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Second City Options (SCO) is a small firm that specializes in option trading. Employing 35 people, SCO is located on LaSalle Street in the Chicago financial district. It is a member firm of the Chicago Board Options Exchange (CBOE), where it trades options on stocks and stock indices. It is also a member firm of the Chicago Mercantile Exchange Group (CME Group), where it trades options on futures and the underlying futures contracts.

SCO trades for itself and a number of corporate and individual clients. In addition, it provides general advice to other clients who trade for themselves. SCO was founded in 1975, two years after the CBOE opened. It was very successful for its first 25 years, but its profits have declined steadily since 2000. SCO’s founder and principal owner, Joseph Jensen, is 67 years old and has been a life-long student of the stock market. Jensen has traded since he was a teenager. By the age of 30 he was a millionaire and an acknowledged expert on analyzing market trends.

Throughout his career Jensen has relied primarily on technical analysis, the evaluation of trends in past market data such as prices and volumes. He has consistently devised option strategies to take advantage of his stock market forecasts. Over the last several years, however, his techniques have grown less profitable. Jensen attributes his declining success to the growing competitiveness of the options market, which has made it increasingly difficult for SCO to implement its trading strategies at attractive prices.

Worried that his firm’s techniques were becoming outmoded, Jensen has conferred recently with his chief trading executive, Bill Harrison, to analyze the situation and discuss possible solutions. Harrison believes that better implementation of option pricing techniques would improve SCO’s trading profits. In particular, there would be two potential benefits for SCO: it could improve its historical strategy of trading based on stock market forecasts, and it could explore other trading

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opportunities related to option mispricing. Consequently, SCO has significantly upgraded and modernized its computer system. Furthermore, it has hired Carla Shilling, a derivatives specialist who recently graduated from a major Midwestern business school. Her mandate is to improve the firm’s trading profits by utilizing her expertise in option valuation.

The Market Outlook

Before making any recommendation regarding strategies, Shilling must finalize her opinion of how the economy and stock market will perform over the next few months. The date is July 2, 2007. Over the last six months the S&P 500 has ranged from 1374.12 to 1539.18, closing at

1503.35 at the end of June. Investors seem worried about an impending credit crunch, even though problems at two Bear Stearns hedge funds that own collateralized debt obligations (CDOs) based on subprime mortgage debt appear to be contained. The economy has slowed with consumer pessimism high, reflecting a weak housing market combined with credit worries. For the last two months personal income has declined after adjusting for inflation. The Federal Reserve has focused on rising inflation. The Federal Open Market Committee, the Fed’s policy-setting arm, left interest rates unchanged in its June 27–28 meeting, keeping the Fed funds target level at 5.25% for the ninth time in the past twelve months. Despite inflation fears, the Federal Reserve did not raise interest rates due to concern that rate hikes would ripple out to home mortgages and hurt borrowers already struggling to make mortgage payments. Economists differ widely in their optimism or pessimism regarding the overall economy. There is no consensus, for example, on whether the stock market will increase or decrease during the next few months. Economists do agree in one key respect, however — they believe there is a large and increasing degree of uncertainty regarding future economic and market behavior.

SCO’s Trading Committee meets weekly to determine the firm’s outlook regarding the overall economy and the stock market. Shilling attended the most recent meeting, and she left that meeting holding the same opinion as the vast majority of her colleagues. There was a strong general belief that market volatility was relatively high, yet it might climb even higher than expected in the near future

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Analysis of Index Option Strategies

For her first task, Shilling must investigate option strategies that can exploit her firm’s consensus opinion regarding stock market uncertainty.

Part I

a) Identify two option strategies that take advantage of high or increasing market volatility. Examine each strategy for both S&P 500 index (SPX) and S&P 100 index (OEX) options. Use data provided in Table 1 of Data.xlsx. At this stage consider only strategies in which all options are held to expiration. If more than one expiration date is possible, use only options with the nearest expiration date, which is August 18, 2007. Also, use options that are at-the-money or closest to at-the-money. Discuss the advantages and disadvantages of each strategy. [5 marks]

b) Provide profit diagrams for each strategy and index using the spreadsheet OptionStrategyAnalyzer10e.xlsm. Let Exhibits 1 and 2 correspond to the first strategy and Exhibits 3 and 4 correspond to the second strategy. These exhibits should contain basic information such as the name of the strategy and index, the maximum profit and maximum loss, and breakeven points. For consistency in these diagrams (and elsewhere throughout this case), use the average of the last bid and ask prices for a given option as the price for that option. Discuss the advantages and disadvantages of using bid-ask midpoints versus bids, asks, or transaction prices. [5 marks]

Shilling knows that if she presents these results to SCO’s management without further analysis, she will be told to come back later when she can provide something more useful. She realizes that she must use valuation methods to strengthen her analysis and support her recommendation. She knows that she can use the BlackScholes-Merton model for SPX options, which are European-style options. She must use another approach, however, to price OEX options, which are American-style options. Moreover, she will need a volatility estimate for each index.

