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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 CMS 2006 Summer Meeting Mathematical Finance Session Calgary, AB, Canada June 3-5, 2006

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Outline Change of Time (CT): Definition and Examples Change of Time Method (CTM): Short History 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 Modeling and Pricing of Variance and Volatility Swaps 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 Example 1 (Time-Changed Brownian Motion): M(t)=B(T(t)), B(t)-Brownian motion, T(t) is change of time Example 2 (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 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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Change of Time: Short History. I. Bochner (1949) -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

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Change of Time: Short History. II. Feller (1966) -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 Clark (1973)-first 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)

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Change of Time: Short History. III. Ikeda & Watanabe (1981)-introduced and studied CTM for the solution of Stochastic Differential Equations Carr, Geman, Madan & Yor (2003)-used subordinated processes to construct SV for Levy Processes (T(t)-business time)

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Geometric Brownian Motion (Black-Scholes Formula by CTM)

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Change of Time Method

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Time-Changed BM is a Martingale

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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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Mean-Reverting Model (Option Pricing Formula by CTM )

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Solution of MRM by CTM

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European Call Option for MRM.I.

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European Call Option (Payoff Function)

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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 BS(T)

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Expression for A(T)

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European Call Option Price for MRM in Real World

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European Call Option for MRM in Risk- Neutral World

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Dependence of ES(t) on T (mean-reverting level L^*=2.569 )

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Dependence of ES(t) on S_0 and T (mean-reverting level L^*=2.569)

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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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Dependence of C_T on T

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Heston Model (Pricing Variance and Volatility Swaps by CTM)

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Explicit Solution for CIR Process: CTM

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Why Trade Volatility?

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Variance Swap for Heston Model

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Volatility Swap for Heston Model

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How Does the Volatility Swap Work?

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Pricing of Variance Swap for Heston Model

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Pricing of Volatility Swap for Heston Model

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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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Conclusions CTM works for: Geometric Brownian motion (to price options in money markets) Mean-Reverting Model (to price options in energy markets) Heston Model (to price variance and volatility swaps) Much More: Covariance and Correlation Swaps

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The End/Fin Thank You!/ Thank You!/ Merci Beaucoup!

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