Extracting Risk-Neutral Densities from Option Prices using Mixture Binomial Trees

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Extracting Risk-Neutral Densities from Option
Pricesusing Mixture Binomial Trees

• Christian Pirkner
• Andreas S. Weigend
• Heinz Zimmermann

Version 1.0
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Outline
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• Motivation
• Butterfly-Spread
• Implied Binomial Tree

Introduction

• Mixture Binomial Tree
• Optimization
• Graph

Model

• Density Extraction 1 Day
• Density Extraction over Time
• Conclusion

Application
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1. Introduction
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Introduction
Model
- Motivation -
Application

• Goal
• What can we learn from market prices of traded
  options?
• ? Extract expectations of market participants
• Use this information for decision making!
• ? Exotic option pricing, risk measurement and
  trading

• An European equity call option (C) is the right
  to
• buy
• an underlying security, S
• for a specified strike price, X
• at time to expiration, T
• ? payoff function max ST - X, 0

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1. Introduction
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Introduction
Model
- a butterfly-spread -
Application
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1. Introduction
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Introduction
Model
- risk-neutral probabilities -
Application
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1. Introduction
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Introduction
Model
- Density extraction techniques -
Application
I. 2nd Derivative of call price function
II. Estimating density directly
III. Recovering parameters of assumed stochastic
process of the underlying security.
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1. Introduction
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Introduction
Model
- Standard implied trees -
Application

• Instead of building a ... standard binomial tree
• starting at time t0
• resting on the assumption of normally distributed
  returns and constant volatility

• We build an implied binomial tree
• starting at time T
• and flexible modeling of end-nodal probabilities

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2. Model
Introduction
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Model
- Mixture binomial tree -
Application
We propose to model end-nodal probabilities with
a mixture of Gaussians ...

• where we optimize for the lowest absolute mean
  squared error in option prices

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2. Model
Introduction
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Model
- Mixture binomial tree -
Application
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3. Application
Introduction
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Model
- Data SP 500 futures options -
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Application
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3. Evaluation Analysis
Introduction
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Model
- February 6, 1 Gauss Error -
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Application
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3. Evaluation Analysis
Introduction
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Model
- February 6, 3 Gauss Error -
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Application
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3. Evaluation Analysis
Introduction
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Model
- February -
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Application
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3. Evaluation Analysis
Introduction
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Model
- May -
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Application
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3. Evaluation Analysis
Introduction
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Model
- July -
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Application
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3. Evaluation Analysis
Introduction
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Model
- August -
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Application
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3. Evaluation Analysis
Introduction
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Model
- October -
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Application
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3. Evaluation Analysis
Introduction
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Model
- January -
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Application
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Conclusion
Introduction
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Model
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Application

• Learning from option prices
• ? Extracting market expectations

• Use information for decision making
• Exotic option pricing
• ? Use extracted kernel to price non-standard
  derivatives consistent with liquid options
• Risk measurement
• ? Calculate Economic Value at Risk
• Trading
• ?Take positions if extracted density differs from
  own view