Relative Implied-Volatility Arbitrage with Index Options · 2016. 2. 27. · Statistical arbitrage, however, be it in deriva-tives or other markets, has received surprisingly little

42 ©2002, AIMR®

Relative Implied-Volatility Arbitrage with Index Options

Manuel Ammann and Silvan Herriger

In the study reported here, we investigated the efficiency of markets as tothe relative pricing of similar risk by using implied volatilities of optionson highly correlated indexes and a statistical arbitrage strategy to profitfrom potential mispricings. We first analyzed the interrelationships overtime of the three most highly correlated and liquid pairs of U.S. stockindexes. Based on this analysis, we derived a relative relationship betweenimplied volatilities for each pair. If this relationship was violated (i.e., if wedetected a relative implied-volatility deviation), we suspected a relativemispricing. We used a simple no-arbitrage barrier to identify significantdeviations and implemented a statistical arbitrage trade each time such adeviation was recorded. We found that, although many deviations can beobserved, only some of them are large enough to be exploited profitably inthe presence of bid–ask spreads and transaction costs.

rbitrage relationships in derivatives mar-kets have been studied extensively. Forexample, option boundary conditions, asderived by Stoll (1969) and Merton (1973),

have been the subject of numerous empirical stud-ies; examples are Gould and Galai (1974), Klemko-sky and Resnick (1979), and Ackert and Tian (1998,1999). Index arbitrage has also been thoroughlyinvestigated. Empirical studies include those ofFiglewski (1984), Chung (1991), Sofianos (1993),and Neal (1996). Figlewski (1989) provided anexample of option arbitrage in imperfect markets.Clearly, the testing of market efficiency in deriva-tives markets by using arbitrage relationships hasdrawn a great deal of interest.

Statistical arbitrage, however, be it in deriva-tives or other markets, has received surprisinglylittle attention in the literature, despite its highpractical relevance. A possible reason may be thenature of the mispricings underlying statisticalarbitrage. Statistical arbitrage is not based on theo-retical, exact pricing relationships but, rather, onempirical, statistically established relationships.Consequently, statistical arbitrage involves risk.Omitting the study of such forms of pricing rela-tionships from research agendas altogether, how-

ever, may lead to an incomplete understanding ofmarket mechanisms and thus of market efficiency.

One study that used the statistical arbitrageapproach to test market efficiency in equity mar-kets is that of Gatev, Goetzmann, and Rouwenhorst(1999), who investigated the relative pricing mech-anisms of securities that are close economic substi-tutes. In addition, motivated by the widespreadintermarket hedging activities in commoditiesmarkets, a number of authors have analyzed vari-ous pricing relationships for commodity spreads.This interest explains the presence of severalpapers, such as Johnson, Zulauf, Irwin, and Gerlow(1991) and Poitras (1997), that applied statisticalarbitrage to such markets.

A statistical arbitrage approach to test the effi-ciency of options markets has not yet beenattempted. Thus, the aim of this study was to deviseand implement a statistical arbitrage strategy fortesting an aspect of market efficiency that the clas-sical boundary conditions for options fail toreveal—namely, the efficiency of markets in pric-ing relative risk in highly correlated markets.

Theory and DataIn what situations is a statistical arbitrage strategypossible and likely to lead to profitable trading? Iftwo indexes are highly correlated (because of asecurities overlap or other reasons), one should beable to calculate the relationship between their vol-atility levels. A similar relationship must also be

Manuel Ammann is professor of finance at the Univer-sity of St. Gallen, Switzerland. Sylvan Herriger is anindex derivatives trader at J.P. Morgan, London.

A


November/December 2002 43

valid for the implied-volatility levels of the respec-tive index options. If the relationship between theimplied volatilities is significantly different fromthe relationship observed between the two indexvolatilities, the option prices are misaligned, whichshould not occur in efficient markets. In such a case,a statistical arbitrage strategy can be implementedto take advantage of the relative implied-volatilitydeviation.

To test these ideas, we used significantlyrelated U.S. equity indexes. Listed options areavailable for 11 stock indexes in the United States,and several of these indexes are closely related.This close relationship is often a result of the samestocks being included in several indexes; for exam-ple, every stock in the S&P 100 Index is alsoincluded in the S&P 500 Index. We studied the timeperiod from January 1995 through February 2000.

Our statistical arbitrage methodology con-sisted of the following consecutive steps:• First, after ensuring that the time-series returns

were stationary, we calculated the correlationsof the various index pairs. We then selected thepairs with the highest correlation coefficientsand no longer considered the other indexes.

• Second, we studied the relationship of the dailyreturns of the index pairs by running OLS(ordinary least-squares) regressions to estab-lish the past relationships between them. Wealso tested the robustness of the relationships.Because the linear relationship between twoindexes is time varying, we estimated statisti-cal boundaries for the OLS coefficients.

• Third, we established a conditional forecast offuture variance based on the past relationshipbetween the indexes’ returns. The reason wasto test, out of sample, the predictive powers ofthe boundaries estimated in the second step.

• Once the predictive capacity of the boundarieswas confirmed for the historical volatilities, weapplied the estimated relationship to impliedvolatilities, for which a similar relationshipshould prevail.

• Based on the implied volatilities and on theriskless rate recorded every trading day, wecalculated the corresponding option prices. Weincorporated bid–ask spreads in the process toensure that, should we identify a deviation ofa certain significant magnitude, we couldimplement and test an option strategy that tookadvantage of the suspected mispricing.

