Electronic copy available at: http://ssrn.com/abstract=1692738

JACOB S. SAGI ROBERT E. WHALEY*

Trading Relative Performance with Alpha Indexes

 Abstract

Relative performance is at the heart of investment management. Stock-picking refers to the practice of attempting to profit from buying stocks that are under- or over-priced relative to the market. Market-timing refers to the practice of attempting to profit from the performance of one asset category versus another. While relative performance is central to investment management, however, complex trading strategies must be devised to capture potential gains because relative performance cannot be traded directly. The purpose of this paper is to introduce a platform for trading the relative performance of various securities. Specifically, we describe a class of relative performance indexes that offer an attractive payoff structures. We then provide a valuation framework for futures and option contracts written on such indexes, and illustrate a variety of ways in which relative performance index products can be a more efficient and cost-effective means of realizing 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 21st Avenue South, Nashville, TN 37203, Telephone: 615-343-7747, Email: whaley@vanderbilt.edu. The authors 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

1   

Trading Relative Performance with Alpha Indexes

Relative performance is at the heart of investment management. Many stock

portfolio managers, for example, focus on identifying under- and over-priced stocks with

the hope of “beating the market.” Commonly referred to as “stock-pickers,” these

individuals take long and short positions in stocks based on their firm-specific analyses

and price predictions. Other stock portfolio managers operate globally and focus on

identifying under- and over-priced stock markets. These managers are also stock pickers,

but of country-specific rather than firm-specific performance. Large institutional

investors, such as pension fund managers and university endowments, spread fund wealth

across many asset categories like stocks, bonds, and real estate. They constantly monitor

the relative performance of the different asset categories in the ongoing decision-making

regarding the allocation of fund wealth.

As these examples illustrate, investment management pits the performance of

individual securities and security portfolios both domestically and internationally against

one another or some benchmark. Relative performance is the overarching theme.

Consequently, it is surprising that relative performance has yet to be actively traded

directly. 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 structure

and 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 wishes

to limit the downside. In this case, the investor could buy a call on the stock and a put on

the market, however, this entails paying unnecessarily for the market volatility embedded

in the call and put option premiums. To avoid this, the investor would have to take

dynamically shifting positions in the stock and a market ETF (or in their respective

derivative products). For the typical institutional or retail investor, constantly migrating

funds from one asset category to another in response to a change in expected performance

is cumbersome and costly. Exchange-traded products on relative performance would

seem to provide a simple and cost-effective means not only of handling existing

2   

return/risk management strategies but also of introducing new return/risk management

strategies to the investment management arsenal.

Currently, the NASDAQ OMX computes and disseminates on a real-time basis

nineteen indexes that measure the relative total return of a single stock (“Target

Component”) against the SPDR ETF (“Benchmark Component”).1 Pending Securities

and Exchange Commission approval, options on these nineteen alpha indexes will be

listed on the NASDAQ OMX PHLXSM within the next few months. The purpose of this

paper is to provide an academic analysis of a complex of relative performance indexes

and associated derivatives (futures and options) which includes the new NASDAQ OMX

Alpha Indexes™. The paper has four main sections. In the first, we supply the mechanics

for calculating the underlying “relative performance index” of a target security versus a

benchmark security. In the second, we describe how futures and option contracts written

on relative performance indexes might be structured and how these contracts can be

valued. The third section provides a set of scenarios in which relative performance index

derivatives are shown to be a more cost effective means for trading relative performance

than are traditional futures and options markets. The fourth section summarizes the key

results of the paper.

I. Relative Performance Indexes

A relative performance index is defined as an index that measures the total return

performance of a target security relative to the adjusted total return performance of a

benchmark like the S&P 500. The total daily security return includes both price

appreciation and dividends and is defined as 1 , 1, 1

t t S tS t

t

S S DR

S+ +

+

− +≡ , where tS is the

target security price at the end of day t, and ,S tD 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 and http://www.nasdaqtrader.com/Micro.aspx?id=Alpha.

3   

fashion, 1 , 1, 1

t t M tM t

t

M M DR

M+ +

+

− +≡ , where tM is the benchmark price level at the end of

day t, and ,M tD is the dividend paid by the benchmark during day t. A family of relative

performance indexes is defined by the updating rule:

( )( )

, 1, 1 ,

, 1

1

1S t

b t b t b

M t

RI I

R+

+

+

+= ×

+ (1)

where b is a relative risk-adjustment coefficient and can be set equal to any value to

reflect the security’s systematic variation with the benchmark.2 Where 0b = , the relative

performance index is a total return index. Where 1b = , the right hand side of (1)

corresponds to the ratio of the target and benchmark returns, and the relative performance

index 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 in

order 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 dividend

yield of 2% or more) unfairly handicaps IBM in a performance comparison. Second, the

index, like the value of an actual portfolio, is always positive. Technically, the index

represents the value of a portfolio that is continuously rebalanced such that, for every

dollar long in security S, the portfolio is short b dollars in security M. Such a portfolio

cannot be synthesized by a buy-and-hold strategy and most of the investment community

would 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 daily updating rule is used for illustration purposes only. In practice the index will be updated continually throughout the trading day. 3 Note that this measure of outperformance is relative performance. The more usual definition of outperformance is the degree to which the price on the target exceeds the price of the benchmark, or absolute performance (see, for example, Margrabe (1978) or Fischer (1978)). Rubinstein (1991) also focuses 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 risk adjustment coefficient is set equal to one (i.e., the exchange rate is acting like 1/benchmark) and only price appreciation is considered.  

