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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!