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

## Sample answers

## Personal notes

### Finance terminology

### Mathematical concepts

#### Martingales

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 $C_0 \propto \frac{S_0}{X}$