Evnine-Vaughan Associates, Inc.

1. Evnine-Vaughan Associates, Inc. A Theory of Non-Gaussian Option Pricing: capturing the smile and the skew Lisa Borland Acknowledgements: Jeremy Evnine Roberto Osorio Jean-Philippe Bouchaud

2. Layout • The Stock Price Model • Option Pricing Theory – the Smile • Results • Option Pricing Theory – the Skew • Results • Conclusions

3. The Standard Stock Price Model

4. The Standard Stock Price Model Fokker-Planck Equation Gaussian Distribution

5. The Generalized Returns Model

6. The Generalized Returns Model

7. The Generalized Returns Model Nonlinear Fokker-Planck Student (Tsallis) Distribution

8. o Empirical --- Gaussian q=1.43 Tsallis Distribution (Osorio et al 2002)

9. Extension to Model: Current Model: More realistic model: eg moving average (work in progress )

11. q=1.5 q=1 SP500 q=1.5

12. SP500 q=1.5 log P d=16 d=16 d=8 d=8 d=4 d=4 d=2 d=2 d=1 d=1 Y(t+d)-Y(t)

13. Option Arbitrage Theorem: Deterministic Risk-Free Portfolio  Return = risk-free rate r Generalized Black-Scholes PDE

14. 1) Exploit PDE’s implied by arbitrage-free portfolios Solve PDE to get option price 2) Convert prices of assets into martingales Take expectations to get option price

15. Martingale: Not a martingale w.r.t. measure F

16. Martingale: Not a martingale w.r.t. measure F Is a martingale w.r.t. measure Q Effectively:

17. Example European Call Stock Price

18. Example European Call Stock Price Must integrate

19. Generalized Feynman-Kac Tsallis (Student) weights Path Integral: Forward Equation: Ansatz: Result:

20. Generalized Feynman-Kac Result:

21. Example European Call Stock Price

22. Example European Call Stock Price Payoff if  q = 1: P is Gaussian q >1 : P is fat tailed Student(Tsallis) dist.

23. Call Price Difference T=0.6 T=0.05 $ =0.3 S(0) = $50, r= 6%,

24. T=0.1 T=0.4 Black-Scholes (q=1) volatilities implied from q=1.5 model

25. q=1.5

26. K=50, T=0.4, sigma=0.3, r=.06

27. q=1 q=1.3 q=1.4 q=1.45 q=1.5 K=50, T=0.4, sigma=0.3, r=.06

28. Volatility Smiles o Empirical implied vols __ q=1.43 implied vols

29. Implied Volatility JY Futures 16 May 2002 T=17 days

30. Implied Volatility JY Futures 16 May 2002 T=37 days

31. Implied Volatility JY Futures 16 May 2002 T=62 days

32. Implied Volatility JY Futures 16 May 2002 T=82 days

33. Implied Volatility JY Futures 16 May 2002 T=147 days

34. Term Structure q=1.4 (Vol Surface) q=1 (BS)

35. Example Currency Futures: (500 options) q Mean square relative pricing error 1. 0.16 1.4 0.008 Benefits of a more parsimonious model: • Better pricing - arbitrage opportunities • Better hedging

36. The Generalized Model with Skew Volatility Leverage Correlation (with Jean-Philippe Bouchaud)

38. The Generalized Model with Skew (with Jean-Philippe Bouchaud)

39. The Generalized Model with Skew small (with Jean-Philippe Bouchaud)

40. Example European Call Integrate using Feynman-Kac and Pade expansion Payoff if

43. Comments • q=1: CEV model of Cox and Ross recovered • Skew model can be mapped onto a higher dimensional free-particle diffusion in cylindrical coordinates • Exact solutions in terms of hyper-geometric functions ?

44. Volatility Skew : T=0.1 T=0.5 T=1.0 alpha = 0. S(0) = 50

45. S0=50 T=0.5 sigma=0.3 r=0.06 q=1.5

46. q=1.5

47. q =1

48. q=1 q=1.5 K=50,T=0.5,sigma=0.3,r=.06

49. SP500 OX q=1.5, alpha = -1. T=.03 T=0.1 T=0.2 T=0.3 T=0.55 Strike K Strike K

50. MSFT Nov 19 2003 ATM = 25.55 T=.082 T=.159 T=.41 T=1.17