CEMLA Course: Financial Mathematics
November 21 - 25, 2022
Videoconference
The Center for Latin American Monetary Studies (CEMLA) conducted the "Financial Mathematics" course in digital format from November 21 to 25, 2022. Its objective was to provide participants with tools from the subject of financial mathematics with a focus on continuous-time stochastic calculus, underscoring applications to central banking using Python. The course was taught by Jaime Casassus, Associate Professor at the Institute of Economics of the Pontificia Universidad Católica de Chile.
The course consisted of theoretical sessions with practical applications in Python. During the first day, Brownian processes and Itô calculus were covered. On the second day, Black and Scholes valuation and risk-neutral valuation were reviewed. Numerical methods, such as Monte Carlo simulations, implicit and explicit finite differences, and applications to option valuation were covered on the third day. Then, on the fourth day, asset pricing models for bonds, futures, options, and swaps were revised and applications were discussed. Finally, the course concluded with Kolmogorov equations and dynamic programming.
Monday, November 21
Brownian processes and Itô calculus
Random variables, definition of a Brownian process, properties of the Brownian process, multivariate Brownian process, simulation of a Brownian process in Python. Definition of the Itô Integral, properties of the Itô Integral, Itô-Doeblin formula, multivariate Itô-Doeblin formula, and applications.
Tuesday, November 22
Black and Scholes valuation, and risk neutral valuation
Stochastic discount factor, the Black and Scholes economy, Black and Scholes extensions. Change of probability measure (Girsanov's theorem), risk neutral valuation and its relationship with the stochastic discount factor, and applications.
Wednesday, November 23
Numerical methods and implementation
Monte Carlo simulation, explicit finite differences, implicit finite differences in Python. Applications to option pricing with mean-reversion and stochastic volatility.
Thursday, November 24
Asset pricing models and applications
Static and dynamic models for bonds, futures, options, options on bonds and swaps. Estimation of linear models in Python. Multifactorial models applied to commodity futures, swaps and options on inflation, bonds with near-zero rates.
Friday, November 25
Kolmogorov equations and dynamic programming
Transition probabilities, Kolmogorov forward equations, Kolmogorov backward equations, application to HACT models. Finite horizon dynamic programming, Infinite horizon dynamic programming, jumps and dynamic programming with regime shifts, application to HACT models. Dynamic programming and numerical methods.
Jaime Casassus
Jaime Casassus works as an Associate Professor at the Institute of Economics of the Pontificia Universidad Católica de Chile. Jaime holds a Ph.D. in Finance from Carnegie Mellon University, a B.S. in Civil Industrial Engineering and a M.S. in Industrial Engineering from the Pontificia Universidad Católica de Chile. He has also been Managing Editor for Quantitative Finance, Associate Editor for the Journal of Commodity Markets, Visiting Professor at the University of California, Berkeley and Member of the Finance Committee that advises the Chilean Ministry of Finance on the investment policy of its sovereign wealth funds. Jaime has publications in Journal of Banking & Finance, Journal of Futures Markets, Review of Financial Studies, European Financial Management, Journal of Finance, among other journals.