IBM stock currently sells for $139. Assume that it follows a GBM

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2. IBM stock currently sells for $139. Assume that it follows a GBM process with a dividend yield d = 3.77%. Interest rates are 1% continuously compounding. If a European option to buy 1000 shares with a strike of $135 3 months from now has just been sold for $9575, what is the implied volatility? How many shares should be held to delta hedge this short position in the call? What is the gamma and vega of the resulting portfolio?3. For a Japanese bank, the option to sell 2 billion yen for 16 million USD on a date 18 months from today is worth 170 million yen. Volatility is 25%, and the continuously compounding interest rates are 3% in USD, and 1% in yen. What are the 2 billion yen worth today in US dollars?4. Your company will require 1 million British pounds (GBP) sterling 2 months from now. Right now, this would cost you 2 million dollars Canadian (CAD), but your firm wants to enter into a range forward to execute that purchase. The spot exchange rate has a volatility of 18%, the continuously compounding CAD interest rate is 3%, while the British rate is 2%. If the minimum that you would pay for the 1 million pounds is 0.30 million CAD less than today’s price, what is the maximum that you would pay?

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