4   

strategy opportunities, and these may be appealing to some segment of the investment

community. We explore these first two issues more deeply in Section III. Third, the ratio

of the levels of the index at two different points in time is easily interpreted, particularly

in the case where 1b = . If the current level of the index is 150 and its level three months

ago was 120, the target security outperformed the benchmark by 25%. Finally, relative

performance indexes can readily be extended to multi-asset benchmarks. An asset pricing

purist, for example, may want to benchmark target security performance to a benchmark

that includes a number of asset classes such as stocks, bonds, real estate, and

commodities. In this case, the returns of asset categories would be included in the

denominator of (1), effectively assigning each category its own relative risk adjustment

coefficient. 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 performance

indexes can be derived analytically under the Black-Scholes (1973)/Merton (1973)

(hereafter, “BSM”) valuation assumptions. Specifically, we assume that markets are

frictionless (e.g., no trading costs or different tax rates on different forms of income) and

that market participants can borrow or lend risklessly at a constant annualized interest

rate r. We also assume now that the total return on the target security and the benchmark

security evolve as multivariate geometric Brownian motion with constant drifts,

and S Mμ μ , volatilities, and S Mσ σ , and instantaneous return correlation, SMρ . Under

these assumptions, a relative performance index with constant relative risk-adjustment

coefficient b will evolve as

2 2

, ,0 , ,exp2

m sb t b S M S S t M M t

bI I b t B b Bσ σμ μ σ σ⎛ ⎞⎛ ⎞−

= − + + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

, (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 ,b tI requires

discounting for risk. To do so and avoid arbitrage in the BSM framework, we replace the

5   

drift terms in (2) by the risk-free interest rate. This results in the following “risk-

adjusted” evolution of the relative performance index:

( )2 2

, ,0 , ,exp 12

m sb t b S S t M M t

bI I b r t B b Bσ σ σ σ⎛ ⎞⎛ ⎞−

= − + + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

. (3)

With the relative performance index dynamics in hand, we now turn to the

valuation of futures and option contracts. We assume that futures and options on the

index expire at the same point in time and are settled in cash. To avoid the complications

of 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 a

portfolio that is continuously rebalanced such that, for every dollar long in security S, the

portfolio is short b dollars in security M. While most of the investment community would

avoid mimicking the index because of excessive trading costs and tracking error, it is

useful to see the mechanics of such a trading strategy in order to develop a better intuition

for 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 as

the relative performance index start at a level of 100 on day 0. The subsequent levels for

the security and the benchmark are set arbitrarily. Note that, at any point in time, the

relative performance index equals 100 times the total return index of the security divided

by the total return index of the benchmark. Over the first day, both the security and the

benchmark advanced—the security by 4.17% and the benchmark by 7.16%. Because the

benchmark return was larger, the relative performance index fell. The objective of the

mimicking portfolio is to match the dollar gain of the relative performance index. On day

1, the dollar gain on the index is –2.79.

The mimicking portfolio has three constituent securities. On day 0, the portfolio is

long 100 dollars of the security, short 100 dollars of the benchmark, and long 100 dollars

in risk-free bonds. Because the sales proceeds from the benchmark exactly offset the

6   

purchase price of the security, the value of the mimicking portfolio equals the value of

the risk-free bonds or 100. Over the first day, the bond position produces 0.070 in interest

income,4 the security position produces a gain (i.e., price appreciation and dividends) of

4.170, and the benchmark position produces a loss of 7.160. The net gain across the three

positions is –2.920. To bring the mimicking portfolio value to the level of the relative

performance index, an additional investment of 0.130 is made in risk-free bonds (i.e., the

mimicking portfolio has a negative payout or dividend). The mimicking portfolio is then

rebalanced. The long position in the security is reduced from 104.17 to 97.21, generating

a gain of 6.96. The short position in the benchmark is, likewise, reduced to the same

dollar value, producing a loss of 9.95. Subtracting the difference, 2.99, from the available

risk-free funds, 100.20, the value of the risk-free bonds in the mimicking portfolio

becomes 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 is

0.068, bringing the balance to 97.278. The long position in the security rises in value

from 97.21 to ( )97.21 108.61/104.17 101.353= for a gain of 4.143, and the short position

in the benchmark rises in value from 97.21 to ( )97.21 111.53 /107.16 101.174= for a loss

of 3.964. To bring the value of the mimicking portfolio into line with the relative

performance 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 from

101.353 to the new index level 97.38, and the short position in the benchmark goes from

101.174 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 the

relative performance index, 97.38.

