Empirical tests of efficiency of the Italian index options marketdirectory.umm.ac.id/Data...

Ž .Journal of Empirical Finance 7 2000 173–193www.elsevier.comrlocatereconbase

Empirical tests of efficiency of the Italian indexoptions market

Laura Cavallo a,b,), Paolo Mammola c,1

a Prime Minister’s Office, Rome, Italyb UniÕersity of Rome ATor Vergata,B Rome, Italy

c Citibank Milan, N.A. Foro Bonaparte 16-20123 Milan, Italy

Accepted 9 May 2000

Abstract

The purpose of this paper is to investigate whether the Italian index option contractŽ . Ž .MIBO30 , recently introduced in the Italian Derivatives Market IDEM , is efficient. Twodifferent methods are used in the analysis. First, we tested on the Italian index optionmarket the validity of the put–call parity conditions, extended to account of transactioncosts associated with replicating and establishing a short hedge on the index. We find thatthe significant deviations from put–call parity are not exploitable when all transaction costsare accounted for. Since the put–call parity is just a weak test for market efficiency, wefurther investigate the possibility to generate profitable positions through the simulation ofan ex-post volatility hedging strategy. This strategy does not allow for systematic abnormalreturns, supporting the hypothesis that option prices are consistent with market efficiency.q 2000 Elsevier Science B.V. All rights reserved.

JEL classification: G13; G15Keywords: Lower-boundary; Put–call parity; Volatility trading; Implied volatilities; Transaction costs

) Corresponding author. Department of Economic Affairs, Prime Minister’s Office, via Barberini 47,00187 Rome, Italy. Tel.: q39-347-6079633; fax: q39-06-23313148.

Ž . ŽE-mail addresses: [email protected] L. Cavallo , [email protected] P. Mam-.mola .

1 Tel. : q39-335-5827738; fax : q39-02-86474362.

0927-5398r00r$- see front matter q2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0927-5398 00 00010-4

( )L. CaÕallo, P. MammolarJournal of Empirical Finance 7 2000 173–193174

1. Introduction

The Italian option on index MIB30 contract or AMIBO30B introduced in theŽ .Italian Derivatives Market IDEM on November 1995 represents one of the most

important steps aimed to improve the efficiency and the liquidity of the Italianfinancial markets. Trading on the IDEM started on November 1994 with theintroduction of the future contract on the same index, the Mib30. The strongsuccess of this contract, which enjoyed a rapid development and is now placedamong the five most traded future contracts in Europe, convinced the ItalianAuthorities to introduce an option contract on the index, the MIBO30, and later onŽ .in February 1996 on single stocks, the ISO a . This paper contains an empiricalanalysis directed towards an investigation of whether the MIBO30 option marketis efficient.

For the purposes of this study, the market is efficient if it does not presentarbitrage opportunities. Since Black and Scholes published their article on optionpricing in 1972, several theoretical and empirical works have been written onoption pricing. However, most of these studies have been conducted on theChicago Board Option Exchange and very few studies have been undertaken totest the efficiency of other option markets. In particular, the Italian Option marketis quite recent and is scarcely investigated2.

To analyse the efficiency of the Italian Index Option Market, the Black andŽ .Scholes 1971,1972 model is probably the simplest valuation model, and evidence

from dealers operating in the market indicates that it is widely used. However,there are several problems in carrying out empirical research based on theBlack–Scholes as on any option pricing models. The first problem is that anystatistical hypothesis about how options are priced has to be a joint hypothesis to

Ž . Ž .the effect that i the model is valid and ii markets are efficient. To distinguishbetween the two hypotheses of market efficiency and model validity, one of thetwo has to be taken as an assumption. A second problem concerns the choice ofthe best estimate of stock price volatility. A third problem is to ensure that data onthe stock price and option price are synchronous.

The current study attempts to overcome the above difficulties in three substan-tive ways. First of all, it uses a very high quality data source, which contains notonly transaction prices but also quoted bid and ask prices. Second, in the first partof the work it employs a test of market efficiency, the put–call parity test, whichdo not rely for its validity on the restrictive Black–Scholes assumptions. More-over, this test can easily be manipulated and extended to take account of the

2 Ž .Two previous studies Barone and Cuoco, 1989, 1991 investigated the premium contracts on theItalian stock exchanges. Trades on premium contracts were substituted by trades on options few months

Ž .after the introduction of contracts. For an analysis of the MIBO30 see Cavallo 1998 and Cavallo et al.Ž .1999 .

( )L. CaÕallo, P. MammolarJournal of Empirical Finance 7 2000 173–193 175

frictions of the market such as transaction costs, so that new conditions are derivedand subjected to empirical test. Since the put–call parity is only a weak test ofmarket efficiency, results obtained are further investigated in the second part of thework, where an ex-post hedging strategy is simulated to verify the possibility toexploit the mispricing evidenced by comparing actual option prices with Black–Scholes prices. This dynamic hedging strategy, taking into account transactioncosts and relaxing some of the assumptions of the Black–Scholes model, allowsverification of the hypothesis that mispricings are due to an inaccuracy of themodel rather than to the inefficiency of the market.

