60.1K

Verified Solution

Link Copied!

1. Ms. Calmind would like to learn more on the option hedging. This time she seeks your advice on how to use the Black-Scholes-Merton (BSM) model for delta and gamma hedging. Coincidentally, you are doing that for one of the stocks in your portfolio. The stock currently trades at a price of $122 and it has a continuously compounded dividend yield of 1.30%. You have used the stock's monthly returns in the past five years to work out its monthly return volatility of 12%. There are two European put options that you consider to use: one with the exercise price (X) of $130 and the other one with X = $140. Both of this stock's options have 110 days to maturity. The continuously compounded interest rate is 4.40% a year and assumed to be constant in the next 30 days.You expect that the stock's price may randomly move to $118.69, $115.47, and $112.34 in the next three days, respectively. (1) Estimate the deltas and gammas of these options on day zero (i.e. when the days to maturity is 110), day one, day two, and day three. [3 marks] (2) Show Ms. Calmind the delta and gamma hedging starting with a long position of 1,700 put options with X = $130. Note that your hedging starts on day zero, is adjusted on days one and two, and finishes on day three without adjustment. Also, clearly explain how many options and stocks you buy or sell and how much they cost or generate on each hedging day. [8 marks] (3) By the end of day three, how much is the hedged portfolio's value, and how far off it is compared to the value of a perfect hedge? [3 marks] (4) Ms. Calmind really appreciates your help but has a final question on this topic: 'Given you know how many units of the stock you are currently holding, can you start your hedging with, say, 1,700 shares of the stock and calculate how many put options with X = $130 and X = $140 to use on day zero? [2 marks] Note: you are required to estimate the options' prices, deltas, and gammas yourself. The textbook software can be used to cross-check only.

Answer & Explanation Solved by verified expert

See Answer