The benchmark model of mathematical finance is the Black-Scholes-Merton model, with Brownian motion as driving noise. This gives a complete market, and leads to the famous Black-Scholes formula. But it does not fit some features of real financial markets -- e.g., it under-estimates the probabilities of extreme events (which are financially particularly important!), because of the extreme thinness of Gaussian tails (log-quadratic tail decay). One thus seeks a generalization of the Black-Scholes set-up, which preserves as much as possible of its lovely structure but gives the flexibility to model more realistic tail-decay, etc. This can be done parametrically (via the generalized hyperbolic distributions, among others). It can also be done semi-parametrically: one preserves a parametric component (mean vector and covariance matrix, required for the Markowitz interpretation of risk and return), and uses a non-parametric component to model the `shape'. The talk is based on joint work with Ruediger Kiesel (Ulm) and Rafael Schmidt (Bonn & LSE).