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
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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!