premium dynamic
Normal Risk factor
Jump Risk factor
189/17
Setup of the Empirics Return Dynamic
- I study two-factor return models that consist of
a Normal and a Jump component - The return dynamic
- The conditional equity premium dynamic
Diffusive variance
Jump intensity
Normal Risk factor
Jump Risk factor
199/17
Setup of the Empirics Return Dynamic
- I study two-factor return models that consist of
a Normal and a Jump component - The return dynamic
- The conditional equity premium dynamic
Diffusive variance
Jump intensity
Normal Risk factor
Jump Risk factor
209/17
Setup of the Empirics Return Dynamic
- I study two-factor return models that consist of
a Normal and a Jump component - The return dynamic
- The conditional equity premium dynamic
Diffusive variance
Jump intensity
Normal Risk factor
Jump Risk factor
219/17
Setup of the Empirics Return Dynamic
- I study two-factor return models that consist of
a Normal and a Jump component - The return dynamic
- The conditional equity premium dynamic
Diffusive variance
Jump intensity
Normal Risk factor
Jump Risk factor
229/17
Setup of the Empirics Return Dynamic
- I study two-factor return models that consist of
a Normal and a Jump component - The return dynamic
- The conditional equity premium dynamic
Diffusive variance
Jump intensity
239/17
Setup of the Empirics Return Dynamic
- I study two-factor return models that consist of
a Normal and a Jump component - The return dynamic
- The conditional equity premium dynamic
Diffusive variance
Jump intensity
Normal Risk factor
Jump Risk factor
2410/17
Setup of the Empirics Other Specifications
- Source of heteroskedasticity
- Time-varying dynamics in the jump and normal
components
2510/17
Setup of the Empirics Other Specifications
- Source of heteroskedasticity
- Time-varying dynamics in the jump and normal
components
2610/17
Setup of the Empirics Other Specifications
- Source of heteroskedasticity
- Time-varying dynamics in the jump and normal
components
2710/17
Setup of the Empirics Other Specifications
- Source of heteroskedasticity
- Time-varying dynamics in the jump and normal
components
- Jump specifications
- I study two types of jump process
- Large and infrequent jumps Merton jump process
(MJ-LGARCH) - Small and high frequency jumps Normal Inverse
Gaussian process (NIG-LGARCH)
2810/17
Setup of the Empirics Other Specifications
- Source of heteroskedasticity
- Time-varying dynamics in the jump and normal
components
- Jump specifications
- I study two types of jump process
- Large and infrequent jumps Merton jump process
(MJ-LGARCH) - Small and high frequency jumps Normal Inverse
Gaussian process (NIG-LGARCH)
2911/17
MLE of daily SP 500 returns Results
Merton jump
NIG jump
No jump (GARCH)
Conditional daily return variance
Jump component of daily returns
Standardized normal component of daily returns
Log likelihood
NIG jump gt MJ jump gtgt No jump
3011/17
MLE of daily SP 500 returns Results
Merton jump
NIG jump
No jump (GARCH)
Conditional daily return variance
Jump component of daily returns
Standardized normal component of daily returns
Log likelihood
NIG jump gt MJ jump gtgt No jump
3111/17
MLE of daily SP 500 returns Results
Merton jump
NIG jump
No jump (GARCH)
Conditional daily return variance
Jump component of daily returns
Standardized normal component of daily returns
Log likelihood
NIG jump gt MJ jump gtgt No jump
3211/17
MLE of daily SP 500 returns Results
Merton jump
NIG jump
No jump (GARCH)
Conditional daily return variance
Jump component of daily returns
Standardized normal component of daily returns
Log likelihood
NIG jump gt MJ jump gtgt No jump
3311/17
MLE of daily SP 500 returns Results
Merton jump
NIG jump
No jump (GARCH)
Conditional daily return variance
Jump component of daily returns
Standardized normal component of daily returns
Log likelihood
NIG jump gt MJ jump gtgt No jump
3411/17
MLE of daily SP 500 returns Results
Merton jump
NIG jump
No jump (GARCH)
Conditional daily return variance
Jump component of daily returns
Standardized normal component of daily returns
Log likelihood
NIG jump gt MJ jump gtgt No jump
3511/17
MLE of daily SP 500 returns Results
Merton jump
NIG jump
No jump (GARCH)
Conditional daily return variance
Jump component of daily returns
Standardized normal component of daily returns
Log likelihood
NIG jump gt MJ jump gtgt No jump
3611/17
MLE of daily SP 500 returns Results
Merton jump
NIG jump
No jump (GARCH)
Conditional daily return variance
Jump component of daily returns
Standardized normal component of daily returns
Log likelihood
NIG jump gt MJ jump gtgt No jump
3711/17
MLE of daily SP 500 returns Results
Merton jump
NIG jump
No jump (GARCH)
Conditional daily return variance