• Finally, we implemented a simple arbitragetrading strategy.1

The 11 stock indexes for which exchange-traded options are available in the United States arethe S&P 500 Index (SPX), the S&P 100 Index (OEX),

the Nasdaq 100 Index (NDX), the NYSE CompositeIndex (NYA), the Philadelphia U.S. TOP 100 Index(PTPX), the Philadelphia Stock Exchange UtilitySector (UTY), the S&P Smallcap 600 Index (SML),the S&P Mid Cap 400 (MID), the Amex Major Mar-ket Index (XMI), the Russell 2000 Index (RTY), andthe Dow Jones Industrial Average (INDU). Theseindexes formed the pool from which we choseindex pairs for further study.

Following Harvey and Whaley (1991), we usedfor this study the implied volatility of at-the-moneyoptions with the shortest maturity. At-the-moneyoptions contain the most information about volatil-ity. Also, we used the “front month” optionsbecause they are the most liquid. If fewer than 20calendar days were left to expiration, we used thenext available series. Thus, the implied volatilitieswere calculated from options ranging from 20 to 50calendar days to expiration, or an average of 35calendar days. Because 35 calendar days representexactly 5 weeks and an average week has 5 tradingdays, an average of 25 trading days was used in thecalculations. We based the calculation of the impliedvolatilities on the closing prices of the options andon the closing price levels of the indexes. We derivedthe volatility from an average of the implied volatil-ities of at-the-money options.

The riskless interest rate was used to calculatethe daily option prices, and because the optionshad, on average, 35 calendar days left to expiration,we used the one-month Eurodollar LIBOR as theriskless rate.

Choosing IndexesThe criteria for selection of the specific indexes tostudy were (1) stationarity of the index returns, (2)high correlation, and (3) liquidity of the market forindex options.

To avoid spurious correlations and regressions,we first tested each index time series for stationarity.The standard stationarity tests revealed that allreturn time series (based on continuously com-pounded returns) can be considered stationaryexcept the Philadelphia UTY. We thus dropped theUTY from further analysis.

We then calculated the correlation coefficientsfor the remaining 10 indexes. This criterion wasmotivated by the conjecture that index pairs withhigh correlations exhibit a strong linear relation-ship between each other. The correlation matrix isin Table 1.

We set a minimum of 0.95 for the correlationcoefficient as a criterion for inclusion in our indexsample, which screened out all but five pairs of

Financial Analysts Journal

44 ©2002, AIMR®

indexes (those in boldface in Table 1) to be consid-ered for further calculations.2 Of these five pairs,we chose the three indexes with the most-liquidoptions markets to ensure that potential arbitragetrading strategies could be executed. The three cho-sen indexes are the SPX, OEX, and NYA.

That the S&P 500 is highly correlated with theS&P 100 is not surprising because the OEX is anintegral part of the SPX. We expected the OEX to bemore volatile than the SPX, however, because oftheir comparative respective constituents (100stocks versus 500 stocks). We also expected theoverlap of the SPX with the NYA to be largebecause many of the stocks in the SPX are listed onthe NYSE. The NYA has more stocks than the SPX,so we expected the SPX to be slightly more volatile.The relationship between the OEX and the NYA ismore surprising than the other relationships.Apparently, the OEX tracks the larger SPX so wellthat it also manages to track the NYA, which is evenlarger. We expected the OEX definitely to be morevolatile than the NYA because of the greater diver-sification of the NYA.

Relationships between Index PairsFor every pair of indexes, we used OLS regressionto regress the daily returns of one index onto thedaily returns of the other:

(1)

where Yi = daily return of Index Y at time tXi = daily return of Index X at time t

= disturbance termThe sample used in every case was half a year

of daily returns, or 125 trading days, from theperiod January 1995 through February 2000.3 Con-sequently, the first regression was made 125 days

into the data, with the use of the past 125 dailyreturns, and for every day after this regression, theregression was run again using the previous 125days. The result was a rolling 125-day regression inwhich the oldest data point was dropped everytime a new one was added.

Because the regressions were rolled day by dayfor a long period for every index pair, a large num-ber of regressions had to be calculated. The panelsin Figure 1 present the resulting graphs of regres-sion coefficient plus the calculated lower andupper boundaries (to be described).

We used t-tests to test the significance of theregression coefficients and found each estimatedslope coefficient to be significantly different fromzero at the 99 percent confidence level. Because theresults regarding the significance of the estimatedslope coefficients were very similar for the severalthousand regressions, we do not present theseresults in detail.

The situation is different for the estimated intercepts. For the large majority, the hypothesisthat they were not different from zero could not berejected at a 95 percent level of confidence. In therare cases in which the estimated intercepts werefound to be statistically significant, the recordedvalue of the intercept was very low (for practicalpurposes, zero).

The boundaries for Figure 1 were set as fol-lows. The lowest and highest recorded R2 valuesfor the coefficients of determination for all regres-sions are in Table 2. The coefficients confirm astrong linear relationship between the selectedindexes. The relationships of the examined indexpairs are not constant, however, over time. Thissimple observation had a profound impact on thisstudy because the relative volatility forecasts hadto be based on these relationships. Time-invariantrelationships between index pairs with consistently

Table 1. Correlation Matrix, 31 March 1995 through 3 December 1999SPX OEX NDX NYA PTPX INDU SML MID XMI RTY

SPX 1 0.988 0.780 0.988 0.798 0.922 0.709 0.804 0.91 0.729

OEX 1 0.770 0.968 0.799 0.933 0.662 0.759 0.923 0.684

NDX 1 0.721 0.563 0.633 0.688 0.745 0.593 0.708

NYA 1 0.785 0.933 0.757 0.846 0.919 0.772

PTPX 1 0.896 0.653 0.748 0.892 0.676

INDU 1 0.637 0.750 0.974 0.651

SML 1 0.922 0.583 0.983

MID 1 0.700 0.922

XMI 1 0.595

RTY 1

Note: Correlation coefficients based on weekly returns.