The paper is organised as follows. Section 2 describes the Italian option marketand the data set used. In Section 3, the put call parity conditions in the presence oftransaction costs are derived and subjected to empirical test on the Italian marketusing infra-day synchronous option and index prices. In Section 4, differentmeasures of volatility are derived and used to simulate a volatility trading strategy,in order to verify the possibility to realise systematic abnormal returns on theItalian option market. Some concluding remarks are offered in Section 5.

2. Market and data

2.1. The AMIBO30B market

The empirical tests of this work are based on data on index options, MIBO30,recently introduced in the IDEM. The index option contracts traded on IDEM arebased on the MIB30 index, which has been proven to be a reliable indicator of theItalian market. The MIB30 index is a capitalisation-weighted index that comprisesthe 30 most liquid and highly capitalised shares traded on the Italian market. Theshares in the index account for over 72% of total market capitalisation, and almost

Ž .75% of trading volume. Its correlation with the Italian general index MIB isabove 0.99. The contract size is set at 10 000 ITL for each point of the index.MIB30 options are European style, which means they can only be exercised atexpiration. The options have cash settlement.

At any given time there are options available for at least five expiration dates:Žthe three nearest expirations dates of the quarterly cycle March, June, September

.and December , which correspond to the expiration dates of the future contract,Žand the two nearest monthly expirations a monthly expiration corresponds to one

.of the quarterly expirations . The expiration day is the third Friday of theexpiration months. Five exercise prices are available for each contract month.They are settled in fixed increments of 500 index points, i.e. 14 500, 15 000,15 500. Trades are matched in real-time by the electronic trading system andguaranteed by the Cassa di Compensazione e Garanzia, the clearing house for theItalian market.


2.2. Data and methodological issues

In this section we describe and motivate some methodological issues used inthe analysis. The empirical tests were carried out only on options with a maturityof 1 month. The main reason of this choice is that the market is still very youngand the thinness of trading makes prices on other expirations not representative.

Ž .The risk-free rate was estimated by the European interbank offered rate Euriborpublished daily by the Sole 24 ore.3 This is the same interest rate used by the

Ž .CONSOB the Italian Securities and Exchange Commission to vigil the dailybehaviour of option prices and the behaviour of market makers in setting the bidand ask quotations. It is then likely that market makers refer to this interest rate inorder to verify whether their quotations are consistent with the theoretical value ofthe option. The value of the Euribor published in the Sole 24 ore is the ask rate;

Ž .the bid interest rates can be obtained subtracting 1r8 0.125% from the ask rates.We eliminated options with fewer than seven calendar days to expiration. Mostdividend payments of the stocks included in the index are concentrated in themonths of May and June. However, since dividend payments occur after theexpiration day of the option, they do not affect the options with 1-month maturityused for the empirical tests.

When, as in our case, the underlying security is not a single option but anindex, arbitrageurs face two alternatives: replicate a portfolio representative of the

Ž .securities included in the index, or use the futures on the index Fib30 . Wedecided to use the first alternative and to replicate the index. Although this choicemay appear more costly, it must be noticed that using the future the trader wouldincur the Abasis risk,B when the expiration of the future does not correspond tothat of the option. Moreover, the MIB30 index includes only 30 securities and isnot very difficult to replicate. The high movements observed in the prices of thesecurities included in the index in proximity to the maturity of the options supportthe assumption that replicating the index is a common practice among traders.Another issue accounted for in the analysis is the cost of short selling the index. Ifthe arbitrage involves a short hedge in the index, the arbitrageur should assume afurther cost represented by the cost of securities lending. The market for securitieslending in Italy is an Over the Counter Market. The characteristics of the contractsare not standardised and can be designed to best satisfy the necessities of the

4 Ž .arbitrageur. Because data on the securities lending market interest rates repo are

3 To match the interest rate to the period of expiry of the options, we used a Relevant Interest Rateobtained by linear interpolation of interest rates for different time horizons.

4 A securities lending contract consists in a double lending: the lender give the borrower thesecuritiesw, and the borrower deposit the proceeds of the short selling to the leader as a gaurantee. Thelender must pay on the deposit an interest rate that is generally lower than the market risk-free rate,because of the opportunity cost for renouncing availability of the securities. The cost of securitieslending is respesented by the differrence between the market risk free rate received on the deposit.


Ž .not available for it is an OTC market , for the empirical tests we used as a proxythe AbidB interest rate. We also account for costs due to the cash settlementprocedure, which distinguish index option from stock and commodities option.The only relevant difference in the results is that the trader has to pay theadditional commission cost to close on the market at maturity the position open onthe underlying asset.

3. Put–call parity and transaction costs

In this section, the put–call parity conditions are derived and subjected toempirical testing on data on index options traded on the IDEM.

