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Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 1 of 34
SECOND CITY OPTIONS:
A Case Study on Index Options
Solution
Don Chance, Brendan Condon, and Michael Hemler1
(Version: May 28, 2015)
Goals 1. Illustrate the importance of volatility in option pricing and the difficulty in estimating volatility 2. Explore relative strengths and weaknesses of the Black-Scholes-Merton and Binomial models 3. Provide the opportunity for students to analyze various strategies, choose the best alternative, and then
defend their choice via written and oral arguments Assumptions These solutions make two key assumptions at the outset to narrow the range of possible answers: 1. To obtain a single option price for constructing profit diagrams and calculating implied volatilities, we use
the arithmetic mean of the corresponding bid and ask prices. 2. All strategies are volatility strategies, i.e., strategies that exploit expected changes in volatility. These assumptions represent an attempt to keep the number of “solutions” manageable. The first assumption is critical. It affects all the calculations that follow. If students use different option prices from the start, then their answers continually differ throughout their entire case analyses. This makes checking answers much more difficult, if not outright impossible. It can be time-consuming and frustrating trying to determine whether slight variations in answers result from different initial data or from subtle methodological errors. The second assumption is also important. Students occasionally pick strategies that do not depend on volatility. For instance, they sometimes recommend bullish or bearish strategies, even arbitrage strategies, based on their reading of the case. Most students, however, choose a strategy based entirely on volatility, which is what the case writers intend. Therefore, we restrict our attention to volatility strategies. Data
We strongly urge professors to discuss features of the data in SecondCityOptionsCase_Data10e.xlsx with their students. It is worthwhile noting pros and cons of using averages of bid and ask prices versus alternatives such as transaction prices, bid prices, or ask prices. (Indeed, we ask students to consider this issue in Section (b) of Part I.) Different choices can impact the analysis significantly. For instance, in Part I a student might analyze a reverse butterfly strategy “conservatively” by buying options at ask prices and selling options at bid prices. This yields profit estimates that are negative for all index values and for both indexes, which generally leads the student to reject this popular volatility strategy immediately. Hence, although it would be simpler to give one price for each option, we give multiple prices so that students can gain insight
1 We thank John Peterson for providing preliminary analysis of Parts I–III. We also thank Jeremy and Justin Dancu for their help reviewing and updating the solutions, especially the exhibits, using software for the 10th edition of the aforementioned textbook by Chance and Brooks.
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from considering the alternative data. Second, in the data analysis that follows, we use software for the 8th edition or higher of An Introduction to Derivatives and Risk Management, 8th ed., by Don M. Chance and Robert Brooks. All references to spreadsheets with “10e” in the file are the 10th edition versions, but earlier versions can be used.
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 SecondCityOptionsCase_Data10e.xlsx.2 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.
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.
Given SCO’s forecast of high and possibly increasing market volatility, students generally select two of the following three strategies: straddle, strangle, and reverse butterfly. For a straddle one buys a call and a put having the same strike price X. For a strangle one buys a put having strike price X1 and a call having strike price X2 where X1<X2. For a reverse butterfly one sells a call having strike price X1, buys two calls having strike price X2, and sells a call having strike price X3 where X1<X2<X3 and X2=(X1+X3)/2. Although one could also construct a reverse butterfly with puts, we consider only calls in the analysis given here. Students should provide complete details for their strategies. They must clearly state which options are bought or sold, whether calls or puts are used, which strike prices are used, etc. Students should also mention key attributes of their recommended strategies. Some examples include:
Unlimited upside as the index increases toward infinity for the straddle and strangle strategies,
Limited upside for the reverse butterfly strategy,
Limited downside to all three strategies, although there is significantly more downside risk for the straddle and strangle strategies than for the reverse butterfly.
Students occasionally want to use a strategy that takes advantage of increasing market volatility, but that is also either bullish or bearish on the underlying index. As a result, they might recommend a strap or a strip. With a strap one buys two calls and one put having the same strike price X. With a strip one buys two puts and one call having the same strike price X. These strategies are quite similar to, yet different from, the corresponding straddle. Although one can make a valid case for choosing a strap or a strip, we do not provide details for them. We restrict our attention to three strategies based solely on volatility – the straddle, strangle, and reverse butterfly. Sometimes students recommend arbitrage strategies based on violations of parity relations when using various combinations of prices from Table 1. Although it is important that students know parity relations such as put-call parity, arbitrage strategies are not realistic alternatives in this situation. Timing is absolutely critical in arbitrage-based trading, and students must be aware that they cannot realistically expect to execute trades at
2The option data in Table 1 of SecondCityOptionsCase_Data10e.xlsx are from Market Data Express.
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prices given in Table 1. The data in Table 1 are available after trading has ended on July 2, 2007. There is no guarantee that those prices would be available the next day. Unlike a buy-and-hold strategy based on a volatility forecast, an arbitrage strategy would depend critically on exploiting a fleeting mispricing opportunity. Hence, an arbitrage strategy based on a violation of a parity relation is not a viable recommendation in this context. The strategies chosen by a student can shed light upon a student’s tolerance for risk. They also can provide insight regarding a student’s expectation that the stock market will move significantly. For example, students often recommend the straddle and strangle as their two strategies. These students typically do not wish to relinquish the “upside potential” of these strategies for the limited profits available from a reverse butterfly. This is certainly justifiable – it depends on the student’s risk tolerance and the student’s probability estimate that a big move in the stock market will occur. Our choice is to recommend the straddle and the reverse butterfly. This provides the most contrast between alternatives. Because each of these three strategies is reasonable, we provide profit diagrams for all of them in this section. In later sections of this analysis, however, we shall restrict our attention solely to the straddle and reverse butterfly. These diagrams, by the way, incorporate a multiplier of 100 to adjust for contract size. To obtain these diagrams one can use the spreadsheet OptionStrategyAnalyzer10e.xlsm. In addition, please note that the generic version of OptionStrategyAnalyzer10e.xlsm has the horizontal axis labeled “Asset Price at Expiration ($).” For Exhibits 1-6 we have changed the label to “Index Value at Expiration” since we are working with index values, not stock prices. Similarly, we have changed the titles of these exhibits from the generic “Profit from Option Strategy” to reflect the particular strategy in question, e.g., an SPX straddle or an OEX strangle. We have often changed labels when appropriate to match specific situations more closely. In other words, exhibits given in this teaching note will not typically match what one obtains from simply plugging data into the relevant spreadsheets. We have frequently modified our exhibits to make them more informative and more relevant for this case study.
Exhibit 1: Profit Diagram for SPX Straddle
(Buy Aug 1520 Call @ $33.90, Buy Aug 1520 Put @ $28.00)
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Profit for SPX Straddle
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(Maximum profit is infinite. Maximum loss is $6,190. Breakeven points are 1,458.10 and 1,581.90.)
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Exhibit 2: Profit Diagram for OEX Straddle
(Buy Aug 700 Call @ $14.65, Buy Aug 700 Put @ $13.00)
(Maximum profit is infinite. Maximum loss is $2,765. Breakeven points are 672.35 and 727.65.)
Exhibit 3: Profit Diagram for SPX Reverse Butterfly
(Sell Aug 1515 Call @ $37.10, Buy 2 Aug 1520 Calls @ $33.90, Sell Aug 1525 Call @ $30.90)
(Maximum profit is $20. Maximum loss is $480. Breakeven points are 1,515.20 and 1,524.80.)
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Profit for OEX Straddle
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Profit for SPX Reverse Butterfly
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Exhibit 4: Profit Diagram for OEX Reverse Butterfly
(Sell Aug 695 Call @ $18.00, Buy 2 Aug 700 Calls @ $14.65, Sell Aug 705 Call @ $11.80)
(Maximum profit is $50. Maximum loss is $450. Breakeven points are 695.50 and 704.50.)
Alternate Exhibit: Profit Diagram for SPX Strangle
(Buy Aug 1515 Put @ $26.20, Buy Aug 1525 Call @ $30.90)
(Maximum profit is infinite. Maximum loss is $5,710. Breakeven points are 1,457.90 and 1,582.10.)
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Profit for OEX Reverse Butterfly
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Alternate Exhibit: Profit Diagram for OEX Strangle
(Buy Aug 695 Put @ $11.10, Buy Aug 705 Call @ $11.80)
(Maximum profit is infinite. Maximum loss is $2,290. Breakeven points are 672.10 and 727.90.)
