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JACOB S. SAGIROBERT E. WHALEY*    Trading Relative Performance with Alpha Indexes                                         ...
Trading Relative Performance with Alpha Indexes       Relative performance is at the heart of investment management. Many ...
return/risk management strategies but also of introducing new return/risk managementstrategies to the investment managemen...
M t +1 − M t + DM ,t +1fashion, RM ,t +1 ≡                                                , where M t is the benchmark pri...
strategy opportunities, and these may be appealing to some segment of the investmentcommunity. We explore these first two ...
drift terms in (2) by the risk-free interest rate. This results in the following “risk-adjusted” evolution of the relative...
purchase price of the security, the value of the mimicking portfolio equals the value ofthe risk-free bonds or 100. Over t...
portfolio that has a constant proportional payout rate or “dividend yield.”5 Note that, insome cases such as when the risk...
European-style option on the relative performance index equals the value of a European-style option on the futures.8 The v...
Cb − Pb = Ib e−δ T − Xe− rT .                         (8)Substituting (7) into (8) and isolating the put value, we get    ...
to the benchmark will not be zero. Among other things, this means that extra care shouldbe taken when delta-hedging index ...
ln ( F / I )                                            σM −                                                     Tσ M     ...
39% over 2010Q4.10 With the index level at 100 and absent a bid/ask spread, an at-the-money alpha-option should sell for $...
AVSPY (i.e., the NASDAQ OMX Alpha Indexes™ that pits the performance of AAPLagainst SPY) rose by 12.9%.14 By construction,...
free bonds, illustrating that there is no limit to the loss that can be incurred from the shortbenchmark-ETF position. On ...
the call value rises at an increasing rate. This is because the benchmark price is in thedenominator of the alpha index.  ...
To illustrate yet another important difference between the call-put strategy aboveand the alpha-call, consider an unexpect...
transaction costs and is still staggeringly large. In other words, even a $1M position in at-the-money calls would require...
return volatility of AAPL is 26.7% and SPY is 17.9%, and that the expected correlationbetween AAPL and SPY returns is 0.71...
example of AAPL vs. SPY, an increase in the correlation from 0.711 to 0.8 correspondsto a decline in the value of a one-ye...
valuation formulae for futures and option contracts on the family of relative performanceindexes. In addition, we illustra...
ReferencesBlack, Fischer. 1976. The pricing of commodity contracts, Journal of FinancialEconomics 3, 167-179.Black, Fische...
Appendix A: Multi-asset relative performance indexes       The benchmark used in the definition of the complex of relative...
Appendix B: Risk metrics for derivatives on relative performance indexes       Under the BSM option valuation assumptions,...
Delta: Based on the futures pricing relation (B-1), the delta of the futures with respect tothe underlying relative perfor...
Ib                                     Γ F , SM = Γ F , MS = −        Δ F ,I .                                            ...
⎛ σ − bρ SM σ M   ⎞ − rT               Vegac ,σ S = I b e−δ T n ( d1 ) T ⎜ S               ⎟ + e N ( d1 ) VegaF ,σ S      ...
Table 1: Simulation of replicating portfolio for the b = 1 relative performance index. At the beginning of each day, the  ...
Table 2: List of NASDAQ OMX Alpha Indexes™ with option contracts pending approval of the SEC. All of the indexes listed ar...
Table 3: Transaction costs and tracking error for delta-hedging portfolios of positions in three-month futures and options...
FFigure 1: Simul lated expiration value of call op  ption on AVSPY index. The ca option has an exercise price o 100 and th...
FFigure 2: Simulaated expiration value of portfolio consisting of lo call option o AAPL and sho call option on SPY. Both c...
FFigure 3: Simulaated expiration value of portfolio consisting of lo call option o AAPL and lo put option on SPY. Both cal...
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Alpha Index Options Explained. These can be used to efficiently convert concentrated employee stock or options positions to a diversified portfolio

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Alpha Index Options Explained. These can be used to efficiently convert concentrated employee stock or options positions to a diversified portfolio by

Jacob Sagi and Robert Whaley

John Olagues
www.truthinoptions.net
olagues@gmail.com
504-875-4825
http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470471921.html

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Transcript of "Alpha Index Options Explained. These can be used to efficiently convert concentrated employee stock or options positions to a diversified portfolio"

  1. 1. JACOB S. SAGIROBERT E. WHALEY* Trading Relative Performance with Alpha Indexes  AbstractRelative performance is at the heart of investment management. Stock-picking refers tothe practice of attempting to profit from buying stocks that are under- or over-pricedrelative to the market. Market-timing refers to the practice of attempting to profit fromthe performance of one asset category versus another. While relative performance iscentral to investment management, however, complex trading strategies must be devisedto capture potential gains because relative performance cannot be traded directly. Thepurpose of this paper is to introduce a platform for trading the relative performance ofvarious securities. Specifically, we describe a class of relative performance indexes thatoffer an attractive payoff structures. We then provide a valuation framework for futuresand option contracts written on such indexes, and illustrate a variety of ways in whichrelative performance index products can be a more efficient and cost-effective means ofrealizing investment objectives than are traditional futures and options markets. Current draft: February 7, 2011*Corresponding author. The Owen Graduate School of Management, Vanderbilt University, 401 21stAvenue South, Nashville, TN 37203, Telephone: 615-343-7747, Email: whaley@vanderbilt.edu. Theauthors are grateful for the financial support from NASDAQ OMX and for comments by Dan Carrigan,Paul Jiganti, Eric Noll, Mark Rubinstein, and Walt Smith. Electronic copy available at: http://ssrn.com/abstract=1692738
  2. 2. Trading Relative Performance with Alpha Indexes Relative performance is at the heart of investment management. Many stockportfolio managers, for example, focus on identifying under- and over-priced stocks withthe hope of “beating the market.” Commonly referred to as “stock-pickers,” theseindividuals take long and short positions in stocks based on their firm-specific analysesand price predictions. Other stock portfolio managers operate globally and focus onidentifying under- and over-priced stock markets. These managers are also stock pickers,but of country-specific rather than firm-specific performance. Large institutionalinvestors, such as pension fund managers and university endowments, spread fund wealthacross many asset categories like stocks, bonds, and real estate. They constantly monitorthe relative performance of the different asset categories in the ongoing decision-makingregarding the allocation of fund wealth. As these examples illustrate, investment management pits the performance ofindividual securities and security portfolios both domestically and internationally againstone another or some benchmark. Relative performance is the overarching theme.Consequently, it is surprising that relative performance has yet to be actively tradeddirectly. If a stock-picker believes that a particular stock will outperform the market,he/she will buy the stock and sell the market using index products such as exchange-traded funds or index futures. But, long stock/short market is only one payoff structureand such a position can entail unlimited downside. Suppose, for instance, that the stock-picker prefers a call-like payoff structure on the relative performance or otherwise wishesto limit the downside. In this case, the investor could buy a call on the stock and a put onthe market, however, this entails paying unnecessarily for the market volatility embeddedin the call and put option premiums. To avoid this, the investor would have to takedynamically shifting positions in the stock and a market ETF (or in their respectivederivative products). For the typical institutional or retail investor, constantly migratingfunds from one asset category to another in response to a change in expected performanceis cumbersome and costly. Exchange-traded products on relative performance wouldseem to provide a simple and cost-effective means not only of handling existing 1   Electronic copy available at: http://ssrn.com/abstract=1692738