The data used to test the put–call parity conditions consist of infra-day prices,captured every 15 min from 10 a.m. to 5.30 p.m. from July 29, 1996 to February18, 1997 for a total of 3642 observations. Data include ask price, bid price and

Ž .transaction price of at the money options call and put and on the MIB30 futures.In our study, we estimated the MIB30 bid and ask prices applying to thetransaction price the same spread observed for the future on the index MIB30.5

Ž .The put–call parity model was first developed by Stoll 1969 , and thenŽ .extended and modified by Merton 1973 . The well-known basic put–call parity

condition, when there is no dividend payment, is the following:

CsPqSyKeyr ŽTyt . . 1Ž .

Ž .The non-arbitrage conditions can be derived from Eq. 1 establishing twoportfolios, both of which result in zero pay-off at expiration. The first portfoliorepresents a long-hedge position, because it involves taking a long position in theunderlying share. The second portfolio represents a short-hedge position. In theabsence of dividends and transaction costs, those conditions are, respectively:

CyPySqKeyr ŽTyt .F0, 2Ž .

PyCqSyKeyr ŽTyt .F0. 3Ž .

Tests based on the put–call parity conditions, though weak tests of marketefficiency, have been widely used in the empirical literature. Some of these studies

Ž .basically supported the theory Nisbet, 1992; Klemkosky and Resnick, 1979 , butŽsome inefficiency where also found to exist Stoll, 1969; Gould and Galai, 1974;

.Evnine and Rudd, 1985; Finucane, 1991 .

5 We are aware that the spread on the stocks and on the index is generally wider than that on thefuture. However, the error induced in the results by this approximation would be less then that inducedusing, as in most empirical studies on index options, the index transaction prices.


In this section, we extend the put–call parity conditions to take into accountfrictions not considered in previous tests, in particular, the costs of replicating theindex and the cost of short selling. Since the option selected for the analysis arethe most liquid and the spread is precisely bounded by the Italian marketregulation, we may be confident that the bid and ask prices used to account for thespread effectively corresponds to a transaction.6 The availability of infra-dayprices ensures a good synchronisation between the option prices and the underly-ing security. Since the Italian Market trades are matched in real-time by theelectronic trading system, it is reasonable to assume that the arbitrage strategycould be implemented at the same prices prevailing when the profit opportunitywere identified. This allows to overcome one of the main problems related toput–call parity tests, that is the possibility that prices used to identify the arbitrageopportunity do not represent tradable prices. The put–call parity conditions are

Ž Ž ..obtained as follows. If the expression representing a long hedge Eq. 2 is notverified, a profit can be made by purchasing the index at its ask price, the put at itsask price, and selling the call at its bid price. The initial investment can be

Ž .financed at the risk free rate. In the case of a short hedge, if condition 3 is notverified, the strategy to profit from the mispricing consists in buying the call,selling the put and short selling the portfolio which replicate the index. Accountingfor all transaction costs involved in the implementation of these strategies, thenon-arbitrage conditions become, respectively, for a long and a short hedge:

C yP y I qKeyr ŽTyt .FTC qTC qTC tŽ .bid ask ask wc bp bi

qTC T eyr ŽTyt .qTk , 4Ž . Ž .si

P yC q I yKeyr pŽTyt .FTC qTC qTC tŽ .bid ask bid wp bc si

qTC T eyr ŽTyt .qTk , 5Ž . Ž .bi

where: TC and TC are, respectively, the cost of writing a call or a put; TCwc wp bc

and TC are the cost of purchasing the call or the put; TC and TC are the costbp bi siŽof purchasingrselling the index when the position is closed at maturity T we

.calculate the present value of the commission cost and Tk are the clearingcommission on the option, usually very small and omitted from the empirical tests.

Note that in the short-hedge condition the present value of K is calculatedŽ .using the repo rp rate. This allows accounting for the assumption that the cost of

the trading will be financed by the funds deposited with the lender of thesecurities.

6 Ž .This contributes to overcome the objection raised by Phillips and Smith 1980 to studies based onbid and ask prices. As the authors pointed out, the average bid–ask spread generally overstatestransaction costs associated with the spread, and most trades occur inside the spread. In this case, thebid and ask quotes would not represent the effective supply and demand prices, and any tests based onbid and ask prices would be biased in favour of the market efficiency hypothesis.


Table 1 summarises the results of the put–call parity tests under differenthypothesis about transaction costs. Cases I and II represent the tests conductedusing, respectively, transaction prices and the bid and ask prices. For both cases Iand II, the table reports three columns with the results obtained under threedifferent assumptions on the level of commission costs. In the first column, tomake the results comparable with studies that ignore transaction costs, these costs

Ž .were omitted Tcs0 . In the second and the third column, we considered thelevel of transaction costs incurred, respectively, by an occasional investor and byan arbitrageur. The position of an arbitrageur involves a lower level of transactioncosts.

Referring to the average market commission, for an option contract we used atransaction cost of 10 000 ITL for the arbitrageur and 15 000 ITL for an individualinvestor. Commissions on replicating the index are expressed in index points, andare 5 and 10 index points, respectively, for an arbitrageur and an individualinvestor. Clearly, the larger the transaction costs, the wider the band within whichprices can fluctuate without creating arbitrage opportunities.

