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

1. Extracting Risk-Neutral Densities from Option Pricesusing Mixture Binomial Trees Christian Pirkner Andreas S. Weigend Heinz Zimmermann Version 1.0

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

3.  Introduction 1. 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]

4. X C Costbsp Payoff if ST = ... DC D(DC) 7 8 9 10 11 12 13 Buy 1 C(X=9) Sell 2 C(X=10) Buy 1 C(X=11) 7 3.354 -0.895 8 2.459 0.106 -0.789 9 1.670 +1.670 0 0 0 1 2 3 4 0.164 -0.625 10 1.045 -2.095 0 0 0 0 -2 -4 -6 0.184 -0.441 11 0.604 +0.604 0 0 0 0 0 1 2 0.162 -0.279 12 0.325 0.118 -0.161 13 0.164 0 0 0 1 0 0 0  Introduction 1. Introduction Model - … a butterfly-spread - Application vj 0.184 S=10

5. X C D(DC) Valuing an option with payoffs jusing vj: vj 7 3.354 Buying all vj’s:  riskless investment 8 2.459 0.106 9 1.670 0.164 Defining Pj’s:  “risk-neutral probabilities”: 10 1.045 0.184 11 0.604 0.162 Alternative way to value derivative: 12 0.325 0.118 13 0.164  Introduction 1. Introduction Model - … risk-neutral probabilities - Application S=10

6. Parametric • Linear • Logit • Polynomial • Gauss • Gamma • Edgeworth expansion • Smoothness • Mixture models • Several tanh Non Parametric • Kernel regression • Kernel density  Introduction 1. Introduction Model - Density extraction techniques - Application II. Estimating density directly I. 2nd Derivative of call price function III. Recovering parameters of assumed stochastic process of the underlying security.

7.  Introduction 1. Introduction Model - Standard & implied trees - Application • 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

8. Subject to constraint: The weights of all mixture components are positive and add up to one ü Introduction 2. Model  Model - Mixture binomial tree - Application … where we optimize for the lowest absolute mean squared error in option prices We propose to model end-nodal probabilities with a mixture of Gaussians ...

9. ü Introduction 2. Model  Model - Mixture binomial tree - Application

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

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

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

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

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

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

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

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

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

19. ü Introduction Conclusion ü Model ü 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