This post outlines my preparation for taking a machine learning quant exam.

## Personal notes#

### Mathematical concepts#

A martingale is

#### Black Scholes Model#

The Black-Scholes-Merton (BSM) model estimates the value of a derivative based on other financial instruments and takes into account time and risk factors. The fundamental assumption is that the distribution of a stock share will follow a log-normal distribution. Further assumptions of the BSM model are that

• No dividends are paid out during the life of the option
• Markets cannot be predicted i.e., they are random

Notation: Let $S_0$ be the stock price, $X$ the exercise price, $r$ the risk-free interest rate, $T$ the time to exercise the option, and $\sigma$ the standard deviation of the log-returns (also known as volatility).

A European call option $C_0$ is given by $$\begin{equation} \label{eq:black_scholes} C_0 = S_0 N(d_1) - X\exp(-rT)N(d_2), \end{equation}$$ where \begin{align} d_1 & = \frac{\log\left(\frac{S_0}{X}\right)+\left(r+\frac{\sigma^2}{2}\right)T}{\sigma\sqrt{T}} \ d_2 & = \frac{\log\left(\frac{S_0}{X}\right)-\left(r+\frac{\sigma^2}{2}\right)T}{\sigma\sqrt{T}} \end{align} and $N$ is the CDF of a unit-Gaussian distribution.

Breaking \eqref{eq:black_scholes} down, we can interpret $\frac{S_0}{X}$ as the ratio of the stock price to the exercise price. The larger this ratio, the more profit we could expect to return. This flows through the model as $C_0 \propto \frac{S_0}{X}$