Financial Markets Modeling

Mathematical finance is an important part of applied mathematics since the 1980s. At the beginning, the main goal was to price derivative products and to provide hedging strategies. Nowadays, the goal is to provide models for prices and interest rates, such that better calibration of parameters can be done. In these pages, we present some basic models. Details can be found in Musiela and Rutkowski (2005).

Models for Prices of Stocks

The first model of prices was elaborated by Louis Bachelier, in his thesis (1900). The idea was that the dynamic of prices has a trend, perturbed by a noise. For this noise, Bachelier set the fundamental properties of the Brownian motion. Bachelier’s prices were of the form

$$S_{t}^{B} = S_{ 0}^{B} +\nu t +\sigma W_{ t}$$

where W is a Brownian motion.

This model, where prices can take negative values, was changed by taking the exponential as in the celebrated Black-Scholes-Merton model (BSM) where

$$S_{t} = {e}^{S_{t}^{B} } = S_{0}\exp (\nu t +\sigma...