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18.642 | Fall 2024 | Undergraduate

# Topics in Mathematics with Applications in Finance

### Lecture 19

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Video description: This lecture provides a comprehensive overview of volatility modeling in finance, covering fundamental concepts such as realized volatility, historical and implied volatility, and various estimation techniques including exponential moving averages and advanced estimators like the Garman-Klass and Yang-Zhang models. It also delves into stochastic process models like geometric Brownian motion, jump diffusion, and time-varying volatility models such as ARCH and GARCH, alongside practical time series forecasting methods and empirical case studies, highlighting their applications and comparative efficiencies.

Volatility Modeling Slides (PDF)

Case Study: Estimating Historical Volatility of the S&P 500 (PDF)

Read:

- Garman, Mark B., and Michael J. Klass. “On the Estimation of Security Price Volatilities from Historical Data.” The Journal of Business 53, no. 1 (1980): 67–78. https://www.jstor.org/stable/2352358.
- Petneházi, Gábor, and Gáll, József. (2019). “Exploring the Predictability of Range‐Based Volatility Estimators Using Recurrent Neural Networks. Intelligent Systems in Accounting.” Finance and Management. 26. 10.1002/isaf.1455.
- Rogers, L. C. G., and S. E. Satchell. “Estimating Variance from High, Low, and Closing Prices.” The Annals of Applied Probability 1, no. 4 (1991): 504–12. https://www.jstor.org/stable/2959703.

### Lecture 20

Guest lecture with Tarek Mansour, Kalshi.com Financial Exchange and Prediction Market

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Video description: This lecture recounts a detailed discussion featuring Tarek Mansour, an MIT alumnus and founder of Kalshi, about building a regulated prediction market exchange in the US. He shares his journey from working at top Wall Street firms to overcoming regulatory challenges as part of building Kalshi business, emphasizing perseverance, innovation in financial markets, and the potential of prediction markets to transform how risks and future events are traded and priced.

## 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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