A New Class of Asset Pricing Models with Lvy processes: Theory and Applications

1
A New Class of Asset Pricing Models with Lévy
processes Theory and Applications
www.ornthanalai.com
FMA Dallas Meeting, Sept 9th 2008

• Chayawat Ornthanalai

2
2/17
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
3
2/17
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
4
3/17
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 ?

5
3/17
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?

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4/17
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?

7
4/17
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?

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5/17
Construction of the Models
9
5/17
Construction of the Models
10
5/17
Construction of the Models
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6/17
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

12
7/17
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

13
7/17
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

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8/17
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

15
8/17
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

16
8/17
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

17
9/17
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
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9/17
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
19
9/17
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
20
9/17
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
21
9/17
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
22
9/17
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
23
9/17
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
24
10/17
Setup of the Empirics Other Specifications

• Source of heteroskedasticity
• Time-varying dynamics in the jump and normal
  components

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10/17
Setup of the Empirics Other Specifications

• Source of heteroskedasticity
• Time-varying dynamics in the jump and normal
  components

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10/17
Setup of the Empirics Other Specifications

• Source of heteroskedasticity
• Time-varying dynamics in the jump and normal
  components

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10/17
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)

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10/17
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)

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11/17
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
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11/17
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
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11/17
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
32
11/17
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
33
11/17
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
34
11/17
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
35
11/17
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
36
11/17
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
37
11/17
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
38
12/17
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
39
12/17
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
40
12/17
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
41
12/17
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
42
13/17
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
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13/17
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 ?
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13/17
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
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14/17
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
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14/17
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
47
15/17
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

48
15/17
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

49
15/17
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

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16/17
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

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17/17
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

52
17/17
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

53
17/17
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

54
17/17
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
55
17/17
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