Part II

a) Table 2 of Data.xlsx contains daily data for the first six months of 2007 for both the SPX and OEX indices. Using this data estimate historical volatility for each index to four decimal places,

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e.g., 0.1234 or 12.34%. For your base estimates, calculate volatility using daily observations for the entire six month period. Then for comparison purposes, calculate historical volatilities over the first three months, the last three months, and all six months combined using both daily and weekly data. Using the Excel spreadsheet HistoricalVolatility10e.xlsm, report these estimates in Exhibit 5. Are these estimates sensitive to the choice of time period or observational frequency? [2 marks]

b) Estimate implied volatility for each index to four decimal places. For these estimates use data in Tables 1 and 3 of Data.xlsx. 5 Restrict your attention to the six call and put options expiring on August 18, 2007, that are at-the-money or closest to at-the-money. Use Excel spreadsheets such as BlackScholesMertonBinomial10e.xlsm or BlackScholesMertonImpliedVolatility10e.xlsm. Give your implied volatility estimates in Exhibit 6. Recall that the Black-Scholes-Merton model can be used for the SPX estimate, but not for the OEX estimate (since OEX options are American). Thus, you must use your best judgment in order to estimate implied volatility for OEX. Whatever methodology you use, explain it clearly. [5 marks]

c) Discuss the advantages and disadvantages of historical estimates versus implied estimates. Then provide one final volatility estimate for each index and explain your reasoning. [3 marks]

Shilling knows that she will ultimately have to use an American option pricing model for the OEX index. She has software for the Binomial model that allows up to 1000 time periods. She knows that this model will capture the early exercise value of an American option. She also knows that if both the Binomial and BlackScholes-Merton models are appropriate in a given situation, then the price from the Binomial model will converge to the price from the Black-Scholes-Merton model as the number of time periods increases to infinity.

Part III

a) Consider the S&P 500 call option that has the shortest time to expiration and is closest to at-the-money. Calculate the theoretical option price based on the Black-Scholes-Merton model. Define n as the number of time periods in the Binomial model, and let n equal 1, 5, 10, 25, 50, 100,

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500, and 1000. For each value of n, calculate the corresponding theoretical option price from the Binomial model. Use the Excel spreadsheet BlackScholesMertonBinomial10e.xlsm to obtain the prices for both models. [4 marks]

b) To obtain a call price C from the Binomial model, one needs five parameters: n, u, d, r, and p. These represent the number of periods, the binomial up factor, the binomial down factor, the risk-free rate, and the risk neutral probability, respectively. (In this scenario, however, one must also incorporate dividends.) Construct a table that shows how the values of u, d, r, p, and C change as n varies (See sample table 4.2 below). Note that the software BlackScholesMertonBinomial10e.xlsm does not report values for u, d, r, and p as n varies; it reports only values of C. Calculate the changing values of u, d, r, and p using the formulae in the file, List of important formulas.pdf. Alternatively, use the formulae in Chapter 4 of “Introduction to Derivatives and Risk Management”, 10th edition, by Don M. Chance and Robert Brooks. Call this table Exhibit 7 and briefly discuss your findings [6 marks]

Sample Table

Source: Chapter 4 “Introduction to Derivatives and Risk Management”, 10th edition, by Don M. Chance and Robert Brooks

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Part IV

a) Using appropriate pricing models and final volatility estimates from Part II, calculate theoretical prices for all options utilized in the strategies recommended in Part I. Determine whether each individual option is overpriced, underpriced, or fairly priced. Present your results in Exhibit 8. [5 marks]

b) Using the information just obtained for individual options, determine which, if any, strategies recommended in Part I are mispriced. Present your results in Exhibit 9. [5 marks]

c) Identify the one strategy and index that seems most attractive of the four possibilities considered. Explain your reasoning. (Hint: In selecting the strategy, consider your tolerance for risk) [5 marks]

Harrison, who has been monitoring Shilling’s work, has been impressed by her analysis. He knows, however, that the firm rarely holds a position until all options in the strategy expire. Noting that Shilling has concentrated on strategies held to expiration, he mentions that there is another strategy

— one in which not all options expire — commonly used in situations where one expects high or increasing volatility.

Part V

a) Identify the strategy to which Harrison refers. Examine this strategy using SPX and OEX options that are at-the-money or closest to at-the-money. Some, but not all, of the options used in this strategy can expire on the nearest expiration date, which is August 18, 2007. Provide Exhibits 10 and 11 for this strategy similar to Exhibits 1 and 2 in Part I. [5 marks]

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