Under the BSM assumptions and continuous (instead of daily) rebalancing, it can

be shown that the payout is a constant proportion of the index level,

( )2M SM S Mrδ σ ρ σ σ= − − . In other words, the relative performance index is the value of a

                                                            4 For illustrative purposes only, the interest income is based on a simple rate of 7 basis points per day.

7   

portfolio that has a constant proportional payout rate or “dividend yield.”5 Note that, in

some cases such as when the risk-free interest rate is low or the correlation is low, the

dividend yield can be negative. In most cases, however, it will be positive. Suppose, for

example, 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 interest

rate is 2% the portfolio that mimics the b = 1 AAPL vs. SPY index would pay a constant

continuous 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 be

derived from calculating the risk-adjusted expected value of the index at the futures

expiration, that is,

( ) ,r Tb bF I e δ−= (4)

where T is the time remaining to expiration of the futures and

( )( )1 22

MM SM S

bbr bσδ σ ρ σ= − + − . The term δ can be viewed as the generalization of

the 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 The

payout rate δ depends on the correlation between the stock and the benchmark. This

feature of the underlying relative performance index would allow one to infer from index

futures prices information about the correlation between the target and benchmark

returns.

C. Valuation of relative performance index options

Under the valuation assumptions listed at the beginning of this section, the

simplest way to value European-style options on relative performance indexes is to first

apply Black’s (1976) futures option formula. Because the futures price and relative

performance 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).

8   

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 relative

performance index futures is

( ) ( )1 2rT

b bC e F N d XN d−= −⎡ ⎤⎣ ⎦ (5)

where X is the exercise price of the option, ( )N d is the cumulative normal density

function with upper integral limit d, and the upper integral limits are

( ) 2

1 2 1

ln / .5 and bF X T

d d d TT

σσ

σ+

= = − .

Because the underlying source of uncertainty is the ratio of two lognormally distributed

prices, the volatility rate in the expressions for 1 2and d d is

2 2 2 2 .S M SM S Mb bσ σ σ ρ σ σ= + − (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 1 2 and d d that

accompany (5) to get

( ) ( )1 2 ,T rTb bC I e N d Xe N dδ− −= − (7)

where

( ) 2

1 2 1

ln / ( .5 ), and .bI X r T

d d d TTδ σ

σσ+ − +

= = −

The value of a European-style put option on a relative performance index follows

straightforwardly from put-call parity. More specifically, the payoff resulting from

purchasing a call option and selling a put option (with the same exercise price and time

remaining 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-style

options:                                                             8 For a proof, see Whaley (2006, p. 198).

9   

.T rTb b bC P I e Xeδ− −− = − (8)

Substituting (7) into (8) and isolating the put value, we get

( ) ( )2 1 .rT Tb bP e XN d I e N dδ− −= − − − (9)

D. Hedging relative performance index futures and options

The valuation equations for the futures on relative performance indexes (4) and

for the options on relative performance indexes (7) and (9) allow us to develop analytical

expressions for the metrics used in risk management (i.e., delta, gamma and vega).

Appendix B contains these expressions. Several results are noteworthy. First, because the

underlying payoffs depend on changes in two distinct securities, delta-risk management

will necessarily require a simultaneous position in both the security S and its

corresponding benchmark M. As one might suspect from the links between portfolio

formation and relative performance indexes, for every dollar of security S used to hedge a

relative performance index derivative, one must take a position of –b dollars in the

benchmark security M. Thus, although a relative performance index derivative might at

first blush seem twice as complicated to hedge as a derivative product on a single

underlying, in practice the hedging position in M is completely determined by the

hedging position in S.

A second noteworthy result is that, because the futures price depends on the

volatilities of S and M (as well as on the correlation between them), the futures vega is

not 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 the

expression for the futures price (4). Finally the gamma with respect to the benchmark and

the cross-gamma (i.e., the sensitivity of the benchmark delta with respect to the

benchmark 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 inversely

proportional to the cumulative performance of the benchmark. Such inverse dependence

is necessarily a convex function, which implies that the gamma of the index with respect

10   

to the benchmark will not be zero. Among other things, this means that extra care should

be 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, we

now turn to providing a series of illustrations that show the potential benefits of these

products. To keep matters simple and realistic, we focus again on the case where the risk-

adjustment coefficient equals one, that is, 1b = . This case is germane because such

indexes are computed on a real-time basis and are disseminated as NASDAQ OMX

Alpha Indexes™.9 Pending Securities and Exchange Commission approval, options on

nineteen alpha indexes pitting the total return of an individual stock against the total

return performance of the SPDR ETF will listed on the NASDAQ OMX PHLXSM within

the next few months. A list of the individual companies, together with the ticker symbols

of the stock and the alpha index, are shown in Table 2. Option products on relative

performance indexes where 1b ≠ are planned, as are futures contracts on alpha indexes.

