Change of Time Method: Applications to Mathematical
Finance. II.
Anatoliy SwishchukMath & Comp Finance LabDept of Math & Stat, U of C
“Lunch at the Lab” TalkNovember 8, 2005
Outline
• Change of Time Method (CTM) (minutes of the previous talk)
• Mean-Reverting Model (MRM)• Solution of MRM by CTM• Option Pricing Formula• Black - Scholes Formula Follows: L=0,
a^*=-r• Numerical Example (AECO Natural Gas
Index)
CTM for Martingales
CTM in General Setting. I .
CTM in General Setting. II.
CTM for SDEs. I.
CTM for SDEs. II.
Connection between phi_t and phi_t^(-1)
Idea of Proof. I.
Idea of Proof. II.
Mean-Reverting Model
Solution of MRM by CTM
Solution of GBM Model (to compare)
Properties of
Explicit Expression for
Explicit Expression for
Explicit Expression for S(t)
Properties of
Properties of
Properties of Eta(t). II.
Properties of MRM S(t). I.
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)
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). II.
European Call Option for MRM
Boundaries for C_T
European Call Option for MRM in Risk-Neutral World
Boundaries for MRM in Risk-Neutral World
Dependence of C_T on T
Paper may be found on the following web page(E-Yellow Series Listing):
http://www.math.ucalgary.ca/research/preprint.php
The End
Thank You for Your Attention and Time!
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