Consistently with expectations, the number of hedges, which would haveprovided a profit opportunity decreases substantially when commission costs andthe bid ask spread are included. When we consider the level of transaction cost

Žincurred by an individual investor, together with the spread bid–ask the last.column of the table , the possibility to realise a profit from the hedge become

Žirrelevant profit opportunities are revealed only in 2% of the simulated long and.short hedges . However, this result is not so surprising, as this is only a weak test

Table 1Number and average values of profitable hedges for cases I and II and different levels of transactioncosts

Profitable hedges Case I — transaction prices Case II — bra spread

Tcs0 TcsTc1 TcsTc2 Tcs0 TcsTc1 TcsTc2

Long hedge Number 1798 1078 546 589 213 82% of the sample 49 30 15 16 6 2

)Average value 19.4 16.34 16.23 13.04 14.45 20.11z 43.9 27.55 16.94 18.91 9.698 6.576

Short hedge Number 1780 1083 596 519 193 70% of the sample 49 30 16 14 5 2Average value 20.65 18.34 18.33 13.62 15.67 23.71z 43.7 29.95 20.8 16.03 8.387 5.567

Case I: Long hedge: Cy P y Iq Keyr ŽTy t .FTc; Short hedge: P yCq Iy Keyr pŽTy t .FTc; CaseII: Long hedge: C y P y I q Keyr ŽTy t .FTc; Short hedge: P yC q I y Keyr pŽTy t .Fbid ask ask bid ask bid

Tc; where Tc is the total transaction cost. In the first column of the table, Tcs0, in the second columnTc1 represents the lower transaction cost level incurred by an arbitrageur, in the third column Tc2 isthe transaction cost incurred by an individual investor.

)Average values are expressed in index points.


of market efficiency. What is interesting to note in the results is that, in contrastŽ . Ž .with the findings of Klemkosky and Resnick 1980 on CBOE and Nisbet 1992

Ž .on the London Traded Option Market LTOM , we do not observe a larger numberof profitable short hedges than profitable long hedges. Different elements can

Ž .contribute to interpret this result: i in contrast with other markets, in the ItalianŽMarket it is not particularly difficult or costly to establish short hedge positions at

. Ž .least for the 30 stocks which constitute the index ; ii previous studies of put callparity, including the ones mentioned above, have not explicitly allowed fortransaction costs associated with the securities lending market when the strategyinvolve a short position in the underlying security. Another explanation is thatmarket makers and options dealers prefer trading on the future contract on theMIB30 index rather than short selling the portfolio, which replicates the index.Moreover, the main insight we can derive from the way we presented the results isthat tests of market efficiency critically depend on the treatment of transactioncosts. This is particularly true for tests based on an option-pricing model, whichrely for their validity on continuous portfolio rebalancing. This problem will beinvestigated further in Section 4, where a stronger test of market efficiency isimplemented.

4. Volatility trading and market efficiency

4.1. The Õolatility trading strategy

The objective of this analysis is to test the efficiency of MIBO30 prices,through the implementation of a volatility trading strategy attempting to exploitdeviations between actual option prices and theoretical prices. More precisely, avolatility trading consists of formulating trading strategies on the basis of one’sown anticipation of the future volatility of the security underlying the option

Ž .contract. To expect a future volatility higher lower than that currently registeredŽ .on the market, corresponds to foreseeing a rise lowering in the price of theŽ .options. The related strategy consists in purchasing selling the options.

The first studies aimed at verifying the efficiency of the options market basedon an analysis of the deviations between theoretical and effective prices were

Ž . Ž .conducted by Black and Scholes 1972 , Galai 1977 , and MacBeth and MervilleŽ .1979 . These authors examined the possibility of realising above-normal returns

Žby purchasing options AundervaluedB by the market under the assumption that the.theoretical price established by the model is the Afair priceB and selling overval-

ued ones. The results of these studies show that such a strategy may lead to therealisation of profits that are significantly higher than the risk-free rate. However,the authors emphasise that the inclusion of transaction costs could diminish the


Ž .possibility of realising such extra-returns. A study of Joo and Dickinson 1993analyses the efficiency of the European Option Market, developing a dynamichedging strategy that takes into account the effects of bid and ask spread costs.

Ž .A study of Xu and Taylor 1995 examines the conditional volatility and theinformational efficiency of currency options.

4.2. Data and Õolatility estimates

A sample consisting of daily data was used for the present analysis. Thissample differs from the one used to test the put–call parity condition, not only forthe period and the frequency of the data, but also because in this sample, apartfrom Aat-the-moneyB options, also AinB and Aout-of-the-moneyB options areavailable. The period surveyed is from December 1996 to September 1997.Synchronisation is essential to obtain an accurate estimate of the implied volatilityand to avoid distortion of the results of the hedging simulations. This is the reasonwhy this sample has been constructed through a very accurate and time-consumingprocedure. From the data on the index, available minute by minute, we selectedthe price at the same time every day. On a daily basis, from the set of prices of allput and call options contracts effectively concluded, continuously quoted, we

Žselected options with three different strike price values the strikes at-the-money,.in-the-money and out-of-the-money closest to the central strike concluded as

close as possible to the time the index was selected.7

The dynamic strategy was repeated using historical and implied estimates forŽ .future volatility. The historic volatility or Historic standard deviation, Hsd was

estimated as the moving average of standard deviation of the logarithmic differ-ences in the index daily prices. The estimation period is 20 days, which corre-sponds to the forecast period, the option time to maturity, measured in tradingdays.8 Annual measures of volatility are obtained multiplying the daily values of

'the standard deviations by 250 .

