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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

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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

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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. T(t)=[M(t)]=\int_0^t\sigma^2(s)ds.

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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))

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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)]=T(t) is that in the SVM case [M(t)]=\int_0^t\sigma^2(s)ds. [M(t)]=\int_0^t\sigma^2(s)ds.

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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

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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

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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)

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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

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CTM for SDEs. I.

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CTM for SDEs. II.

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Idea of Proof. I.

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Idea of Proof. II.

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Geometric Brownian Motion

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Change of Time Method

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Solution for GBM Equation Using Change of Time

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Option Pricing

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European Call Option Pricing (Pay-Off Function)

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European Call Option Pricing

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Black-Scholes Formula

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Risk-Neutral Stock Price

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Explicit Expression for

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European Call Option Through

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Derivation of Black - Scholes Formula I

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Derivation of Black-Scholes Formula II (continuation)

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Derivation of Black - Scholes Formula III (continuation)

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Derivation of Black - Scholes Formula IV (continuation)

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Mean-Reverting Model

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Solution of MRM by CTM

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Solution of GBM Model (just to compare with solution of MRM)

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Properties of

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Explicit Expression for

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Explicit Expression for S(t)

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Properties of

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Properties of Eta (t). I.

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Properties of Eta(t). II.

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Properties of MRM S (t)

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Dependence of ES(t) on T

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Dependence of ES(t) on S_0 and T

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Properties of MRM S(t). II.

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

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

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European Call Option for MRM.I.

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European Call Option. II.

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Expression for y_0 for MRM

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Expression for C_T C_T=BS(T)+A(T) ( Black-Scholes Part+Additional Term due to mean-reversion )

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

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Expression for BS(T)

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Expression for A(T).I.

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

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Expression for A(T). III.

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European Call Option for MRM

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Boundaries for C_T

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European Call Option for MRM in Risk- Neutral World

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Boundaries for MRM in Risk-Neutral World

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Dependence of C_T on T

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Heston Model

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Explicit Solution for CIR Process: CTM

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Proof. I.

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Proof. II.

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Properties of

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Variance Swap for Heston Model. I.

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Variance Swap for Heston Model. II.

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Volatility Swap for Heston Model. II.

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Volatility Swap for Heston Model. I.

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Why Trade Volatility?

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How Does the Volatility Swap Work?

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Pricing of Variance Swap in Heston Model. I.

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Pricing of Variance Swap in Heston Model. II.

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Proof

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Pricing of Volatility Swap for Heston Model. I.

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Pricing of Volatility Swap for Heston Model. II.

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Proof. I.

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Proof. II.

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Proof. III.

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Proof. IV.

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Proof. V.

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Brockhaus and Long Results Brockhaus & Long (2000) obtained the same results for variance and volatility swaps for Heston model using another technique (analytical rather than probabilistic), including inverse Laplace transform

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Statistics on Log Returns of S&P Canada Index (Jan 1997-Feb 2002)

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Histograms of Log-Returns for S&P60 Canada Index

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Convexity Adjustment

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S&P60 Canada Index Volatility Swap

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References. I.

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References. II.

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References. III.

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References. IV.

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References. V.

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References. VI.

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References. VII.

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References. VIII.

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References. IX.

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References. X. Swishchuk, A. (2005): Modeling and Pricing of Variance Swaps for Stochastic Volatility with Delay, Wilmott Magazine, September Issue, 19, No 2., 63-73. Swishchuk, A. (2005): Modeling and Pricing of Variance Swaps for Stochastic Volatility with Delay, Wilmott Magazine, September Issue, 19, No 2., 63-73. Swishchuk, A. (2006): Change of Time Method in Mathematical Finance, 2006 Stochastic Modeling Symposium, Toronto, April 3-4, 2006, paper for presentation Swishchuk, A. (2006): Change of Time Method in Mathematical Finance, 2006 Stochastic Modeling Symposium, Toronto, April 3-4, 2006, paper for presentation

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The End Thank You for Your Attention! Thank You for Your Attention!

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