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Change of Time Method in Mathematical 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. Outline.

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## Change of Time Method in Mathematical Finance

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**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**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**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.**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))**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.**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**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**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)**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**Solution of GBM Model (just to compare with solution of MRM)****Expression for C_T**C_T=BS(T)+A(T) (Black-Scholes Part+Additional Term due to mean-reversion)

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