  3. 3. return/risk management strategies but also of introducing new return/risk managementstrategies to the investment management arsenal. Currently, the NASDAQ OMX computes and disseminates on a real-time basisnineteen indexes that measure the relative total return of a single stock (“TargetComponent”) against the SPDR ETF (“Benchmark Component”).1 Pending Securitiesand Exchange Commission approval, options on these nineteen alpha indexes will belisted on the NASDAQ OMX PHLXSM within the next few months. The purpose of thispaper is to provide an academic analysis of a complex of relative performance indexesand associated derivatives (futures and options) which includes the new NASDAQ OMXAlpha Indexes™. The paper has four main sections. In the first, we supply the mechanicsfor calculating the underlying “relative performance index” of a target security versus abenchmark security. In the second, we describe how futures and option contracts writtenon relative performance indexes might be structured and how these contracts can bevalued. The third section provides a set of scenarios in which relative performance indexderivatives are shown to be a more cost effective means for trading relative performancethan are traditional futures and options markets. The fourth section summarizes the keyresults of the paper. I. Relative Performance Indexes A relative performance index is defined as an index that measures the total returnperformance of a target security relative to the adjusted total return performance of abenchmark like the S&P 500. The total daily security return includes both price St +1 − St + DS ,t +1appreciation and dividends and is defined as RS ,t +1 ≡ , where St is the Sttarget security price at the end of day t, and DS ,t is the dividend (or other distribution)paid by the security during day t. The total daily benchmark return is defined in a similar                                                            1 See http://www.nasdaqtrader.com/TraderNews.aspx?id=fpnews2010-044 andhttp://www.nasdaqtrader.com/Micro.aspx?id=Alpha. 2  
  4. 4. M t +1 − M t + DM ,t +1fashion, RM ,t +1 ≡ , where M t is the benchmark price level at the end of Mtday t, and DM ,t is the dividend paid by the benchmark during day t. A family of relativeperformance indexes is defined by the updating rule: I b ,t +1 = I b ,t × (1 + R ) S ,t +1 (1) (1 + R ) b M ,t +1where b is a relative risk-adjustment coefficient and can be set equal to any value toreflect the security’s systematic variation with the benchmark.2 Where b = 0 , the relativeperformance index is a total return index. Where b = 1 , the right hand side of (1)corresponds to the ratio of the target and benchmark returns, and the relative performanceindex is an outperformance index.3 The family of relative performance indexes (1) has several noteworthy features.First, the performance measure is based on dividends as well as price appreciation inorder to put all stocks on an equal footing. AAPL, for example, does not pay dividends.Comparing its price appreciation to that of, say, IBM (which often distributes a dividendyield of 2% or more) unfairly handicaps IBM in a performance comparison. Second, theindex, like the value of an actual portfolio, is always positive. Technically, the indexrepresents the value of a portfolio that is continuously rebalanced such that, for everydollar long in security S, the portfolio is short b dollars in security M. Such a portfoliocannot be synthesized by a buy-and-hold strategy and most of the investment communitywould avoid mimicking the index due to excessive trading costs and/or tracking error.Futures and option contracts on relative performance indexes would create buy-and-hold                                                            2 The value of b can be set to the security’s price elasticity or beta with respect to the benchmark. The dailyupdating rule is used for illustration purposes only. In practice the index will be updated continuallythroughout the trading day.3 Note that this measure of outperformance is relative performance. The more usual definition ofoutperformance is the degree to which the price on the target exceeds the price of the benchmark, orabsolute performance (see, for example, Margrabe (1978) or Fischer (1978)). Rubinstein (1991) alsofocuses on the valuation of absolute performance options. More closely related is the work Reiner (1992)who values foreign equity options struck in a domestic currency. This is a special case of (1) where the riskadjustment coefficient is set equal to one (i.e., the exchange rate is acting like 1/benchmark) and only priceappreciation is considered.  3  
  5. 5. strategy opportunities, and these may be appealing to some segment of the investmentcommunity. We explore these first two issues more deeply in Section III. Third, the ratioof the levels of the index at two different points in time is easily interpreted, particularlyin the case where b = 1 . If the current level of the index is 150 and its level three monthsago was 120, the target security outperformed the benchmark by 25%. Finally, relativeperformance indexes can readily be extended to multi-asset benchmarks. An asset pricingpurist, for example, may want to benchmark target security performance to a benchmarkthat includes a number of asset classes such as stocks, bonds, real estate, andcommodities. In this case, the returns of asset categories would be included in thedenominator of (1), effectively assigning each category its own relative risk adjustmentcoefficient. Appendix A provides the multi-factor version of (1). II. Futures and Options on Relative Performance Indexes Valuation equations for futures and option contracts on relative performanceindexes can be derived analytically under the Black-Scholes (1973)/Merton (1973)(hereafter, “BSM”) valuation assumptions. Specifically, we assume that markets arefrictionless (e.g., no trading costs or different tax rates on different forms of income) andthat market participants can borrow or lend risklessly at a constant annualized interestrate r. We also assume now that the total return on the target security and the benchmarksecurity evolve as multivariate geometric Brownian motion with constant drifts,μS and μM , volatilities, σ S and σ M , and instantaneous return correlation, ρSM . Underthese assumptions, a relative performance index with constant relative risk-adjustmentcoefficient b will evolve as ⎛⎛ bσ 2 − σ s2 ⎞ ⎞ I b ,t = I b ,0 exp ⎜ ⎜ μ S − bμ M + m ⎟ t + σ S BS ,t − bσ M BM ,t ⎟ , (2) ⎝⎝ 2 ⎠ ⎠where BS,t and BS,t are the Brownian motion variables associated with securities S and M,respectively. Calculating the present value of payoff derivatives on I b,t requiresdiscounting for risk. To do so and avoid arbitrage in the BSM framework, we replace the 4  
