# Selected Courses in Financial Mathematics

21-270 Introduction to Mathematical Finance
Spring: 9 units

This is a first course for those considering majoring or minoring in Computational Finance. The theme of this course is pricing derivative securities by replication. The simplest case of this idea, static hedging, is used to discuss net present value of a non-random cash flow, internal rate of return, and put-call option parity. Pricing by replication is then considered in a one-period random model. Risk-neutral probability measures, the Fundamental Theorems of Asset Pricing, and an introduction to expected utility maximization and mean-variance analysis are presented in this model. Finally, replication is studied in a multi-period binomial model. Within this model, the replicating strategies for European and American options are determined. 3 hours lecture.

Introduction to Mathematical Finance is a prerequisite for 21-370 Discrete-Time Finance.

21-370 Discrete-Time Finance
Fall: 9 units

This course introduces the Black-Scholes option pricing formula, shows how the binomial model provides a discretization of this formula, and uses this connection to fit the binomial model to data. It then sets the stage for Continuous-Time Finance by discussing in the binomial model the mathematical technology of filtrations, martingales, Markov processes and risk-neutral measures. Additional topics are American options, expected utility maximization, the Fundamental Theorems of Asset Pricing in a multi-period setting, and term structure modeling, including the Heath-Jarrow-Morton model. 3 hours lecture.

Prerequisite: 21-270 and (21-256 or 21-259)

Co-requisite: 70-207, 21-325, 36-225, or 36-217.

Students in 21-370 are expected to read and write proofs.

Discrete-Time Finance is a prerequisite for 21-420 Continuous-Time Finance.

Note that 70-207 Probability and Statistics for Business Applications is an adequate probability co-requisite for this course but is not an adequate pre-requisite for 21-420 Continuous-Time Finance, which requires one of the calculus-based probability courses 21-325, 36-225 or 36-217.

21-420 Continuous-Time Finance
Spring: 9 units

This course begins with Brownian motion, stochastic integration,and Ito's formula from stochastic calculus. This theory is used to develop the Black-Scholes option pricing formula and the Black-Scholes partial differential equation. Additional topics may include models of credit risk, simulation, and expected utility maximization. 3 hours lecture.

Prerequisites: 21-260 and 21-370 and (21-325, 36-225 or 36-217).

70-207 is not sufficient preparation in probability for this course.

21-420 is a prerequisite for 45-816 Studies in Financial Engineering.

46-977 Studies in Financial Engineering
Spring: 6 units

This course focuses on the use of financial engineering and derivative securities in solving practical business problems. Students will work through business cases and give in-class simulated sales pitches to hypothetical clients. The cases highlight the design, valuation and hedging of structured products. In addition, we will look at real options and at derivative pricing with exotic "underlyings" such as energy, weather, and credit.

Prerequisite: 21-420