1. 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

2. 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

3. DMF 99 1. 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

4. DMF 99 1. Introduction - … a butterfly-spread -  Introduction Model X 7 8 9 10 11 12 13 C 3.354 2.459 1.670 1.045 0.604 0.325 0.164 +1.670 -2.095 +0.604 0001234 0000-2-4-6 0000012 Payoff if S T =... 78910111213 00010000.184 CC -0.895 -0.789 -0.625 -0.441 -0.279 -0.161  (  C) 0.106 0.164 0.184 0.162 0.118 Cost bsp Buy 1 C(X=9) Sell 2 C(X=10) Buy 1 C(X=11) S=10 Application vjvj

5. DMF 99 1. Introduction - … risk-neutral probabilities -  Introduction Model S=10 Application X 7 8 9 10 11 12 13 C 3.354 2.459 1.670 1.045 0.604 0.325 0.164  (  C) 0.106 0.164 0.184 0.162 0.118 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”:

6. DMF 99 1. 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

7. DMF 99 1. 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

8. DMF 99 2. 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

9. DMF 99 2. Model - Mixture binomial tree - Introduction Model  Application

10. DMF 99 3. Application - Data: S&P 500 futures options - Introduction Model  Application

11. DMF 99 3. Evaluation & Analysis - February 6, 1 Gauss & Error - Introduction Model  Application

12. DMF 99 3. Evaluation & Analysis - February 6, 3 Gauss & Error - Introduction Model  Application

13. DMF 99 3. Evaluation & Analysis - February - Introduction Model  Application

14. DMF 99 3. Evaluation & Analysis - May - Introduction Model Application 

15. DMF 99 3. Evaluation & Analysis - July - Introduction Model Application 

16. DMF 99 3. Evaluation & Analysis - August - Introduction Model Application 

17. DMF 99 3. Evaluation & Analysis - October - Introduction Model Application 

18. DMF 99 3. Evaluation & Analysis - January - Introduction Model Application 

19. 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