Part II
a) Table 2 of SecondCityOptionsCase_Data10e.xlsx contains daily data for the first six months of 2007 for both the SPX and OEX indices.3 Using this data estimate historical volatility for each index to four decimal places, 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.4 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?
b) Estimate implied volatility for each index to four decimal places. For these estimates use data in Tables 1 and 3 of SecondCityOptionsCase_Data10e.xlsx.5 Restrict your attention to the six call and put options
3The stock index data in Table 2 of SecondCityOptionsCase_Data10e.xlsx are from the Chicago Board Options Exchange (CBOE) at www.cboe.com/LearnCenter/pricehistory.xls. 4In calculating volatilities use the following conventions: For daily calculations use all 124 daily index observations over six months. This yields 123 daily returns. Use the first 60 returns to estimate volatility for the first three months. Use the last 63 returns to estimate volatility for the last three months. For weekly calculations begin with the first Friday in the data, January 5, 2007, and go to the last Friday in the data, June 29, 2007. This yields 25 weekly returns. The first 12 returns correspond to the first three months, and the last 13 returns correspond to the last three months. Because there is no observation on Friday, April 6, replace it with the preceding day’s observation. 5 The stock index return data in Table 3 of SecondCityOptionsCase_Data10e.xlsx are from the Center for Research in Security Prices (CRSP). The interest rate data are from Federal Reserve Economic Data (FRED) at www.research.stlouisfed.org. The daily dividend estimates calculated in Table 3 are not actual forecasts of daily dividends that could be made on July 2, 2007, which are what one should use as inputs for the option pricing models. One reason is that the SPX dividend estimates in Table 3 consist of realized, not estimated, dividends for the dates in question. They are calculated using SPX returns from CRSP. Another reason is that when estimating the OEX dividends in Table 3, the
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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. Thus, you must use your best judgment in order to estimate implied volatility for OEX. Whatever methodology you use, explain it clearly.
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.
Historical Volatility The historical volatility calculations are straightforward using HistoricalVolatility10e.xlsm. The most likely reason why calculations might differ across students is that different students might use slightly different sets of observations. For example, consider the entire six month period. Using daily data students should have 124 daily index observations corresponding to 123 daily returns. What happens when one divides the six month period into two subperiods, the first three months and the last three months? Some students use the first 61 observations for the first three months, and then use the last 63 observations for the last three months. In that case, they analyze 60 daily returns for the first three months and 62 daily returns for the last three months. This means that they analyze a total of 122 daily returns, which is one less than they analyze over six months. They exclude the return from March 30 to April 2. As another example, suppose one considers weekly data rather than daily data. One must then choose a convention for which day of the week a return begins and which day of the week a return ends. Assume one student starts with the first data point, which corresponds to a Wednesday, but another student ends with the last data point, which corresponds to a Friday. These two students are not using the same weekly returns, and they do not get the same volatility estimates. Our conventions are as follows: For daily calculations we use all 124 daily index observations over six months. This yields 123 daily returns. We use the first 60 returns to estimate volatility for the first three months. We use the last 63 returns to estimate volatility for the last three months. For weekly calculations we begin with the first Friday in the data, January 5, 2007, and go to the last Friday in the data, June 29, 2007. This yields 25 weekly returns. The first 12 returns correspond to the first three months, and the last 13 returns correspond to the last three months. Because there is no observation on Friday, April 6, the preceding day’s observation is used for that week. Given the above conventions, the following table gives the desired historical volatilities for both SPX and OEX. As noted earlier, some students will use one less observation in the last three months than we consider most appropriate. Therefore, we report volatilities estimated using both conventions for the last three months. See Appendix A for sample calculations.
percentage of total return due to dividends for SPX proxies for the analogous percentage of total return due to dividends for OEX. The rationale is that SPX returns, unlike OEX returns, are directly available from CRSP. To obtain accurate dividend estimates in practice, proprietary market making firms such as Chicago Trading Company (a lead market maker for SPX and OEX options at the CBOE) utilize financial information services companies such as Markit. For the illustrative purposes of this case, however, the dividend estimates in Table 3 suffice.
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Exhibit 5: Historical Volatilities
Frequency (# of Returns) Time Period SPX Volatility OEX Volatility
Daily (123 Returns) All 6 Months 11.35% 11.18%
Daily (60 Returns) 1st 3 Months 12.28% 12.16%
Daily (63 Returns) 2nd 3 Months 10.44% 10.16%
Daily (62 Returns) 2nd 3 Months 10.52% 10.24%
Weekly (25 Returns) All 6 Months 11.83% 11.62%
Weekly (12 Returns) 1st 3 Months 14.43% 13.96%
Weekly (13 Returns) 2nd 3 Months 9.24% 9.06%
Weekly (12 Returns) 2nd 3 Months 9.29% 9.18%
There are several observations one can make regarding these estimates. Volatilities are similar across indices regardless of the frequency and time period used. Volatilities for the entire six month period are similar regardless of the frequency used. Volatilities for the first three months are always higher than those for the second three months regardless of the index and frequency used. This last observation is somewhat surprising. Indeed, the case emphasizes that volatility could be increasing. Note that estimates are sensitive to whether one uses the first or second three month period, and this is especially true with weekly observations where the sample size is relatively small. Students sometimes argue in favor of historical volatility over implied volatility by claiming that historical volatility is more objective. Certainly, implied volatility estimates are subjective in the sense that they rely on a joint assumption that an underlying model is correct and that theoretical prices equal market prices. On the other hand, historical volatility estimates are also subjective in the sense that they depend on the time period and observational frequency used. Given all these historical volatilities, one still desires a single historical volatility estimate for each index. Given the small sample sizes associated with the weekly estimates, we prefer estimates based on daily data. Our first choice would be estimates based on all six months of data, and our second choice would be estimates based on the last three months of data. This reflects two preferences: more data is better than less data, and more recent data is better than less recent data. Therefore, our historical volatility estimates are 11.35% for SPX and 11.18% for OEX. Implied Volatility Implied volatility calculations for SPX options are straightforward using the spreadsheet BlackScholesMertonBinomial10e.xlsm. Because these options are European, one can use the Black-Scholes- Merton model. In this setting one needs six inputs: the stock price, the exercise price, the time to expiration, the volatility, the continuously compounded risk-free rate, and the dividends for the relevant time period. To compute implied volatility, one begins by plugging all inputs except volatility into the Black-Scholes-Merton formula. One then uses “trial and error” to find a volatility such that the model price equals the market price. Implied volatility calculations for OEX options are less obvious. Because OEX options are American, the Black-Scholes-Merton model is inappropriate and should not be used. Students use various methods for obtaining OEX implied volatilities. One common approach is to consider historical volatility estimates. Students often take a relationship that holds for certain historical volatility estimates, and then assume that the same relationship holds for implied volatility estimates. For instance, based on the historical volatility results using all six months of data, students might assume that an implied volatility estimate for an OEX
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option is 0.2% less than an implied volatility estimate for the corresponding SPX option. In our opinion, the most reasonable approach for OEX options is to follow the same procedure outlined above for SPX options, except that one replaces the Black-Scholes-Merton model with the Binomial model. To insure reasonable accuracy one should use a large number of time periods, e.g., 1000. To perform the calculations, we use the software program BlackScholesMertonBinomial10e.xlsm and data available in SecondCityOptionsCase_Data10e.xlsx. For each index we use the six options that are closest to expiration and closest to at-the-money. Thus, we use August options having strike prices of 1515, 1520, and 1525 for SPX and 695, 700, and 705 for OEX. Since there are 47 days from July 2 to August 18, the time to expiration is 0.128767. The continuously compounded risk-free rate is 4.8922%, which is based on the three month T-bill discount rate of 4.81 quoted on July 2. (One obtains this rate if one assumes that the discount rate 4.81 applies to a 47-day horizon, not just a 90-day horizon. Alternatively, one could solve for the continuously compounded rate r such that exp[r(90/365)]=100-(90/360)4.81. Solving for r in this equation yields 4.9064%. Results will be extremely similar regardless of which rate, 4.8922% or 4.9064%, is used.) For the underlying stock price, we use index values of 1519.43 for SPX and 699.49 for OEX. The relevant dividend information is from Table 3 of SecondCityOptionsCase_Data10e.xlsx. There are 33 dividends for each index starting on July 3 and ending on August 17. See Appendix B for a list of the dividends used. As noted earlier, we use the average of bid and ask prices to obtain a market price for each option. Finally, we use 1000 periods when implementing the Binomial model for OEX options. Using the relevant data and model for each of the twelve cases considered, one obtains the following table of implied volatility estimates. See Appendix C for sample calculations.
