Download presentation

Presentation is loading. Please wait.

Published byJanae Twining Modified over 2 years ago

1
Girsanov’s Theorem: From Game Theory to Finance Anatoliy Swishchuk Math & Comp Finance Lab Dept of Math & Stat, U of C “Lunch at the Lab” Talk December 6, 2005

2
Outline Simplest Case: Girsanov’s Theorem in Game Theory GT for Brownian Motion Applications GT in Finance Discrete-Time (B,S)-Security Markets Continuous-Time (B,S)-Security Markets Other Models in Finance: Merton (Poisson), Jump-Diffusion, Diffusion with SV General Girsanov’s Theorem Conclusion

3
Original Girsanov’s Paper Girsanov, I. V. (1960) On transforming a certain class of stochastic processes by absolutely continuous substitution of measures. Theory Probability and Its Applications, 5, 285-301. Extension of Cameron-Martin Theorem (1944) for multi-dimensional shifted Brownian motion

4
Cameron-Martin Theorem

5
Girsanov’s Theorem

6
Game Theory. I.

7
Game Theory. II.

8
Girsanov’s Theorem in Game Theory Take p=1/2-probability of success or to win- to make game fair, or (the same) to make total gain X_n martingale in nth game p=1/2 is a martingale measure (simpliest)

9
Discrete-Time (B,S)-Security Market. I.

10
Discrete-Time (B,S)-Security Market. II.

11
Discrete-Time (B,S)-Security Market. III.

12
GT for Discrete-Time (B,S)-SM Change measure from p to p^*=(r-a) / (b-a). Here: p^* is a martingale measure (discounted capital is a martingale)

13
GT for Discrete-Time (B,S)-SM: Density Process

14
Continuous-Time (B,S)-Security Market. I.

15
Continuous-Time (B,S)-Security Market. II.

16
GT for Continuous-Time (B,S)- SM. I.

17
GT for Continuous-Time (B,S)- SM. II.

18
GT for Other Models. I: Merton (Poisson) Model

19
GT for Other Models. II: Diffusion Model with Jumps

20
GT for Other Models. II: Diffusion Model with Jumps (contd)

21
GT for Other Models. III. Continuous- Time (B,S)-SM with Stochastic Volatility

22
GT for Other Models. III. Continuous- Time (B,S)-SM with Stochastic Volatility (contd)

23
General Girsanov’s Theorem (Transformation of Drift)

24
The End Thank You for Your Attention and Time! Merry Christmas!

Similar presentations

OK

Stability of Financial Models Anatoliy Swishchuk Mathematical and Computational Finance Laboratory Department of Mathematics and Statistics University.

Stability of Financial Models Anatoliy Swishchuk Mathematical and Computational Finance Laboratory Department of Mathematics and Statistics University.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Small ppt on water conservation Ppt on active directory services Ppt on dc power supply Ppt on palm island in dubai Blood vessel anatomy and physiology ppt on cells Ppt on acid base and salt Ppt on endangered species of wildlife in india Ppt on internal auditing process flowchart Ppt on value of pie in math Ppt on conservation of environment biodiversity