Change of Time Method: Applications to Mathematical Finance. II. Anatoliy Swishchuk Math & Comp...

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!