Exhibit 6: Implied Volatilities
Index Type of Option Exercise Price Implied Volatility
SPX Call 1515 14.77%
SPX Put 1515 14.33%
SPX Call 1520 14.44%
SPX Put 1520 14.08%
SPX Call 1525 14.21%
SPX Put 1525 13.82%
OEX Call 695 14.28%
OEX Put 695 14.32%
OEX Call 700 13.62%
OEX Put 700 13.90%
OEX Call 705 13.17%
OEX Put 705 13.34%
On a technical note, it is important that when one calculates the implied volatilities for the OEX options, one must use the output corresponding to American, not European, call and put options. Empirically, this distinction is not critical for call options. Dividends are so small that American calls are not exercised early, and the software gives the same price for European and American calls. That is definitely not true with puts, however. For the parameter values used here, American puts are more valuable than European puts. Hence,
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if one inadvertently uses European puts rather than American puts, one obtains OEX implied volatilities that are significantly higher than those reported here. Once implied volatilities have been calculated for all these individual options, there is still the issue of obtaining a single implied volatility estimate for each index. Several alternatives are available, and each has its supporters and detractors. Some advocate using only the implied volatilities from at-the-money options. Noting that such options generally have the highest trading volume, they argue that the prices of such options contain the best information available regarding current market conditions. Others believe that one should not ignore information contained in the prices of options that are not at-the-money. They advocate using averages of implied volatilities. For instance, they might use an average where the weights reflect relative trading volume. That is, the more highly traded the option, the greater the weight given that option. For simplicity, we use the simple weighted average of implied volatilities to get an implied volatility for each index. This yields an implied volatility of 14.27% for SPX and an implied volatility of 13.77% for OEX. Interestingly, note that the SPX estimate is slightly larger than the OEX estimate, which is also true for the corresponding historical volatility estimates. Given that we have both historical and implied volatility estimates available, how do we obtain final volatility estimates for each index? Some prefer historical volatility estimates, claiming that implied volatilities are too subjective and too dependent on heroic assumptions regarding option pricing models and market efficiency. Others prefer implied volatility estimates. They note that implied volatility estimates, unlike historical volatility estimates, are “forward looking” and incorporate market expectations. They also note that traders depend far more on implied volatilities than historical volatilities. Finally, some prefer simple or weighted averages of the two types of estimates. We definitely recommend implied volatility estimates over historical volatility estimates. We are confident that most practitioners share that sentiment. Hence, we believe that 14.27% and 13.77% are good final estimates for SPX and OEX volatility, respectively, and we expect many students to use those values. Given that these values are so close together and that their average is 14.02%, we shall use 14% as our final volatility estimate for both indexes. Regardless of whether one considers historical or implied volatilities, the two indexes have volatility estimates that are very close to each other. So using the same estimate for both indices seems reasonable. Given the variation in the individual implied volatility estimates and the sensitivity of those estimates to changes in the inputs, rounding our final volatility estimate to 14% also seems plausible. This decision might seem arbitrary, but we believe that it is reasonable given the inherent estimation error. In our opinion, rounding to 14% is preferable to giving a volatility estimate to four or more decimal places and assuming that one can estimate volatility that precisely. Consequently, we use a volatility of 14% in future calculations.
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, 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.
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. 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 Chapter 4 of An Introduction to Derivatives and Risk Management, 10th edition, by Don M. Chance and Robert Brooks,
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which also contains a table, Table 4.2, similar to the one requested here. Call this table Exhibit 7 and discuss your findings.
As the number of time periods approaches infinity, prices from the Binomial model converge to the corresponding price from the Black-Scholes-Merton model. This assumes, of course, that both models are applicable in this setting, which is true because SPX call options are European. To demonstrate this convergence we use the software program BlackScholesMertonBinomial10e.xlsm. To compute the Black-Scholes-Merton model price, one needs six inputs: the stock price, the exercise price, the time to expiration, the volatility, the continuously compounded risk-free rate, and the dividends for the relevant time period. The stock price is 1519.43. The exercise price is 1520. The time to expiration is 0.128767. The volatility is 0.14. The continuously compounded risk-free rate is 0.048922. As noted in the previous section, there are 33 dividends starting on July 3 and ending on August 17. These dividends are available in Appendix B. When one incorporates all the above information, one obtains a Black-Scholes- Merton price of $32.9447. If one omits the dividends but incorporates all the other information correctly, then one obtains a price of $35.0577. Students occasionally ignore dividends completely or use the discrete dividends to estimate a dividend yield which can be input more easily. Ignoring dividends completely has a significant impact on the Black-Scholes-Merton price, although convergence will still occur for the inputs used. To generate the Binomial model prices, 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. One also incorporates the same dividend information used earlier for the Black- Scholes-Merton model. Using the formulae in Chapter 4 of An Introduction to Derivatives and Risk Management, 10th ed., by Don M. Chance and Robert Brooks, one obtains:
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Exhibit 7: Convergence of Binomial Model Prices to Black-Scholes-Merton Model Prices
Note that the software BlackScholesMertonBinomial10e.xlsm does not calculate the values of u, d, and r. Students must calculate those values using the aforementioned formulae. Students should understand how these parameter values vary over time as the number of time periods increases to infinity. Note also that the values of zero given for the risk-free rate are actually non-zero; they appear to be zero when reported to only four decimal places. Most importantly, note the convergence of the Binomial model prices to the corresponding Black-Scholes-Merton model price. The convergence is quite good. As the number of time periods ranges from 100 to 500 to 1000, the difference between the two prices is less than five cents, two cents, and one cent, respectively. See Appendix D for sample calculations.
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.
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.
c) Identify the one strategy and index that seems most attractive of the four possibilities considered. Explain your reasoning.
The main goal of this section is to determine whether individual options and the strategies chosen in Section I are mispriced and, if so, by how much and in what direction. Thus, we compute theoretical values for the options and strategies under consideration and then compare those values to the corresponding market values. To obtain the theoretical values, we use the basic assumptions and methodology given earlier in Section III. In particular, we use the Black-Scholes-Merton model for SPX options, the Binomial model for OEX options, and a 14% volatility estimate for both indexes. Given those theoretical values, we then compute the difference of the market price minus the theoretical price. This difference, multiplied by 100 to adjust for contract size, is tabulated below. This adjusted difference represents the amount of mispricing present. Positive adjusted differences correspond to overpriced options; negative adjusted differences correspond to underpriced options. The results are as follows:
Exhibit 8: Mispricing of Individual Options
n u D r p C
1 1.0515 0.9510 0.0063 0.5503 $40.2758
5 1.0227 0.9778 0.0013 0.5224 $34.3883
10 1.0160 0.9842 0.0006 0.5159 $32.6852
25 1.0101 0.9900 0.0003 0.5100 $33.1888
50 1.0071 0.9929 0.0001 0.5071 $32.9885
100 1.0050 0.9950 0.0001 0.5050 $32.9924
500 1.0022 0.9978 0.0000 0.5022 $32.9567
1000 1.0016 0.9984 0.0000 0.5016 $32.9414
Index Option Exercise Theoretical Market Difference
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 14 of 34
Given the above information regarding individual options, it is straightforward to determine the analogous information for straddles and butterflies using both the SPX and OEX indexes. For the moment we consider butterflies rather than reverse butterflies, despite the fact that we chose reverse butterflies in Part I. The reason is that it is less confusing to talk about strategies being underpriced or overpriced when one is buying, not selling, them. Once we know whether straddles and butterflies are underpriced or overpriced, we can determine which strategies of Part I are attractively priced. The straddle of Part I is attractively priced if the straddle is underpriced. On the other hand, the reverse butterfly of Part I is attractively priced if the butterfly is overpriced. Using the information in the previous table, we obtain the following:
Exhibit 9: Mispricing of Option Strategies
Index Strategy Theoretical Market Difference
Price Price (Mult. By 100)
SPX Straddle 60.77 61.90 113
SPX Butterfly 0.14 0.20 6
OEX Straddle 28.13 27.65 -48
OEX Butterfly 0.29 0.50 21
Consider the results. For SPX the straddle is overpriced, which implies that it is unattractive. The butterfly is barely overpriced, which implies that the reverse butterfly is very slightly attractive. For OEX the straddle is underpriced, which implies that it is attractive. The butterfly is overpriced, which implies that the reverse butterfly is attractive. Overall, both strategies appear more attractive for OEX than for SPX when considered solely from a pricing perspective. Students have a relatively clear choice depending on their tolerance for risk. If they prefer more risk and are willing to bear a larger maximum loss in return for greater upside potential, then they should choose the OEX straddle. This strategy appears more favorably priced than the SPX straddle. By way of contrast, if they prefer less risk and are willing to bear smaller upside potential in return for a smaller maximum loss, then they should choose the OEX reverse butterfly. This strategy appears more favorably priced than the SPX reverse butterfly. Given a choice between the OEX straddle and the OEX butterfly, it is our experience that students typically prefer the OEX straddle. The OEX reverse butterfly typically seems unattractive given its maximum profit of $50 versus its maximum loss of $450. That profit/loss ratio of 1/9 is not particularly appealing even though the breakeven points are extremely close to each other. Given that we must recommend one strategy, we also would choose the OEX straddle.
