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# Pricing Options with Mathematical Models

### Overview

Class Central Tips
This is an introductory course on options and other financial derivatives, and their applications to risk management. We will start with defining derivatives and options, continue with discrete-time, binomial tree models, and then develop continuous-time, Brownian Motion models. A basic introduction to Stochastic, Ito Calculus will be given. The benchmark model will be the Black-Scholes-Merton pricing model, but we will also discuss more general models, such as stochastic volatility models. We will discuss both the Partial Differential Equations approach, and the probabilistic, martingale approach. We will also cover an introduction to modeling of interest rates and fixed income derivatives. I teach the same class at Caltech, as an advanced undergraduate class. This means that the class may be challenging, and demand serious effort. On the other hand, successful completion of the class will provide you with a full understanding of the standard option pricing models, and will enable you to study the subject further on your own, or otherwise. Prerequisites. A basic knowledge of calculus based probability/statistics. Some exposure to stochastic processes and partial differential equations is helpful, but not mandatory. It is strongly recommended you take the prerequisites test available in Unit 0, to see if your mathematical background is strong enough for successfully completing the course. If you get less than 70% on the test, it may be more useful to work further on your math skills before taking this course. Or you can just do a part of the course.

### Syllabus

• Unit 0: Pre-course
• Since this is a quantitative course, a certain level of mathematical background is necessary for a student to master the course material. In this unit, I would like to invite you to take the prerequisites assessment.
• Unit 1. Stocks, Bonds, Derivatives
• Unit 2. Interest Rates, Forward Rates, Bond Yields
• Unit 3. No-Arbitrage Pricing Relations
• Unit 4: Pricing in Discrete Time Models
• Unit 5. Brownian Motion and Ito Calculus
• Unit 6. Pricing in Black-Scholes-Merton model
• Unit 7. Extensions of Black-Scholes-Merton
• Unit 8. Hedging
• Unit 9. Beyond Black-Scholes-Merton
• Unit 10. Pricing in Fixed Income Markets
• Final Exam (number of attempts is limited)

Jaksa Cvitanic

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