7 Ž .Harvey and Whaley 1991 suggest that the use of more than one value for every day partlymitigates distortion of the estimate of volatility due to considering the effective transaction prices,ignoring the bid and ask prices. In order to reduce measurement errors, we selected also a secondobservation every day to use to calculate implied volatilities. Calculating implied volatilities using calland put prices on several strikes, in turn calculated as the average of two daily values, we believe thatwe have considerably reduced the measurement problems evidenced by Harvey and Whaley.

8 The opportunity to utilise trading days rather than calendar days when calculating volatility isŽ .recommended in the well-known observation Fama, 1965; French, 1980; French and Roll, 1986 that

on days when the stock exchange is open, volatility is noticeably higher than it is on days when thestock exchange is closed, and that, in a certain sense, the trading itself seems to cause some of thevolatility.


Implied volatility estimates are obtained as a weighted average of singleŽ .implied volatilities. Following Chiras and Manaster 1978 , the weights used are

the elasticity of the individual prices to volatility:

n EW sj jISDÝ j

Es Wj jjs1WISDs ,n EW sj jÝ

Es Wj jjs1

where WISD is the AWeighted Implied Standard DeviationB for the index Mib30on the observation date, ISD the AImplied Standard DeviationB of option j, W isj

Ž .Ž .the option price and dW rds s rW is the price elasticity of option j withj j j jŽ .respect to the index standard deviation s . We calculated three measures of

Ž .implied volatility: the first WISDc is constructed using the three call optionŽ . Ž .prices respectively AatB, AinB and AoutB of the money . The second WISDp is

Ž .computed using the three put option prices. The last WISDpc is a weightedaverage of the first two. Some descriptive evidence on the time series properties ofthe implied volatilities is provided in Table 2.

All implied volatilities evidence persistence in the level of volatility, all theseries presenting significant positive serial correlation.9 The decline of autocorrela-tions at longer lags is an indication of stationarity. Looking at differenced series,estimated coefficients show negative serial correlation at lag 1 for all implied

Ž .volatility estimates. This result, consistent with the findings of French et al. 1987Ž .on S&P 500 volatility and Harvey and Whaley 1992 on S&P 100 index options,

may indicate mean reversion and predictability in volatility changes.10 The tableshows that the average implied volatilities of the calls are always significantlylower than the average volatilities of the puts.11 This result, also evidenced in Figs.

Ž .1 and 2, is consistent with the finding of Harvey and Whaley 1992 on impliedŽ .volatilities of S&P100 index options, and of Gemmill 1996 on implied volatili-

ties of the FTSE 100 index options. Fig. 1 presents the smile of implied volatilitiesfrom call and put options. Both smiles show a left-skewed pattern around the

Ž .at-the-money strike price k . This result is consistent with expectations, con-firmed by the literature on asymmetric GARCH models: as the market falls, stock

9 We can observe that the implied volatility series derived from in the money call options presents asensibly lower level of autocorrelation, which tend to disappear at lag 3. A possible explanation is thatin the money call options are more likely to be affected by measurement problems, as we will precisein discussing volatility trading strategy results.

10 Ž .Harvey and Whaley 1992 underline that negative serial correlation is only a weak evidenceagainst the hypothesis that volatility changes are unpredictable. In fact, it may be spuriously induced byasynchronous observation of the index and the option and by the bid ask spread.

11 Mean of the difference tests are not reported in the table, but are available on request.

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Table 2Summary statistics and autocorrelations for the index MIB30 options implied annualised volatilities, based on daily data for the period December 1996,through September 1997

Mean Standard Min Max Autocorrelationsdeviation r r r r r r r r1 2 3 4 5 10 20 30

( )Annualised call options implied Õolatilities %At the money levels 21.639 3.237 15.830 36.370 0.805 0.76 0.701 0.632 0.565 0.432 0.121 0.002

18 difference 0.025 2.009 y13.260 14.010 y0.39 0.037 0.027 y0.012 y0.087 y0.039 y0.087 0.136In the money levels 22.258 4.301 12.640 47.270 0.229 0.193 0.07 y0.167 y0.083 0.007 0.02 0.019

18 difference 0.034 4.453 y 24.590 26.180 y0.476 0.056 0.073 y0.208 y0.051 y0.118 y0.071 0.028Out of the money levels 21.473 3.125 15.710 32.760 0.895 0.838 0.79 0.727 0.658 0.497 0.129 y0.03

18 difference 0.024 1.408 y8.930 9.740 y0.241 y0.024 0.059 0.02 y0.096 y0.004 y0.067 0.056