Type Price Price Price (Mult. by 100)
SPX Call 1515 35.58 37.10 152
SPX Put 1515 25.50 26.20 70
SPX Call 1520 32.94 33.90 96
SPX Put 1520 27.83 28.00 17
SPX Call 1525 30.44 30.90 46
SPX Put 1525 30.30 29.90 -40
OEX Call 695 17.73 18.00 27
OEX Put 695 10.79 11.10 31
OEX Call 700 15.03 14.65 -38
OEX Put 700 13.10 13.00 -10
OEX Call 705 12.62 11.80 -82
OEX Put 705 15.71 15.05 -66
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 15 of 34
As noted in Part I, students might consider strategies other than a straddle or reverse butterfly, with a strangle being a prime candidate. If so, they should analyze their strategies similarly. That is, they should first determine whether their alternate strategy is attractively priced. After considering factors such as the amount of mispricing and their tolerance for risk, they should make their recommendations accordingly.
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.
b) In terms of pricing, how does this strategy compare to the two strategies analyzed in Part IV? If you must recommend one final strategy and index from all six of the combinations investigated in Parts IV and V, what do you recommend and why?
The obvious strategy is a reverse calendar spread. For SPX one buys the August 1520 call and sells the September 1520 call. For OEX one buys the August 700 call and sells the September 700 call. In both cases there is a cash inflow when one initiates the strategy. Assuming that one uses the average of bid and ask prices for the option price, these initial cash flows (adjusted for a contract size of 100) are $1370 and $625 for SPX and OEX, respectively. These positions are profitable as long as there is a sufficiently large movement in the underlying index. Profit diagrams can be obtained as before from the program OptionStrategyAnalyzer10e.xlsm. Because the September options must be priced prior to expiration, one now must input the risk-free rate, the time remaining to expiration when the position is closed, and volatility, which are unnecessary in Part I. These inputs are 0.048922, 0.095890, and 0.14, respectively. Note that 0.095890 corresponds to the time from August expiration until September expiration, which equals 35 days. It is used only for the September call option. Note also that this software uses the Black-Scholes-Merton model to price options prior to expiration. Thus, it implicitly assumes that the options are European. Because OEX options are American, this introduces error into the OEX graph. Furthermore, this software only allows one to input a continuous dividend yield. One cannot input discrete dividends, which affects both the SPX and OEX graphs. Given that the discrete dividends are very small in this case, we input a continuous dividend yield of zero. For these reasons the graphs in this section should be considered approximate, especially in the case of OEX. For example, breakeven points for the OEX graph will differ slightly from breakeven points obtained analytically using the Binomial model to price the September option at the earlier expiration date of August 18. Nonetheless, we use OptionStrategyAnalyzer10e.xlsm to generate the following graphs:
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 16 of 34
Exhibit 10: Profit Diagram for SPX Reverse Calendar Spread
Exhibit 11: Profit Diagram for OEX Reverse Calendar Spread
-20
-15
-10
-5
0
5
10
15
20
1 3 2 0
1 3 6 0
1 4 0 0
1 4 4 0
1 4 8 0
1 5 2 0
1 5 6 0
1 6 0 0
1 6 4 0
1 6 8 0
P ro
fi t
( $ 1 0 0 s )
Stock Index at Expiration
SPX Reverse Calendar Spread
-10
-8
-6
-4
-2
0
2
4
6
8
5 0 0
5 4 0
5 8 0
6 2 0
6 6 0
7 0 0
7 4 0
7 8 0
8 2 0
8 6 0
P ro
fi t
( $ 1 0 0 s )
Stock Index at Expiration
OEX Reverse Calendar Spread
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 17 of 34
Rather than computing the maximum profit, maximum loss, and breakeven points analytically, we simply get approximate values from the graphs themselves. By changing the scale of the horizontal axis, one can get reasonably accurate values quickly and easily. For SPX we get the following: the maximum profit is $1,370, the maximum loss is $1,623.54, and the breakeven points are 1,483.13 and 1,573.81. For OEX we get the following: the maximum profit is $625.00, the maximum loss is $753.60, and the breakeven points are 682.84 and 725.12. See Appendix E for sample calculations. We now check both reverse calendar spreads for mispricing. As in Part IV, we compute theoretical and market prices for the individual options in the spreads and the spreads themselves. Consider the difference of the market price minus the theoretical price. If the difference is positive, then the calendar spread is overpriced and the reverse calendar spread is attractive. If the difference is negative, then the calendar spread is underpriced and the reverse calendar spread is unattractive. The option price calculations differ slightly from earlier ones. With September options the time to expiration is 0.224658 rather than 0.128767 as was used for August options. Moreover, with September options there are 62 dividends rather than 33 dividends as was used for August options. See Appendix B for a list of the dividends used. First consider the SPX case. The market price of a calendar spread (not a reverse calendar spread!) is $13.70. The theoretical price of the August call is $32.94 from Part IV. Using the methodology described in Part IV, one can use the Black-Scholes-Merton model to obtain a theoretical price of $44.54 for the September call. (See Appendix F.) Thus, the calendar spread has a theoretical price of $11.60. It appears that the calendar spread is overpriced by $210 (adjusted for the 100 multiplier), which implies that the reverse calendar spread is attractively priced. Now consider the OEX case. The market price of a calendar spread is $6.25. The theoretical price of the August call is $15.03 from Part IV. Using the methodology described in Part IV, one can use the Binomial model to obtain a theoretical price of $20.45 for the September call. (See Appendix F.) Thus, the calendar spread has a theoretical price of $5.42. It appears that the calendar spread is overpriced by $83 (adjusted for the 100 multiplier), which implies that the reverse calendar spread is attractively priced. Based on the above analysis, both reverse calendar spreads are viable options. How do they compare to the OEX straddle, which is our favorite strategy considered in Part IV? For simplicity, we shall compare the OEX reverse calendar spread to the OEX straddle since those strategies have the same underlying index. First, compare maximum loss. It equals 2765 for the OEX straddle and $753.60 for the OEX reverse calendar spread. So the reverse calendar spread looks better from that perspective. The OEX straddle has a positive cash outflow at the start; the OEX reverse calendar spread has a positive cash inflow at the start. Some might consider that to be a slight advantage for the OEX reverse calendar spread. The OEX straddle has breakeven points of 672.35 and 727.65. The OEX reverse calendar spread has breakeven points of 682.84 and 725.12. So the OEX reverse calendar spread has the advantage in that respect. On the other hand, there is more upside potential with the OEX straddle in the sense that profits are unbounded, but how likely is it that the underlying index OEX will move so much that profits from the OEX straddle exceed profits from the OEX reverse calendar spread? To summarize, the OEX reverse calendar spread has three comparative advantages over the OEX straddle. The former has a smaller maximum loss, a positive cash flow at the start, and breakeven points that are closer to the strike price. Conversely, the OEX straddle has one advantage over the OEX reverse calendar spread. The former has unbounded profits; the latter strategy has bounded profits. There is no obvious winner between the two. One could argue for either strategy. For many the choice probably hinges on the importance of the unbounded profit potential of the OEX straddle.