Jump component of daily returns
Standardized normal component of daily returns
Log likelihood
NIG jump gt MJ jump gtgt No jump
3812/17
Joint Options Returns MLE Methodology
- Weekly Wednesday index call options from
1996-2005, with the total of 21,718 contracts - Daily index returns from 1995-2005
- I use IV loss function as the option pricing
error - Joint estimation is done by maximizing
Risk-neutral return distribution
Realized return distribution
3912/17
Joint Options Returns MLE Methodology
- Weekly Wednesday index call options from
1996-2005, with the total of 21,718 contracts - Daily index returns from 1995-2005
- I use IV loss function as the option pricing
error - Joint estimation is done by maximizing
Risk-neutral return distribution
Realized return distribution
4012/17
Joint Options Returns MLE Methodology
- Weekly Wednesday index call options from
1996-2005, with the total of 21,718 contracts - Daily index returns from 1995-2005
- I use IV loss function as the option pricing
error - Joint estimation is done by maximizing
Risk-neutral return distribution
Realized return distribution
4112/17
Joint Options Returns MLE Methodology
- Weekly Wednesday index call options from
1996-2005, with the total of 21,718 contracts - Daily index returns from 1995-2005
- I use IV loss function as the option pricing
error - Joint estimation is done by maximizing
Risk-neutral return distribution
Realized return distribution
4213/17
Joint MLE Results The Implied Equity Premium
- Decomposition of the long-run return risk
premium implied by different models
What do investors demand in excess return for
bearing the market risks ?
- Comparison of the Log Likelihood values
NIG jump gtgtgt MJ jump gtgt No jump
115599 112880 112207
4313/17
Joint MLE Results The Implied Equity Premium
- Decomposition of the long-run return risk
premium implied by different models
What do investors demand in excess return for
bearing the market risks ?
4413/17
Joint MLE Results The Implied Equity Premium
- Decomposition of the long-run return risk
premium implied by different models
What do investors demand in excess return for
bearing the market risks ?
- Comparison of the Log Likelihood values
NIG jump gtgtgt MJ jump gtgt No jump
115599 112880 112207
4514/17
Lévy GARCH versus Standard GARCH models
- Standard GARCH models produce excessively high
conditional volatility - Lévy GARCH models do not suffer from this problem
News Impact Curve
Percentage change in next periods volatility
4614/17
Lévy GARCH versus Standard GARCH models
- Standard GARCH models produce excessively high
conditional volatility - Lévy GARCH models do not suffer from this problem
News Impact Curve
Percentage change in next periods volatility
4715/17
Conclusions and Future Research
- I introduce a new class of discrete-time asset
pricing models - Easy to implement
- Lead to simple valuation of derivatives
- Can capture the key stylized facts of asset
returns - Empirical studies on the index options show that
- Jumps in the market index are small and arrive at
high frequency - Jump risk factor is economically important
- The framework has many applications in empirical
asset pricing - Study of jump specifications in equity returns
- Price credit derivative products, i.e. CDS
- Joint estimation using options and returns data
4815/17
Conclusions and Future Research
- I introduce a new class of discrete-time asset
pricing models - Easy to implement
- Lead to simple valuation of derivatives
- Can capture the key stylized facts of asset
returns - Empirical studies on the index options show that
- Jumps in the market index are small and arrive at
high frequency - Jump risk factor is economically important
- I find that jump risk premium is priced at 3-5
in annualized term - The framework has many applications in empirical
asset pricing - Study of jump specifications in equity returns
- Price credit derivative products, i.e. CDS
- Joint estimation using options and returns data
4915/17
Conclusions and Future Research
- I introduce a new class of discrete-time asset
pricing models - Easy to implement
- Lead to simple valuation of derivatives
- Can capture the key stylized facts of asset
returns - Empirical studies on the index options show that
- Jumps in the market index are small and arrive at
high frequency - Jump risk factor is economically important
- I find that jump risk premium is priced at 3-5
in annualized term - The framework has many applications in empirical
asset pricing - Study of jump specifications in equity returns
- Price credit derivative products, i.e. CDS
- Joint estimation using options and returns data
5016/17
Future Research Why use Joint Estimation?
