CEMLA Course: Financial Mathematics (introductory)

The theoretical component of the Course on Financial Mathematics was delivered online over two days. The course was designed for participants with diverse professional backgrounds and levels of training. Rather than beginning with a collection of formulas, the aim was to develop a conversation around the questions that gave rise to the discipline: how to compare amounts of money received or paid at different dates, how to represent uncertainty, and how to value contracts whose payments depend on prices, interest rates, or exchange rates.

The starting point was very concrete. We discussed the time value of money, returns, volatility, and risk, as well as how derivatives can be used to modify a financial exposure. Forwards, futures, options, and swaps were initially presented as contracts that redistribute risk. This perspective allowed the mathematics to emerge as a natural response to an economic problem, rather than as a collection of techniques disconnected from practice.

From Random Movements to Continuous Time Models

We then discussed why an ordinary differential equation, although useful as a benchmark, is not sufficient to describe the irregular evolution of financial prices. Starting with random walks and the accumulation of small disturbances, we arrived at Brownian motion. The central idea was that paths can be continuous while being so irregular that they do not possess an ordinary derivative. This observation explains why a different form of calculus is needed.

With this motivation, stochastic differential equations and Itô calculus were introduced. We reviewed simple models such as arithmetic and geometric Brownian motion, as well as mean-reverting processes and models in which volatility itself evolves randomly. Rather than attempting to cover every technical detail, the objective was to show that each choice of drift and volatility reflects a different way of understanding the behavior of the underlying asset.

The Bridge Between Probability and Valuation

On the second day, the Feynman–Kac theorem was presented as a bridge connecting two languages: stochastic processes and partial differential equations. This connection makes it possible to represent the value of a financial contract as a discounted expected value, provided that the appropriate probability measure is used. At this point, an important distinction was emphasized: the probability measure used for valuation need not coincide with the historical probability measure governing actual price movements.

Finally, all the pieces were brought together in the Black–Scholes formula. The equation was explained through a replicating portfolio and the principle of no arbitrage. In this way, the formula ceased to appear as an isolated result and instead emerged as the consequence of a sequence of ideas: modeling uncertainty, constructing a hedge, and requiring consistency between strategies that generate the same cash flows.

The Overall Message

The discussion concluded with European, digital, and American options. In the American case, the possibility of exercising before maturity provided an intuitive introduction to the comparison between immediate exercise and continuation, as well as to the idea of a free boundary. This prepared the ground for the in-person component of the course, in which these problems were revisited using the binomial model, numerical methods, simulation, and applied exercises.

The message connecting all the sessions was straightforward: in financial mathematics, the essential task is to understand which risk is being modeled, what cash flows the contract generates, which assumptions make its valuation possible, and which aspects of reality remain outside the model. Mathematics provides precision, but its usefulness depends on preserving this economic interpretation.