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 18 of 34
Our choice is to recommend the OEX straddle as we did at the end of Part IV. We prefer having the unbounded profit potential of the straddle. In our opinion, there is little difference in the breakeven points for the strategies. The fact that the reverse calendar spread generates a positive initial cash flow is nice, but not that important to us. The biggest disadvantage of the straddle is that its maximum loss is significantly greater than that for the reverse calendar spread. This is misleading, however. One can actually lose more than $753.60 with the reverse calendar spread. The maximum loss for the reverse calendar spread is a loss that is estimated based on the theoretical value of the September call option on August 18, 2007. That theoretical value is based on a volatility of 14%. What happens if volatility increases sharply as we believe it might? Then the actual loss will be much higher than estimated. That is, assume one initiates the reverse calendar spread on July 3 and holds the position until August 18. Meanwhile, volatility increases from 14% to some significantly higher value. To close the position, one must buy back the September call option, and that call option will be priced much higher than its current theoretical price. In short, the $753.60 maximum loss for the reverse calendar spread is an estimated maximum loss. One can actually lose more than $753.60 with that strategy.
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 19 of 34
Appendix A: Historical Volatilities Exhibit A1 – SPX Historical Volatility Using All Data and Weekly Observations (25 Returns)
Inputs: Results: Enter the data frequency: Periodic Annualized
Variance 0.000269 0.014002
Standard Deviation 1.64% 11.83%
Mean 0.26% 13.38%
Obs. Date
Asset
Price
Simple
Rate of
Return
Continuously
Compounded
Rate of Return
Squared
Deviation
1 5-Jan 1409.71 NA NA NA
2 12-Jan 1430.73 1.49% 1.48% 0.000150
3 19-Jan 1430.50 -0.02% -0.02% 0.000007
4 26-Jan 1422.18 -0.58% -0.58% 0.000071
5 2-Feb 1448.39 1.84% 1.83% 0.000246
6 9-Feb 1438.06 -0.71% -0.72% 0.000095
7 16-Feb 1455.54 1.22% 1.21% 0.000090
8 23-Feb 1451.19 -0.30% -0.30% 0.000031
9 2-Mar 1387.17 -4.41% -4.51% 0.002274
10 9-Mar 1402.85 1.13% 1.12% 0.000075
11 16-Mar 1386.95 -1.13% -1.14% 0.000195
12 23-Mar 1436.11 3.54% 3.48% 0.001041
13 30-Mar 1420.86 -1.06% -1.07% 0.000176
14 5-Apr 1443.76 1.61% 1.60% 0.000180
15 13-Apr 1452.85 0.63% 0.63% 0.000014
16 20-Apr 1484.35 2.17% 2.14% 0.000356
17 27-Apr 1494.07 0.65% 0.65% 0.000016
18 4-May 1505.62 0.77% 0.77% 0.000026
19 11-May 1505.85 0.02% 0.02% 0.000006
20 18-May 1522.75 1.12% 1.12% 0.000074
21 25-May 1515.73 -0.46% -0.46% 0.000052
22 1-Jun 1536.34 1.36% 1.35% 0.000120
23 8-Jun 1507.67 -1.87% -1.88% 0.000458
24 15-Jun 1532.91 1.67% 1.66% 0.000197
25 22-Jun 1502.56 -1.98% -2.00% 0.000509
26 29-Jun 1503.35 0.05% 0.05% 0.000004
=============================================== ==========
Totals 25 0.006462
HISTORICAL VOLATILITY
ESTIMATION 10e
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 20 of 34
Appendix A: Historical Volatilities (Continued)
Exhibit A2 – SPX Historical Volatility Using 1st Three Months of Data and Weekly Observations (12 Returns)
Appendix A: Historical Volatilities (Continued)
Exhibit A3 – SPX Historical Volatility Using 2nd Three Months of Data and Weekly Observations (13 Returns)
Inputs: Results: Enter the data frequency: Periodic Annualized
Variance 0.000401 0.020832
Standard Deviation 2.00% 14.43%
Mean 0.07% 3.41%
Obs. Date
Asset
Price
Simple
Rate of
Return
Continuously
Compounded
Rate of Return
Squared
Deviation
1 5-Jan 1409.71 NA NA NA
2 12-Jan 1430.73 1.49% 1.48% 0.000200
3 19-Jan 1430.50 -0.02% -0.02% 0.000001
4 26-Jan 1422.18 -0.58% -0.58% 0.000042
5 2-Feb 1448.39 1.84% 1.83% 0.000310
6 9-Feb 1438.06 -0.71% -0.72% 0.000061
7 16-Feb 1455.54 1.22% 1.21% 0.000131
8 23-Feb 1451.19 -0.30% -0.30% 0.000013
9 2-Mar 1387.17 -4.41% -4.51% 0.002095
10 9-Mar 1402.85 1.13% 1.12% 0.000112
11 16-Mar 1386.95 -1.13% -1.14% 0.000145
12 23-Mar 1436.11 3.54% 3.48% 0.001168
13 30-Mar 1420.86 -1.06% -1.07% 0.000128
=============================================== ==========
Totals 12 0.004407
HISTORICAL VOLATILITY
ESTIMATION 10e
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 21 of 34
Inputs: Results: Enter the data frequency: Periodic Annualized
Variance 0.000164 0.008541
Standard Deviation 1.28% 9.24%
Mean 0.43% 22.57%
Obs. Date
Asset
Price
Simple
Rate of
Return
Continuously
Compounded
Rate of Return
Squared
Deviation
1 30-Mar 1420.86 NA NA NA
2 5-Apr 1443.76 1.61% 1.60% 0.000136
3 13-Apr 1452.85 0.63% 0.63% 0.000004
4 20-Apr 1484.35 2.17% 2.14% 0.000293
5 27-Apr 1494.07 0.65% 0.65% 0.000005
6 4-May 1505.62 0.77% 0.77% 0.000011
7 11-May 1505.85 0.02% 0.02% 0.000018
8 18-May 1522.75 1.12% 1.12% 0.000047
9 25-May 1515.73 -0.46% -0.46% 0.000080
10 1-Jun 1536.34 1.36% 1.35% 0.000084
11 8-Jun 1507.67 -1.87% -1.88% 0.000537
12 15-Jun 1532.91 1.67% 1.66% 0.000150
13 22-Jun 1502.56 -1.98% -2.00% 0.000592
14 29-Jun 1503.35 0.05% 0.05% 0.000015
=============================================== ==========
Totals 13 0.001971
HISTORICAL VOLATILITY
ESTIMATION 10e
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 22 of 34
Appendix A: Historical Volatilities (Continued)
Exhibit A4 – SPX Historical Volatility Using Last Three Months of Data and Weekly Observations (12 Returns)
Inputs: Results: Enter the data frequency: Periodic Annualized
Variance 0.000166 0.008623
Standard Deviation 1.29% 9.29%
Mean 0.34% 17.53%
Obs. Date
Asset
Price
Simple
Rate of
Return
Continuously
Compounded
Rate of Return
Squared
Deviation
1 5-Apr 1443.76 NA NA NA
2 13-Apr 1452.85 0.63% 0.63% 0.000008
