Key Largo Bank holds the following portfolio of options on the ASX 200

Type/Maturity	Position	Delta of Option	Gamma of Option	Vega of Option
Call 2-months	-10,000	0.5	0.002	732
Call 6-months	-5000	0.8	0.006	1001
Put 1- month	-20,000	-0.40	0.013	841

Next, Key Largo Bank adds to its option portfolio by writing an additional 5,000 European put options on the ASX 200. They all have a strike price of 5400 and a maturity of 3 months. Currently the ASX 200 is at 5200, the annualized risk free rate in Australia is at 2%, the dividend yield on the ASX 200 is 3.2% p.a., (both risk free rate and the dividend yield are continuously compounded) and the volatility is expected to be 30% per annum.

Assume that each option is written on 1$ times the index value

NOTE:

For all answers show your calculations.

For underlying asset units and number of options contracts, round to the nearest integer.

In part a and b, round d1 and d2 to 2 decimal digits only when you need to compute the normal probabilities from the Normal Table. For all other calculations, including d1 and d2, round to 4 decimal digits at each step.

1. Using the BSM model, compute the delta, gamma and vega of the banks position in the additional options (the puts with a strike of 5400 that the bank has written).

2. Verify that gamma for the newly traded options (again, the puts with a strike of 5400) is approximately correct by recomputing their delta following an upward move of 5 points in the ASX 200.

3. How does the new trade in European puts (the puts with a strike of 5400 that the bank has written) change the overall delta, gamma and vega of Key Largo Bank? Specifically, compute the delta, gamma and vega of the banks overall position after the trade and carefully explain what each number means.

4. The bank is especially concerned about an increase in the volatility of the ASX 200 and wants to hedge its option portfolio against such risk, while making the portfolio delta neutral as well. To this end, the bank considers yet another trade in ASX options. Specifically, they look at a 2-month call with a strike of 5300. This option has a delta of 0.4538 and a vega of 837.2696. What should Key Largo Bank do in order to achieve its hedging goals? Again, assume that each option is written on 1$ times the index value.