Transcript Change of Time Method in Mathematical Finance

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 nonnegative 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.
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 logprice 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)
Change of Time Method
Time-Changed BM is a Martingale
Option Pricing
European Call Option Pricing
(Pay-Off Function)
European Call Option Pricing
Black-Scholes Formula
Mean-Reverting Model
(Option Pricing Formula by CTM)
Solution of MRM by CTM
European Call Option for MRM.I.
European Call Option
(Payoff Function)
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 BS(T)
Expression for A(T)
European Call Option Price for MRM
in Real World
European Call Option for MRM in RiskNeutral World
Dependence of ES(t) on T
(mean-reverting level L^*=2.569)
Dependence of ES(t) on S_0 and T
(mean-reverting level L^*=2.569)
Dependence of Variance of S(t) on S_0 and T
Dependence of Volatility of S(t) on S_0 and T
Dependence of C_T on T
Heston Model
(Pricing Variance and Volatility Swaps by CTM)
Explicit Solution for CIR Process: CTM
Why Trade Volatility?
Variance Swap for Heston Model
Volatility Swap for Heston Model
How Does the Volatility Swap Work?
How Does the Volatility Swap Work?
Pricing of Variance Swap for Heston Model
Pricing of Volatility Swap for Heston Model
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
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!/
Merci Beaucoup!