  6. 6. drift terms in (2) by the risk-free interest rate. This results in the following “risk-adjusted” evolution of the relative performance index: ⎛⎛ bσ m − σ s2 ⎞ 2 ⎞ I b ,t = I b ,0 exp ⎜ ⎜ (1 − b ) r + ⎟ t + σ S BS ,t − bσ M BM ,t ⎟ . (3) ⎝⎝ 2 ⎠ ⎠ With the relative performance index dynamics in hand, we now turn to thevaluation of futures and option contracts. We assume that futures and options on theindex expire at the same point in time and are settled in cash. To avoid the complicationsof early exercise, we consider only European-style index options.A. The relative performance index as a dynamically rebalanced portfolio Earlier we mentioned that the relative performance index represents the value of aportfolio that is continuously rebalanced such that, for every dollar long in security S, theportfolio is short b dollars in security M. While most of the investment community wouldavoid mimicking the index because of excessive trading costs and tracking error, it isuseful to see the mechanics of such a trading strategy in order to develop a better intuitionfor how relative performance indexes help to complete the market. Table 1 contains a simple example of the dynamic rebalancing rule in the case b =1. In the illustration, the total return indexes for the security and the benchmark as well asthe relative performance index start at a level of 100 on day 0. The subsequent levels forthe security and the benchmark are set arbitrarily. Note that, at any point in time, therelative performance index equals 100 times the total return index of the security dividedby the total return index of the benchmark. Over the first day, both the security and thebenchmark advanced—the security by 4.17% and the benchmark by 7.16%. Because thebenchmark return was larger, the relative performance index fell. The objective of themimicking portfolio is to match the dollar gain of the relative performance index. On day1, the dollar gain on the index is –2.79. The mimicking portfolio has three constituent securities. On day 0, the portfolio islong 100 dollars of the security, short 100 dollars of the benchmark, and long 100 dollarsin risk-free bonds. Because the sales proceeds from the benchmark exactly offset the 5  
  7. 7. purchase price of the security, the value of the mimicking portfolio equals the value ofthe risk-free bonds or 100. Over the first day, the bond position produces 0.070 in interestincome,4 the security position produces a gain (i.e., price appreciation and dividends) of4.170, and the benchmark position produces a loss of 7.160. The net gain across the threepositions is –2.920. To bring the mimicking portfolio value to the level of the relativeperformance index, an additional investment of 0.130 is made in risk-free bonds (i.e., themimicking portfolio has a negative payout or dividend). The mimicking portfolio is thenrebalanced. The long position in the security is reduced from 104.17 to 97.21, generatinga gain of 6.96. The short position in the benchmark is, likewise, reduced to the samedollar value, producing a loss of 9.95. Subtracting the difference, 2.99, from the availablerisk-free funds, 100.20, the value of the risk-free bonds in the mimicking portfoliobecomes 97.21, exactly the level of the relative performance index. On day 2, the interest income from the investment of 97.21 in risk-free bonds is0.068, bringing the balance to 97.278. The long position in the security rises in valuefrom 97.21 to 97.21(108.61/104.17 ) = 101.353 for a gain of 4.143, and the short positionin the benchmark rises in value from 97.21 to 97.21(111.53 /107.16 ) = 101.174 for a lossof 3.964. To bring the value of the mimicking portfolio into line with the relativeperformance index level, 0.075 is paid out, bringing the risk-free bond balance to 97.203.The mimicking portfolio is then rebalanced. The long position in the security goes from101.353 to the new index level 97.38, and the short position in the benchmark goes from101.174 to 97.38. Adding the difference, 0.179, to the value of the risk-free bonds,97.203, the new risk-free bond balance settles, and not coincidently, at the level of therelative performance index, 97.38. Under the BSM assumptions and continuous (instead of daily) rebalancing, it canbe shown that the payout is a constant proportion of the index level,δ = r − (σ M − ρ SM σ Sσ M ) . In other words, the relative performance index is the value of a 2                                                            4 For illustrative purposes only, the interest income is based on a simple rate of 7 basis points per day. 6  
  8. 8. portfolio that has a constant proportional payout rate or “dividend yield.”5 Note that, insome cases such as when the risk-free interest rate is low or the correlation is low, thedividend yield can be negative. In most cases, however, it will be positive. Suppose, forexample, that the total return volatilities of AAPL and SPY are 26.7% and 17.9%respectively, and that the correlation between the two is 71.1%.6 If the risk-free interestrate is 2% the portfolio that mimics the b = 1 AAPL vs. SPY index would pay a constantcontinuous dividend yield of 2.19%.B. Valuation of relative performance index futures The value of a futures contract written on a relative performance index can bederived from calculating the risk-adjusted expected value of the index at the futuresexpiration, that is, Fb = Ib e( r −δ )T , (4)where T is the time remaining to expiration of the futures and bσ Mδ = br − 2 ( (1 + b ) σ M − 2 ρ SM σ S ) . The term δ can be viewed as the generalization ofthe payout rate in the previous subsection to the case where b can be different from 1.Note that (4) is the usual cost of carry relation for a stock with a payout yield of δ .7 Thepayout rate δ depends on the correlation between the stock and the benchmark. Thisfeature of the underlying relative performance index would allow one to infer from indexfutures prices information about the correlation between the target and benchmarkreturns.C. Valuation of relative performance index options Under the valuation assumptions listed at the beginning of this section, thesimplest way to value European-style options on relative performance indexes is to firstapply Black’s (1976) futures option formula. Because the futures price and relativeperformance index level are the same at futures/option expiration, the value of a                                                            5 Merton (1973) was the first to value securities using the constant proportional dividend yield assumption.6 The figures correspond to daily returns for the calendar year 2010 used in generating Table 2.7 For a development of the cost of carry relation, see Whaley (2006, pp. 125-127). 7  
  9. 9. European-style option on the relative performance index equals the value of a European-style option on the futures.8 The value of a European-style call option on a relativeperformance index futures is Cb = e − rT ⎡ Fb N ( d1 ) − XN ( d 2 ) ⎤ ⎣ ⎦ (5)where X is the exercise price of the option, N ( d ) is the cumulative normal densityfunction with upper integral limit d, and the upper integral limits are ln ( Fb / X ) + .5σ 2T d1 = and d 2 = d1 − σ T . σ TBecause the underlying source of uncertainty is the ratio of two lognormally distributedprices, the volatility rate in the expressions for d1 and d2 is σ = σ S + b2σ M − 2bρSM σ Sσ M . 2 2 (6)Then, to value a European-style call option on a relative performance index, substitute (4)into (5) as well as into the expressions for the upper integral limits d1 and d2 thataccompany (5) to get Cb = Ibe−δ T N ( d1 ) − Xe− rT N ( d2 ) , (7)where ln ( Ib / X ) + (r − δ + .5σ 2 )T d1 = , and d 2 = d1 − σ T . σ T The value of a European-style put option on a relative performance index followsstraightforwardly from put-call parity. More specifically, the payoff resulting frompurchasing a call option and selling a put option (with the same exercise price and timeremaining to expiration) equals the value of the index at expiration less the exercise price.The present value of these payoffs yields the put-call parity relation for European-styleoptions:                                                            8 For a proof, see Whaley (2006, p. 198). 8  
