Exercise

# Options pricing and the underlying asset

Options are essentially *bets* on the future evolution of the underlying asset's price.

For example, a **put** option is valuable when the spot (market) price falls *below* the option's strike price. The option holder may *exercise* the option to sell the underlying at the strike \(X\), and buy it back at the spot \(S < X\), yielding profit \(X - S\).

In this exercise you'll value and visualize a European **put** option on `IBM`

stock, again applying the Black-Scholes pricing formula, *as the spot \(S\) changes*.

The strike `X`

= 140, the time to maturity `T`

is 1/2 a year, and the risk-free interest rate is 2%.

The annualized volatility of `IBM`

is available as `sigma`

, and the plotting axis `option_axis`

is available to add your plot.

You can find the source code of the `black_scholes()`

function here.

Instructions

**100 XP**

- Set
`IBM_spot`

to be the first 100 observations from the`IBM`

spot price time series data. - Compute the Numpy array
`option_values`

, by iterating through an enumeration of`IBM_spot`

and by using the`black_scholes()`

pricing formula. - Plot
`option_values`

to see the relationship between spot price changes (in blue) and changes in the option value (in red).