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

1
A New Class of Asset Pricing Models with Lévy
processes Theory and Applications(Technical
Slides General Framework)
www.ornthanalai.com
Nov 2008

• Chayawat Ornthanalai

2
Methodological Contribution New Framework

• I provide a general solution for the pricing
  transform of a large class of asset return
  dynamics
• The setup allows for a wide variety of asset
  return specifications
• Closed-form valuation of various derivatives

What do returns look like?
Shocks to return
Lévy processes Normal, Poisson, Gamma, VG,
NIG, CGMY, IG, LS, Meixner, TS
How does the return volatility behave?
Affine GARCH dynamics

• Persistence long memory?
• Jumps in volatility?

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The General Framework

• Assume d-dimensional contemporaneously
  independent Levy shocks in the return
• Each of the Levy shocks consists of a
  parameter that controls the dynamic of
  its variance and higher moments
• I use affine GARCH dynamics to drive
  for i 1..d

The Return Dynamic under the Physical Measure
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The General Solution to the Pricing Transform

• The MGF of asset return at time T conditional on
  current period t is

• The general solution to the MGF has an
  exponential affine form

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The General Solution to the Pricing Transform

• The MGF of asset return at time T conditional on
  current period t is

• The general solution to the MGF has an
  exponential affine form

• I provide a general solution for the affine
  coefficients in Proposition 1

6
A New Class of Asset Pricing Models with Lévy
processes Theory and Applications(Technical
Slides MLIS/Estimation)
www.ornthanalai.com
October 2008

• Chayawat Ornthanalai

7
Affine GARCH(1,1) dynamic

• The return dynamic
• The affine GARCH(1,1) dynamic
• We need to filter out zt expost of Rt ,
  equivalently
• Note that the filtering problem of
  are equivalent

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Why is MLE easier with Lévy GARCH ?
Continuous-time
The density function involves triple integration
over three sources of randomness
Lévy GARCH
The density function involves one integration as
diffusive variance and the jump intensity are
deterministic
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Estimation of Lévy GARCH Particle Filter

• I use Particle Filtering (PF) in the estimation
  
• It can handle nonlinearity such as jumps
• It is relatively fast and lends itself naturally
  to MLE
• MLE PF gt Maximum Likelihood Importance
  Sampling, see Pitt (2002), and Neil and Sheppard
  (2000)
• The sampling technique is advantageous because
  many Lévy processes do not have analytical
  density function

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Estimation of Lévy GARCH Particle Filter

• Note that the filtering problem of
  
• are equivalent
• The filtering problem of yt involves computing
• The sampling density of yt is
• The sampling technique is advantageous because
  many Lévy processes do not have analytical
  density function

Sampling
Re-sampling
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Estimation of Lévy GARCH MLIS

• The log likelihood function is computed from
• This probability function is the same as the one
  used in the particle filtering of yt (see last
  slide)
• The likelihood is already computed as part of
  the PF procedure
• Relatively fast, efficient, and does not rely on
  the analytical likelihood function