  10. 10. Cb − Pb = Ib e−δ T − Xe− rT . (8)Substituting (7) into (8) and isolating the put value, we get Pb = e− rT XN ( −d2 ) − Ibe−δ T N ( −d1 ) . (9)D. Hedging relative performance index futures and options The valuation equations for the futures on relative performance indexes (4) andfor the options on relative performance indexes (7) and (9) allow us to develop analyticalexpressions for the metrics used in risk management (i.e., delta, gamma and vega).Appendix B contains these expressions. Several results are noteworthy. First, because theunderlying payoffs depend on changes in two distinct securities, delta-risk managementwill necessarily require a simultaneous position in both the security S and itscorresponding benchmark M. As one might suspect from the links between portfolioformation and relative performance indexes, for every dollar of security S used to hedge arelative performance index derivative, one must take a position of –b dollars in thebenchmark security M. Thus, although a relative performance index derivative might atfirst blush seem twice as complicated to hedge as a derivative product on a singleunderlying, in practice the hedging position in M is completely determined by thehedging position in S. A second noteworthy result is that, because the futures price depends on thevolatilities of S and M (as well as on the correlation between them), the futures vega isnot zero. Indeed, we must consider a new type of vega here, corresponding to price-sensitivity to changes in the correlation between S and M. This is apparent from theexpression for the futures price (4). Finally the gamma with respect to the benchmark andthe cross-gamma (i.e., the sensitivity of the benchmark delta with respect to thebenchmark and its sensitivity with respect to the stock) of the futures price are not zero.The intuition for this is as follows. The current value of the index is inverselyproportional to the cumulative performance of the benchmark. Such inverse dependenceis necessarily a convex function, which implies that the gamma of the index with respect 9  
  11. 11. to the benchmark will not be zero. Among other things, this means that extra care shouldbe taken when delta-hedging index derivatives against large benchmark movements. III. Using Relative Performance Index Products With the relative performance index product valuation mechanics in hand, wenow turn to providing a series of illustrations that show the potential benefits of theseproducts. To keep matters simple and realistic, we focus again on the case where the risk-adjustment coefficient equals one, that is, b = 1 . This case is germane because suchindexes are computed on a real-time basis and are disseminated as NASDAQ OMXAlpha Indexes™.9 Pending Securities and Exchange Commission approval, options onnineteen alpha indexes pitting the total return of an individual stock against the totalreturn performance of the SPDR ETF will listed on the NASDAQ OMX PHLXSM withinthe next few months. A list of the individual companies, together with the ticker symbolsof the stock and the alpha index, are shown in Table 2. Option products on relativeperformance indexes where b ≠ 1 are planned, as are futures contracts on alpha indexes. To begin, it is worthwhile to note that, under the assumption that b = 1, thedividend yield term in the various valuation equations is δ = r − σ M (σ M − ρ SM σ S ) . Thefutures price, therefore, can be rewritten as σ M (σ M − ρSM σ S )T F = Ie , (10)which implies that, depending upon whether σ M is greater than or less than ρ SM σ S , thefutures may trade at a premium or a discount relative to the underlying relativeperformance index. It is also worthwhile to note that, if a relative performance indexfutures is actively traded, its price implies the level of correlation between the stock andthe benchmark returns via                                                            9 Most of the discussion in this section also applies qualitatively to other cases in which b > 0 . 10  
  12. 12. ln ( F / I ) σM − Tσ M ρ SM = . (11) σSBecause the level of the relative performance index and the time remaining to theexpiration of the futures are known, and σ S and σ M can be estimated using stock optionand index option prices, the correlation is uniquely determined and, by definition, isforward-looking. Likewise, the implied beta of a security can also be inferred from alpha indexderivative prices because its definition is σS β SM ≡ ρ SM . (12) σMSuch an estimation approach may be particularly useful given that current approaches toestimating beta involve using a long time-series of past return data and assuming the betais constant over the entire time-series history. In other words, typical beta estimates areinherently backward-looking and stable through time. At the same time, finance theoryhas long recognized that the beta of a stock changes with the nature of a firm’s business,financial and operating leverage, the macro-economy, and other factors—an obviousconundrum. To demonstrate the potential value of alpha index options as a source ofinformation about correlations, consider the example of CSCO (Cisco Systems, Inc.) vs.SPY. The realized correlation between the daily returns of these assets was 66% in2010Q3 and 34% in 2010Q4. This suggests that realized correlation in one quarter maybe a poor forecast correlation in the subsequent quarter and underscores the need forforward-looking estimates of correlations. One obvious concern is whether the Alpha-option bid-ask spreads will be too large to allow for useful inference of option-impliedcorrelations. To examine this, assume it is October 1, 2010 and that the true correlationbetween CSCO and SPY over the next quarter is 34%. Suppose further that thevolatilities of CSCO and SPY are well-estimated from standard options to be 11% and 11  
  13. 13. 39% over 2010Q4.10 With the index level at 100 and absent a bid/ask spread, an at-the-money alpha-option should sell for $7.25. If bid/ask spreads provided a price range of$7.14 to $7.36 (a 3% difference), then the corresponding range in implied correlationwould be 30% to 38%.11 This is significantly more accurate than the 66% correlationestimate from the previous quarter’s realized correlation. Moreover, even if the previousquarter’s realized correlation happened to be 34%, estimation error would lead to a 90%-confidence interval of 15% to 52% for the historical correlation forecast, which is nearlyfour times larger than the interval implied by the alpha-option price.12 In summary, underreasonable assumptions for option bid/ask spreads, we expect that option-impliedcorrelations will be both forward-looking and more accurate than estimates based onhistorical time-series.13A. Efficiency gains to trading relative performance using index derivatives Absent a specific view on individual stock performance, an equities investorshould hold a well-diversified stock portfolio. With a strong view that a particular stockwill outperform the market, on the other hand, an investor may want to devote a largeportion of portfolio wealth to the individual stock rather than the market. Buying thestock directly, however, is not a “clean” way to implement the view that the stock willoutperform the market. To illustrate, consider an investor who, on April 21, 2010,believed that AAPL’s shares would outperform the market over the remaining part of thesecond quarter of the year. Buying AAPL’s shares on April 21 and holding until June 30would have produced a disappointing return of –3.0%. Does that mean the investor waswrong? The answer is no. The price of SPY ETF shares (a proxy for the stock market)fell by 14.0% over the same period. While AAPL outperformed the market as theinvestor expected, it also declined with the rest of the market. Over the same period,                                                            10 We are using the actual realized volatilities in 2010Q4.11   Standard options on CSCO and SPY feature a bid/ask spread of roughly 1.5%, half of the spread weassume in the example. 12 To arrive at this confidence interval we employ a Fisher transformation and the fact that there were 64days in 2010Q3.13 Implied correlations and betas, by definition, apply to the life of the alpha option. Thus one can deduce aterm structure of implied correlations/betas from alpha index options of varying maturities, correspondingto the market’s forecast of how a firm’s systematic risk is forecasted to change over time. 12  
