Exercise

# Using options for hedging

Suppose that you have an investment portfolio with one asset, IBM. You'll hedge the portfolio's risk using *delta hedging* with a **European put option** on IBM.

First, value the European put option using the **Black-Scholes** option pricing formula, with a strike `X`

of 80 and a time to maturity `T`

of 1/2 a year. The risk-free interest rate is 2% and the spot `S`

is *initially* **70**.

Then create a *delta hedge* by computing the `delta`

of the option with the `bs_delta()`

function, and use it to hedge against a *change* in the stock price to **69.5**. The result is a **delta neutral** portfolio of both the option and the stock.

Both of the functions `black_scholes()`

and `bs_delta()`

are available in your workspace.

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

and `bs_delta()`

functions here.

Instructions

**100 XP**

- Compute the price of a European put option at the spot price
**70**. - Find the
`delta`

of the option using the provided`bs_delta()`

function at the spot price**70**. - Compute the
`value_change`

of the option when the spot price falls to**69.5**. - Show that the sum of the spot price change and the
`value_change`

weighted by 1/`delta`

is (close to) zero.