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2 two questions

Newspaper Prices

For this question, we are using the Newspaper data pdf file for SPI 200 index options. The underlying asset for these options is the SPI 200 futures, so these are options on futures, and they are American options. The maturity date is the same as for the corresponding futures contract, with the same maturity month as the option. If you can't read the headings in the pdf file, the headings are the same as in the previous question.

Note that the spreadsheets are designed to price options on an asset paying a dividend yield,

q.See Hull p. 370, Section 17.8, to see how to use this spreadsheet to price options on futures.

Using the spreadsheet Puts&Dvd.xls, we want to find implied volatilities for some of the options listed.

For the risk-free rate, use the continuously compounded risk-free rates calculated in the previous question.

We now want to use Excel's Solver to find implied volatilities for each option. See question 1 above for a description of how to use Solver to find implied volatilities.

Find the option implied volatilities for the September 17 put options with strike prices 5475, 5500, 5575, 5600, 5625.

(a)What does the 17 refer to? Show me all your inputs for this problem (i.e., the cells with red font). This is in order to make it easier for me to mark the question.

(b)Show me a table of strike prices and implied volatilities for these options. Is there any evidence of a volatility skew or smile?

(c)Graph the implied volatilities as a function of the strike prices. Is the pattern interesting?

Now, find the option implied volatilities for the December 17 put options with strike prices 5575, 5600, 5625, 5650, 5675.

(d)Again, show me your inputs for this problem (i.e., the cells with red font). This is in order to make it easier for me to mark the question.

(e)Show me a table of strike prices and implied volatilities for these options. Is there any evidence of a volatility skew or smile?

(f)Graph the implied volatilities as a function of the strike prices. Is the pattern interesting?

(g)How do all the option implied volatilities compare to the range of historical volatilities found in question 2 above? Use your judgement; e.g., compare the range of historical volatilities to the range of implied volatilities from this question and see if they overlap substantially.

2.Portfolios

Using Portfolio.xls, assume that r = 6% (compounded continuously), q = 0, S = 26.40, and ? =

0.30.Suppose the following options are available on this stock:

Call/PutTime to Maturity (days)Strike Price

Call3026.00

Call3026.50

Call6026.00

Call6026.50

Call6027.00

Call9026.50

Put3026.50

Put3027.00

Put6026.50

Put9026.00

There may be space on the spreadsheet for more options. You can ignore those other rows if you just assume that the number of contracts you hold in those options (column B) is equal to zero. In column C you enter the type of option; so, e.g., if you type call, the spreadsheet calculates the value of the call (column G) and each of the Greek letters for the call (columns H-L).

For the maturity dates, enter =TODAY + 30, etc in column D. The spreadsheet is designed to continuously update today's date, so you won't be able to type in actual maturity dates for this question. Enter the strike prices in column E. Note that the spreadsheet also allows for different implied volatilities (column F), but at the moment, I've set each implied volatility

=SIGMA, so if you enter the volatility into cell D7, you don't need to worry about the other implied volatilities.

When using the graphs, if either T1 or T2 are greater than one of your maturity dates, then that option will not appear in that graph. Also, in general, you may have to click on the graph's x-axis to re-set the minimum and maximum values. Note that there are two graphs, one using a non-smoothed plot and the other (to the right of the first) using a smoothed plot.

The market convention in Australia for equity options is that one option contract is actually for options on 100 shares, usually. This is shown in column A.

(a)Assume that you buy 500 of each of the above options contracts (but no shares or futures). Report the value of this portfolio, and of all the portfolio "Greeks". Also, report how the value of the portfolio changes when various inputs change.

(b)How many shares of the underlying asset must you buy or sell if you want the above portfolio to be (approximately) delta-neutral? (You should round off to the nearest integer.) With this delta-neutral portfolio, what is the effect of increasing or decreasing S by 1% (see the portfolio changes in row 7 of the spreadsheet)? Is it (approximately) symmetrical? Was it symmetrical in the previous case? If not, why not? What would be the effect of increasing or decreasing S by 1% if the portfolio

delta was still zero but gamma was the negative of its current value? What is the simplest way to form such a negative gamma portfolio?

(c)Now assume that you own 50,000 shares of the underlying asset as well as 500 of each of the option contracts. We want to use the call option with the largest delta, and the put option with most negative delta. With these two options use Solver to find out how many of these options you have to hold (i.e., be long or short) in order to be both delta and gamma-neutral. [HINT: you can set gamma equal to zero subject to the constraint that delta equals zero. You do this while changing two cells in column

B. Again, round off to the nearest integer number of options.]

(d)Hand in a screenshot of the first page of this spreadsheet showing the delta and gamma-neutral portfolio. (This is to make it easier for me to mark.)

Graphing the payoff of a combination of options.

(e)Assume that you buy 1000 put option contracts with strike price 26.50 and 60 days to maturity. How many 60-day 27.00 call option contracts must you sell in order to pay for this put option? (In other words, you want the initial value of this combination of put and calls to be equal to zero dollarsassuming there are no other assets in this portfolio.)

(f)Explain briefly why a hedger might want this combination of options.

(g)Consider the graph of the value of this combination after T1 = 30 days, and after T2 =59.99 days. Use either the non-smoothed graph or the smoothed graph (to the right of the first graph)whichever you prefer. Hand in a copy of your graph.