1. Change of Time Method: Applications to Mathematical Finance. II. Anatoliy Swishchuk Math & Comp Finance Lab Dept of Math & Stat, U of C “Lunch at the Lab” Talk November 8, 2005

2. Outline Change of Time Method (CTM) (minutes of the previous talk) Mean-Reverting Model (MRM) Solution of MRM by CTM Option Pricing Formula Black - Scholes Formula Follows: L=0, a^*=-r Numerical Example (AECO Natural Gas Index)

3. CTM for Martingales

4. CTM in General Setting. I.

5. CTM in General Setting. II.

6. CTM for SDEs. I.

7. CTM for SDEs. II.

8. Connection between phi_t and phi_t^(-1)

9. Idea of Proof. I.

10. Idea of Proof. II.

11. Mean-Reverting Model

12. Solution of MRM by CTM

13. Solution of GBM Model (to compare)

14. Properties of

15. Explicit Expression for

17. Explicit Expression for S(t)

18. Properties of

20. Properties of Eta(t). II.

21. Properties of MRM S(t). I.

22. Dependence of ES(t) on T

23. Dependence of ES(t) on S_0 and T

24. Properties of MRM S(t). II.

25. Dependence of Variance of S(t) on S_0 and T

26. Dependence of Volatility of S(t) on S_0 and T

27. European Call Option for MRM.I.

28. European Call Option. II.

29. Expression for y_0 for MRM

30. Expression for C_T C_T=BS(T)+A(T)

31. Expression for C_T=BS(T)+A(T).II.

32. Expression for BS(T)

33. Expression for A(T).I.

34. Expression for A(T).II. Characteristic function of Eta(T):

35. Expression for A(T). II.

36. European Call Option for MRM

37. Boundaries for C_T

38. European Call Option for MRM in Risk- Neutral World

39. Boundaries for MRM in Risk-Neutral World

42. Dependence of C_T on T

43. Paper may be found on the following web page (E-Yellow Series Listing): http://www.math.ucalgary.ca/research/preprint.php

44. The End Thank You for Your Attention and Time!