Change of Time MethodChange of Time Methodinin

Mathematical FinanceMathematical Finance

Anatoliy SwishchukAnatoliy SwishchukMathematical & Computational Finance LabMathematical & Computational Finance Lab

Department of Mathematics & Statistics Department of Mathematics & Statistics University of Calgary, Calgary, Alberta, CanadaUniversity of Calgary, Calgary, Alberta, Canada

CMS 2006 Summer MeetingCMS 2006 Summer MeetingMathematical Finance Session Mathematical Finance Session

Calgary, AB, CanadaCalgary, AB, CanadaJune 3-5, 2006June 3-5, 2006

OutlineOutline

Change of Time (CT)Change of Time (CT): Definition and Examples: Definition and Examples Change of Time Method (CTM):Change of Time Method (CTM): Short HistoryShort History Black-Scholes by CTMBlack-Scholes by CTM (i.e., CTM for (i.e., CTM for GBMGBM)) Explicit Option Pricing FormulaExplicit Option Pricing Formula (EOPF) for (EOPF) for

Mean-Reverting ModelMean-Reverting Model (MRM) by CTM (MRM) by CTM Black-Scholes Formula as a Particular Case Black-Scholes Formula as a Particular Case

of EOPF for MRMof EOPF for MRM Modeling and Pricing of Variance and Modeling and Pricing of Variance and

Volatility Swaps Volatility Swaps by CTMby CTM

Change of Time: Definition and ExamplesChange of Time: Definition and Examples

Change of Time-Change of Time-change time from t to a non-change time from t to a non-negative process with non-decreasing sample negative process with non-decreasing sample pathspaths

Example 1Example 1 ( (Time-Changed Brownian MotionTime-Changed Brownian Motion): ): M(t)=B(T(t)), B(t)-Brownian motion, T(t) is M(t)=B(T(t)), B(t)-Brownian motion, T(t) is change of timechange of time

Example 2 Example 2 ((SubordinatorSubordinator): X(t) and T(t)>0 are ): X(t) and T(t)>0 are some processes, then X(T(t)) is subordinated to some processes, then X(T(t)) is subordinated to X(t); T(t) is change of timeX(t); T(t) is change of time

Example 3Example 3 ( (Standard Stochastic Volatility Standard Stochastic Volatility ModelModel ( (SVMSVM)) ): M(t)=\int_0^t\sigma(s)dB(s), ): 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.

Change of Time: Short History. I.Change of Time: Short History. I.

BochnerBochner (1949) - (1949) -introduced the notion of introduced the notion of change of time (CT) (time-changed Brownian change of time (CT) (time-changed Brownian motion)motion)

BochnerBochner (1955) (‘Harmonic Analysis and the (1955) (‘Harmonic Analysis and the Theory of Probability’, UCLA Press, 176)-Theory of Probability’, UCLA Press, 176)-further further development of CTdevelopment of CT

Change of Time: Short History. II.Change of Time: Short History. II.

FellerFeller (1966) -introduced subordinated (1966) -introduced subordinated processes X(T(t)) with Markov process X(t) and processes X(T(t)) with Markov process X(t) and T(t) as a process with independent increments T(t) as a process with independent increments (i.e., Poisson process); T(t) was called (i.e., Poisson process); T(t) was called randomized operational timerandomized operational time

ClarkClark (1973)-first (1973)-first introduced Bochner’s (1949) introduced Bochner’s (1949) time-changed Brownian motion into financial time-changed Brownian motion into financial economics:economics: he wrote down a model for the log-he wrote down a model for the log-price M as M(t)=B(T(t)), where B(t) is Brownian price M as M(t)=B(T(t)), where B(t) is Brownian motion, T(t) is time-change (B and T are motion, T(t) is time-change (B and T are independent)independent)

Change of Time: Short History. III.Change of Time: Short History. III.

Ikeda & WatanabeIkeda & Watanabe (1981)- (1981)-introduced and introduced and studied CTM for the solution of Stochastic studied CTM for the solution of Stochastic Differential EquationsDifferential Equations

Carr, Geman, Madan & Yor (2003)-Carr, Geman, Madan & Yor (2003)-used used subordinated processes to construct SV for subordinated processes to construct SV for Levy Processes (T(t)-business time)Levy Processes (T(t)-business time)

Geometric Brownian MotionGeometric Brownian Motion(Black-Scholes Formula by CTM)(Black-Scholes Formula by CTM)

Change of Time MethodChange of Time Method

Time-Changed BM is a MartingaleTime-Changed BM is a Martingale

Option PricingOption Pricing

European Call Option PricingEuropean Call Option Pricing(Pay-Off Function)(Pay-Off Function)

European Call Option PricingEuropean Call Option Pricing

Black-Scholes FormulaBlack-Scholes Formula

Mean-Reverting Model Mean-Reverting Model (Option Pricing Formula by CTM(Option Pricing Formula by CTM))

Solution of MRM by CTMSolution of MRM by CTM

European Call Option for MRM.I.European Call Option for MRM.I.

European Call OptionEuropean Call Option(Payoff Function)(Payoff Function)

Expression for y_0 for MRMExpression for y_0 for MRM

Expression for C_TExpression for C_T

C_T=BS(T)+A(T)C_T=BS(T)+A(T)((Black-Scholes Part+Additional TermBlack-Scholes Part+Additional Term

due to mean-reversiondue to mean-reversion))

Expression for BS(T)Expression for BS(T)

Expression for A(T)Expression for A(T)

European Call Option Price for MRMEuropean Call Option Price for MRMin Real Worldin Real World

European Call Option for MRM in Risk-European Call Option for MRM in Risk-Neutral WorldNeutral World

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

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

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

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

Dependence of C_T on TDependence of C_T on T

Heston ModelHeston Model(Pricing Variance and Volatility Swaps by CTM)(Pricing Variance and Volatility Swaps by CTM)

Explicit Solution for CIR Process: CTMExplicit Solution for CIR Process: CTM

Why Trade Volatility?Why Trade Volatility?

Variance Swap for Heston ModelVariance Swap for Heston Model

Volatility Swap for Heston ModelVolatility Swap for Heston Model

How Does the Volatility Swap Work?How Does the Volatility Swap Work?

How Does the Volatility Swap Work?How Does the Volatility Swap Work?

Pricing of Variance Swap for Heston ModelPricing of Variance Swap for Heston Model

Pricing of Volatility Swap for Heston ModelPricing of Volatility Swap for Heston Model

Brockhaus and Long ResultsBrockhaus and Long Results

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

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

Histograms of Log-Returns Histograms of Log-Returns for S&P60 Canada Indexfor S&P60 Canada Index

Convexity AdjustmentConvexity Adjustment

S&P60 Canada Index Volatility SwapS&P60 Canada Index Volatility Swap

ConclusionsConclusions CTM works for:CTM works for: Geometric Brownian motion (to price Geometric Brownian motion (to price

options in money markets)options in money markets) Mean-Reverting Model (to price options in Mean-Reverting Model (to price options in

energy markets)energy markets) Heston Model (to price variance and Heston Model (to price variance and

volatility swaps)volatility swaps) Much More: Covariance and Correlation Much More: Covariance and Correlation

SwapsSwaps

The End/FinThe End/Fin

Thank You!/Thank You!/

Merci Beaucoup!Merci Beaucoup!