Quantitative Finance Collector is a blog on Quantitative finance analysis, methods in mathematical finance focusing on derivative pricing, quantitative trading and quantitative risk management.

To price and hedge derivative securities, it is crucial to have a good model of the probability distribution of the underlying product. The most famous continuous-time model is the celebrated Black Scholes model, which uses the Normal distribution to fit the log returns of the underlying.

As we know from empirical research, one of the main problems with the Black–Scholes model is that the data suggest that the log returns of stocks/indices are not Normally distributed as in the Black–Scholes model. The log returns of most financial assets do not follow a Normal law. They are skewed and have an actual kurtosis higher than that of the Normal distribution. Other more flexible distributions are needed.

Moreover, not only do we need a more flexible static distribution, but in order to model the behaviour through time we need more flexible stochastic processes (which generalize Brownian motion). Looking at the definition of Brownian motion, we would like to have a similar,i.e. with independent and stationary increments, process, based on a more general distribution than the normal. However, in order to define such a stochastic process with independent and stationary increments, the distribution has to be infinitely divisible, such processes are called Lévy processes, one example of such process is normal inverse gaussian (NIG).