Yi β1 β2Xi ui,+ +=

ui

β2

β1



Figure 1. Regression Coefficient and Boundaries, 7 May 1991 through 10 February 2000

A. SPX−OEX

0.80

β2

0.85

0.90

0.95

1.00

1.05

1.10

7/M

ay/91

25/Aug/92

14/Dec

/93

5/Apr/

95

25/Ju

l/96

12/Nov/97

9/M

ar/99

10/Feb

/00

B. SPX−NYA

0.97

β2

1.02

1.07

1.12

1.17

1.22

1.27

7/M

ay/91

17/Aug/92

26/Nov/93

16/M

ar/95

9/Aug/96

19/Nov/97

10/M

ar/99

10/Feb

/00

C. OEX−NYAβ2

0.87

0.92

0.97

1.02

1.07

1.12

1.17

1.22

1.27

1.32

7/M

ay/91

17/Aug/92

26/Nov/93

16/M

ar/95

8/Aug/96

18/Nov/97

5/M

ar/99

10/Feb

/00

Regression Slope Boundaries


46 ©2002, AIMR®

high R2 values would have been ideal. Then, estab-lishing their interrelationships once would havesufficed to predict the relative volatilities for anyfuture time interval. With both varying interrela-tionships and varying goodness of fit of the models,however, an alternative method had to be found toaccount for these instabilities.

The slope coefficients that were estimated fromthe previous 125 trading days were not sufficientas a prediction tool. Such point estimates are sub-ject to estimation error because coefficients havebeen found to vary over time. Thus, an upper anda lower boundary are needed to render the esti-mated slope coefficients more robust as a predic-tion tool. Applying the method of intervalestimation would be inappropriate because ordi-nary interval estimation makes a statement aboutthe confidence level with which the calculatedinterval will contain the true slope coefficient; thetrue slope coefficient is thus assumed to be con-stant. In our study, because we were dealing withrolling regressions, the degree of variance of theslope coefficients over time also had to be consid-ered when establishing boundaries for the estima-tors. Consequently, we chose, instead, an empiricalboundary calculation based on a simple minimum–maximum approach.

To construct the boundaries, we recorded thelargest variations of the relevant parameters duringa time span that matched the forecasting horizon.This method reflects the market situation: In a vol-atile situation, when the index relationships varygreatly, the boundaries are wider; when the rela-

tionships are relatively constant, the boundariesare narrower.

Because 25 trading days formed the horizon ofthe various forecasting calculations, we recordedthe largest percentage change of the estimated slopecoefficients measured in any preceding 25-dayinterval during the preceding 250 trading days.4

These extreme changes of the beta coefficient werethen used for establishing minimum and maximumboundaries at each point in time to ensure robustbeta forecasts. The boundary estimation methodol-ogy is illustrated in Figure 2.

The following equations were used to calculatethe boundaries of the estimated slope coefficientsat time t:

Lower boundary:

, (2)

Higher boundary:

, (3)

where is the estimated slope coefficient attime t based on the sample t–124 to t and is the largest percentage change of the estimatedslope coefficients measured over any 25-trading-day period during the subsample (250 tradingdays) preceding the time of estimation. 5

Volatility Relationships For the purpose of this study, the relationships thatwere established between the daily returns of indexpairs needed to be transformed into relationshipsbetween the respective volatilities. For the relation-ship between random variables X and Y such that

Y = a + bX + u, (4)

we obtained

var(Y) = var(a + bX + u). (5)

Table 2. Minimum and Maximum R2s

Index Pair Minimum R2 Maximum R2

SPX–OEX 0.891 0.988

SPX–NYA 0.935 0.996

OEX–NYA 0.831 0.976

β2 low t( ) β2 t( ) max ∆β2( )β2 t( )–=

β2 high t( ) β2 t( ) max ∆β2( )β2 t( )+=

β2 t( )max ∆β2( )

Figure 2. Beta Boundary Estimation

Note: Because betas were estimated using 125 trading days of data, the first boundary estimationavailable was 375 days into the data.

Rolling Regressions to Estimate Beta

max ∆β

∆β(t)

Forecasting Horizon

−t 375 −t 250 −t 125−t 225 t t + 25



By basic statistics, if a and b are constants and X andu are independent random variables,

var(Y) = b2 var(X) + var(u). (6)

Applied to the regression outputs, this expressioncan be written as

(7)

Based on this relationship, we attempted aforecast of the future interrelationships of the indexpairs. Equation 7 indicates that two inputs need tobe estimated to forecast the variance of RYi—theslope coefficients, and the variance of the sam-ple disturbance term, . Estimates of the slopecoefficients were described previously. To estimate

, we chose an approach similar to the estimationof : Both the smallest and the largest 25-dayvariance of the residuals during the 250 tradingdays preceding a point of estimation wererecorded. Formally,

Lower boundary:

(8)

Higher boundary:

(9)

Once all necessary parameters had been esti-mated, we proceeded to test their forecastingpower.

Out-of-Sample Tests of VolatilityBoundaries. To test the volatility boundariesaround calculated previously, we used theboundaries to make daily rolling forecasts of thevariable to be explained. In other words, we usedthe established relationships between the indexesto forecast their relative future volatilities over a 25-trading-day period and then compared this fore-cast with the actual recorded volatility levels. Weapplied the statistical properties introduced inEquations 6 and 7. For example, to test the estab-lished SPX–OEX relationship at time t, we appliedthe following equation:

(10a)

where var(OEX) and var(SPX) are the realizedfuture 25-day variances.

