Exercise

# Black-Scholes options pricing

Options are the world's most widely used *derivative* to help manage asset price risk. In this exercise you'll price a European call option on IBM's stock using the **Black-Scholes** option pricing formula. `IBM_returns`

data has been loaded in your workspace.

First you'll compute the volatility `sigma`

of `IBM_returns`

, as the annualized standard deviation.

Next you'll use the function `black_scholes()`

, created for this and the following exercises, to price options for two different volatility levels: `sigma`

and two times `sigma`

.

The strike price `K`

, i.e. the price an investor has the right (but not the obligation) to buy IBM, is 80. The risk-free interest rate `r`

is 2% and the market spot price `S`

is 90.

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

function here.

Instructions

**100 XP**

- Compute the volatility of
`IBM_returns`

as the**annualized**standard deviation`sigma`

(you annualized volatility in Chapter 1). - Calculate the Black-Scholes European call option price
`value_s`

using the`black_scholes()`

function provided, when volatility is`sigma`

. - Next find the Black-Scholes option price
`value_2s`

when volatility is instead 2 *`sigma`

. - Display
`value_s`

and`value_2s`

to examine how the option price changes with an*increase*in volatility.