( )Annualised put options implied Õolatilities %At the money levels 22.185 3.151 14.900 33.010 0.829 0.808 0.77 0.735 0.685 0.505 0.147 y0.08

18 difference 0.030 1.823 y8.690 9.980 y0.449 0.045 y0.009 0.039 y0.037 y0.036 0.065 y0.102In the money levels 21.776 3.378 15.960 30.620 0.816 0.769 0.769 0.694 0.653 0.536 0.151 y0.078

18 difference 0.050 1.996 y8.280 7.710 y0.383 y0.121 0.213 y0.106 y0.119 0.135 y0.028 y0.108Out of the money levels 22.853 2.951 17.270 30.320 0.861 0.796 0.787 0.75 0.696 0.552 0.22 0.028

18 difference 0.023 1.544 y6.550 7.350 y0.271 y0.199 0.095 0.065 y0.093 0.008 0.008 y0.066

Weighted implied ÕolatilitiesWISDc levels 21.718 3.365 15.730 35.280 0.816 0.769 0.749 0.671 0.64 0.443 0.086 y0.024

18 difference 0.025 2.025 y11.890 12.830 y0.378 y0.07 0.155 y0.131 0.033 y0.077 y0.035 0.037WISDpc levels 22.600 2.926 17.180 30.290 0.876 0.827 0.807 0.782 0.737 0.569 0.2 y0.038

18 difference 0.026 1.440 y6.720 6.900 y0.313 y0.112 0.014 0.078 y0.096 0.082 0.012 y0.105WISDp levels 22.290 2.956 16.780 30.100 0.905 0.868 0.853 0.791 0.761 0.559 0.174 y0.015

18 difference 0.027 1.258 y 5.240 5.400 y0.319 y0.111 0.249 y0.185 0.075 y0.095 0.031 0.011

Weights used to calculate weighted implied volatilities are the elasticities of the single options to volatility. WISDc and WISDp are obtained, respectively,Ž .from the three call and put option prices the at-the-money price and the prices of the two option with strike price around the at the money , WISDpc is

obtained using the three call prices and the three put prices.


Fig. 1. Volatility smiles. The figure evidences the smile for one-month options. K denotes the Aat theŽ . Ž .money strike price,B k q500 and k y500 denote the exercise prices which are, respectively, 500

index points above and below this price.

returns become more volatile.12 Fig. 2 evidences that term structure of impliedvolatilities for call and put options is upward sloping. Empirical evidence indicatesthat upward sloping term structure is a characteristic of markets with lower

Ž . Ž .volatility as in the United States , while high volatility markets notably Japanusually present downward-sloping term structures.13

4.3. Methodology

ŽFirst of all, we identify the presence of eventual mispricings taking the value.of Black and Scholes as the theoretical price using the historical and implied

volatilities estimated in ty1 as a forecast of volatility at time t. Once an optionappears to be undervalued–overvalued by the market, a Adelta hedgingB strategy issimulated in an attempt to exploit the mispricing. The AdeltaB is recalculated on a

Ž .daily basis up to maturity excluding Bank Holidays and weekends . The hedgingŽ .strategy takes place when the absolute value of the percentage deviation d of the

Ž . Ž .actual price of the option OP from the theoretical price OTP is more than 15%,< <in other words, when d )0.15. Each hedge is carried out on 10 contracts. It is

assumed that the strategy to profit from the mispricing detected at time t takes

12 Ž .Christie 1982 explains that when prices are low companies become more leveraged increasingboth the required return and its variance.

13 See Goldman Sachs, Equity Dervatives Research, Various years.


Fig. 2. Term structure of volatility. The Figure evidences the variation of implied volatility as afunction of option maturity for at-the-money options.

Ž .place exactly at time t and at the same prices this is an Aex-postB test . Thepositions thus built up are maintained until maturity of the options. If it appearsprofitable within maturity to open more than one position, the equal sign positionsare accumulated and opposite sign ones counterbalance one another to get the netposition. The MIB30 index positions are then closed at maturity, at the settlementprice of the contract, fixed, on the basis of a CONSOB deliberations, as equal tothe value of the index calculated on the opening prices of its securities.14 It isassumed that the purchase of the option or of the index is financed by borrowingat the risk-free rate, and therefore involves the payment of interest. As evidencedin the previous analysis, the short sale of the option or of the index involves a costderiving from the difference between the market interest rate and the repo rate atwhich the sale proceeds can be invested. We used as a proxy of this interest ratethe bid rate. Apart from the cost due to payment of interest, the commission costson the index and the options must also be considered. We recall from Section 3that commission varies according to whether a trader of a certain importance or anindividual investor is involved. It is assumed that, if the simulated strategies wereactually implemented by professional arbitrageurs, the tariffs would be 10 000 ITLfor option contracts and 5 index points for basket trading.

14 The selling or purchasing of the index Mib30 on the electronic trading system at the opening priceand at maturity allows to eliminate the so called Adivergency riskB represented by a difference in thesettlement price of the call and the price of trading in the index. In fact, the settlement price of the callis calculated at the value of the index calculated at the opening price of the assets constituting theindex.


