Change of Time Method in Mathematical Finance

1. Change of Time MethodinMathematical Finance Anatoliy Swishchuk Mathematical & Computational Finance Lab Department of Mathematics & Statistics University of Calgary, Calgary, Alberta, Canada 2006 Stochastic Modeling Symposium Toronto, ON, Canada April 3, 2006

2. Outline • Change of Time (CT): Definition and Examples • Interpretation of CTM • Change of Time Method (CTM): Short History • CTM for Stochastic Differential Equations • 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 • Variance and Volatility Swaps (VarSw and VolSw) • Modeling and Pricing of VarSw and VolSw 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 • Example1 (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 2 (Time-Changed Brownian Motion): M(t)=B(T(t)), B(t)-Brownian motion • 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.

4. Interpretation of CT. I. • If M(t) is fair game process (another name-martingale) • Then M(t)=B(T(t)) (Dambis-Dubins-Schwartz Theorem) • Time-change is the quadratic variation process [M(t)] • Then M(t) can be written as a SVM process (martingale representation theorem, Doob (1953))

5. Interpretation of CT. II. • This implies that time-changed BMs are canonical in continuous sample path price processes and SVMs are special cases of this class • A consequence of the fact that for continuous sample path time changed BM, [M(t)]=T(t) is that in the SVM case [M(t)]=\int_0^t\sigma^2(s)ds.

6. Change of Time: Short History. I. • Bochner (1949) (‘Diffusion Equation and Stochastic Process’, Proc. N.A.S. USA, v. 35)-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 • Feller (1966) (‘An Introduction to Probability Theory’, vol. II, NY: Wiley)-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

7. Change of Time: Short History. II. • Clark (1973) (‘A Subordinated Stochastic Process Model with Fixed Variance for Speculative Prices’, Econometrica, 41, 135-156)-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) • Johnson (1979) (‘Option Pricing When the Variance Rate is Changing’, working paper, UCLA)-introduced time-changed SVM in continuous time • Johnson & Shanno (1987) (‘Option Pricing When the Variance is Changing’, J. of Finan. & Quantit. Analysis, 22, 143-151)-studied the pricing of options using time-changing SVM

8. Change of Time: Short History. III. • Ikeda & Watanabe (1981) (‘SDEs and Diffusion Processes’, North-Holland Publ. Co)-introduced and studied CTM for the solution of SDEs • Carr, Geman, Madan & Yor (2003) (‘SV for Levy Processes’, mathematical Finance, vol.13)-used subordinated processes to construct SV for Levy Processes (T(t)-business time)

9. Time-Changed Models and SVMs • The probability literature has demonstrated that SVMs and their time-changed BM relatives and time-changed models are fundamentals • Shephard (2005): Stochastic Volatility, working paper, University of Oxford • Shephard (2005): Stochastic Volatility: Selected Readings, Oxford, Oxford University Press

10. CTM for SDEs. I.

11. CTM for SDEs. II.

12. Idea of Proof. I.

13. Idea of Proof. II.

14. Geometric Brownian Motion

15. Change of Time Method

16. Solution for GBM EquationUsing Change of Time

17. Option Pricing

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

19. European Call Option Pricing

20. Black-Scholes Formula

21. Risk-Neutral Stock Price

22. Explicit Expression for

23. European Call Option Through

24. Derivation of Black - Scholes Formula I

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

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

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

28. Mean-Reverting Model

29. Solution of MRM by CTM

30. Solution of GBM Model (just to compare with solution of MRM)

31. Properties of

32. Explicit Expression for

33. Explicit Expression for

34. Explicit Expression for S(t)

35. Properties of

36. Properties of Eta (t). I.

37. Properties of Eta(t). II.

38. Properties of MRM S (t)

39. Dependence of ES(t) on T

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

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

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

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

44. European Call Option for MRM.I.

45. European Call Option. II.

46. Expression for y_0 for MRM

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

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

49. Expression for BS(T)

50. Expression for A(T).I.