- Estimation of time-series relation between
expected return and risk is difficult - Empirical studies are still not in agreement, see
Bali and Engle (2008) for review of the
literature - Option prices and their underlying assets share
the same risk factors - Options data can be used to increase the
precision in the estimation of the risk-return
relation - Options data are rich cross-sectional and
time-series - Joint estimation can be used to estimate the
pricing kernel - How are the risk factors priced in the ICAPM
framework? - Comparing this method to the Fama-Macbeth
regression
5117/17
Motivations and Literature Review
- The growing literature that supports the use of
Lévy processes to model stock returns and price
derivatives - Carr and Wu (2003ab, 2004), Carr, Geman, Madan
and Yorr (2002), Bakshi, Carr and Wu (2008),
Huang and Wu (2004), Li, Wells and Yu (2008)
...... - The success of time-changed Lévy processes in
derivative valuation Time-changed Lévy
processes Lévy processes SV - The models are difficult to implement and
estimate - Empirical research in this area is hindered by
econometric difficulty - Bates (2008) is the first to estimate
time-changed Lévy processes using returns data
5217/17
Motivations and Literature Review
- The growing literature that supports the use of
Lévy processes to model stock returns and price
derivatives - Carr and Wu (2003ab, 2004), Carr, Geman, Madan
and Yorr (2002), Bakshi, Carr and Wu (2008),
Huang and Wu (2004), Li, Wells and Yu (2008)
...... - The success of time-changed Lévy processes in
option valuation Time-changed Lévy processes
Lévy processes SV - The models are difficult to implement and
estimate - Empirical research in this area is hindered by
econometric difficulty - Bates (2008) is the first to estimate
time-changed Lévy processes using returns data
5317/17
Motivations and Literature Review
- The growing literature that supports the use of
Lévy processes to model stock returns and price
derivatives - Carr and Wu (2003ab, 2004), Carr, Geman, Madan
and Yorr (2002), Bakshi, Carr and Wu (2008),
Huang and Wu (2004), Li, Wells and Yu (2008)
...... - The success of time-changed Lévy processes in
option valuation Time-changed Lévy processes
Lévy processes SV - The models are difficult to implement and
estimate - Empirical research in this area is hindered by
econometric difficulty - Bates (2008) is the first to estimate
time-changed Lévy processes using returns data
5417/17
Motivations and Literature Review
- The growing literature that supports the use of
Lévy processes to model stock returns and price
derivatives - Carr and Wu (2003ab, 2004), Carr, Geman, Madan
and Yorr (2002), Bakshi, Carr and Wu (2008),
Huang and Wu (2004), Li, Wells and Yu (2008)
...... - The success of time-changed Lévy processes in
option valuation Time-changed Lévy processes
Lévy processes SV - The models are difficult to implement and
estimate - Empirical research in this area is hindered by
econometric difficulty - Bates (2008) is the first to estimate
time-changed Lévy processes using returns data
The Lévy GARCH models offer the versatility
that equates to time-changed Lévy processes, but
are simpler to implement
5517/17
Application of Lévy GARCH models
- Joint MLE of SP 500 index Options and Returns
- This method helps with the identification of
the pricing kernel that links the two markets - I obtain precise estimates of the long-run
market risk premia - The most important issue in empirical option
pricing research - I use 10 years of time-series cross-sectional
data - The most extensive joint estimation in the
literature - Pan (2002) and Eraker (2004) also apply joint
estimation method on smaller data sets