To begin, it is worthwhile to note that, under the assumption that b = 1, the

dividend yield term in the various valuation equations is ( )M M SM Srδ σ σ ρ σ= − − . The

futures price, therefore, can be rewritten as

( )M M SM S TF Ieσ σ ρ σ−= , (10)

which implies that, depending upon whether Mσ is greater than or less than SM Sρ σ , the

futures may trade at a premium or a discount relative to the underlying relative

performance index. It is also worthwhile to note that, if a relative performance index

futures is actively traded, its price implies the level of correlation between the stock and

the benchmark returns via

                                                            9 Most of the discussion in this section also applies qualitatively to other cases in which 0b > .

11   

( )ln /

.M

MSM

S

F IT

σσρ

σ

−= (11)

Because the level of the relative performance index and the time remaining to the

expiration of the futures are known, and and S Mσ σ can be estimated using stock option

and index option prices, the correlation is uniquely determined and, by definition, is

forward-looking.

Likewise, the implied beta of a security can also be inferred from alpha index

derivative prices because its definition is

.SSM SM

M

σβ ρσ

≡ (12)

Such an estimation approach may be particularly useful given that current approaches to

estimating beta involve using a long time-series of past return data and assuming the beta

is constant over the entire time-series history. In other words, typical beta estimates are

inherently backward-looking and stable through time. At the same time, finance theory

has 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 obvious

conundrum.

To demonstrate the potential value of alpha index options as a source of

information about correlations, consider the example of CSCO (Cisco Systems, Inc.) vs.

SPY. The realized correlation between the daily returns of these assets was 66% in

2010Q3 and 34% in 2010Q4. This suggests that realized correlation in one quarter may

be a poor forecast correlation in the subsequent quarter and underscores the need for

forward-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-implied

correlations. To examine this, assume it is October 1, 2010 and that the true correlation

between CSCO and SPY over the next quarter is 34%. Suppose further that the

volatilities of CSCO and SPY are well-estimated from standard options to be 11% and

12   

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 correlation

would be 30% to 38%.11 This is significantly more accurate than the 66% correlation

estimate from the previous quarter’s realized correlation. Moreover, even if the previous

quarter’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 nearly

four times larger than the interval implied by the alpha-option price.12 In summary, under

reasonable assumptions for option bid/ask spreads, we expect that option-implied

correlations will be both forward-looking and more accurate than estimates based on

historical time-series.13

A. Efficiency gains to trading relative performance using index derivatives

Absent a specific view on individual stock performance, an equities investor

should hold a well-diversified stock portfolio. With a strong view that a particular stock

will outperform the market, on the other hand, an investor may want to devote a large

portion of portfolio wealth to the individual stock rather than the market. Buying the

stock directly, however, is not a “clean” way to implement the view that the stock will

outperform 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 the

second quarter of the year. Buying AAPL’s shares on April 21 and holding until June 30

would have produced a disappointing return of –3.0%. Does that mean the investor was

wrong? 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 the

investor 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 we assume in the example. 12 To arrive at this confidence interval we employ a Fisher transformation and the fact that there were 64 days in 2010Q3. 13 Implied correlations and betas, by definition, apply to the life of the alpha option. Thus one can deduce a term structure of implied correlations/betas from alpha index options of varying maturities, corresponding to the market’s forecast of how a firm’s systematic risk is forecasted to change over time.

13   

AVSPY (i.e., the NASDAQ OMX Alpha Indexes™ that pits the performance of AAPL

against SPY) rose by 12.9%.14 By construction, the relative performance index is less

exposed to events that move the entire market.

We now turn to comparing how alpha index futures and option values change in

reaction to changes in relative performance with those of alternative buy-and-hold

strategies that use existing exchange-traded products. Our aim is to highlight the

differences, thereby helping to point out how these new instruments help to “complete the

market” for investors interested in relative performance.

Long stock/short benchmark: Consider an investor who has 100 dollars to

invest and wants to speculate that the price of a particular stock will rise relative to a

benchmark. 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 the

benchmark ETF.15 Assuming the 100 dollars is invested in risk-free bonds, the overall

value 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 dollar

amount of alpha-futures.16 Over a very short horizon, the benefits of both of these

strategies are the same. Over longer periods of time, however, the passive long

stock/short ETF position can become unbalanced, exposing the investor to more

downside risk. Suppose, for example, that the stock, the benchmark ETF, and the alpha

index are priced at 100 at the beginning of the investment horizon. The long stock/short

benchmark-ETF position has a value of 100 (i.e., the risk-free bonds), as does the fully

collateralized alpha-futures position. Now, suppose that, over the investment horizon, the

stock falls to 50, and the benchmark rises to 200. The gain on the long stock/short

benchmark-ETF position is –150, while the gain on the alpha-futures is –75. The total

value 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 maintain full 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.