  14. 14. AVSPY (i.e., the NASDAQ OMX Alpha Indexes™ that pits the performance of AAPLagainst SPY) rose by 12.9%.14 By construction, the relative performance index is lessexposed to events that move the entire market. We now turn to comparing how alpha index futures and option values change inreaction to changes in relative performance with those of alternative buy-and-holdstrategies that use existing exchange-traded products. Our aim is to highlight thedifferences, thereby helping to point out how these new instruments help to “complete themarket” for investors interested in relative performance. Long stock/short benchmark: Consider an investor who has 100 dollars toinvest and wants to speculate that the price of a particular stock will rise relative to abenchmark. One possible trading strategy that uses currently traded securities is to buy$100-worth of stock, financing its purchase by selling an equal dollar amount of thebenchmark ETF.15 Assuming the 100 dollars is invested in risk-free bonds, the overallvalue of this three-security, passive position is 100. If alpha index futures (hereafter,“alpha-futures”) were also traded, the investor could also form a similarly-purposed,passive strategy by buying 100 dollars of risk-free bonds and buying an equal dollaramount of alpha-futures.16 Over a very short horizon, the benefits of both of thesestrategies are the same. Over longer periods of time, however, the passive longstock/short ETF position can become unbalanced, exposing the investor to moredownside risk. Suppose, for example, that the stock, the benchmark ETF, and the alphaindex are priced at 100 at the beginning of the investment horizon. The long stock/shortbenchmark-ETF position has a value of 100 (i.e., the risk-free bonds), as does the fullycollateralized alpha-futures position. Now, suppose that, over the investment horizon, thestock falls to 50, and the benchmark rises to 200. The gain on the long stock/shortbenchmark-ETF position is –150, while the gain on the alpha-futures is –75. The totalvalue of the long/short strategy is negative because the loss exceeds the value of the risk-                                                            14 Neither AAPL nor SPY paid dividends during the period.15 This assumes, of course, that the investor can short the benchmark ETF at low cost and largely maintainfull use of the cash proceeds.16 This type of position involving an exact dollar futures overlaid on risk-free bonds is called a fully-collateralized futures position. 13  
  15. 15. free bonds, illustrating that there is no limit to the loss that can be incurred from the shortbenchmark-ETF position. On the other hand, the value of the fully collateralized alpha-futures position is 25, well above its minimum level of 0. Long stock-call, short benchmark-call: Oftentimes investors prefer to useoption-like structures in their trading strategies. To capture relative performance, aninvestor could buy an at-the-money call option on the stock and sell an at-the-money calloption on the benchmark. Using the illustration above, the long stock-call would expireworthless at the end of the investment horizon since the stock price, 50, sank below theexercise price, 100. On other hand, the benchmark call is 100 dollars in the money atexpiration. Since the investor is short the call, he loses 100. As an alternative strategy, theinvestor could choose to buy an at-the-money call on the alpha index (hereafter, “alpha-call”). In the example, at the end of the call option’s life the index is at 25 and the callexpires worthless. Obviously, the alpha-call has less potential downside exposure. To examine the potential upside of the alpha-call, we reverse the price movementsby letting the stock price go from 100 to 200 and the benchmark to go from 100 to 33.33.The long stock-call/short benchmark-call strategy would have a payoff of 100 atexpiration because the stock-call is 100 in the money and the benchmark call isworthless. On the other hand, the alpha index rises to a level of 600 leaving the alpha-call500 in the money. Obviously, the alpha-call has greater upside potential. Naturally, the decision about which strategy to use must be based on investorpreferences and other portfolio considerations. Our purpose is to illustrate how a staticposition in an alpha-call differs from a position in existing instruments. To this end,Figure 1 shows the value of the alpha-call as a function of the stock price and thebenchmark price at expiration. The call option has an exercise price of 100 and threemonths remaining to expiration. The volatility rates of the stock (AAPL) and thebenchmark (SPY) are 26.7% and 17.9%, respectively. The correlation between thereturns is 0.711. The risk-free interest rate is 2%. Note that, as the stock price falls andthe benchmark price rises, the call goes to its lowest possible value of 0. On the otherhand, as the stock price rises, the call value rises, however, as the benchmark price falls, 14  
  16. 16. the call value rises at an increasing rate. This is because the benchmark price is in thedenominator of the alpha index. Figure 2 shows the value surface of the long stock-call/short benchmark-callportfolio. The value of this portfolio position is scaled to match the value of the at-the-money alpha-call option to ensure equal dollar investment. In general, the scale factor isgreater than one because the proceeds from the sale of the benchmark-call are used tooffset the purchase price of the stock-call. Like the alpha-call, the value of the long stock-call/short benchmark-call portfolio falls in value as the stock price falls and/or thebenchmark price rises. Unlike the alpha-call, however, the portfolio value may becomenegative. Moreover, as the stock price rises and the benchmark price falls, the portfoliovalue rises, albeit not to the same levels as the alpha-call. Long stock-call/long benchmark-put: Another option strategy that is designedto capture relative performance is to buy an at-the-money stock-call and buy an at-the-money benchmark-put. Returning to our illustration, suppose the stock price falls from100 to 50 and the index level rises from 100 to 200. The long stock-call expires out of themoney, as does the long benchmark put. Hence, the “straddle” expires worthless, similarto the alpha-call. If the reverse happens and the stock price rises to 200 and thebenchmark falls to 50, the stock-call is 100 in the money, the put is 50 in the money, andthe straddle value is 150. The alpha-call, on the other hand, has a value of 300. Again, the decision about which strategy to use is based on investor preferences.Figure 3 shows the value surface of the long stock-call/long benchmark-put portfolio atexpiration which can be compared with the corresponding alpha-call surface in Figure 1.Here too the call-put position is scaled so that its initial value matches that of the alpha-call. In general, the scale factor is less than one because two options are being purchased,and the stock-call alone has a higher premium than the alpha-call. Like the alpha-call, thevalue of the long stock-call/long benchmark-put portfolio falls in value as the stock pricefalls and/or the benchmark price rises. And, also like the alpha-call, the call-put positionnever falls below 0. On the other hand, as the stock price rises and the benchmark pricefalls, the call-put position value does not rise to the same levels as the alpha-call. 15  
