Asset Pricing with Lvy Jump Processes

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Asset Pricing with Lévy Jump Processes
www.ornthanalai.com
May 12th, 2009 Doctoral Thesis Defense
Chayawat Ornthanalai McGill University
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Introduction
Lévy Processes
Definition
Any stochastic processes that have stationary
and independent increment, i.e. Brownian motion,
Poisson processes
A Lévy process can be decomposed into three
components (1) a drift, (2) a diffusion
component, (3) a pure jump component
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Introduction
Lévy Processes
Definition
Any stochastic processes that have stationary
and independent increment, i.e. Brownian motion,
Poisson processes
A Lévy process can be decomposed into three
components (1) a drift, (2) a diffusion
component, (3) a pure jump component
Why is this important?

• Asset prices jump! Stock markets crash!
• Pricing insurance contracts, i.e. options,
  credit derivatives
• Risk management

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Introduction
Daily Log Returns of SP 500 Jan 85 March 09
Manic Monday
Percentage Return
9/11
LTCM
Fall-08 crisis
East Asian Crisis
Black Monday
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The thesis comprises of three essays
Introduction
Asset Pricing with Lévy Jump Processes

• Exploring time-varying jump intensities
  evidence from SP 500 returns and options with
  Peter Christoffersen, and Kris Jacobs
• A new class of asset pricing models with Lévy
  processes theory and applications Job market
  paper
• The market crash risk implicit in individual
  equity options with Redouane Elkamhi

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First Essay
Exploring Time-Varying Jump Intensities
Evidence from SP 500 Returns and Options
Objectives of this paper

• Develop a new discrete-time model that is based
  on the continuous-time jump-diffusion framework
• Jumps follow a Compound Poisson process
• The model can be estimated using standard MLE
• Investigate various joint specifications between
  jump and normal components for returns fitting?
• Main Question Should jump intensity be
  time-varying?
• Study the effect that different jump
  specifications have on the option pricing
  performance.

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First Essay
Main findings

• Strong evidence for time-varying jump intensities
• Parsimony is the key. Option pricing results
  favour the model with jump intensity that is
  linear with the variance of the normally
  distributed return component
• The presence of jump risk premia is very
  important in option pricing

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Second Essay
A New Class of Asset Pricing Models with Lévy
Processes Theory and Applications
Objectives

• 1) Develop a new class of discrete-time asset
  pricing models
• Capturing the key stylized facts of asset
  returns (1) nonnormality, (2)
  heteroskedasticity, (3) leverage effect, (4)
  jumps in volatility
• 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
• 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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Second Essay
Theory Construction of the Models
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Second Essay
Theory Construction of the Models
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Second Essay
Theory Construction of the Models
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Second Essay
Empirics Two-factor Lévy GARCH model

• In this essay, I study two-factor index return
  models

• Three jump specifications
• No jump
• Jumps are large and rare events Merton jump
  process
• Jumps are hyperactive events Normal Inverse
  Gaussian process

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Second Essay
Results MLE of daily SP 500 returns
Jumps are large and rare
Jumps are hyperactive events
No jump
Jump component of daily returns
Standardized normal component of daily returns
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Second Essay
Results MLE of daily SP 500 returns
Jumps are large and rare
Jumps are hyperactive events
No jump
Jump component of daily returns
Standardized normal component of daily returns
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Second Essay
Results Joint MLE of options and returns

• What is the economic role of jumps?
• Models without jump risk will yield implausibly
  high equity premium

Call options implied equity premium
No jump
Hyperactive jump
Large jump
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Third Essay
The Market Crash Risk Implicit in Individual
Equity Options

• Objectives
• To provide evidence that the market crash risk is
  priced in the cross section of individual equity
  options as well as in their underlying returns.
• To show that the market volatility and jump risk
  factors explain the differential price structures
  among individual equity options

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Introduction
Third Essay
Motivations

• Studies in index option pricing consistently
  show that the market volatility and crash risk
  are priced in the index
• Does this result also extend to individual
  equities?
• The need for a risk-based explanation for the
  differential price structures among individual
  equity options
• Growing literature on the demand-based
  arguments for pricing equity options
• Our explanation is grounded on valuation
  theories, namely the risk premia

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Third Essay
The Model
The return of each jth stock is modeled as
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Third Essay
Data and Estimation

• We study 32 stocks which are among the largest
  listed on the SP 500 index (Jan 1996- Dec 2005)
• Joint MLE for each firm (1) index returns, (2)
  equity returns, (3) equity options

Extraction of the return premium
Implicit market volatility risk premium
Implicit market jump risk premium
Excess Return
Firm-specific risk premium
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Third Essay
Main results

• The market volatility and jump risk factors are
  priced in equity options
• 3.18 for market jump risk
• 5.53 for market volatility risk
• 2. The market volatility and jump risk premia
  explain different dimensions of the price
  structure of individual equity options
• Market volatility risk premium ? IV level
• Market jump risk premium ? IV slope
• Equity specific risk is priced
• The premium is on average positive.

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Conclusions
Concluding remarks

• Asset prices jump
• Jumps arrive at time-varying rate
• Jumps are best modeled as hyperactive events
• Jump risk is economically important
• It is priced in the index
• It is priced in the cross section of individual
  equity options and returns
• The fear of a market-wide crash risk is embedded
  in the price of structure of index and individual
  equity options
• 3. Jump risk is real and inevitable. It should
  be taken seriously in the practice of risk
  management

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www.ornthanalai.com
THANK YOU
Co-supervisors Peter Christoffersen, Kris
Jacobs External committee members Eric Jacquier,
Jean-Guy Simon to McGill faculties and staffs
Adolfo Demotta, Saibal Ray, Jan-Ericsson, Vihang
Errunza, Benjamin Croitoru, Jianguo Xu, Sue
Lovasik, Stella Scalia All the PhD students in
the faculty of management , and the attendees of
this Thesis defense