14   

free bonds, illustrating that there is no limit to the loss that can be incurred from the short

benchmark-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 use

option-like structures in their trading strategies. To capture relative performance, an

investor could buy an at-the-money call option on the stock and sell an at-the-money call

option on the benchmark. Using the illustration above, the long stock-call would expire

worthless at the end of the investment horizon since the stock price, 50, sank below the

exercise price, 100. On other hand, the benchmark call is 100 dollars in the money at

expiration. Since the investor is short the call, he loses 100. As an alternative strategy, the

investor 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 call

expires worthless. Obviously, the alpha-call has less potential downside exposure.

To examine the potential upside of the alpha-call, we reverse the price movements

by 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 at

expiration because the stock-call is 100 in the money and the benchmark call is

worthless. On the other hand, the alpha index rises to a level of 600 leaving the alpha-call

500 in the money. Obviously, the alpha-call has greater upside potential.

Naturally, the decision about which strategy to use must be based on investor

preferences and other portfolio considerations. Our purpose is to illustrate how a static

position 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 the

benchmark price at expiration. The call option has an exercise price of 100 and three

months remaining to expiration. The volatility rates of the stock (AAPL) and the

benchmark (SPY) are 26.7% and 17.9%, respectively. The correlation between the

returns is 0.711. The risk-free interest rate is 2%. Note that, as the stock price falls and

the benchmark price rises, the call goes to its lowest possible value of 0. On the other

hand, as the stock price rises, the call value rises, however, as the benchmark price falls,

15   

the call value rises at an increasing rate. This is because the benchmark price is in the

denominator of the alpha index.

Figure 2 shows the value surface of the long stock-call/short benchmark-call

portfolio. 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 is

greater than one because the proceeds from the sale of the benchmark-call are used to

offset 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 the

benchmark price rises. Unlike the alpha-call, however, the portfolio value may become

negative. Moreover, as the stock price rises and the benchmark price falls, the portfolio

value rises, albeit not to the same levels as the alpha-call.

Long stock-call/long benchmark-put: Another option strategy that is designed

to 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 from

100 to 50 and the index level rises from 100 to 200. The long stock-call expires out of the

money, as does the long benchmark put. Hence, the “straddle” expires worthless, similar

to the alpha-call. If the reverse happens and the stock price rises to 200 and the

benchmark falls to 50, the stock-call is 100 in the money, the put is 50 in the money, and

the 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 at

expiration 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, the

value of the long stock-call/long benchmark-put portfolio falls in value as the stock price

falls and/or the benchmark price rises. And, also like the alpha-call, the call-put position

never falls below 0. On the other hand, as the stock price rises and the benchmark price

falls, the call-put position value does not rise to the same levels as the alpha-call.

16   

To illustrate yet another important difference between the call-put strategy above

and 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 to

market 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 features

of alpha index derivatives. These products can simultaneously provide downside

protection while reducing exposure to market volatility. While existing exchange-traded

products 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 its

derivatives products. To do so would require dynamic trading strategies, and dynamic

strategies are prohibitively expensive, at least for retail investors.

To get a rough sense for the trading costs in replicating alpha-products, we

perform a set of simulations. In the simulations, the investor is assumed to pay a

proportional cost of $0.01 per share, a fixed cost of $1.00 for trading any quantity of the

stock, benchmark, or bond, and an additional 0.20% net fee for a short sale.18 The trading

strategy aims to replicate a $1,000 investment in a three-month, at-the-money alpha-call

option on the b = 1 AAPL vs. SPY index using standard delta-hedging techniques.19 The

volatility and correlation parameters are drawn from Table 2.

The simulation results are quite striking. In order to achieve a tracking standard

error no larger than about 5%, the average trading costs are around 110% of the value of

the position! The reason is that hedging an at-the-money call requires frequent

rebalancing (about 300 times during the three months to achieve the 5% standard tracking

error). Each time the portfolio is rebalanced the investor must pay $1 per trade, or $3 in

total to trade in risk-free bonds, AAPL, and SPY, amounting to about $900 over the life

of 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’s idiosyncratic volatility. 18 We assume that the proportional costs for trading the short-term bond are negligible. Our cost assumptions 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.

17   

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 in

trading costs if the standard tracking error is to be kept below 5%.

Table 3 documents simulation results of tracking errors and trading costs for

various three-month AAPL vs. SPY alpha-call options. The initial investment is assumed

to be $10,000, and the replicating portfolio is assumed to be rebalanced daily (90 trading

periods 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 earlier

illustration. This is because the replicating portfolio is rebalanced only 90 times as

opposed to 300. Out-of-the-money calls are the most difficult and costly to replicate. The

reason is that the high leverage implicit in such options requires large positions in the

stock and benchmark relative to the option value. Even small fluctuations in the

underlying prices can greatly unbalance the portfolio. In addition, the large trades

required for rebalancing entail higher costs. The 10% out-of-the-money alpha-call has an

average tracking error of 34% and an average total trading cost of 36% of the option’s

value. Overall, the lesson from this exercise is that, while forward/futures positions on an

alpha 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 risk

means 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 sum

of market risk and idiosyncratic risk), while relative performance index options are

essentially based on the difference in risk between the stock and the benchmark. Because

option value varies directly with volatility, relative performance index options are

cheaper.