3 20-Apr 1484.35 2.17% 2.14% 0.000327
4 27-Apr 1494.07 0.65% 0.65% 0.000010
5 4-May 1505.62 0.77% 0.77% 0.000019
6 11-May 1505.85 0.02% 0.02% 0.000010
7 18-May 1522.75 1.12% 1.12% 0.000061
8 25-May 1515.73 -0.46% -0.46% 0.000064
9 1-Jun 1536.34 1.36% 1.35% 0.000103
10 8-Jun 1507.67 -1.87% -1.88% 0.000493
11 15-Jun 1532.91 1.67% 1.66% 0.000175
12 22-Jun 1502.56 -1.98% -2.00% 0.000546
13 29-Jun 1503.35 0.05% 0.05% 0.000008
=============================================== ==========
Totals 12 0.001824
HISTORICAL VOLATILITY
ESTIMATION 10e
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 23 of 34
Appendix B: Estimated Daily Dividends for SPX and OEX
Observation DATE SPX Dividend OEX Dividend Time to Ex Dividend
1 20070703 0.30236657 0.13919851 0.00273973
2 20070705 0.01677357 0.00773223 0.00821918
3 20070706 0.43473900 0.20030370 0.01095890
4 20070709 0.00918264 0.00422700 0.01917808
5 20070710 0.00459555 0.00211674 0.02191781
6 20070711 0.18272452 0.08424141 0.02465753
7 20070712 0.01366884 0.00630540 0.02739726
8 20070713 0.01857240 0.00857688 0.03013699
11 20070718 0.16113448 0.07459088 0.04383562
12 20070719 0.03710808 0.01715808 0.04657534
13 20070720 0.14443644 0.06678423 0.04931507
14 20070723 0.01994330 0.00922896 0.05753425
15 20070724 0.00000000 0.00000000 0.06027397
16 20070725 0.01511040 0.00701200 0.06301370
17 20070726 0.00303618 0.00141188 0.06575342
18 20070727 0.21795102 0.10140648 0.06849315
19 20070730 0.03355585 0.01562022 0.07671233
20 20070731 0.00884346 0.00411054 0.07945205
21 20070801 0.09313728 0.04324800 0.08219178
22 20070802 0.34739697 0.16159845 0.08493151
23 20070803 0.08833200 0.04107060 0.08767123
24 20070806 0.02722814 0.01268098 0.09589041
25 20070807 0.05577146 0.02597908 0.09863014
26 20070808 0.49617456 0.23110752 0.10136986
27 20070809 0.26805071 0.12496706 0.10410959
28 20070810 0.00290618 0.00135204 0.10684932
29 20070813 0.22240692 0.10336986 0.11506849
30 20070814 0.16127412 0.07487061 0.11780822
31 20070815 0.43509470 0.20250780 0.12054795
32 20070816 0.05626800 0.02623320 0.12328767
33 20070817 0.00000000 0.00000000 0.12602740
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 24 of 34
Appendix B (cont’d): Estimated Daily Dividends for SPX and OEX
Observation DATE SPX Dividend OEX Dividend Time to Ex Dividend
34 20070820 0.00000000 0.00000000 0.13424658
35 20070821 0.00578220 0.00269648 0.13698630
36 20070822 0.08972144 0.04177870 0.13972603
37 20070823 0.01903291 0.00885664 0.14246575
38 20070824 0.15648750 0.07295688 0.14520548
39 20070827 0.00887622 0.00413682 0.15342466
40 20070828 0.06600555 0.03081330 0.15616438
41 20070829 0.53570264 0.25020226 0.15890411
42 20070830 0.08489808 0.03961516 0.16164384
43 20070831 0.02769516 0.01292874 0.16438356
44 20070904 0.02358384 0.01099952 0.17534247
45 20070905 0.66874958 0.31170927 0.17808219
46 20070906 0.25470617 0.11878699 0.18082192
47 20070907 0.00000000 0.00000000 0.18356164
48 20070910 0.05378135 0.02507638 0.19178082
49 20070911 0.00290340 0.00135608 0.19452055
50 20070912 0.39877379 0.18635315 0.19726027
51 20070913 0.08535048 0.03992836 0.20000000
52 20070914 0.00296790 0.00139000 0.20273973
53 20070917 0.01038975 0.00486066 0.21095890
54 20070918 0.01771980 0.00829344 0.21369863
55 20070919 0.04407362 0.02059232 0.21643836
56 20070920 0.32568339 0.15198189 0.21917808
57 20070921 0.00000000 0.00000000 0.22191781
58 20070924 0.00000000 0.00000000 0.23013699
59 20070925 0.02731914 0.01278252 0.23287671
60 20070926 0.27764943 0.13002516 0.23561644
61 20070927 0.05796596 0.02713428 0.23835616
62 20070928 0.01990794 0.00931398 0.24109589
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 25 of 34
Appendix C: Implied Volatilities Exhibit C1 – Implied Volatility for SPX 1520 Call
Inputs: Black-Scholes-Merton Model Binomial Model
Asset price (S0) 1519.43 European European Steps: 1,000
Exercise price (X) 1520 Call Put European European
Time to expiration (T) 0.1288 Price 33.8953 28.7840 Call Put
Standard deviation (s) 14.44% Delta (D) 0.5363 -0.4637 Price 33.8985 29.0126
Risk-free rate (r or rc) 4.89% Gamma (G) 0.0051 0.0051 Delta (D) 0.5351 -0.4649
Dividends: 0.00% Theta (Q) -159.2534 -85.3589 Gamma (G) 0.0050 0.0050
continuous yield (dc) or discrete dividends below: Vega 216.0654 216.0654 Theta (Q) -158.7724 -86.6179
Rho 100.2961 -94.2008 American American
Call Put
d1 0.0911 Price 33.8985 29.3407
d2 0.0393 Delta (D) 0.5351 -0.4720
N(d1) 0.5363 Gamma (G) 0.0050 0.0052
N(d2) 0.5157 Theta (Q) -158.7724 -89.7157
PV of divs 3.8640
PV of strike 1510.4548
Dividend # Dividend Time to Present Value S - PV divs 1515.5660
1 0.3024 0.0027 0.3023
2 0.0168 0.0082 0.0168
3 0.4347 0.0110 0.4345
4 0.0092 0.0192 0.0092
5 0.0046 0.0219 0.0046
6 0.1827 0.0247 0.1825
7 0.0137 0.0274 0.0137
8 0.0186 0.0301 0.0185
9 0.0000 0.0384 0.0000
10 0.0000 0.0411 0.0000
11 0.1611 0.0438 0.1608
12 0.0371 0.0466 0.0370
13 0.1444 0.0493 0.1441
14 0.0199 0.0575 0.0199
15 0.0000 0.0603 0.0000
16 0.0151 0.0630 0.0151
17 0.0030 0.0658 0.0030
18 0.2180 0.0685 0.2172
19 0.0336 0.0767 0.0334
20 0.0088 0.0795 0.0088
21 0.0931 0.0822 0.0928
22 0.3474 0.0849 0.3460
23 0.0883 0.0877 0.0880
24 0.0272 0.0959 0.0271
25 0.0558 0.0986 0.0555
26 0.4962 0.1014 0.4937
27 0.2681 0.1041 0.2667
28 0.0029 0.1068 0.0029
29 0.2224 0.1151 0.2212
30 0.1613 0.1178 0.1603
31 0.4351 0.1205 0.4325
32 0.0563 0.1233 0.0559
33 0.0000 0.1260 0.0000
========= ======= ======= ===========
Sum 3.8640
In lieu of a continuously compounded yield, place below up to one hundred discrete dividends and the time in years to each ex-dividend date. Leave all unused cells blank.
Set the yield above to zero. If yield is not set to zero, all discrete dividends are disregarded.
