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

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!