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, Extension of Cameron-Martin Theorem (1944) for multi-dimensional shifted Brownian motion

Cameron-Martin Theorem

Girsanov’s Theorem

Game Theory. I.

Game Theory. II.

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)

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

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

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

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

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

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

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

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

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

GT for Other Models. II: Diffusion Model with Jumps

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

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)

General Girsanov’s Theorem (Transformation of Drift)

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