To illustrate, return once more to the AAPL vs. SPY example. Suppose an

investor decides to buy an at-the-money call option with an exercise price of 100 and

three months remaining to expiration. The investor estimates that the expected future

18   

return volatility of AAPL is 26.7% and SPY is 17.9%, and that the expected correlation

between AAPL and SPY returns is 0.711. The risk-free interest is 2%. Under the BSM

assumptions, the value of an at-the-money call option on AAPL is $11.53. At the same

time, the value of an at-the-money call option on AVSPY is $7.24, corresponding to a

discount of 37%. The volatility used in the valuation of the relative performance index

option, given by equation (6), is 18.8%. As long as the correlation between the assets is

positive and S Mσ σ> , the volatility of the index (i.e., σ ) is guaranteed to be smaller than

the volatility of the stock (i.e., Sσ ), and the price an alpha-call will likewise be

guaranteed 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 the

performances of two securities, one should consider their total return, which includes

income distribution as well as price appreciation. There are other advantages to using

total returns in measuring performance. Standard exchange-traded stock futures and

option contracts are based only on the price appreciation of the underlying stock. Because

a relative performance index futures incorporates dividends into the performance

calculation, its payoffs implicitly include the automatic reinvestment of security income

without incurring transaction costs. It also means that relative performance index options

are not susceptible to the trading games played in the stock option market when deep in-

the-money options remain unexercised.20

D. Trading correlation

According to the valuation equations derived in Section II, the values of relative

performance index futures and options depend on the correlation between the security S

and the benchmark M. A higher correlation implies lower index futures and call options

prices. A trader who believes correlations will increase (decrease) can sell (buy) index

futures 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 to exercise outstanding call option positions on stocks when it is optimal to do so (just prior to the ex-dividend day) and, as a result, forfeit the ex-dividend call option price drop to market makers and proprietary firms. 

19   

example of AAPL vs. SPY, an increase in the correlation from 0.711 to 0.8 corresponds

to a decline in the value of a one-year at-the-money call option from $7.24 to $6.08. The

trader could sell the option and hope that the new information would be reflected in

prices soon after (at which point he or she would close out the position for a profit). The

risk here is that the index will simultaneously move higher, cancelling the decline in the

option value. To make a cleaner investment, the trader can hedge the written option using

the deltas calculated in Appendix B and using a correlation value of 0.711 to calculate the

hedging positions. The net position will be insensitive to changes in the index itself, but

once the new correlation value of 0.8 is absorbed by the market, the option value will

sink below that of the hedging portfolio and the trader can close out the portfolio at a

profit.21 The opposite strategy can be employed if the trader believes correlations will

decline. This illustrates how investors can trade on their views concerning correlations

between the index components.

In general, the construction of a portfolio that moves only with the correlation

between two assets requires model-calculated positions in relative performance index

derivatives and the index constituents (and potentially their derivatives). What is

important to emphasize is that such trades could not be contemplated without traded

index derivatives (just as one could not trade volatility without traded stock derivatives).

Thus, just as stock options have enabled markets to trade stock volatility, relative

performance index products have the potential of opening up an entirely new market for

trading correlations.

IV. Summary

The purpose of this paper is to introduce a new complex of relative performance

indexes, tracking the performance of a target security versus that of a benchmark

security. In particular, we assess how derivatives (options and futures) on these indexes

might 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 account for changing volatility in constructing the hedging portfolio.

20   

valuation formulae for futures and option contracts on the family of relative performance

indexes. In addition, we illustrate a variety of ways in which the new index products

could be a more efficient and cost-effective means of realizing certain investment

objectives than are traditional futures and options markets.

The new indexes essentially track a portfolio that is long on a target stock and

short on a benchmark asset, such that the ratio of the two dollar positions is constant. As

the target asset outperforms its benchmark, the equivalent long-short position increases.

Likewise, if the target asset underperforms its benchmark over a period, then the

equivalent portfolio long-short position is decreased. This behavior ensures downside

protection which could only otherwise be implemented through dynamic trading – an

exercise 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. We

also show that they offer a different payoff structure than related positions using standard

calls and puts on the underlying target and benchmark securities. In particular, the new

index call options simultaneously have downside protection and an accelerated upside

relative to static positions in existing instruments. At the same time, such options would

be creating investment opportunities otherwise only available to typical investors at great

costs. Finally, an exciting aspect of the new index options is that they present investors

with 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 option

implied target CAPM betas if the benchmark is a market index) are both forward-looking

and potentially more informative about future correlations than historical estimates.

21   

References Black, Fischer. 1976. The pricing of commodity contracts, Journal of Financial Economics 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 the valuation 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 Economics and Management Science 1, 141-183.

