Change of Time Method: Applications to Mathematical Finance. II. - PowerPoint PPT Presentation

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  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

  16. Explicit Expression for

  17. Explicit Expression for S(t)

  18. Properties of

  19. 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

  40. Dependence of C_T on T

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

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