## The Black-Scholes formula

Certainly one of the biggest breakthroughs in quantitative finance, the Black-Scholes (pronounced like “Black-Shoales” not “Black-Skoales”) model was introduced in 1973 and provides a mathematically principled approach to options pricing. While the original models relies on partial differential equations it subsequently found a different interpretation through stochastic processes (martingales) by describing stock prices through a … Continue reading The Black-Scholes formula

## Regularization as robust regression

Ridge regression In this post I want to write about the connection of Ridge regression and robust regression. Ridge regression (also know as Thikonov regularization) is a form of regularization or shrinkage, where the parameters of linear regression are shrunk towards 0. There are several reason why one might want to use methods like this. … Continue reading Regularization as robust regression

## The Sharpe ratio and loss probability

Assume we have a portfolio with Sharpe ratio of \displaystyle S_r = 1 . What is the probability of the portfolio losing value over a 4 year time horizon?

## Stepsize for gradient descent in linear regression

Consider the setting of least squares regression and suppose we are given a data matrix together with a vector of labels . The solution to the problem is given by that solves the equation . Using matrix inversion the solution can be easily calculated as the well known least squares estimator . However, for large … Continue reading Stepsize for gradient descent in linear regression