Black-Scholes-Merton and Binomial
Option Pricing 10e
Run Binomial Model
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 26 of 34
Appendix C: Implied Volatilities (Continued) Exhibit C2 – Implied Volatility for SPX 1520 Put
Inputs: Black-Scholes-Merton Model Binomial Model
Asset price (S0) 1519.43 European European Steps: 1,000
Exercise price (X) 1520 Call Put European European
Time to expiration (T) 0.1288 Price 33.1175 28.0063 Call Put
Standard deviation (s) 14.08% Delta (D) 0.5367 -0.4633 Price 33.1145 28.0032
Risk-free rate (r or rc) 4.89% Gamma (G) 0.0052 0.0052 Delta (D) 0.5367 -0.4633
Dividends: 0.00% Theta (Q) -156.2902 -82.3957 Gamma (G) 0.0052 0.0052
continuous yield (dc) or discrete dividends below: Vega 216.0451 216.0451 Theta (Q) -156.3012 -82.4067
Rho 100.4754 -94.0215 American American
Call Put
d1 0.0921 Price 33.1145 28.3517
d2 0.0416 Delta (D) 0.5367 -0.4712
N(d1) 0.5367 Gamma (G) 0.0052 0.0054
N(d2) 0.5166 Theta (Q) -156.3012 -85.7312
PV of divs 3.8640
PV of strike 1510.4548
Dividend # Dividend Time to Present Value S - PV divs 1515.5660
1 0.3024 0.0027 0.3023
2 0.0168 0.0082 0.0168
3 0.4347 0.0110 0.4345
4 0.0092 0.0192 0.0092
5 0.0046 0.0219 0.0046
6 0.1827 0.0247 0.1825
7 0.0137 0.0274 0.0137
8 0.0186 0.0301 0.0185
9 0.0000 0.0384 0.0000
10 0.0000 0.0411 0.0000
11 0.1611 0.0438 0.1608
12 0.0371 0.0466 0.0370
13 0.1444 0.0493 0.1441
14 0.0199 0.0575 0.0199
15 0.0000 0.0603 0.0000
16 0.0151 0.0630 0.0151
17 0.0030 0.0658 0.0030
18 0.2180 0.0685 0.2172
19 0.0336 0.0767 0.0334
20 0.0088 0.0795 0.0088
21 0.0931 0.0822 0.0928
22 0.3474 0.0849 0.3460
23 0.0883 0.0877 0.0880
24 0.0272 0.0959 0.0271
25 0.0558 0.0986 0.0555
26 0.4962 0.1014 0.4937
27 0.2681 0.1041 0.2667
28 0.0029 0.1068 0.0029
29 0.2224 0.1151 0.2212
30 0.1613 0.1178 0.1603
31 0.4351 0.1205 0.4325
32 0.0563 0.1233 0.0559
33 0.0000 0.1260 0.0000
========= ======= ======= ===========
Sum 3.8640
In lieu of a continuously compounded yield, place below up to one hundred discrete dividends and the time in years to each ex-dividend date. Leave all unused cells blank.
Set the yield above to zero. If yield is not set to zero, all discrete dividends are disregarded.
Black-Scholes-Merton and Binomial
Option Pricing 10e
Run Binomial Model
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 27 of 34
Appendix C: Implied Volatilities (Continued) Exhibit C3 – Implied Volatility for OEX 700 Call
Inputs: Black-Scholes-Merton Model Binomial Model
Asset price (S0) 699.49 European European Steps: 1,000
Exercise price (X) 700 Call Put European European
Time to expiration (T) 0.1288 Price 14.6542 12.5610 Call Put
Standard deviation (s) 13.62% Delta (D) 0.5342 -0.4658 Price 14.6524 12.5592
Risk-free rate (r or rc) 4.89% Gamma (G) 0.0117 0.0117 Delta (D) 0.5343 -0.4657
Dividends: 0.00% Theta (Q) -70.1462 -36.1158 Gamma (G) 0.0117 0.0117
continuous yield (dc) or discrete dividends below: Vega 99.5123 99.5123 Theta (Q) -70.1529 -36.1226
Rho 46.1088 -43.4621 American American
Call Put
d1 0.0859 Price 14.6524 12.7214
d2 0.0370 Delta (D) 0.5343 -0.4740
N(d1) 0.5342 Gamma (G) 0.0117 0.0121
N(d2) 0.5148 Theta (Q) -70.1529 -37.6686
PV of divs 1.7926
PV of strike 695.6042
Dividend # Dividend Time to Present Value S - PV divs 697.6974
1 0.1392 0.0027 0.1392
2 0.0077 0.0082 0.0077
3 0.2003 0.0110 0.2002
4 0.0042 0.0192 0.0042
5 0.0021 0.0219 0.0021
6 0.0842 0.0247 0.0841
7 0.0063 0.0274 0.0063
8 0.0086 0.0301 0.0086
9 0.0000 0.0384 0.0000
10 0.0000 0.0411 0.0000
11 0.0746 0.0438 0.0744
12 0.0172 0.0466 0.0171
13 0.0668 0.0493 0.0666
14 0.0092 0.0575 0.0092
15 0.0000 0.0603 0.0000
16 0.0070 0.0630 0.0070
17 0.0014 0.0658 0.0014
18 0.1014 0.0685 0.1011
19 0.0156 0.0767 0.0156
20 0.0041 0.0795 0.0041
21 0.0432 0.0822 0.0431
22 0.1616 0.0849 0.1609
23 0.0411 0.0877 0.0409
24 0.0127 0.0959 0.0126
25 0.0260 0.0986 0.0259
26 0.2311 0.1014 0.2300
27 0.1250 0.1041 0.1243
28 0.0014 0.1068 0.0013
29 0.1034 0.1151 0.1028
30 0.0749 0.1178 0.0744
31 0.2025 0.1205 0.2013
32 0.0262 0.1233 0.0261
33 0.0000 0.1260 0.0000
========= ======= ======= ===========
Sum 1.7926
In lieu of a continuously compounded yield, place below up to one hundred discrete dividends and the time in years to each ex-dividend date. Leave all unused cells blank.
Set the yield above to zero. If yield is not set to zero, all discrete dividends are disregarded.
Black-Scholes-Merton and Binomial
Option Pricing 10e
Run Binomial Model
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 28 of 34
Appendix C: Implied Volatilities (Continued) Exhibit C4 – Implied Volatility for OEX 700 Put
Inputs: Black-Scholes-Merton Model Binomial Model
Asset price (S0) 699.49 European European Steps: 1,000
Exercise price (X) 700 Call Put European European
Time to expiration (T) 0.1288 Price 14.9329 12.8396 Call Put
Standard deviation (s) 13.90% Delta (D) 0.5339 -0.4661 Price 14.9305 12.8372
Risk-free rate (r or rc) 4.89% Gamma (G) 0.0114 0.0114 Delta (D) 0.5340 -0.4660
Dividends: 0.00% Theta (Q) -71.2079 -37.1775 Gamma (G) 0.0114 0.0114
continuous yield (dc) or discrete dividends below: Vega 99.5186 99.5186 Theta (Q) -71.2169 -37.1865
Rho 46.0466 -43.5243 American American
Call Put
d1 0.0852 Price 14.9305 12.9989
d2 0.0353 Delta (D) 0.5340 -0.4741
N(d1) 0.5339 Gamma (G) 0.0114 0.0118
N(d2) 0.5141 Theta (Q) -71.2169 -38.7221
PV of divs 1.7926
PV of strike 695.6042
Dividend # Dividend Time to Present Value S - PV divs 697.6974
1 0.1392 0.0027 0.1392
2 0.0077 0.0082 0.0077
3 0.2003 0.0110 0.2002
4 0.0042 0.0192 0.0042
5 0.0021 0.0219 0.0021
6 0.0842 0.0247 0.0841
7 0.0063 0.0274 0.0063
8 0.0086 0.0301 0.0086
9 0.0000 0.0384 0.0000
10 0.0000 0.0411 0.0000
11 0.0746 0.0438 0.0744
12 0.0172 0.0466 0.0171
13 0.0668 0.0493 0.0666
14 0.0092 0.0575 0.0092
15 0.0000 0.0603 0.0000
16 0.0070 0.0630 0.0070
17 0.0014 0.0658 0.0014
18 0.1014 0.0685 0.1011
19 0.0156 0.0767 0.0156
20 0.0041 0.0795 0.0041
21 0.0432 0.0822 0.0431
22 0.1616 0.0849 0.1609
23 0.0411 0.0877 0.0409
24 0.0127 0.0959 0.0126
25 0.0260 0.0986 0.0259
26 0.2311 0.1014 0.2300
27 0.1250 0.1041 0.1243
28 0.0014 0.1068 0.0013
29 0.1034 0.1151 0.1028
30 0.0749 0.1178 0.0744
31 0.2025 0.1205 0.2013
32 0.0262 0.1233 0.0261
33 0.0000 0.1260 0.0000
========= ======= ======= ===========
Sum 1.7926
In lieu of a continuously compounded yield, place below up to one hundred discrete dividends and the time in years to each ex-dividend date. Leave all unused cells blank.
Set the yield above to zero. If yield is not set to zero, all discrete dividends are disregarded.
