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Change of Time Method: Applications to Mathematical Finance. II. Anatoliy Swishchuk Math & Comp Finance Lab Dept of Math & Stat, U of C “Lunch at the Lab” Talk November 8, 2005

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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)

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CTM for Martingales

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CTM in General Setting. I.

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CTM in General Setting. II.

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CTM for SDEs. I.

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CTM for SDEs. II.

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Connection between phi_t and phi_t^(-1)

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Idea of Proof. I.

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Idea of Proof. II.

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Mean-Reverting Model

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Solution of MRM by CTM

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Solution of GBM Model (to compare)

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Properties of

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Explicit Expression for

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Explicit Expression for S(t)

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Properties of

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Properties of Eta(t). II.

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Properties of MRM S(t). I.

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Dependence of ES(t) on T

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Dependence of ES(t) on S_0 and T

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Properties of MRM S(t). II.

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Dependence of Variance of S(t) on S_0 and T

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Dependence of Volatility of S(t) on S_0 and T

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European Call Option for MRM.I.

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European Call Option. II.

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Expression for y_0 for MRM

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Expression for C_T C_T=BS(T)+A(T)

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Expression for C_T=BS(T)+A(T).II.

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Expression for BS(T)

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Expression for A(T).I.

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Expression for A(T).II. Characteristic function of Eta(T):

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Expression for A(T). II.

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European Call Option for MRM

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Boundaries for C_T

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European Call Option for MRM in Risk- Neutral World

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Boundaries for MRM in Risk-Neutral World

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Dependence of C_T on T

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Paper may be found on the following web page (E-Yellow Series Listing): http://www.math.ucalgary.ca/research/preprint.php

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The End Thank You for Your Attention and Time!

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Explicit Option Pricing Formula for A Mean-Reverting Asset Anatoliy Swishchuk “Lunch at the Lab” Talk March 10, 2005.

Explicit Option Pricing Formula for A Mean-Reverting Asset Anatoliy Swishchuk “Lunch at the Lab” Talk March 10, 2005.

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