Explicit Option Pricing Formula for a MeanReverting Asset in Energy Markets


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1 Explicit Option Pricing Formula for a MeanReverting 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 1215, 2007 This research is supported by MITACS and NSERC
2 Outline MeanReverting 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) MeanReverting Model (MRM) Option Pricing Formula Drawback of OneFactor 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 SemiSpectral Method for Pricing an Option on a MeanReverting Asset, Quantit. Finance J. (PDE approach, semispectral 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 continuoustime 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 MeanReversion 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 nonevent levels and we observe them go back to more or less the levels they started from
7 The MeanReverting Deterministic Process
8 MeanReverting Plot (a=4.6,l=2.5)
9 Meaning of MeanReverting Parameter The greater the meanreverting 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 equilibriumin days
10 MeanReverting Stochastic Process
11 MeanReverting Models in Financial Markets Stock (asset) Prices follow geometric Brownian motion The Variance of Stock Price follows MeanReverting Models Example: Heston Model
12 MeanReverting Models in Energy Markets Asset Prices follow MeanReverting Stochastic Processes Example: ContinuousTime GARCH Model (or Pilipovic OneFactor Model)
13 MeanReverting Models in Energy Markets
14 Change of Time: Definition and Examples Change of Timechange time from t to a nonnegative process T(t) with nondecreasing 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 (TimeChanged Brownian Motion): M(t)=B(T(t)), B(t)Brownian motion Example 3 (Product Process):
15 TimeChanged 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) (timechanged 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 logprice 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 timechanged 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 timechanging SVM
18 Change of Time: Short History. II. Ikeda & Watanabe (1981) ( SDEs and Diffusion Processes, NorthHolland Publ. Co)introduced and studied CTM for the solution of SDEs BarndorffNielsen, 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 timechronometer) 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 timechanged 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 MeanReverting 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 RiskNeutral World
55
56
57 Dependence of C_T on T
58 Comparison of Three Solutions Heston Model MeanReverting Model BlackScholes 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 MeanReverting 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 OneFactor Mean Reverting Models The longterm 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 modelimplied volatility term structure that has the volatilities going to zero as expiration time increases (spot volatilities have to be increased to nonintuitive levels so that the long term options do not lose all the volatility valueas in the marketplace they certainly do not)
65 Future Work Change of Time Method for Two Factor ContinuousTime GARCH Model
66 The End Thank You for Your Attention and Time!
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