If the realized SPX variance was within theforecasted boundaries, we deemed the forecastingsuccessful. This test was rolled daily, similarly to

the regressions. We took the percentage of success-ful forecasts as an indication of the forecastingpower of the established boundaries.

The results of this out-of-sample test for theboundaries with four tolerance levels are in Table3. We used tolerance levels to widen the boundariesand thus make the forecasts more robust. For agiven tolerance ψ, the forecast was successful if

(10b)

Clearly, the wider the boundaries are, thehigher the probability is that the future volatilitywill fall between those boundaries. This fact isimportant when evaluating the power of such aforecasting test: The ideal would be a narrowboundary that includes all future volatility read-ings, but the narrower the boundary, the smallerthe probability, all else being equal, of including thefuture volatility. This opposition in the goals is whywe included tolerance levels in Table 3; because weare dealing with statistical arbitrage, the forecast-ing need not be perfect, but it must be sufficientlyclose to allow for arbitrage trades to be triggeredwhen a deviation has been detected and thus amispricing is suspected.

Keep in mind the interpretation of Figure 1. Aresult of 100 percent with no tolerance and a narrowboundary band would indicate, in effect, the exist-ence of a quasi-deterministic relationship betweenthe considered indexes; that is, knowing the vola-tility of one would be enough to determine pre-cisely and repeatedly the volatility of the other.That such a result is not shown in Table 3 is notsurprising, but the table does indicate that, with a1 percent tolerance level, the percentage of success-ful forecasts is more than 90 percent for all threeindex pairs. Considering the relatively narrowboundaries depicted in Figure 1, this degree ofprecision is satisfactory.

The next step is to apply the inferences madeso far regarding relative future volatility of indexpairs to their relative implied-volatility levels.

Arbitrage Boundaries for Relative ImpliedVolatilities. In the previous sections, we devel-oped a method that, based on the past relationshipof daily returns, provides a robust forecast of thefuture relative volatility of three index pairs. In thissection, the lower and higher boundaries of theserelationships are used to identify relative implied-volatility deviations in the option markets.

var RYi( ) β22 var RXi( ) var ui( ).+=

β2,ui

uiβ2

var ut( )low

min var25 days ui( ) ,=

var ut( )high

max var25 days ui( ) .=

β2

β2 low t( )2var OEX( ) var ut( )

low+

var SPX( )≤

≤ β2 high t( )2var OEX( ) var ut( )

high,+

β2 low t( )2var OEX( ) var ut( )

lowψ–+

≤ var SPX( )

≤ β2 high t( )2var OEX( ) var ut( )

highψ.+ +


48 ©2002, AIMR®

We want to stress that we are not postulatingany relationship between the magnitude of the his-torical volatility and the implied volatility. 6 We areaddressing, rather, the relationship between therelative volatilities—that is, the relative differencein volatility levels, whether historical or implied. Inother words, we are establishing a relationship notbetween implied and realized volatility butbetween the implied volatility of options on twodistinct indexes. Therefore, once we have estab-lished a precise relationship between the relativefuture volatility of index pairs, this relationshipmust also hold for the relative implied volatility ofoptions on the two indexes. The boundaries calcu-lated for the relative future volatility must also holdfor relative implied volatility, and Equation 10amust also apply to implied volatilities.

Using the previous example of OEX and SPXoptions, the relationship we were examining isgiven by

(11)

where varimpl (OEX) and varimpl (SPX) are theobserved implied variances (squared implied vola-tilities) of at-the-money options with 25 tradingdays left to expiration. If such a boundary wasviolated, a theoretical mispricing was suspected.7

The nature of this mispricing is not absolute; it is arelative mispricing (i.e., the relative difference of theimplied volatilities of a particular index pair at aspecific time differs significantly from the calcu-lated relative future volatility difference betweenthe two indexes).

The size of the theoretical deviation (suspectedmispricing) was defined as the difference betweenthe violated boundary and the observed impliedvolatility. The deviations for call and put options forthe three combinations and for every trading dayare presented in Figure 3, Figure 4, and Figure 5.

In testing the use of deviations in possible arbi-trage trading strategies, because of transaction

costs, we introduced a security margin similar tothe tolerance level in Table 3 to identify significantdeviations before an arbitrage trade was initiated.This security margin can be interpreted as a formof no-arbitrage barrier. Its magnitude was fixed attwo times the bid–ask spread of at-the-moneyoptions. Whenever we observed such a significantdeviation, we simulated a statistical arbitragetrade. In other words, the relative implied volatilityfalling outside the bounds implied by Equation 11was not sufficient to signal a trade. The deviationalso had to be of a certain minimum size before itwas considered a statistical arbitrage opportunity.This rule underscores the conservative approachwe took to identifying potential trading opportuni-ties. The no-arbitrage security margins are identi-fied by the shaded lines in Figures 3, 4, and 5.

Somewhat surprisingly, the number of theo-retical deviations (thus, suspected mispricings)shown in Figures 3–5 is rather large. Most devia-tions failed to surpass the no-arbitrage barrier,however, and were thus not considered substantialenough to represent arbitrage opportunities.

Note that the deviations seem to have occurredin clusters. In certain periods, one index was per-sistently over/undervalued relative to the other asto its implied volatility.8 A particularly clear exam-ple is the OEX–NYA relationship depicted in Fig-ure 5. In such periods, the market seems to bepersistently mistaken by failing to recognize thecorrect relationship between the future volatilitiesof the indexes. Under our conservative rules, how-ever, this persistent deviation could not be elimi-nated by arbitrage as long as it stayed inside the no-arbitrage barrier formed by the security margins.

