Aan de slagGa gratis aan de slag

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.

Deze oefening maakt deel uit van de cursus

Quantitative Risk Management in Python

Cursus bekijken

Oefeninstructies

  • 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.

Praktische interactieve oefening

Probeer deze oefening eens door deze voorbeeldcode in te vullen.

# Compute the annualized standard deviation of `IBM` returns
sigma = np.sqrt(252) * IBM_returns.std()

# Compute the Black-Scholes value at IBM spot price 70
value = black_scholes(S = ____, X = 80, T = 0.5, r = 0.02, 
                      sigma = sigma, option_type = "put")
# Find the delta of the option at IBM spot price 70
delta = bs_delta(S = ____, X = 80, T = 0.5, r = 0.02, 
                 sigma = sigma, option_type = "put")

# Find the option value change when the price of IBM falls to 69.5
value_change = ____(S = 69.5, X = 80, T = 0.5, r = 0.02, 
                             sigma = sigma, option_type = "put") - ____

print( (69.5 - 70) + (1/delta) * ____ )
Code bewerken en uitvoeren