Pool, Veronika, Hans R. Stoll, and Robert E. Whaley. 2008. Failure to exercise call options: 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, John Wiley & Sons, Inc.: Hoboken, New Jersey.  

22   

Appendix A: Multi-asset relative performance indexes

The benchmark used in the definition of the complex of relative performance

indexes in (1) can be extended to include multiple-asset benchmarks. Suppose, for

example, the desired benchmark has n different asset classes (i.e., stocks, bonds, real

estate, and commodities) and each asset class constitutes w% of the overall benchmark

portfolio, where, of course, 1

1n

ii

w=

=∑ . Under these assumptions, the updating rule for the

multi-asset relative performance index is

( )( )

, 1, , 1 , ,

, 11

1

1 i

S tb w t b w t n w b

i ti

RI I

R

++

+=

+= ×

+∏.

where b is as before and w is a vector of benchmark asset allocation weights (i.e.,

, 1,...,iw i n= ) . With a relative performance so defined, a change in the log-index

corresponds to the difference between the instantaneous performance of security S versus

the weighted and scaled instantaneous performances of the benchmark assets, that is,

( ) ( ), , 1 , , , 1 , 11

ln ln ln 1 ln 1n

b w t b w t S t i i ti

I I R b w R+ + +=

− = + − +∑

The futures and option valuation equations on these multi-factor benchmarks are also

analytically tractable and available from the authors upon request.

23   

Appendix B: Risk metrics for derivatives on relative performance indexes

Under the BSM option valuation assumptions, we showed that the value of a

futures contract written on a relative performance index with a constant relative risk-

adjustment coefficient of b is

( ) ,r T

b bF I e δ−= (B-1)

where and b bF I are the futures price and the relative performance index level, r is the

annualized risk-free interest rate, T is the futures time remaining to expiration in years,

( )( )1 22

MM SM S

bbr bσδ σ ρ σ= − + − , and S Mσ σ are the volatility rates of the security

and the benchmark, and SMρ is the correlation between the returns of the security and the

benchmark. We also showed that the values of the European-style call and put options on

the relative performance index are

( ) ( )1 2 ,T rTb bC I e N d Xe N dδ− −= − (B-2)

and

( ) ( )2 1 .rT Tb bP e XN d I e N dδ− −= − − − (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 upper

integral limit d, 2 2 2 2S M SM S Mb bσ σ σ ρ σ σ= + − , and the upper integral limits are

( ) 2

1 2 1

ln / ( .5 ), and .bI X r T

d d d TTδ σ

σσ+ − +

= = −

The put-call parity relation for European-style options is

.T rTb b bC P I e Xeδ− −− = − (B-4)

Based on these equations, the risk metrics (i.e., delta, gamma and vega) of the futures and

options are as follows.

24   

Delta: Based on the futures pricing relation (B-1), the delta of the futures with respect to

the underlying relative performance index is

( ),

r TF I e δ−Δ = .

The deltas with respect to the prices of the security and the benchmark are

( ),

r TbF S

I eS

δ−Δ = and ( ),

r TbF M

I eM

δ−Δ = − .

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 performance

index are

( ), 1T

c I e N dδ−Δ = and ( ), 1T

p I e N dδ−Δ = − − .

Since the alpha options may be hedged using shares of the security or the benchmark, the

deltas with respect to the per share security price, S, are

, , , , and b bc S c I p S p I

I IS S

Δ = Δ Δ = − Δ

and the deltas with respect to the per share benchmark price, M, are

, , , , and b bc M c I p M p I

I IM M

Δ = − Δ Δ = Δ .

Gamma:

(B-5) shows that the delta is not a function of the relative performance index level, so the

gamma of a futures with respect to the index is 0. This is also true of the gamma with

respect to the security, S. The gamma with respect to the benchmark price is not zero,

however, and is given by

( ), 22 .r Tb

F MMI e

Mδ−Γ =

The cross gammas are

25   

, , , .bF SM F MS F I

ISM

Γ = Γ = − Δ

The gamma of a call option with respect to the underlying index is the same as that for

the put option:

( )1, ,

TI c I p I

b

n de

I Tδ

σ−Γ = Γ = Γ = ,

where ( ) 21 /2

112

dn d eπ

−= is the normal density function evaluated at 1d . The gammas of

the call option with respect to the underlying security and benchmark prices are given by

2

, 2b

c S IIS

Γ = Γ , ( ), 2

2b b I Ic M

I IMΓ + Δ

Γ = and ( )

, , b b I Ic SM c MS

I ISMΓ + Δ

Γ = Γ = − .

