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Girsanov’s Theorem: From Game Theory to Finance

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. Outline. Simplest Case: Girsanov’s Theorem in Game Theory GT for Brownian Motion Applications GT in Finance

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Girsanov’s Theorem: From Game Theory to Finance

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  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!

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