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

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A New Class of Asset Pricing Models with Lévy
processes Theory and Applications
www.ornthanalai.com
August 21st, 2009 EFA Conference, Bergen,
Norway

• Chayawat Ornthanalai

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The Issues in Modeling Asset Prices

• Presence of jumps in returns /conditional
  non-normality
• Return volatility is time-varying
• Leverage effect
• Volatility also jumps ?

Stylized facts of asset returns

• Asset return dynamics that price these
  derivatives well are different from the
  historical return dynamics
• How to economically link the risk-neutral and
  realized return dynamics together?

Derivative valuation
Realized Returns
Derivative Prices
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Objectives of this paper

• 1) Develop a new class of discrete-time asset
  pricing models
• Capturing the key stylized facts of asset
  returns
• Simple valuation of derivative securities
• Ease of implementation

• 2) Conduct empirical studies on index returns and
  options
• Extraction of the risk premia implicit in the
  SP 500 index using joint estimation of options
  and returns
• What is the economic role of jumps ?
• Specification analysis of different jump
  structures
• How should jumps in the index returns jump?

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Construction of the Models
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Construction of the Models
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Construction of the Models
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Literature Review
Models with Time-varying Lévy Jumps

• Discrete-time
• Bollerslev and Forsberg (02), Christoffersen,
  Jacobs and Heston (06), Stentoft (07)
• Continuous-time Time-changed Lévy Processes
• Carr and Wu (04), Carr, Madan, Geman and Yor
  (03), Huang and Wu (04), Bakshi, Carr and Wu
  (08), Bates (08)

Models with Lévy Jumps (Non-Poisson type jumps)
Small and High Frequency Carr and Wu (03), Carr,
Geman, Madan and Yorr (02), Li, Wells and Yu
(08)
Models with the Classical Poisson Jumps Large
and Rare Jumps
Bakshi, Cao and Chen (97), Bates (96, 00, 06),
Duffie, Pan and Singleton (00), Pan (02),
Johannes (04), Eraker, Johannes and Polson (03),
Eraker (04), Chernov, Gallant, Ghysels, Tauchen
(03), Andersen, Benzoni and Lund (02), Duan,
Ritchken and Sun (06), Maheu and McCurdy (04),
Kou (02), Broadie, Chernov and Johannes (07, 08),
Santa-Clara and Yan (08) etc
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Literature Review
Models with Time-varying Lévy Jumps (All types)

• Discrete-time
• Bollerslev and Forsberg (02), Christoffersen,
  Jacobs and Heston (06), Stentoft (07)
• Continuous-time Time-changed Lévy Processes
• Carr and Wu (04), Carr, Madan, Geman and Yor
  (03), Huang and Wu (04), Bakshi, Carr and Wu
  (08), Bates (08)

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Literature Review
Models with Time-varying Lévy Jumps (All types)

• Discrete-time
• Continuous-time Time-changed Lévy Processes
• Carr and Wu (04), Carr, Madan, Geman and Yor
  (03), Huang and Wu (04), Bakshi, Carr and Wu
  (08), Bates (08)

Lévy GARCH
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Application Two-factor Lévy GARCH model

• In this paper, I study two-factor return model
• The shocks
• The dynamics of and are driven
  by an affine GARCH(1,1) of Heston and Nandi
  (2000)
• The conditional equity premium dynamic

Diffusive variance
Jump intensity
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Jump Specifications

• Three jump specifications
• No jump (GARCH)
• Finite-Activity Jumps Merton jump process
  (Merton)
• Infinite-Activity Jumps Normal Inverse Gaussian
  process (NIG)

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Empirics (1) Joint Estimation

• Joint MLE of SP 500 index Options and Returns
• I use time-series cross-sectional Call options
  data from 1996-2005
• This is the most extensive joint estimation in
  the jump literature
• Related studies, i.e. Pan (2002) and Eraker
  (2004), use only ATM and short-term to maturity
  options
• How options and returns can be jointly used to
  estimate the risk factors?

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Joint MLE Methodology

• I maximize the joint likelihood function
• I use the implied volatility (IV) loss function
  for fitting options data
• Assuming that IV residuals are normally
  distributed

Time line
1995
2006

• Fitting daily returns filtering the state
  variables
• Fitting a cross section of options on each
  Wednesday

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Joint MLE Results The economic role of jumps

• What do investors demand in excess return for
  bearing the market risk factors ?
• The long-run return risk premium implied by
  different models

Annual excess Return ()
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Robustness check What if we use put options?

• Can jump risk explain the structure of put
  option prices?
• Same exercise, but using put options

Call options implied equity premium
Put options implied equity premium
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Empirics (2) MLE of SP 500 returns
Daily Log Return of SP 500 index
Manic Monday
Percentage Return
9/11
LTCM
Fall-08 crisis
East Asian Crisis
Black Monday
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MLE of daily SP 500 returns Jan 85 Mar 09
Merton Jumps are large and rare
NIG Jumps are hyperactive events
No jump
Jump component of daily returns
Standardized normal component of daily returns
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MLE of daily SP 500 returns Jan 85 Mar 09
Merton Jumps are large and rare
NIG Jumps are hyperactive events
No jump
Jump component of daily returns
Standardized normal component of daily returns
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MLE of daily SP 500 returns Jan 85 Mar 09
Merton Jumps are large and rare
NIG Jumps are hyperactive events
No jump
Jump component of daily returns
Standardized normal component of daily returns
Active flows of information
Hyperactive jumps
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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
• Jump risk (crash fear) is economically important,
  consistent with Reitz (88), Barro (05), Gabaix
  (08)
• Jumps in the index returns are small but arrive
  at very high frequency (infinite-activity)

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Conclusions and Future Research (continued..)

• The framework has many applications in empirical
  asset pricing
• Specification analysis using other types of Lévy
  jump processes NIG and MJ are just two of many
  possible choices
• Price credit derivative products, i.e. CDS
• Joint estimation using options and returns data

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Lévy GARCH versus Standard GARCH models

• Standard GARCH models produce excessively high
  conditional volatility
• Chen and Ghysels (2008) show that bad news
  increase volatility while good news do not

News Impact Curve
Percentage change in next periods volatility
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Why do we need to model index returns with
Jumps? MLE of the SP 500 index returns (1785
31st March 2009)
Daily log returns of SP 500 index
GARCH-without jumps
Standardized Normal Residuals
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Why do we need to model index returns with
Jumps? MLE of the SP 500 index returns (1785
31st March 2009)
Daily log returns of SP 500 index
GARCH with Levy jumps
Standardized Normal component of Residuals
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MLE of daily SP 500 returns Jan 85 Mar 09
No jump (GARCH)
Merton jump
NIG jump
Conditional daily return variance
Jump component of daily returns
Standardized normal component of daily returns