We have not been able to identify factors thatcould cause the concentration of volatility devia-tions in phases. The explanation that simply linksan increase of deviations to an increase in marketvolatility, although intuitively appealing, is unsat-isfactory for two reasons. First, although deviations(significant or not) appear to be slightly more fre-quent in highly volatile markets, they can also beobserved in periods of low volatility. Second, link-ing the concentration of deviations to volatility

Table 3. Successful Out-of-Sample Relative-Volatility Forecasts from Established Relationship Boundaries, 7 May 1991 through 10 February 2000

Tolerance Level

Index Pair Zero 0.25 0.50 1.00

SPX–OEX 68.1% 83.6% 90.8% 97.7%

SPX–NYA 60.6 82.6 92.4 97.3

OEX–NYA 68.0 76.7 84.5 91.8

β2 low t( )2varimpl OEX( ) var ut( )low+

≤ varimpl SPX( )

≤ β2 high t( )2varimpl OEX( ) var ut( )high ,+



levels fails to account for the fact that the observeddeviations are persistent overvaluations of oneindex vis-à-vis the other in some phases and under-valuations in other phases. An unknown factorseems to be influencing the subjective relative riskperception of market participants in phases.

Arbitrage Trading StrategyTo test whether the suspected mispricings can beused for profitable arbitrage trading, we imple-mented an arbitrage strategy using at-the-moneyoptions. Option bid–ask prices were calculated fromimplied-volatility data by using a bid–ask spread (in

terms of implied-volatility percentage) around thevolatility implied by the closing price of the options.The bid–ask spread we used was 1 percent, whichfor near-month at-the-money options in liquidoptions markets is a conservative assumption.9 Weused 35 calendar days to expiration and the relevantone-month Eurodollar LIBOR. Because the SPXoptions are European-style options and thus notexercisable prior to maturity, we used the Black–Scholes option-pricing model for them. For theAmerican-style OEX and NYA options, we used abinomial tree that incorporated the early exercisefeature.

Figure 3. SPX–OEX Relative Implied-Volatility Deviations, 3 January 1995 through 10 February 2000

Mispricing Magnitude (%)A. Calls

4

3

2

1

0

1

2

3

4

5

6

7

8

9

3/Jan/95 22/Jan/96 23/Jan/97 28/Jan/98 4/Feb/99 10/Feb/00−

−

−−

Mispricing Magnitude (%)B. Puts

43/Jan/95 27/Feb/96 15/Jan/97 25/Feb/98 19/Jan/99 10/Feb/00−

3−

2−

1−

0

1

2

3

4

5

6

7


50 ©2002, AIMR®

When a volatility deviation was identified assignificant, we sold one at-the-money option of theovervalued index and bought the point-equivalent,

-adjusted amount of at-the-money options of theundervalued index.10

Analytically, for the SPX–OEX example, ifeither

(12a)

or

(12b)

we opened an arbitrage trade. If the first (second)condition was satisfied, we entered into a short(long) position of at-the-moneyoptions on the OEX and a long (short) position ofone at-the-money option on the SPX.11 The termsSPXt and OEXt represent the index levels, and ψ isthe security margin (in volatility percentage). A

Figure 4. SPX–NYA Relative Implied-Volatility Deviations, 3 January 1995 through 10 February 2000


43/Jan/95 9/Feb/96 29/Jan/97 20/Jan/98 12/Jan/99 10/Feb/00−

3−

2−

1−

0

1

2

3

4

5

6


43/Jan/95 18/Mar/96 23/Jan/97 19/Feb/98 19/Mar/99 10/Feb/00−

3−

2−

1−

0

1

2

3

4

5

6

β2

β2 low t( )2varimpl OEX( )t var ut( )

low+

– varimpl SPX( )t ψ%≥

varimpl SPX( )t

– β2 high t( )2varimpl OEX( )t var ut( )

high+

≥ ψ%,

β2 SPX OEX⁄( )



simple example in Exhibit 1 explains this proce-dure.

Translated from the equations, whenever weobserved a significant deviation, we sold one at-the-money option (put or call) on the relativelyovervalued index at the bid price (short position)and bought the point-equivalent, -adjustedamount of at-the-money options on the underval-ued index at the ask price (long position).12 Weheld the two positions without rebalancing fordelta neutrality until either (1) the deviation disap-peared (i.e., the observed implied volatility of theindexes went back inside the calculated bound-

aries) or (2) the options expired before the devia-tion disappeared.

If the volatility deviation disappeared beforeexpiration, we unwound the position by buyingback the previously overvalued option at the askprice and selling the previously undervaluedamount of options at the bid price. If the deviationpersisted until expiration of the options, we calcu-lated and summed up the intrinsic values of theoptions. The total cash flows upon opening andclosing the positions (interest on cash flows wasneglected) could then be calculated.

