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 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 MeanReverting 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 Change of Time: Definition and Examples Change of Time-change time from t to a nonnegative 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. Interpretation of CT. I. If M(t) is fair game process (another name-martingale) Then M(t)=B(T(t)) (Dambis-DubinsSchwartz 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)) 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)]=\int_0^t\sigma^2(s)ds. 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 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 timechanging SVM 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) 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 CTM for SDEs. I. CTM for SDEs. II. Idea of Proof. I. Idea of Proof. II. Geometric Brownian Motion Change of Time Method Solution for GBM Equation Using Change of Time Option Pricing European Call Option Pricing (Pay-Off Function) European Call Option Pricing Black-Scholes Formula Risk-Neutral Stock Price Explicit Expression for European Call Option Through Derivation of Black - Scholes Formula I Derivation of Black-Scholes Formula II (continuation) Derivation of Black - Scholes Formula III (continuation) Derivation of Black - Scholes Formula IV (continuation) Mean-Reverting Model Solution of MRM by CTM Solution of GBM Model (just to compare with solution of MRM) Properties of Explicit Expression for Explicit Expression for Explicit Expression for S(t) Properties of Properties of Eta (t). I. Properties of Eta(t). II. Properties of MRM S (t) Dependence of ES(t) on T Dependence of ES(t) on S_0 and T Properties of MRM S(t). II. Dependence of Variance of S(t) on S_0 and T Dependence of Volatility of S(t) on S_0 and T European Call Option for MRM.I. European Call Option. II. Expression for y_0 for MRM Expression for C_T C_T=BS(T)+A(T) (Black-Scholes Part+Additional Term due to mean-reversion) Expression for C_T=BS(T)+A(T).II. Expression for BS(T) Expression for A(T).I. Expression for A(T).II. Characteristic function of Eta(T): Expression for A(T). III. European Call Option for MRM Boundaries for C_T European Call Option for MRM in RiskNeutral World Boundaries for MRM in Risk-Neutral World Dependence of C_T on T Heston Model Explicit Solution for CIR Process: CTM Proof. I. Proof. II. Properties of Properties of Variance Swap for Heston Model. I. Variance Swap for Heston Model. II. Volatility Swap for Heston Model. II. Volatility Swap for Heston Model. I. Why Trade Volatility? How Does the Volatility Swap Work? How Does the Volatility Swap Work? Pricing of Variance Swap in Heston Model. I. Pricing of Variance Swap in Heston Model. II. Proof Pricing of Volatility Swap for Heston Model. I. Pricing of Volatility Swap for Heston Model. II. Proof. I. Proof. II. Proof. III. Proof. IV. Proof. V. 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 Convexity Adjustment S&P60 Canada Index Volatility Swap References. I. References. II. References. III. References. IV. References. V. References. VI. References. VII. References. VIII. References. IX. 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. (2006): Change of Time Method in Mathematical Finance, 2006 Stochastic Modeling Symposium, Toronto, April 3-4, 2006, paper for presentation The End Thank You for Your Attention!