By virtue of the put-call parity relation (B-4), the gammas of the European-style put

options are related through the gamma of the futures price:

, ,p S c SΓ = Γ , , , ,rT

p M c M F Me−Γ = Γ − Γ and , , , , .rT

p SM p MS c MS F MSe−Γ = Γ = Γ − Γ

Vega:

Unlike the usual cost of carry model, the futures pricing equation for a relative

performance index is a function of volatility. The vegas with respect to , , and S M SMσ σ ρ

are:

, SF SM M bVega b TFσ ρ σ= − ,

( ), 1MF M SM S bVega b b TFσ σ ρ σ= + −⎡ ⎤⎣ ⎦ ,

and

, SMF S M bVega b TFρ σ σ= − .

The vegas of the European-style call option with respect to , , and S M SMσ σ ρ are

26   

( ) ( ), 1 1 ,S S

T rTS SM Mc b F

bVega I e n d T e N d VegaT

δσ σ

σ ρ σσ

− −−⎛ ⎞= +⎜ ⎟⎝ ⎠

( ) ( ) ( ), 1 1 ,M M

M SM ST rTc b F

b bVega I e n d T e N d Vega

Tδ

σ σ

σ ρ σσ

− −−⎛ ⎞= +⎜ ⎟

⎝ ⎠

and

( ) ( ), 1 1 ,SM SM

T rTS Mc b F

bVega I e n d T e N d VegaT

δρ ρ

σ σσ

− −⎛ ⎞= − +⎜ ⎟⎝ ⎠

.

Note that, by the put-call parity relation (B-4), the vegas of the call, put and futures are

related by , , ,rT

c i p i F iVega Vega e Vega−− = , where , ,S M SMi σ σ ρ= . Hence, the vegas of the

European-style put option with respect to , , and S M SMσ σ ρ are

, , ,S S S

rTp c FVega Vega e Vegaσ σ σ

−= − ,

, , ,M M M

rTp c FVega Vega e Vegaσ σ σ

−= − ,

and

, , ,SM SM SM

rTp c FVega Vega e Vegaρ ρ ρ

−= − .

27  

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

 

28  

Table 2: List of NASDAQ OMX Alpha Indexes™ with option contracts pending approval of the SEC. All of the indexes listed are outperformance indexes with the benchmark being the SPDR ETF (ticker symbol SPY). The annualized volatility of each stock and correlation with SPY are calculated using daily return data for the calendar year 2010. The data were collected from Datastream. Over the sample 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%

29  

Table 3: Transaction costs and tracking error for delta-hedging portfolios of positions in three-month futures and options positions written 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, with initial share values for AAPL and SPY taken to be $340 and $130, respectively. Short selling costs are assumed to be 0.20% of the amount borrowed. The AAPL volatility rate is assumed to be 26.7%, the SPY volatility rate is 17.9%, and the correlation between the two returns is 71.1%. 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. The table also reports a breakdown of the accrued trading costs at the option’s expiration. The fixed costs can accrue to more than $273 because costs are capitalized. They can also be below that value for deep out-of-the-money options because certain price paths lead to essentially zero option 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%

 

Frc

 

Figure 1: Simulremaining to expcorrelation betwe

lated expiration piration. The voleen the returns is 0

 

B

value of call oplatility rates of th0.711. The risk-fr

50

70

90

110

Benchmark price

ption on AVSPYhe security (AA

free interest rate is

110

130

150

150

30

Y index. The caPL) and the bens 2%.

110

120

130

140

all option has an nchmark (SPY) a

6070809010

0

Security price

exercise price oare 26.7% and 1

-100

-50

0

50

100

150

200

50

Opt

ion

valu

e

of 100 and three 17.9%, respective

months ely. The

 

 

 

Fham

Figure 2: Simulahave an exercise are 26.7% and 17money options at

ated expiration vprice of 100 and 7.9%, respectivelorigination is sca

Be

value of portfoliothree months rem

ly. The correlatioaled to match the

50

70

90

110

enchmark price

o consisting of lomaining to expiraon between the re value of the at-th

110

130

150 140

150

31

ong call option oation. The volatilieturns is 0.711. The-money alpha c

100

110

120

130

140

on AAPL and shoity rates of the seThe risk-free intecall option.

506070809010

0

Security price

ort call option oecurity (AAPL) aerest rate is 2%. P

-150-100-50050100150

200

50

Opt

ion

valu

e

n SPY. Both calland the benchmarPortfolio value o

l options rk (SPY) of at-the-

 

 

Fo(t

Figure 3: Simulaoptions have an e(SPY) are 26.7% the-money option

ated expiration vexercise price of 1

and 17.9%, respns at origination i

Be

value of portfoli100 and three moectively. The cors scaled to match

50

70

90

110

enchmark price

o consisting of loonths remaining torrelation between h the value of the

110

130

150 140

150

32

ong call option oo expiration. Thethe returns is 0.7at-the-money alp

10

0

110

120

130

140

on AAPL and lo volatility rates o

711. The risk-freepha call option.

506070809010

0

Security price

ong put option onof the security (AAe interest rate is 2

 

-100

-50

0

50

100

150

200

50

Opt

ion

valu

e

n SPY. Both callAPL) and the ben2%. Portfolio valu

l and put nchmark ue of at-