Presentation on theme: "Change of Time Method in Mathematical Finance Anatoliy Swishchuk Mathematical & Computational Finance Lab Department of Mathematics & Statistics University."— Presentation transcript:
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
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
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.
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
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)
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)
Geometric Brownian Motion (Black-Scholes Formula by CTM)
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
Statistics on Log Returns of S&P Canada Index (Jan 1997-Feb 2002)
Histograms of Log-Returns for S&P60 Canada Index
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
The End/Fin Thank You!/ Thank You!/ Merci Beaucoup!