If the method used to define the interrelation-ship of the indexes and to forecast their relative

Figure 5. OEX–NYA Relative Implied-Volatility Deviations, 3 January 1995 through 10 February 2000


3/Jan/95 1/Mar/96 12/Mar/97 7/Jan/98 20/Jan/99 10/Feb/006−

5−

4−

3−

2−

1−

0

1

2

3

4

5

6


43/Jan/95 8/Feb/96 28/Jan/97 16/Jan/98 7/Jan/99 10/Feb/00−

3−

2−

1−

0

1

2

3

4

5

6

β2


52 ©2002, AIMR®

volatility relationship is correct, the sum of the cashflows should, on average, be positive. This statisti-cal arbitrage strategy may not be profitable forevery single trade, however, for two reasons. First,the option positions were constructed to be deltaand gamma neutral under the assumption that theunderlying indexes vary according to the statisticalinferences made about their relationships. Whenthe index pairs failed to behave as predicted, theoption position gained in delta exposure, which canresult in a losing trade even though the observedimplied-volatility deviation may have disap-peared. Second, an inherent risk in statistical arbi-trage is that an observed deviation, however large,may persist or even increase. The statistical arbi-trageur must always be aware of this risk whendeciding on the size of an arbitrage trade.

Table 4 shows the results of the statistical arbi-trage strategy. Only 10 SPX–OEX statistical arbi-trage trades were triggered over the observed timeperiod; trades involving NYA options occurredmore often. The reason could be that the relation-ship between the SPX and the OEX indexes is moreobvious and thus more widely monitored by mar-ket participants.

For every index pair, the average profit/lossper trade was positive. Although some trades gen-erated losses, as expected for a statistical arbitrage

strategy, most were profitable. Because we usedonly daily closing prices and implicitly assumed anability to trade at the signal price, the profits of anactual trading strategy could be different from ourresults.

One trade needs to be singled out because ofits magnitude: As Figure 3 (volatility deviations ofthe SPX–OEX pair) shows, a significant deviationwas identified on 27 October 1997, when the SPXand the OEX lost 7.11 percent and 7.09 percent,respectively, followed by a bounce on the followingday of 4.99 percent and 5.61 percent, respectively.Clearly, in such times, the probability of a relativedeviation increases. Bid–ask spreads are obviouslywider in such times, but even if one were to doublethe spread on these days, a significant deviationwould have been identified and the ensuing tradewould have remained profitable. If we assumedthat no trading in options was possible at all duringthose two days, the average profit/loss for the SPX–OEX index pair shown in Table 4 would fall greatly.The average would be $0.61 achieved from 8instead of 10 trades (on 27 October 1997, the strat-egy called for entering into both call and put posi-tions). Nevertheless, the value from the strategywould still be significantly positive and of a mag-nitude comparable to that achieved by the othertwo index pairs.

Exhibit 1.Example Offsetting Hedge Strategy

The procedure was as follows: Assume that• Index A has 100 points, • Index B has 1,000 points, • one at-the-money call on Index A is worth $8 (35 days to expiration), • one at-the-money call on Index B is worth $170 (35 days to expiration), • the relative-volatility differential (the estimated slope coefficient, ) of Index A to Index B is 2 (for every percentage A moves,

B will move by 2 percent), and• Index A and Index B are perfectly correlated. If the implied volatility on Index B is overvalued relative to the implied volatility on Index A (meaning that the relative implied-volatility differential is higher than 2), the following position is taken: One sells 1 Index B call (+$170). Then, to render the positions point equivalent, one must buy 10 (i.e., 1,000/100) calls on Index A (because A is relatively undervalued); the result is 10 × –$8 = –$80. This point-equivalent position then has to be adjusted by the relative-volatility differential (the adjustment) to keep the offsetting hedge delta and gamma neutral. This adjustment is achieved by multiplying the long position by the relative-volatility differential (2 in this example). A net long position of –$160 and a net short position of +$170 would be the result of this trade.

β2

β2

Table 4. Profitability of Arbitrage Trades, 3 January 1995 through 10 February 2000

Index PairNumber of

TradesNumber of

Winning TradesAverage

ProfitNumber of

Losing TradesAverage

LossAverage

Time HeldTotal Average

Profit/Loss

SPX–OEX 10 9 $3.62 1 $–0.12 2.9 days $3.24

SPX–NYA 37 31 2.00 6 –1.57 9.6 1.42

OEX–NYA 28 22 1.07 6 –0.84 3.6 0.66

Notes: Trades effectuated every time a mispricing was larger than 2 percent. Profit per profitable position and average profit perarbitrage trade were normalized to a position of one option of the first-named index of the pair.



Of the two potential risks involved with thisstrategy (persistence of volatility deviation andvarying relationship of indexes), persistence, sur-prisingly, never materialized. The theoretical devi-ations seem to have been strongly mean reverting,with the mean defined as the historical relationshipbetween relative volatility levels. No significantrelative deviation lasted longer than 35 days.

The experience in this study of the other poten-tial risk, however, illustrates the difficulty of mak-ing exact statistical inferences about the futurerelationships of stock indexes. All the losing posi-tions were caused by indexes varying in a mannerdifferent from what was predicted by a linear rela-tionship. Those positions ended up in the loss col-umn even though the implied-volatility deviationdisappeared. The reason is that we took the optionpositions in such a way as to ensure they remaineddelta neutral as long as the indexes varied relativeto one another according to their past relationship.If they ceased to vary this way during the time theoption position was open, a delta exposure arose,introducing a new risk. Because the chosen indexesare strongly correlated, however, and because ofthe robust statistical inferences (rolling regressionswith min–max boundaries), the impact of this riskwas kept to a minimum.

Although the amount and magnitude of vola-tility deviations tended to increase when the under-lying stock indexes were volatile, the majority ofsignificant deviations recorded were not linked tosuch extraordinary market conditions.

