Explicit Option Pricing Formula for a Mean-Reverting Asset in Energy Markets

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1 Explicit Option Pricing Formula for a Mean-Reverting Asset in Energy Markets Anatoliy Swishchuk Mathematical & Computational Finance Lab Dept of Math & Stat, University of Calgary, Calgary, AB, Canada QMF 2007 Conference Sydney, Australia December 12-15, 2007 This research is supported by MITACS and NSERC

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 Wilmott, Javaheri & Haug (2002) Model Wilmott, Javaheri & Haug (GARCH and Volatility Swaps, Wilmott Magazine, 2002)- volatility swap for -continuous-time GARCH(1,1) model

5 M. Yor s Results M. Yor On some exponential functions of Brownian motion, Adv. In Applied Probab., v. 24, No. 3, (1992), started the research for the integral of an exponential Brownian motion H. Matsumoto, M. Yor Exponential Functionals of Brownian motion, I: Probability laws at fixed time, Probability Surveys, v. 2 (2005), there is still no closed form probability density function, while the best result is a function with a double integral.

6 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

7 The Mean-Reverting Deterministic Process

8 Mean-Reverting Plot (a=4.6,l=2.5)

9 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

10 Mean-Reverting Stochastic Process

11 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

12 Mean-Reverting Models in Energy Markets Asset Prices follow Mean-Reverting Stochastic Processes Example: Continuous-Time GARCH Model (or Pilipovic One-Factor Model)

13 Mean-Reverting Models in Energy Markets

14 Change of Time: Definition and Examples Change of Time-change time from t to a nonnegative process T(t) 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 (Product Process):

15 Time-Changed Brownian Motion by Bochner 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

16 Change of Time: First Intro into Financial Economics Clark (1973) ( A Subordinated Stochastic Process Model with Fixed Variance for Speculative Prices, Econometrica, 41, )-introduced Bochner s (1949) timechanged 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 timechange (B and T are independent)

17 Change of Time: Short History. I. 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 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, )-studied the pricing of options using time-changing SVM

18 Change of Time: Short History. II. Ikeda & Watanabe (1981) ( SDEs and Diffusion Processes, North-Holland Publ. Co)-introduced and studied CTM for the solution of SDEs Barndorff-Nielsen, Nicolato & Shephard (2003) ( Some recent development in stochastic volatility modelling )-review and put in context some of their recent work on stochastic volatility (SV) modelling, including the relationship between subordination and SV (random time-chronometer) 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)

19 CT and Embedding Problem Embedding Problem was first terated by Skorokhod (1965)-sum of any sequence of i.r.v. with mean zero and finite variation could be embedded in Brownian motion (BM) using stopping time Dambis (1965) and Dubis and Schwartz (1965)-every continuous martingale could be time-changed BM Huff (1969)-every processes of pathwise bounded variation could be embedded in BM Monroe (1972)-every right continuous martingale could be embedded in a BM Monroe (1978)-local martingale can be embedded in BM

20 Change of Time: Simplest (Martingale) Case

21 Change of Time: Ito Integral s Case

22 Change of Time: SDE s Case

23 Geometric Brownian Motion SVM

24 Change of Time Method

25 Connection between phi_t and phi_t^(-1)

26 Solution for GBM Equation Using Change of Time

27 Explicit Expression for

28 Mean-Reverting SV Model

29 Solution of MRM by CTM

30 Explicit Expression for

31 Explicit Expression for

32 Comparison: Solution of GBM & MRM -GBM -MRM

33 Explicit Expression for S(t) where

34 Properties of

35 Properties of

36 Properties of eta(t)

37 Properties of Eta(t). II.

38 Mean Value of MRM S(t)

39 Dependence of ES(t) on T

40 Dependence of ES(t) on S_0 and T

41 Variance for S(t)

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 C_T in the case of MRM C_T=BS(T)+A(T)

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

48 Expression for BS(T)

49 Expression for y_0 for MRM

50 Expression for A(T).I.

51 Moment generating) function of Eta(T)

52 Expression for A(T)

53 European Call Option for MRM (Explicit Formula)

54 European Call Option for MRM in Risk-Neutral World

55

56

57 Dependence of C_T on T

58 Comparison of Three Solutions Heston Model Mean-Reverting Model Black-Scholes Model

59 Comparison: Heston Model (1993)

60 Explicit Solution for CIR Process: CTM

61 Comparison: Solutions to the Three Models -GBM -MRM -Heston model

62 Summary GBM Model 1. -martingale Mean-Reverting Model 2. Heston Model -sum of two martingales 3. -martingale

63 Problem -explicit expression? To calculate an option price for Heston model, for example We know all the moments at this moment, though

64 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)

65 Future Work Change of Time Method for Two- Factor Continuous-Time GARCH Model

66 The End Thank You for Your Attention and Time!

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