Greeks

Consider a one-year European call option on a stock with time-to-maturity of 6-months when the stock price is $48, the strike price is $50, the risk-free rate is 5% (measured with continuous compounding), and the volatility is 30% per annum. The underlying stock pays no dividends.

What is the call option premium, the delta, the gamma, and the vega, according tothe Black-Scholes-Merton (BSM) model?

Given your answers in (a), compute the delta-approximated option price when thestock price is equal to $45, $47, $49, and $51, respectively.

How does your answer in (b) change if you compute the delta-gamma approximationof the predicted option price change when the stock price is equal to $45, $47, $49, and

$51, respectively? Calculate and compare the relative prediction error (in %) of the delta and the delta-gamma approximations obtained in answers (b) and (c). How is the relative prediction error related to the change of the underlying stock price? Does

the delta approximation tend to predict a greater or a lower price change than the actual change in option prices? Explain! Hint: The relative prediction error is defined as |PriceApproximation PriceBS|/PriceBS.

Consider a second 3-month European put option on the same stock, with strike price equal to $56. Compute the options corresponding option premium, the delta, and the gamma. How can you use the put option, the call option from part (a) of this question, together with the stock, to gamma and delta neutralize the portfolio? Please indicate clearly whether you go long or short in the stock/options. What is the value of your total portfolio? Hint: You can normalize your portfolio to have 1 share of the stock, and solve for the positions in the call and put options. Note that non-integer units of the option are allowed to solve this exercise.

You have delta-gamma hedged your portfolio (in part (d) of this question), and you areaway on sick leave. After two months, you come back to your trading desk, and see that the stock price has moved up to $58 (while all other risk factors remain unchanged). Recalculate the value of your delta-gamma hedged portfolio from (d),

3 Go to the website of TMX to find the new option prices on that date.

assuming that you have not rebalanced your portfolio holdings or your hedge. Quantify the change in portfolio value and explain the reasoning behind it.

What is the approximated price change of the call option over one week when volatility increases by 2 percentage points, and the stock increases by $1.5?