The results of Table 4 do not account for trans-action costs. Even retail transaction costs would below, however, compared with the arbitrage profitsdisplayed in Table 4. Commissions are usuallycharged per traded contract, with one contract typ-ically representing 100 options. Assuming rates of$1.25 per contract ($0.0125 per option),13 the resultis transaction costs of approximately $0.05 perroundtrip arbitrage trade (4 × $0.0125).14 Such lowcosts would have enabled even the retail investorto take advantage of the observed deviations. Pro-fessional market participants, who benefit fromlower transaction costs, could have profited to aneven greater extent.

Some might argue that our simulated arbitragestrategy might not be fully replicating real marketconditions because the option prices calculatedfrom the implied-volatility data are not exact. Forexample, because the option prices used for simu-lated arbitrage trades were theoretical prices, notreal market prices, the simulated arbitrage oppor-tunities might not have been, in fact, real arbitrageopportunities. But our analysis was based onimplied volatilities computed from real market

option prices. Therefore, by computing optionprices from implied volatilities again, we were per-forming a simple reverse transformation withoutdistorting actual arbitrage opportunities.

Nevertheless, the statistical arbitrage opportu-nities we detected might not have been implement-able in some cases for other reasons. First, theoption prices were calculated from the closingprices of the indexes, which as Harvey and Whaleypointed out, is an imprecise method. Becauseoption markets close 15 minutes after stock mar-kets, calculated option prices may be different fromreal option prices. Second, we corrected the impliedvolatilities used to calculate the bid and ask optionprices by a bid–ask implied-volatility spread.Although we chose this spread to be conservativelylarge, compared with normal market circum-stances, a more exact approach would be to use theactual intraday option bid and ask prices for thesimulated trades. For example, bid–ask spreadsmay occasionally be much larger than usual, inwhich case, although an arbitrage opportunitywould be detected, the arbitrage could not beimplemented (or would at least be less profitablethan we found). Third, when an arbitrage opportu-nity is detected, an arbitrage trade taking advan-tage of this opportunity has to go through at thenext price. That price may be different from theinitial price and may no longer allow arbitrage.Furthermore, an arbitrage trade consists of twolegs. Often, these legs cannot be executed at thesame time. We assumed that trading at the signalprice is possible. Thus, our results regarding theprofitability of the arbitrage strategy should beinterpreted with caution.

Under the assumption that the results in Table4 could have been attained in practice, can weconclude that option markets are inefficient? Notreally. Statistical arbitrage is not riskless. It returnsa profit during normal market situations but mayresult in losses during unusual circumstances. Per-haps the unusual market circumstances nevermaterialized in our sample of only a few dozenarbitrage transactions. To determine conclusivelywhether the profits of a statistical arbitrage strategyarise from market inefficiency, a much larger sam-ple of transactions would be needed.

ConclusionWe illustrated and tested an aspect of option mar-ket efficiency that cannot be tested by using con-ventional exact arbitrage pricing relationships. Weused options on several pairs of highly correlatedstock indexes to establish the efficiency with whichmarket participants value the relative risk (implied


54 ©2002, AIMR®

volatility) of these indexes. We calculated bound-aries on the basis of the historical covariance of theindex-pair volatilities. Whenever the relativeimplied volatilities violated such a boundary, arelative implied-volatility deviation was identifiedand a mispricing suspected.

Each of the three investigated index pairs wasfound to have had a large number of such devia-tions, which seemed to be concentrated in phasesduring the time of the study. After we includedbid–ask spreads in the calculations, however, onlya small fraction of the deviations could be identifiedas an arbitrage opportunity.

We thus found some indication that deviationsoccasionally occur that can be exploited profitablyby statistical arbitrage. Their number is small, how-ever, and the possibility exists that not all of thearbitrage opportunities we detected could havebeen executed because of various arbitrage barriersin certain market situations and our assumptionthat the arbitrage trade could go through at the

signal price. To test the presence and persistence ofpotential relative mispricings and to determineconclusively whether a statistical arbitrage strategybased on relative implied volatility could havebeen implemented profitably in the period westudied would require further research using intra-day tick data.

An interesting area for future research wouldbe an analysis of why relative-volatility deviationstend to appear in clusters. Common factors causingsuch deviations could possibly be identified.

Additionally, other highly correlated markets,such as certain currency or commodity markets,could be investigated with the methodology pre-sented here.

All data used in this study were provided by BloombergProfessional information service through Wegelin andCompany. We thank David Rey and Maud Capelle forhelpful comments.

Notes1. Unless indicated otherwise, the term “arbitrage” here refers

to statistical arbitrage.2. Although our choice of the minimum limit for the correla-

tion coefficients is somewhat arbitrary, it was guided bytwo factors. First, we wanted the coefficient to be as high aspossible, but second, we needed several indexes to be leftin the sample for further examination.

3. Because the regressions were used to establish the relation-ships between the index pairs to allow forecasts of relativevolatility, an optimal sample size had to be found. Samplesizes of 3, 6, and 12 months were tested. Sample sizes of 3months were unsatisfactory because they resulted in com-paratively unstable relationships between index pairs. Thecause could be “seasonal” relationships, which wouldprove to be unreliable in forecasting future relative devel-opments in a different “season.” Relationships betweenindex pairs do change structurally over time, however, andthe longer the sampling period, the slower the reaction ofthe forecast to this change. The 12-month sampling periodproved to be too slow in adapting to such changes. Thus,we used a 6-month sample, which was long enough not tobe seasonally biased but short enough to remain somewhatflexible as to structural changes.

4. Deciding how many trading days to use as a subsample toestablish such maximum variations is a delicate matter. Thelonger the sampling period, the more conservative theresult, because a large variation will widen the boundariesfor a long period of time. The boundaries are designed toreflect current variation, however, not the variation longpast that is no longer reflected in the market situation. So,we chose 250 trading days, approximately one trading year,as an informal compromise between conservatism and therelevance of information.

