Change of Time Method: Applications to Mathematical Finance. II. Anatoliy Swishchuk Math & Comp...
-
Upload
angeline-shakespeare -
Category
Documents
-
view
217 -
download
3
Transcript of Change of Time Method: Applications to Mathematical Finance. II. Anatoliy Swishchuk Math & Comp...
![Page 1: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/1.jpg)
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
![Page 2: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/2.jpg)
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)
![Page 3: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/3.jpg)
CTM for Martingales
![Page 4: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/4.jpg)
CTM in General Setting. I .
![Page 5: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/5.jpg)
CTM in General Setting. II.
![Page 6: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/6.jpg)
CTM for SDEs. I.
![Page 7: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/7.jpg)
CTM for SDEs. II.
![Page 8: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/8.jpg)
Connection between phi_t and phi_t^(-1)
![Page 9: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/9.jpg)
Idea of Proof. I.
![Page 10: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/10.jpg)
Idea of Proof. II.
![Page 11: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/11.jpg)
Mean-Reverting Model
![Page 12: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/12.jpg)
Solution of MRM by CTM
![Page 13: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/13.jpg)
Solution of GBM Model (to compare)
![Page 14: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/14.jpg)
Properties of
![Page 15: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/15.jpg)
Explicit Expression for
![Page 16: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/16.jpg)
Explicit Expression for
![Page 17: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/17.jpg)
Explicit Expression for S(t)
![Page 18: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/18.jpg)
Properties of
![Page 19: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/19.jpg)
Properties of
![Page 20: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/20.jpg)
Properties of Eta(t). II.
![Page 21: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/21.jpg)
Properties of MRM S(t). I.
![Page 22: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/22.jpg)
Dependence of ES(t) on T
![Page 23: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/23.jpg)
Dependence of ES(t) on S_0 and T
![Page 24: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/24.jpg)
Properties of MRM S(t). II.
![Page 25: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/25.jpg)
Dependence of Variance of S(t) on S_0 and T
![Page 26: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/26.jpg)
Dependence of Volatility of S(t) on S_0 and T
![Page 27: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/27.jpg)
European Call Option for MRM.I.
![Page 28: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/28.jpg)
European Call Option. II.
![Page 29: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/29.jpg)
Expression for y_0 for MRM
![Page 30: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/30.jpg)
Expression for C_T
C_T=BS(T)+A(T)
![Page 31: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/31.jpg)
Expression for C_T=BS(T)+A(T).II.
![Page 32: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/32.jpg)
Expression for BS(T)
![Page 33: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/33.jpg)
Expression for A(T).I.
![Page 34: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/34.jpg)
Expression for A(T).II.
Characteristic function of Eta(T):
![Page 35: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/35.jpg)
Expression for A(T). II.
![Page 36: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/36.jpg)
European Call Option for MRM
![Page 37: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/37.jpg)
Boundaries for C_T
![Page 38: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/38.jpg)
European Call Option for MRM in Risk-Neutral World
![Page 39: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/39.jpg)
Boundaries for MRM in Risk-Neutral World
![Page 40: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/40.jpg)
![Page 41: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/41.jpg)
![Page 42: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/42.jpg)
Dependence of C_T on T
![Page 43: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/43.jpg)
Paper may be found on the following web page(E-Yellow Series Listing):
http://www.math.ucalgary.ca/research/preprint.php
![Page 44: 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”](https://reader036.fdocuments.in/reader036/viewer/2022070308/551be1b6550346b9588b5e35/html5/thumbnails/44.jpg)
The End
Thank You for Your Attention and Time!