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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!

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