Black Scholes and Greeks

Black-Scholes Model

In the Black-Scholes model, the stock price follows a Geometric Brownian motion

$$dS_t = \mu \, S_t \, dt + \sigma \, S_t \, dW_t,$$

where $\mu$ is the drift coefficient and $\sigma^2$ corresponds to the volatility.

To price the spread option of two futures contracts see Kirk's approximation

Greeks and Delta-Hedging

The Greeks give the sensitivity of the price of derivatives to a change in different parameters. They help to quantify the risk. In this notebook, we implement the Greeks for Call and Put Options.

We also implement Delta-Hedging, the Delta of an option $V$ is given by

$$\Delta = \frac{\partial V}{\partial S}$$

where $S$ is the stock price. At each step we caluclate the Delta and buy/sell the given amount of shares so that the portfolio's Delta is zero.