  17. 17. To illustrate yet another important difference between the call-put strategy aboveand the alpha-call, consider an unexpected change to market volatility. Roughly, the call-put position value will be highly affected because both the call and the put are exposed tomarket risk and move in the same direction in reaction to news about market volatility.By contrast, the alpha-call value will be less affected for most stocks.17 The three strategy comparisons provided above illustrate the distinctive featuresof alpha index derivatives. These products can simultaneously provide downsideprotection while reducing exposure to market volatility. While existing exchange-tradedproducts can be combined to create a passive, directional bet on relative performance,they cannot be combined to create a passive, directional bet on the alpha index or itsderivatives products. To do so would require dynamic trading strategies, and dynamicstrategies are prohibitively expensive, at least for retail investors. To get a rough sense for the trading costs in replicating alpha-products, weperform a set of simulations. In the simulations, the investor is assumed to pay aproportional cost of $0.01 per share, a fixed cost of $1.00 for trading any quantity of thestock, benchmark, or bond, and an additional 0.20% net fee for a short sale.18 The tradingstrategy aims to replicate a $1,000 investment in a three-month, at-the-money alpha-calloption on the b = 1 AAPL vs. SPY index using standard delta-hedging techniques.19 Thevolatility and correlation parameters are drawn from Table 2. The simulation results are quite striking. In order to achieve a tracking standarderror no larger than about 5%, the average trading costs are around 110% of the value ofthe position! The reason is that hedging an at-the-money call requires frequentrebalancing (about 300 times during the three months to achieve the 5% standard trackingerror). Each time the portfolio is rebalanced the investor must pay $1 per trade, or $3 intotal to trade in risk-free bonds, AAPL, and SPY, amounting to about $900 over the lifeof the option. The remaining $200 in average costs is generated by the proportional                                                            17 In particular, if the stock’s beta is fixed at one then the alpha-call value only depends on the stock’sidiosyncratic volatility.18 We assume that the proportional costs for trading the short-term bond are negligible. Our costassumptions are conservative and understate the true trading costs currently faced by retail investors.19 Specific details regarding the delta-hedge simulation are available from the authors upon request. 16  
  18. 18. transaction costs and is still staggeringly large. In other words, even a $1M position in at-the-money calls would require an expenditure of roughly 20% of the portfolio value intrading costs if the standard tracking error is to be kept below 5%. Table 3 documents simulation results of tracking errors and trading costs forvarious three-month AAPL vs. SPY alpha-call options. The initial investment is assumedto be $10,000, and the replicating portfolio is assumed to be rebalanced daily (90 tradingperiods after the initial portfolio setup) over the life of the option. Note that the at-the-money alpha-call now has a tracking error of 9% as opposed to the 5% in the earlierillustration. This is because the replicating portfolio is rebalanced only 90 times asopposed to 300. Out-of-the-money calls are the most difficult and costly to replicate. Thereason is that the high leverage implicit in such options requires large positions in thestock and benchmark relative to the option value. Even small fluctuations in theunderlying prices can greatly unbalance the portfolio. In addition, the large tradesrequired for rebalancing entail higher costs. The 10% out-of-the-money alpha-call has anaverage tracking error of 34% and an average total trading cost of 36% of the option’svalue. Overall, the lesson from this exercise is that, while forward/futures positions on analpha index might be relatively inexpensive to implement through dynamic trading,replicating alpha-options is not.B. Relative performance index options are generally less expensive The fact that relative performance index products are less exposed to market riskmeans that relative performance index options are less expensive than standard options.The reason is simple. Stock options are based on the total risk of the stock (i.e., the sumof market risk and idiosyncratic risk), while relative performance index options areessentially based on the difference in risk between the stock and the benchmark. Becauseoption value varies directly with volatility, relative performance index options arecheaper. To illustrate, return once more to the AAPL vs. SPY example. Suppose aninvestor decides to buy an at-the-money call option with an exercise price of 100 andthree months remaining to expiration. The investor estimates that the expected future 17  
  19. 19. return volatility of AAPL is 26.7% and SPY is 17.9%, and that the expected correlationbetween AAPL and SPY returns is 0.711. The risk-free interest is 2%. Under the BSMassumptions, the value of an at-the-money call option on AAPL is $11.53. At the sametime, the value of an at-the-money call option on AVSPY is $7.24, corresponding to adiscount of 37%. The volatility used in the valuation of the relative performance indexoption, given by equation (6), is 18.8%. As long as the correlation between the assets ispositive and σ S > σ M , the volatility of the index (i.e., σ ) is guaranteed to be smaller thanthe volatility of the stock (i.e., σ S ), and the price an alpha-call will likewise beguaranteed to be cheaper than the price on a target-call.C. Relative performance index products are based on total return Earlier, we argued that in order to make a fair comparison between theperformances of two securities, one should consider their total return, which includesincome distribution as well as price appreciation. There are other advantages to usingtotal returns in measuring performance. Standard exchange-traded stock futures andoption contracts are based only on the price appreciation of the underlying stock. Becausea relative performance index futures incorporates dividends into the performancecalculation, its payoffs implicitly include the automatic reinvestment of security incomewithout incurring transaction costs. It also means that relative performance index optionsare not susceptible to the trading games played in the stock option market when deep in-the-money options remain unexercised.20D. Trading correlation According to the valuation equations derived in Section II, the values of relativeperformance index futures and options depend on the correlation between the security Sand the benchmark M. A higher correlation implies lower index futures and call optionsprices. A trader who believes correlations will increase (decrease) can sell (buy) indexfutures or sell (buy) index call options to benefit from his or her insights. With the earlier                                                            20 Pool, Stoll, and Whaley (2008), for example, show that many long call option holders unwittingly fail toexercise outstanding call option positions on stocks when it is optimal to do so (just prior to the ex-dividendday) and, as a result, forfeit the ex-dividend call option price drop to market makers and proprietary firms.  18  
