The option greeks are used describe the different dimensions of risk involved in taking an options position in the market. These dimensions are referred to as the option greeks because they are represented by greek symbols. In plain english, the greeks describe how the option price changes when each of the parameters are changed (holding all others constant).

\[\Delta = \frac{\partial V}{\partial S}, \hspace{0.5cm} \theta = \frac{\partial V}{\partial \tau}, \hspace{0.5cm} \nu = \frac{\partial V}{\partial \sigma}, \hspace{0.5cm} \rho = \frac{\partial V}{\partial r}\]

The purpose of this article is explore the first order greeks for a European call and put option where the underlying does not pay dividends. This article also demonstrates a simple implementation framework using Python.

The delta of an option describes how the price of the option \(V\) changes with respect to changes in the underlying asset \(S\).

\[\Delta = \frac{\partial V}{\partial S}\]

This article explains how the delta for a European option which does not pay dividends can be derived by evaluating the partial derivative of the of the value with respect to the underlying price. The same methodology can be used to derive the other option greeks, however this article is only intended to provide an example derivation for the delta.

A Copula is a multivariate cumulative distribution function which describe the dependence between random distributions. Copulas are often used in quantitative finance to model the tail-risk or returns of a set of correlated distributions (Marginal Distributions). There are many types of Copula which all have different application. The purpose of this article is to demonstrate how to simulate from a Gaussian Copula and the underlying mathematics.

An important topic in statistics, especially when understanding the behaviour between variables or distributions, is the concept of dependence (or independence). One of the most useful measures of dependence is Pearson’s \(\rho\). Pearson’s correlation is well known and used in the industry, however it does have its limitation; particularly when measuring the relationship between non-linear distributions or where distributions are fat-tailed.

\[\begin{align}\notag \text{Independence} &\Rightarrow \text{No Correlation} \\ \text{No Correlation} &\nRightarrow \text{Independence} \end{align}\]

This article discusses the connection between dependence and correlation in detail and provides an example to show that having no correlation does not imply independence.

The movement of stock prices are generally considered to be a random walk which are very difficult to accurately forecast. Because of the randomness associated with stock price movements, they need to be simulated using Stochastic Differential Equations (SDE). Geometric Brownian Motion (GBM) is one of the most common models for simulating the dynamics of stock prices because of the following properties:

• Log-Normal random path meaning that the simulated stock price cannot become negative;
• There is a drift and diffusion component which can be calibrated to historical rates;
• Closed-form solutions exist which makes simulating forward paths computationally efficient.

The equation for Geometric Brownian Motion is

\[dS_t = \mu S_t dt + \sigma S_t dW_t\]

where \(S\) is the stock price, \(\mu\) is the drift coefficient, \(\sigma\) is the diffusion coefficient, and \(W_t\) is the Brownian Motion.