Black-Scholes-Merton and Binomial
Option Pricing 10e
Run Binomial Model
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 29 of 34
Appendix D: Convergence of Binomial Model Prices to Black-Scholes-Merton Model Price
Inputs: Black-Scholes-Merton Model Binomial Model
Asset price (S0) 1519.43 European European Steps: 1,000
Exercise price (X) 1520 Call Put European European
Time to expiration (T) 0.1288 Price 32.9447 27.8334 Call Put
Standard deviation (s) 14.00% Delta (D) 0.5368 -0.4632 Price 32.9414 27.8302
Risk-free rate (r or rc) 4.89% Gamma (G) 0.0052 0.0052 Delta (D) 0.5368 -0.4632
Dividends: 0.00% Theta (Q) -155.6319 -81.7375 Gamma (G) 0.0052 0.0052
continuous yield (dc) or discrete dividends below: Vega 216.0404 216.0404 Theta (Q) -155.6440 -81.7495
Rho 100.5161 -93.9808 American American
Call Put
d1 0.0924 Price 32.9414 28.1791
d2 0.0421 Delta (D) 0.5368 -0.4711
N(d1) 0.5368 Gamma (G) 0.0052 0.0054
N(d2) 0.5168 Theta (Q) -155.6440 -85.0800
PV of divs 3.8640
PV of strike 1510.4548
Dividend # Dividend Time to Present Value S - PV divs 1515.5660
1 0.3024 0.0027 0.3023
2 0.0168 0.0082 0.0168
3 0.4347 0.0110 0.4345
4 0.0092 0.0192 0.0092
5 0.0046 0.0219 0.0046
6 0.1827 0.0247 0.1825
7 0.0137 0.0274 0.0137
8 0.0186 0.0301 0.0185
9 0.0000 0.0384 0.0000
10 0.0000 0.0411 0.0000
11 0.1611 0.0438 0.1608
12 0.0371 0.0466 0.0370
13 0.1444 0.0493 0.1441
14 0.0199 0.0575 0.0199
15 0.0000 0.0603 0.0000
16 0.0151 0.0630 0.0151
17 0.0030 0.0658 0.0030
18 0.2180 0.0685 0.2172
19 0.0336 0.0767 0.0334
20 0.0088 0.0795 0.0088
21 0.0931 0.0822 0.0928
22 0.3474 0.0849 0.3460
23 0.0883 0.0877 0.0880
24 0.0272 0.0959 0.0271
25 0.0558 0.0986 0.0555
26 0.4962 0.1014 0.4937
27 0.2681 0.1041 0.2667
28 0.0029 0.1068 0.0029
29 0.2224 0.1151 0.2212
30 0.1613 0.1178 0.1603
31 0.4351 0.1205 0.4325
32 0.0563 0.1233 0.0559
33 0.0000 0.1260 0.0000
========= ======= ======= ===========
Sum 3.8640
In lieu of a continuously compounded yield, place below up to one hundred discrete dividends and the time in years to each ex-dividend date. Leave all unused cells blank.
Set the yield above to zero. If yield is not set to zero, all discrete dividends are disregarded.
Black-Scholes-Merton and Binomial
Option Pricing 10e
Run Binomial Model
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 30 of 34
Appendix E: Profit Diagram Details for SPX Reverse Calendar Spread
Exhibit E1 – Maximum Profit and Maximum Loss for SPX Reverse Calendar Spread
Exhibit E1 – Maximum Profit and Maximum Loss for SPX Reverse Calendar Spread
Asset price at
end of
holding
period Profitability of Each Component of the Strategy
Option 1 Option 2 Overall
1,120.00 -3,390.00 4,760.00 1,370.00
1,136.00 -3,390.00 4,760.00 1,370.00
1,152.00 -3,390.00 4,760.00 1,370.00
1,168.00 -3,390.00 4,760.00 1,370.00
1,184.00 -3,390.00 4,760.00 1,370.00
1,200.00 -3,390.00 4,760.00 1,370.00
1,216.00 -3,390.00 4,760.00 1,370.00
1,232.00 -3,390.00 4,760.00 1,370.00
1,248.00 -3,390.00 4,759.99 1,369.99
1,264.00 -3,390.00 4,759.98 1,369.98
1,280.00 -3,390.00 4,759.92 1,369.92
1,296.00 -3,390.00 4,759.73 1,369.73
1,312.00 -3,390.00 4,759.18 1,369.18
1,328.00 -3,390.00 4,757.71 1,367.71
1,344.00 -3,390.00 4,754.07 1,364.07
1,360.00 -3,390.00 4,745.79 1,355.79
1,376.00 -3,390.00 4,728.34 1,338.34
1,392.00 -3,390.00 4,694.16 1,304.16
1,408.00 -3,390.00 4,631.72 1,241.72
1,424.00 -3,390.00 4,525.00 1,135.00
1,440.00 -3,390.00 4,353.74 963.74
1,456.00 -3,390.00 4,094.79 704.79
1,472.00 -3,390.00 3,724.51 334.51
1,488.00 -3,390.00 3,221.87 -168.13
1,504.00 -3,390.00 2,571.59 -818.41
Asset price at
end of
holding
period Profitability of Each Component of the Strategy
Option 1 Option 2 Overall
1,520.00 -3,390.00 1,766.46 -1,623.54
1,536.00 -1,790.00 808.37 -981.63
1,552.00 -190.00 -292.31 -482.31
1,568.00 1,410.00 -1,518.66 -108.66
1,584.00 3,010.00 -2,849.87 160.13
1,600.00 4,610.00 -4,264.09 345.91
1,616.00 6,210.00 -5,740.74 469.26
1,632.00 7,810.00 -7,262.05 547.95
1,648.00 9,410.00 -8,813.81 596.19
1,664.00 11,010.00 -10,385.37 624.63
1,680.00 12,610.00 -11,969.25 640.75
1,696.00 14,210.00 -13,560.46 649.54
1,712.00 15,810.00 -15,155.84 654.16
1,728.00 17,410.00 -16,753.50 656.50
1,744.00 19,010.00 -18,352.35 657.65
1,760.00 20,610.00 -19,951.82 658.18
1,776.00 22,210.00 -21,551.57 658.43
1,792.00 23,810.00 -23,151.46 658.54
1,808.00 25,410.00 -24,751.42 658.58
1,824.00 27,010.00 -26,351.40 658.60
1,840.00 28,610.00 -27,951.39 658.61
1,856.00 30,210.00 -29,551.39 658.61
1,872.00 31,810.00 -31,151.39 658.61
1,888.00 33,410.00 -32,751.39 658.61
1,904.00 35,010.00 -34,351.39 658.61
General Inputs:
Asset price when transaction is initiated 0 (necessary only if buying or selling the asset)
# of units of asset (>0 buy, < 0 sell) 0 (necessary only if buying or selling the asset)
Asset price at expiration 1520 (required for centering graph)
Lowest asset price to graph 1120 (required)
Highest asset price to graph 1904 (required)
Option-Specific Inputs: Option 1 Option 2
Choose "Call", "Put" or "No option" Call Call
Exercise Price (X) 1520 1520
Original option price 33.9000 47.6000
# of units (> 0 buy, < 0 sell) 100 -100
Time to expiration when position closed 0.0000 0.0959 Risk-free rate (rc) 4.89% 4.89%
Volatility (s) 14.00% 14.00%
Dividend yield (dc) 0.00% 0.00%
Output: Option value and profit at close of position for asset price chosen above
Value at close of position 0.0000 29.9354
Profit at close of position -3,390.00 1,766.46
Overall profit at close of position -1,623.54
Data Table for Graph Results
OPTION STRATEGY
ANALYZER
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 31 of 34
Appendix E: Profit Diagram Details for SPX Reverse Calendar Spread (Continued) Exhibit E2 – Left Breakeven Point for SPX Reverse Calendar Spread
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 32 of 34
Appendix E: Profit Diagram Details for SPX Reverse Calendar Spread (Continued) Exhibit E3 – Right Breakeven Point for SPX Reverse Calendar Spread
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 33 of 34
Appendix F: Theoretical Prices of September Call Options for SPX and OEX Exhibit F1 – Theoretical Price for SPX September 1520 Call Option
Version: 5/28/15 SCO Case Solutions (Chance-Condon-Hemler) p. 34 of 34
Appendix F: Theoretical Prices of September Call Options for SPX and OEX (Continued) Exhibit F2 – Theoretical Price for OEX September 700 Call Option