  20. 20. example of AAPL vs. SPY, an increase in the correlation from 0.711 to 0.8 correspondsto a decline in the value of a one-year at-the-money call option from $7.24 to $6.08. Thetrader could sell the option and hope that the new information would be reflected inprices soon after (at which point he or she would close out the position for a profit). Therisk here is that the index will simultaneously move higher, cancelling the decline in theoption value. To make a cleaner investment, the trader can hedge the written option usingthe deltas calculated in Appendix B and using a correlation value of 0.711 to calculate thehedging positions. The net position will be insensitive to changes in the index itself, butonce the new correlation value of 0.8 is absorbed by the market, the option value willsink below that of the hedging portfolio and the trader can close out the portfolio at aprofit.21 The opposite strategy can be employed if the trader believes correlations willdecline. This illustrates how investors can trade on their views concerning correlationsbetween the index components. In general, the construction of a portfolio that moves only with the correlationbetween two assets requires model-calculated positions in relative performance indexderivatives and the index constituents (and potentially their derivatives). What isimportant to emphasize is that such trades could not be contemplated without tradedindex derivatives (just as one could not trade volatility without traded stock derivatives).Thus, just as stock options have enabled markets to trade stock volatility, relativeperformance index products have the potential of opening up an entirely new market fortrading correlations. IV. Summary The purpose of this paper is to introduce a new complex of relative performanceindexes, tracking the performance of a target security versus that of a benchmarksecurity. In particular, we assess how derivatives (options and futures) on these indexesmight provide investors with new opportunities to trade relative performance. We provide                                                            21 This assumes that volatilities are constant. If this is not the case, then the trader would have to accountfor changing volatility in constructing the hedging portfolio. 19  
  21. 21. valuation formulae for futures and option contracts on the family of relative performanceindexes. In addition, we illustrate a variety of ways in which the new index productscould be a more efficient and cost-effective means of realizing certain investmentobjectives than are traditional futures and options markets. The new indexes essentially track a portfolio that is long on a target stock andshort on a benchmark asset, such that the ratio of the two dollar positions is constant. Asthe target asset outperforms its benchmark, the equivalent long-short position increases.Likewise, if the target asset underperforms its benchmark over a period, then theequivalent portfolio long-short position is decreased. This behavior ensures downsideprotection which could only otherwise be implemented through dynamic trading – anexercise typically beyond the reach of most investors. The introduction of options on the new indexes poses several additional benefits.We show that such options are generally cheaper than options on the target security. Wealso show that they offer a different payoff structure than related positions using standardcalls and puts on the underlying target and benchmark securities. In particular, the newindex call options simultaneously have downside protection and an accelerated upsiderelative to static positions in existing instruments. At the same time, such options wouldbe creating investment opportunities otherwise only available to typical investors at greatcosts. Finally, an exciting aspect of the new index options is that they present investorswith the ability to trade the correlation between the target and benchmark securities.Indeed, we show that option implied correlations (which could be used to calculate optionimplied target CAPM betas if the benchmark is a market index) are both forward-lookingand potentially more informative about future correlations than historical estimates. 20  
  22. 22. ReferencesBlack, Fischer. 1976. The pricing of commodity contracts, Journal of FinancialEconomics 3, 167-179.Black, Fischer and Myron Scholes. 1973. The pricing of options and corporate liabilities,Journal of Political Economy 81, 637-659.Fischer, Stanley. 1978. Call option pricing when the exercise price is uncertain and thevaluation of index bonds, Journal of Finance 33, 169-176.Margrabe, William. 1978. The value of an option to exchange one asset for another,Journal of Finance 33, 177-186.Merton, Robert C. 1973. Theory of rational option pricing, Bell Journal of Economicsand Management Science 1, 141-183.Pool, Veronika, Hans R. Stoll, and Robert E. Whaley. 2008. Failure to exercise calloptions: An anomaly and a trading game, Journal of Financial Markets 11, 1-35.Reiner, Eric. 1992. Quanto mechanics, RISK 5, 59-63.Rubinstein, Mark. 1991. One for another, RISK 4, 30-32.Whaley, Robert E. 2006. Derivatives: Markets, Valuation, and Risk Management, JohnWiley & Sons, Inc.: Hoboken, New Jersey.  21  
  23. 23. Appendix A: Multi-asset relative performance indexes The benchmark used in the definition of the complex of relative performanceindexes in (1) can be extended to include multiple-asset benchmarks. Suppose, forexample, the desired benchmark has n different asset classes (i.e., stocks, bonds, realestate, and commodities) and each asset class constitutes w% of the overall benchmark nportfolio, where, of course, ∑ w = 1 . Under these assumptions, the updating rule for the i =1 imulti-asset relative performance index is I b , w,t +1 = I b , w,t × (1 + R ) S ,t +1 . n ∏ (1 + R ) wi b i ,t +1 i =1where b is as before and w is a vector of benchmark asset allocation weights (i.e.,wi , i = 1,..., n ) . With a relative performance so defined, a change in the log-indexcorresponds to the difference between the instantaneous performance of security S versusthe weighted and scaled instantaneous performances of the benchmark assets, that is, n ln I b , w ,t +1 − ln I b , w ,t = ln (1 + RS ,t +1 ) − b ∑ wi ln (1 + Ri ,t +1 ) i =1The futures and option valuation equations on these multi-factor benchmarks are alsoanalytically tractable and available from the authors upon request. 22  
  24. 24. Appendix B: Risk metrics for derivatives on relative performance indexes Under the BSM option valuation assumptions, we showed that the value of afutures contract written on a relative performance index with a constant relative risk-adjustment coefficient of b is Fb = Ib e( r −δ )T , (B-1)where Fb and Ib are the futures price and the relative performance index level, r is theannualized risk-free interest rate, T is the futures time remaining to expiration in years, bσ Mδ = br − 2 ( (1 + b ) σ M − 2 ρ SM σ S ) , σ S and σ M are the volatility rates of the securityand the benchmark, and ρSM is the correlation between the returns of the security and thebenchmark. We also showed that the values of the European-style call and put options onthe relative performance index are Cb = Ibe−δ T N ( d1 ) − Xe− rT N ( d2 ) , (B-2)and Pb = e− rT XN ( −d2 ) − Ibe−δ T N ( −d1 ) . (B-3)where X is the exercise price of the option, T is the time remaining to option expiration(i.e., same time as futures), N ( d ) is the cumulative normal density function with upperintegral limit d, σ = σ S + b2σ M − 2bρSM σ Sσ M , and the upper integral limits are 2 2 ln ( Ib / X ) + (r − δ + .5σ 2 )T d1 = , and d 2 = d1 − σ T . σ TThe put-call parity relation for European-style options is Cb − Pb = Ib e−δ T − Xe− rT . (B-4)Based on these equations, the risk metrics (i.e., delta, gamma and vega) of the futures andoptions are as follows. 23  
  25. 25. Delta: Based on the futures pricing relation (B-1), the delta of the futures with respect tothe underlying relative performance index is Δ F , I = e( r −δ )T .The deltas with respect to the prices of the security and the benchmark are I b ( r −δ )T I r −δ T Δ F ,S = e and Δ F , M = − b e( ) . S M Based upon the valuation equations (B-2) and (B-3), the deltas of the European-style call and put options with respect to a change in the underlying relative performanceindex are Δc, I = e−δ T N ( d1 ) and Δ p, I = −e−δ T N ( −d1 ) .Since the alpha options may be hedged using shares of the security or the benchmark, thedeltas with respect to the per share security price, S, are Ib I Δc,S = Δ c , I and Δ p , S = − b Δ p , I S Sand the deltas with respect to the per share benchmark price, M, are Ib I Δc,M = − Δ c , I and Δ p , M = b Δ p , I . M MGamma: (B-5) shows that the delta is not a function of the relative performance index level, so thegamma of a futures with respect to the index is 0. This is also true of the gamma withrespect to the security, S. The gamma with respect to the benchmark price is not zero,however, and is given by I b ( r −δ )T Γ F , MM = 2 e . M2The cross gammas are 24  
