1. Explicit Option Pricing Formula for Mean-Reverting Asset Anatoliy Swishchuk Math & Comp Finance Lab Dept of Math & Stat, U of C MITACS Project Meeting McMaster University, Hamilton November 12, 2005

2. Outline Mean-Reverting Models (MRM): Deterministic vs. Stochastic MRM in Finance Markets: Variances or Volatilities (Not Asset Prices) MRM in Energy Markets: Asset Prices Change of Time Method (CTM) Mean-Reverting Model (MRM) Option Pricing Formula Drawback of One-Factor Models Future Work

3. Motivations for the Work Paper: Javaheri, Wilmott and Haug (2002) ”GARCH and Volatility Swaps”, Wilmott Magazine, Jan Issue (they applied PDE approach to find a volatility swap for MRM and asked about the possible option pricing formula Paper: Bos, Ware and Pavlov (2002) “On a Semi-Spectral Method for Pricing an Option on a Mean-Reverting Asset”, Quantit. Finance J. (PDE approach, semi-spectral method to calculate numerically the solution)

4. Mean-Reversion Effect Guitar String Analogy: if we pluck the guitar string, the string will revert to its place of equilibrium To measure how quickly this reversion back to the equilibrium location would happen we had to pluck the string Similarly, the only way to measure mean reversion is when the variances of asset prices in financial markets and asset prices in energy markets get plucked away from their non-event levels and we observe them go back to more or less the levels they started from

5. The Mean-Reverting Deterministic Process

6. Mean-Reverting Plot (a=4.6,L=2.5)

7. Meaning of Mean-Reverting Parameter The greater the mean-reverting parameter value, a, the greater is the pull back to the equilibrium level For a daily variable change, the change in time, dt, in annualized terms is given by 1/365 If a =365, the mean reversion would act so quickly as to bring the variable back to its equilibrium within a single day The value of 365/ a gives us an idea of how quickly the variable takes to get back to the equilibrium-in days

8. Mean-Reverting Stochastic Process

9. Mean-Reverting Models in Financial Markets Stock (asset) Prices follow geometric Brownian motion The Variance of Stock Price follows Mean-Reverting Models Example: Heston Model

10. Mean-Reverting Models in Energy Markets Asset Prices follow Mean- Reverting Stochastic Processes Example: Pilipovic Model

11. Mean-Reverting Models in Energy Markets

12. CTM for Martingales

13. CTM for SDEs. I.

14. CTM for SDEs. II.

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

16. Idea of Proof. I.

17. Idea of Proof. II.

18. Mean-Reverting Model

19. Solution of MRM by CTM

20. Solution of GBM Model (to compare)

21. Properties of

22. Explicit Expression for

24. Explicit Expression for S(t)

25. Properties of

27. Properties of Eta(t). II.

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

29. Dependence of ES(t) on T

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

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

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

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

34. Drawback of One-Factor Mean- Reverting Models The long-term mean L remains fixed over time: needs to be recalibrated on a continuous basis in order to ensure that the resulting curves are marked to market The biggest drawback is in option pricing: results in a model-implied volatility term structure that has the volatilities going to zero as expiration time increases (spot volatilities have to be increased to non-intuitive levels so that the long term options do not lose all the volatility value-as in the marketplace they certainly do not)

35. European Call Option for MRM.I.

36. European Call Option. II.

37. Expression for y_0 for MRM

38. Expression for C_T in the case of MRM C_T=BS(T)+A(T)

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

40. Expression for BS(T)

41. Expression for A(T).I.

42. Expression for A(T).II. Characteristic (moment generating) function of Eta(T):

43. Expression for A(T). II.

44. European Call Option for MRM (Explicit Formula)

45. Boundaries for C_T

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

47. Boundaries for MRM in Risk-Neutral World

50. Dependence of C_T on T

51. Paper may be found on the following web page (E-Yellow Series Listing, Dept of Math & Stat, U of C, Calgary, AB): http://www.math.ucalgary.ca/research/preprint.php

52. Future work. I: Analytical Approach (Integro – Partial DE) (Joint Working Paper with T. Ware)

53. Future Work.II: Probabilistic Approach ( Change of Time Method ).

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