5. For example, assume that during the 250 trading dayspreceding time t of the ABC–XYZ index-pair sample, thelargest change of the estimated slope coefficient over any25-trading-day period was x percent. At time t , the esti-mated slope coefficient is β2t; thus, at that time, the lower

boundary around this estimated slope coefficient would beβ2t – (β2t x%) and the higher boundary would beβ2t + (β2t x%).

6. For an analysis of such a relationship, see, for example,Ncube (1994).

7. In the context of statistical arbitrage, a mispricing can neverbe identified with certainty. Consequently, a statistical arbi-trage trade attempting to take advantage of a suspectedmispricing is not guaranteed to be profitable.

8. The terms “undervalued” and “overvalued” are used herein reference to the relative implied volatility valuation ofthe indexes, not to their values measured in index points.

9. Bid–ask spreads were recorded on a sample of days whenrealized volatility was relatively high and a sample of dayswhen realized volatility was relatively low so as to give arealistic reflection of such spreads in different market con-ditions. In relatively volatile markets, bid–ask spreads ofapproximately 1 percent were recorded for near-month at-the-money options for all three indexes, with slightly lowerreadings for the OEX. In less volatile markets, the spreadstended to narrow, so the decision to use 1 percent as anoverall spread in the calculations can be regarded as aconservative assumption.

10. Point equivalency refers to the index points of the underly-ing indexes.

11. Because the idea behind this offsetting hedge strategy wasan approximate immunization of the entire position againstall parameter changes, except for the normalization of theobserved implied volatility mispricing, taking positionswith equivalent “weight” and adjusting them to reflect therelative volatility differential was crucial.

12. In this study, we implicitly assumed that trading at thesignal prices was possible (i.e., that the mispricing persistedlong enough to allow a trade). Because only end-of-dayvolatility data were available for the study, however,whether the persistence in the mispricings was sufficient toallow for the implementation of a trade remains unclear. Atwo-tick persistence (the first tick for the signal and the



second for the trade) may be sufficient to implement sucha trade.

13. Rates charged by the U.S. discount broker Ameritrade as ofMarch 2000.

14. The opening and closing of the offsetting trades involves thebuying and selling of approximately two options (pointequivalency and adjustments would increase or decreasethis number somewhat).

ReferencesAckert, L., and Y. Tian. 1998. “The Introduction of Toronto IndexParticipation Units and Arbitrage Opportunities in the Toronto35 Index Option Market.” Journal of Derivatives, vol. 5, no. 4(Summer):44–54.

———. 1999. “Efficiency in Index Options Markets and Tradingin Stock Baskets.” Working paper series, Federal Reserve Bankof Atlanta, vol. 99, no. 5.

Chung, P. 1991. “A Transactions Data Test of Stock IndexFutures Market Efficiency and Index Arbitrage Profitability.”Journal of Finance, vol. 46, no. 5 (November):1791–1809.

Fama, E. 1965. “The Behavior of Stock-Market Prices.” Journal ofBusiness, vol. 38, no. 1 (January):34–105.

Figlewski, S. 1984. “Hedging Performance and the Basis Risk inStock Index Futures.” Journal of Finance, vol. 39, no. 3 (July):657–669.

———. 1989. “Options Arbitrage in Imperfect Markets.” Journalof Finance, vol. 44, no. 5 (December):1289–1311.

Gatev, E., W. Goetzmann, and K. Rouwenhorst. 1999. “PairsTrading: Performance of a Relative Value Arbitrage Rule.”Working Paper 7032, National Bureau of Economic Research(March).

Gould, J., and D. Galai. 1974. “Transactions Costs and theRelationship between Put and Call Prices.” Journal of FinancialEconomics, vol. 1, no. 2 (July):107–129.

Harvey, C., and R. Whaley. 1991. “S&P 100 Index OptionVolatility.” Journal of Finance, no. 46, no. 4 (September):1551–61

Johnson, R., C. Zulauf, S. Irwin, and M. Gerlow. 1991. “TheSoybean Complex Spread: An Examination of Market Efficiencyfrom the Viewpoint of the Production Process.” Journal of FuturesMarkets, vol. 11, no. 1 (February):25–37.

Klemkosky, R., and B. Resnick. 1979. “Put–Call Parity andMarket Efficiency.” Journal of Finance, vol. 34, no. 5(December):1141–55.

Merton, R. 1973. “Theory of Rational Option Pricing.” BellJournal of Economics and Management Science, vol. 4, no. 1(Spring):141–183.

Ncube, M. 1994. “Modeling Implied Volatility with OLS andPanel Data Models.” Discussion Paper No. 194, LSE FinancialMarkets Group Discussion Paper Series, London.

Neal, R. 1996. “Direct Tests of Index Arbitrage Models.” Journalof Financial and Quantitative Analysis , vol. 31, no. 4(December):541–562.

Poitras, G. 1997. “Turtles, Tails and Stereos: Arbitrage and theDesign of Futures Spread Trading Strategies.” Journal ofDerivatives, vol. 5, no. 2 (Winter):71–88.

Sofianos, G. 1993. “Index Arbitrage Profitability.” Journal ofDerivatives, vol. 1, no. 1 (Fall):6–20.

Stoll, H. 1969. “The Relationship between Put and Call OptionsPrices.” Journal of Finance, vol. 24, no. 5 (December):801–822.

β2

Documents

Relative Implied-Volatility Arbitrage with Index Options · 2016. 2. 27. · Statistical arbitrage, however, be it in deriva-tives or other markets, has received surprisingly little