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18.642 | Fall 2024 | Undergraduate

# Topics in Mathematics with Applications in Finance

### Lecture 13

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Video description: This lecture provides a comprehensive overview of portfolio management, focusing on the practical aspects of asset allocation, risk measurement, and investment sizing beyond traditional modern portfolio theory, highlighting its limitations and proposing improved approaches such as gain-loss ratios. It also explores behavioral finance concepts like crowding behavior and power law distributions, emphasizing the importance of dynamic rebalancing and understanding market influences from powerful agents such as governments and large funds.

Portfolio Management Slides (PDF)

### Lecture 14

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Video description: Brownian motion is a fundamental stochastic process characterized by continuous, random, and independent increments with normally distributed changes over time, whose variance grows proportionally with the time interval. It serves as a mathematical model for diverse natural and financial phenomena, exhibiting key properties such as the Markov property, reflection principle, quadratic variation, and extensions like Brownian motion with drift, reflected and absorbed Brownian motions, and the Brownian bridge, all crucial for understanding random dynamics and derivative pricing.

Stochastic Processes II Slides (PDF)

Two-Dimensional Brownian Motion (PDF)

Diffusion Normal Density (PDF)

Limiting Random Walk (PDF)

Reflected Brownian Motion (PDF)

Maximum Brownian Motion (PDF)

Hitting Time Density (PDF)

See the Wikipedia article Reflection Principle (Wiener process)

Gamblers Ruin (PDF)

Quadratic Variation (PDF)

See Problem Set 4

## Course Info

##### Instructors

- Dr. Peter Kempthorne
- Dr. Vasily Strela
- Dr. Jake Xia

##### Departments

- Mathematics

##### As Taught In

Fall 2024

##### Level

Undergraduate

##### Topics

- Business
  - Finance
- Mathematics
  - Applied Mathematics
  - Probability and Statistics

##### Learning Resource Types

notes Lecture Notes
theaters Lecture Videos
assignment Problem Sets
assignment Activity Assignments
group_work Projects
Readings

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