Table 3Number of mispricings at various thresholds using different volatility estimates

< <d )5% )10% )15% )20% )25% )30%

Ž .a Call options

Call at the moneyHsd 113 70 44 23 14 5

61% 38% 24% 13% 8% 3%WISDc 52 10 3 2 2 2

28% 5% 2% 1% 1% 1%WISDpc 62 13 3 2 1 1

34% 7% 2% 1% 1% 1%

Call out of the moneyHsd 137 111 83 60 47 37

74% 60% 45% 33% 26% 20%WISDc 80 34 16 8 5 4

43% 18% 9% 4% 3% 2%WISDpc 104 54 31 20 9 6

57% 29% 17% 11% 5% 3%

Call in the moneyHsd 81 36 16 13 10 7

44% 20% 9% 7% 5% 4%WISDc 40 20 16 13 10 8

22% 11% 9% 7% 5% 4%WISDpc 46 20 15 14 8 7

25% 11% 8% 8% 4% 4%

Ž .b Put options

Put at the moneyHsd 133 87 63 48 37 28

72% 47% 34% 26% 20% 15%WISDp 76 27 12 10 4 3

41% 15% 7% 5% 2% 2%WISDpc 63 19 9 6 6 3

34% 10% 5% 3% 3% 2%

Put out of the moneyHsd 147 133 116 97 83 72

80% 72% 63% 53% 45% 39%WISDp 89 53 31 17 12 10

48% 29% 17% 9% 7% 5%WISDpc 107 54 40 29 20 13

58% 29% 22% 16% 11% 7%

Put in the moneyHsd 94 44 25 16 4 2

51% 24% 14% 9% 2% 1%WISDp 53 16 8 5 2 2

29% 9% 4% 3% 1% 1%


Ž .Table 3 continued

< <d )5% )10% )15% )20% )25% )30%

Ž .b Put options

WISDpc 43 12 9 4 2 223% 7% 5% 2% 1% 1%

The Table reports the number of mispricings both in absolute value and as a percentage of the totalnumber of observations. Hsd is the Historical standard deviation; WISDc, WISDp are, respectively, the

Ž .weighted implied volatilities calculated using three call option prices at different strike prices andthree put option prices. WISDpc is the weighted implied volatilit y obtained using the three call priceand the three put prices.

Running a trading simulation as described, we tested the MIBO30 marketefficiency, verifying the possibility of achieve profits reducing the effects of theMIB30 index variations on the results of the strategy. The simulation undertakenpermitted to overcome some of the already mentioned limitations inherent to the

Ž .Black and Scholes model. In particular: i the Black and Scholes model was usedin the strategy exclusively to identify overpriced and underpriced call options.Moreover, the threshold level for the difference between calculated and effective

Ž .prices was purposely fixed at a high level 15% in order to account forŽ .differences attributable to any inefficiency of the model. ii The dynamic hedging

allows to account for eventual variations in the volatility of the index returnsŽ .during the period and for the effective distribution of these returns; iii during the

course of the trial, the transaction costs of both call trading and dealings in MIB30indices were taken into account.15 The simulation also permitted to have someindication of the forecasting capacity of the two forecasts of the volatility utilised.

Ž < <Table 3a and b give the number of mispricings at various thresholds d s<Ž . < .OTPyOP rOP )5%, 10%, 15%, 20%, 25%, 30% . It can be noted that, forboth call and put options, the highest number of arbitrage opportunities isdiscovered for options out of the money and the lowest number for options in themoney. Moreover, the number of cases in which the mispricing exceeds 15% isquite restricted.

4.4. Empirical eÕidence

The simulation of the arbitrage strategies are based on estimates of historicaland weighted implied volatilities. Table 4a and b gives the number of options,respectively, call and put, on which hedging was applied. The strategy consists inassuming a longrshort position in the option if it results to be undervalued orovervalued by the market.

15 Ž .Cavallo 1999 compares the results obtained using different option pricing models which explicityaccount for transaction costs on a similar trading strategy.


