1. Change of Time Method in Mathematical Finance Anatoliy Swishchuk Mathematical & Computational Finance Lab Department of Mathematics & Statistics University of Calgary, Calgary, Alberta, Canada CMS 2006 Summer Meeting Mathematical Finance Session Calgary, AB, Canada June 3-5, 2006

2. Outline  Change of Time (CT): Definition and Examples  Change of Time Method (CTM): Short History  Black-Scholes by CTM (i.e., CTM for GBM)  Explicit Option Pricing Formula (EOPF) for Mean-Reverting Model (MRM) by CTM  Black-Scholes Formula as a Particular Case of EOPF for MRM  Modeling and Pricing of Variance and Volatility Swaps by CTM

3. Change of Time: Definition and Examples  Change of Time-change time from t to a non- negative process with non-decreasing sample paths  Example 1 (Time-Changed Brownian Motion): M(t)=B(T(t)), B(t)-Brownian motion, T(t) is change of time  Example 2 (Subordinator): X(t) and T(t)>0 are some processes, then X(T(t)) is subordinated to X(t); T(t) is change of time  Example 3 (Standard Stochastic Volatility Model (SVM) ): M(t)=\int_0^t\sigma(s)dB(s), T(t)=[M(t)]=\int_0^t\sigma^2(s)ds. T(t)=[M(t)]=\int_0^t\sigma^2(s)ds.

4. Change of Time: Short History. I.  Bochner (1949) -introduced the notion of change of time (CT) (time-changed Brownian motion)  Bochner (1955) (‘Harmonic Analysis and the Theory of Probability’, UCLA Press, 176)-further development of CT

5. Change of Time: Short History. II.  Feller (1966) -introduced subordinated processes X(T(t)) with Markov process X(t) and T(t) as a process with independent increments (i.e., Poisson process); T(t) was called randomized operational time  Clark (1973)-first introduced Bochner’s (1949) time-changed Brownian motion into financial economics: he wrote down a model for the log- price M as M(t)=B(T(t)), where B(t) is Brownian motion, T(t) is time-change (B and T are independent)

6. Change of Time: Short History. III.  Ikeda & Watanabe (1981)-introduced and studied CTM for the solution of Stochastic Differential Equations  Carr, Geman, Madan & Yor (2003)-used subordinated processes to construct SV for Levy Processes (T(t)-business time)

7. Geometric Brownian Motion (Black-Scholes Formula by CTM)

8. Change of Time Method

9. Time-Changed BM is a Martingale

10. Option Pricing

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

12. European Call Option Pricing

13. Black-Scholes Formula

14. Mean-Reverting Model (Option Pricing Formula by CTM )

15. Solution of MRM by CTM

16. European Call Option for MRM.I.

17. European Call Option (Payoff Function)

18. Expression for y_0 for MRM

19. Expression for C_T C_T=BS(T)+A(T) ( Black-Scholes Part+Additional Term due to mean-reversion )

20. Expression for BS(T)

21. Expression for A(T)

22. European Call Option Price for MRM in Real World

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

26. Dependence of ES(t) on T (mean-reverting level L^*=2.569 )

27. Dependence of ES(t) on S_0 and T (mean-reverting level L^*=2.569)

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

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

30. Dependence of C_T on T

31. Heston Model (Pricing Variance and Volatility Swaps by CTM)

32. Explicit Solution for CIR Process: CTM

33. Why Trade Volatility?

34. Variance Swap for Heston Model

35. Volatility Swap for Heston Model

36. How Does the Volatility Swap Work?

38. Pricing of Variance Swap for Heston Model

39. Pricing of Volatility Swap for Heston Model

40. Brockhaus and Long Results  Brockhaus & Long (2000) obtained the same results for variance and volatility swaps for Heston model using another technique (analytical rather than probabilistic), including inverse Laplace transform

41. Statistics on Log Returns of S&P Canada Index (Jan 1997-Feb 2002)

42. Histograms of Log-Returns for S&P60 Canada Index

43. Convexity Adjustment

44. S&P60 Canada Index Volatility Swap

45. Conclusions  CTM works for:  Geometric Brownian motion (to price options in money markets)  Mean-Reverting Model (to price options in energy markets)  Heston Model (to price variance and volatility swaps)  Much More: Covariance and Correlation Swaps

46. The End/Fin Thank You!/ Thank You!/ Merci Beaucoup!