Data Mining in Finance, 1999 March 8, 1999 Extracting Risk-Neutral Densities from Option Prices using Mixture Binomial Trees Christian Pirkner Andreas S. Weigend Heinz Zimmermann Version 1.0

DMF 99 Outline Introduction Model Application Motivation Butterfly-Spread Implied Binomial Tree Mixture Binomial Tree Optimization Graph Density Extraction: 1 Day Density Extraction over Time Conclusion Part 1 Part 2 Part 3  Introduction Application Model

DMF Introduction - Motivation -   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 [S T - X, 0]  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 Introduction Application Model

DMF Introduction - … a butterfly-spread -  Introduction Model X C Payoff if S T = CC  (  C) Cost bsp Buy 1 C(X=9) Sell 2 C(X=10) Buy 1 C(X=11) S=10 Application vjvj

DMF Introduction - … risk-neutral probabilities -  Introduction Model S=10 Application X C  (  C) vjvj Valuing an option with payoffs  j using v j : Buying all v j ’s:  riskless investment Alternative way to value derivative: Defining P j ’s:  “risk- neutral probabilities”:

DMF Introduction - Density extraction techniques -  Parametric Non Parametric I. 2nd Derivative of call price function II. Estimating density directly Linear Logit Polynomial Several tanh Kernel regression Gauss Gamma Edgeworth expansion Smoothness Mixture models Kernel density III. Recovering parameters of assumed stochastic process of the underlying security. Introduction Model Application

DMF Introduction - Standard & implied trees -  Introduction Model  Instead of building a... standard binomial tree – starting at time t=0 – 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 Application

DMF Model - Mixture binomial tree - … where we optimize for the lowest absolute mean squared error in option prices Subject to constraint: The weights of all mixture components are positive and add up to one Introduction Model We propose to model end-nodal probabilities with a mixture of Gaussians...  Application

DMF Model - Mixture binomial tree - Introduction Model  Application

DMF Application - Data: S&P 500 futures options - Introduction Model  Application

DMF Evaluation & Analysis - February 6, 1 Gauss & Error - Introduction Model  Application

DMF Evaluation & Analysis - February 6, 3 Gauss & Error - Introduction Model  Application

DMF Evaluation & Analysis - February - Introduction Model  Application

DMF Evaluation & Analysis - May - Introduction Model Application 

DMF Evaluation & Analysis - July - Introduction Model Application 

DMF Evaluation & Analysis - August - Introduction Model Application 

DMF Evaluation & Analysis - October - Introduction Model Application 

DMF Evaluation & Analysis - January - Introduction Model Application 

DMF 99 Conclusion Introduction Model  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 Application