1. Change of Time Method: Application to Mathematical Finance. I. Anatoliy Swishchuk Math & Comp Finance Lab Dept of Math & Stat, U of C ‘Lunch at the Lab’ Talk October 18, 2005

2. Outline Change of Time Method (CTM) for Martingale (Wiener Process) CTM in General Setting CTM for SDEs Geometrical Brownian Motion and CTM: Solution Black-Scholes Formula by CTM Cox-Ingersoll-Ross Process and CTM: Solution Variance and Volatility Swaps by CTM

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. Idea of Proof. I.

9. Idea of Proof. II.

10. Geometric Brownian Motion

11. Change of Time Method for GBM

12. Solution for GBM Equation Using Change of Time

13. Properties of the Process

14. Properties of the Solution of GBM Using Change of Time Method

15. Option Pricing

16. European Call Option Pricing (Pay-Off Function)

17. European Call Option Pricing

18. Black-Scholes Formula

19. Stock Price under Risk-Neutral Measure

20. Explicit Expression for

21. European Call Option Through

22. Derivation of Black - Scholes Formula I

23. Derivation of Black-Scholes Formula II (continuation)

24. Derivation of Black - Scholes Formula III (continuation)

25. Derivation of Black - Scholes Formula IV (continuation)

26. Heston Model (Stochastic Volatility Model)

27. Explicit Solution for CIR Process: CTM

28. Proof. I.

29. Proof. II.

30. Properties of

32. Heston Model

33. Variance Swap for Heston Model. I.

34. Variance Swap for Heston Model. II.

35. Pricing of Variance Swap in Heston Model. I.

36. Pricing of Variance Swap in Heston Model. II.

37. Proof

38. Volatility Swap for Heston Model. I.

39. Volatility Swap for Heston Model. II.

40. Pricing of Volatility Swap for Heston Model. I.

41. Pricing of Volatility Swap for Heston Model. II.

42. Proof. I.

43. Proof. II.

44. Proof. III.

45. Proof. IV.

46. Proof. V.

47. References. I.

48. References. II.

49. References. III.

50. References. IV.

51. References. V.

52. References. VI.

53. References. VII.

54. References. VIII.

55. References. IX.

56. References. X. Elliott, R., Chan, L. and T. K. Siu (2005) “Pricing Volatility Swaps Under Heston's Volatility Model with Regime Switching ”

57. The End Thank you for your Attention!