  26. 26. Ib Γ F , SM = Γ F , MS = − Δ F ,I . SMThe gamma of a call option with respect to the underlying index is the same as that forthe put option: n ( d1 ) Γ I = Γ c , I = Γ p , I = e −δ T , I bσ T 1 − d12 / 2where n ( d1 ) = e is the normal density function evaluated at d1 . The gammas of 2πthe call option with respect to the underlying security and benchmark prices are given by I b2 I ( I Γ + 2Δ I ) I ( I Γ + ΔI ) Γ c,S = 2 Γ I , Γc,M = b b I 2 and Γ c , SM = Γ c , MS = − b b I . S M SMBy virtue of the put-call parity relation (B-4), the gammas of the European-style putoptions are related through the gamma of the futures price: Γ p ,S = Γc ,S , Γ p,M = Γc,M − e− rT ΓF ,M and Γ p ,SM = Γ p,MS = Γc,MS − e− rT Γ F ,MS .Vega: Unlike the usual cost of carry model, the futures pricing equation for a relativeperformance index is a function of volatility. The vegas with respect to σ S , σ M , and ρSMare: VegaF ,σ S = −b ρ SM σ M TFb , VegaF ,σ M = b ⎡(1 + b ) σ M − ρ SM σ S ⎤ TFb , ⎣ ⎦and VegaF , ρSM = −bσ Sσ M TFb .The vegas of the European-style call option with respect to σ S , σ M , and ρSM are 25  
  27. 27. ⎛ σ − bρ SM σ M ⎞ − rT Vegac ,σ S = I b e−δ T n ( d1 ) T ⎜ S ⎟ + e N ( d1 ) VegaF ,σ S ⎝ σ T ⎠ ⎛ b ( bσ M − ρ SM σ S ) ⎞ − rT Vegac ,σ M = I b e −δ T n ( d1 ) T ⎜ ⎟ + e N ( d1 ) VegaF ,σ M ⎝ σ T ⎠and ⎛ bσ σ ⎞ − rT Vegac , ρSM = − Ib e−δ T n ( d1 ) T ⎜ S M ⎟ + e N ( d1 ) VegaF , ρSM . ⎝ σ T ⎠Note that, by the put-call parity relation (B-4), the vegas of the call, put and futures arerelated by Vegac,i − Vega p,i = e− VegaF ,i , where i = σ S , σ M , ρSM . Hence, the vegas of the rTEuropean-style put option with respect to σ S , σ M , and ρSM are Vega p ,σ S = Vegac,σ S − e− rTVegaF ,σ S , Vega p ,σ M = Vegac,σ M − e− rTVegaF ,σ M ,and Vega p, ρSM = Vegac, ρSM − e− rTVegaF , ρSM . 26  
  28. 28. Table 1: Simulation of replicating portfolio for the b = 1 relative performance index. At the beginning of each day, the portfolio is rebalanced so that the position in cash and the target security is set equal to the level of the index, while the portfolio position in the benchmark is short an amount equal to the level of the index. The table documents the daily income from these positions, as well as the payout that results from rebalancing.    Relative Hedge portfolio Mimicking Total return indexes performance index Interest Dollar income Net portfolio Day Security Benchmark Level Gain income Security Benchmark gain Payout value 0 100 100 100 100 1 104.17 107.16 97.21 -2.79 0.070 4.170 7.160 -2.920 -0.130 97.21 2 108.61 111.53 97.38 0.17 0.068 4.143 3.964 0.247 0.075 97.38 3 110.68 110.36 100.29 2.91 0.068 1.856 -1.022 2.946 0.038 100.29 4 111.01 113.83 97.52 -2.77 0.070 0.299 3.153 -2.784 -0.017 97.52 5 112.62 110.70 101.73 4.21 0.068 1.414 -2.682 4.164 -0.048 101.73   27  
  29. 29. Table 2: List of NASDAQ OMX Alpha Indexes™ with option contracts pending approval of the SEC. All of the indexes listed areoutperformance indexes with the benchmark being the SPDR ETF (ticker symbol SPY). The annualized volatility of each stock andcorrelation with SPY are calculated using daily return data for the calendar year 2010. The data were collected from Datastream. Over thesample period, SPY daily returns had a realized annualized volatility of 17.9%.      Alpha Stock return Stock index Correlation Stock name ticker ticker Volatility with SPY return Amazon.com, Inc. AMZN ZVSPY 32.6% 60.4% Apple Inc. AAPL AVSPY 26.7% 71.1% Cisco Systems, Inc. CSCO CVSPY 31.9% 61.7% Ford Motor Company F FVSPY 38.1% 70.6% General Electric Company GE LVSPY 27.4% 82.3% Google Inc. GOOG UVSPY 27.8% 65.1% Hewlett Packard Company HPQ HVSPY 24.9% 68.0% International Business Machines IBM IVSPY 17.8% 78.3% Intel Corporation INTC JVSPY 25.3% 77.3% Coca-Cola Company KO KVSPY 15.5% 63.7% Merck & Company, Inc. MRK NVSPY 20.6% 65.7% Microsoft Corporation MSFT MVSPY 22.0% 73.0% Oracle Corporation ORCL OVSPY 24.4% 67.5% Pfizer Inc. PFE PVSPY 21.3% 66.2% Research in Motion Limited RIMM RVSPY 38.5% 44.9% AT&T Inc. T YVSPY 15.1% 70.7% Target Corporation TGT XVSPY 20.2% 66.0% Verizon Communications Inc. VZ VVSPY 16.0% 60.3% Wal-Mart Stores, Inc. WMT WVSPY 14.0% 50.3% 28  
  30. 30. Table 3: Transaction costs and tracking error for delta-hedging portfolios of positions in three-month futures and options positionswritten on the AVSPY alpha index. Following the initial setup, the hedging portfolio is rebalanced 90 times in equal intervals (“days”)subsequent to the initial investment of $10,000. Fixed costs are set at $1 per asset per trade. Proportional costs are set to $0.01 per share, withinitial share values for AAPL and SPY taken to be $340 and $130, respectively. Short selling costs are assumed to be 0.20% of the amountborrowed. The AAPL volatility rate is assumed to be 26.7%, the SPY volatility rate is 17.9%, and the correlation between the two returns is71.1%. The index is set to 100 when the derivative positions are initiated. The table reports the standard deviation of the tracking error,defined as the difference between the option value and the value of the tracking portfolio at expiry in the absence of transaction costs. Thetable also reports a breakdown of the accrued trading costs at the option’s expiration. The fixed costs can accrue to more than $273 becausecosts are capitalized. They can also be below that value for deep out-of-the-money options because certain price paths lead to essentially zerooption value well before the option’s expiration.   Tracking Average fixed Average proportional Average total Derivative standard error (%) costs ($) costs ($) costs (%) Futures 0.1% $274 $30 3% Call option X=80 0.5% $274 $160 4% X=90 2% $274 $400 7% X=100 9% $272 $1,300 16% X=110 34% $268 $3,300 36% 29 
  31. 31. FFigure 1: Simul lated expiration value of call op ption on AVSPY index. The ca option has an exercise price o 100 and three months Y all ofrremaining to exppiration. The vol latility rates of th security (AAPL) and the ben he nchmark (SPY) a 26.7% and 1 are 17.9%, respective The ely.ccorrelation betwe the returns is 0.711. The risk-fr interest rate is 2%. een 0 free s   200 150 100 Option value 50 0 -50 -100 50 50 70 60 70 90 80 90 100 110 110 120 130 130 140 Benchmark price B 150 Security price      30 
  32. 32. FFigure 2: Simulaated expiration value of portfolio consisting of lo call option o AAPL and sho call option on SPY. Both call options v o ong on ort lhhave an exercise price of 100 and three months rem maining to expiraation. The volatili rates of the se ity ecurity (AAPL) a the benchmar (SPY) and rka 26.7% and 17are 7.9%, respectivel The correlatio between the re ly. on eturns is 0.711. T risk-free inte The erest rate is 2%. P Portfolio value o at-the- ofmmoney options at origination is sca to match the value of the at-th aled he-money alpha c option. call   200 150 100 Option value 50 0 -50 -100 50 50 70 60 70 -150 90 80 90 100 110 110 120 130 130 140 150 Be enchmark price Security price 31 
  33. 33. FFigure 3: Simulaated expiration value of portfolio consisting of lo call option o AAPL and lo put option on SPY. Both call and put v ong on ong n lo onths remaining to expiration. The volatility rates o the security (AAoptions have an exercise price of 100 and three mo e 1 o of APL) and the ben nchmark((SPY) are 26.7% and 17.9%, respectively. The cor rrelation between the returns is 0.7 711. The risk-free interest rate is 2 Portfolio valu of at- e 2%. uetthe-money option at origination is scaled to match the value of the at-the-money alp call option. ns h pha   200 150 100 Option value 50 0 -50 -100 50 50 70 60 70 90 80 90 100 110 110 120 130 130 140 150 Be enchmark price Security price   32 

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