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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. 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 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 • 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
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
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)
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)
GT for Other Models. III. Continuous-Time (B,S)-SM with Stochastic Volatility
GT for Other Models. III. Continuous-Time (B,S)-SM with Stochastic Volatility (contd)
The End Thank You for Your Attention and Time! Merry Christmas!