girsanov’s theorem: from game theory to finance anatoliy swishchuk math & comp finance lab...
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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
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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
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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
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Cameron-Martin Theorem
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Girsanov’s Theorem
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Game Theory. I.
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Game Theory. II.
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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)
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Discrete-Time (B,S)-Security Market. I.
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Discrete-Time (B,S)-Security Market. II.
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Discrete-Time (B,S)-Security Market. III.
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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)
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GT for Discrete-Time (B,S)-SM: Density Process
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Continuous-Time (B,S)-Security Market. I.
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Continuous-Time (B,S)-Security Market. II.
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GT for Continuous-Time (B,S)-SM. I.
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GT for Continuous-Time (B,S)-SM. II.
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GT for Other Models. I: Merton (Poisson) Model
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GT for Other Models. II: Diffusion Model with Jumps
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GT for Other Models. II: Diffusion Model with Jumps (contd)
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GT for Other Models. III. Continuous-Time (B,S)-SM with Stochastic Volatility
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GT for Other Models. III. Continuous-Time (B,S)-SM with Stochastic Volatility
(contd)
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General Girsanov’s Theorem (Transformation of Drift)
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The End
Thank You for Your Attention and Time!
Merry Christmas!