Table 4Number of profitable long and short hedges

Position Dec Jan Feb Mar Apr May June July Aug Sept

Ž .a Call options

At the moneyHsd Long 0 0 3 0 0 0 0 0 0 2

Short 4 2 0 7 2 5 14 4 0 1WISDc Long 0 0 1 0 1 0 0 0 0 0

Short 0 0 0 0 1 0 0 0 0 0WISDpc Long 0 0 0 0 2 0 0 0 0 0

Short 0 0 0 0 1 0 0 0 0 0

Out of the moneyHsd Long 0 0 7 0 0 0 0 1 0 6

Short 5 6 0 14 5 9 19 6 3 2WISDc Long 1 1 2 1 1 1 1 1 0 1


Short 0 0 0 0 1 0 1 0 1 0

In the moneyHsd Long 1 0 9 0 0 0 0 0 1 0

Short 0 0 4 1 0 0 0 0 0 0WISDc Long 1 0 8 1 1 0 0 0 1 0


Short 0 0 4 0 0 0 0 0 0 0

Ž .b Put options

At the moneyHsd Long 0 0 4 0 1 0 0 0 0 2

Short 4 9 1 9 1 3 20 10 2 2WISDp Long 0 2 2 0 2 1 0 1 0 1


Short 0 0 1 0 0 0 2 0 0 1

Out of the moneyHsd Long 0 0 6 0 1 0 0 0 0 2

Short 7 13 1 17 12 14 22 19 7 5WISDp Long 0 0 2 0 2 0 0 1 0 1


Short 4 7 3 4 4 0 6 4 2 1

In the moneyHsd Long 0 0 1 1 1 0 0 0 0 0

Short 0 1 0 1 0 0 15 4 1 1WISDp Long 0 2 1 1 2 0 0 0 0 0

Short 0 0 1 0 1 0 0 0 0 0


Ž .Table 4 continued

Position Dec Jan Feb Mar Apr May June July Aug Sept

Ž .b Put options

WISDpc Long 0 2 3 1 2 0 0 0 0 0Short 0 0 1 0 0 0 0 0 0 0

Hsd is the Historical standard deviation; WISDc, WISDp are, respectively, the weighted impliedŽ .volatilities calculated using three call option prices at different strike prices and three put option

prices. WISDpc is the weighted implied volatility obtained using the three call price and the three putprices.

Overall results of the strategies are presented synthetically in Table 5a and b.Despite being quite controversial, these results do not indicate the possibility ofrealising systematic profits using a delta hedging strategy, whatever is the estimateused to forecast the effective volatility.16

Despite the strategy played on both Ain-the-moneyB call options and Aout-of-the-moneyB put options always give positive results at the end of the whole periodconsidered, monthly results present a high variability, with positive and negativesigns. Moreover, results obtained for in the money call option, positive for almostall the sub-periods, have to be taken with caution. We can observe that themajority of the profits are concentrated in the month of February, when the indexregistered a strong and unexpected rise. This evidence indicates the sensibility ofthe portfolio to movements in the index. It should be kept in mind that the deltarepresents only a rough estimate of the price variation of the option as comparedto the variation of the underlying security. For in the money options, the hedgingerror is particularly high. Therefore, these options are subjected to a widerdistortion resulting from the movements of the MIB30 index. To this we must addthat, although the risk of a lack of synchrony between the option price and theindex price is minimised using a high-quality data set, this risk is higher when inthe money options are involved. In fact, in the money options are the least liquid,with effective exchanges and related prices taking place intermittently. For theseoptions, therefore, it is possible that the time lapse between the moment ofdisclosure of the index and that of the price of the option may be sufficient todetermine an error in the calculation of the theoretical price. The risk of error inthe case of in the money calls is such that it compensates for the profits obtainableby the arbitrage. In fact, even if in this case the error was favourable and led toprofits, a different path of the index price movements could have led to different

Ž .results. Taken as a whole, the two effects hedging error and lack of synchronyseem to diminish the results, without refuting the validity of the hypothesis of

16 Details on the different voices constituting the overall results are available upon request.


Tab

le5

Ž.

Ove

rall

resu

lts

ofth

evo

lati

lity

trad

ing

stra

tegy

valu

esex

pres

sed

inin

dex

poin

ts

Dec

Jan

Feb

Mar

Apr

ilM

ayJu

neJu

lyA

ugS

ept

Tot

al

Ž. a

Cal

lop

tion

s

At

the

mon

eyH

sdy

9902

.479

3.11

2y

5268

.417

52.1

535

02.1

537

52.9

3y

155.

41y

738.

050

206.

497

y60

57.4

24W

ISD

c0

0y

2600

026

39.9

80

00

00

39.9

39W

ISD

pc0

00

020

71.5

10

00

00

2071

.512

Out

ofth

em

oney

Hsd

y43

63.4

y15

234.

1y

1112

0.1

4191

.86

6380

.47

1183

8.8

y31

319

y84

99.5

y23

5034

57.1

23y

4701

7.3

WIS

Dc

26.8

y51

56.5

994.

127

3.97

918

42.2

y10

26.6

y10

82.1

3174

.47

28.8

83y

1336

.00

y22

60.8

WIS

Dpc

1669

.921

309.

6y

2449

.1y

395.

2686

7.90

5y

1026

.653

61.6

713

565.

228

.883

038

932.

3

Inth

em

oney

Hsd

3807

.80

3322

5.1

317.

014

00

00

1549

.20

3889

9.18

WIS

Dc

3517

.80

3251

0.7

583.

71y

1182

.60

00

1549

.20

3697

8.96

WIS

Dpc

3517

.80

3229

4.3

934.

979

00

00

1549

.20

3829

6.35

Ž. b

Put

opti

ons

At

the

mon

eyH

sdy

1033

6y

3466

.4y

6383

.728

92.9

129

15.3

633

12.5

9y

851.

8515

4.41

3y

85.5

413

71.2

63y

1047

6.9

WIS

Dp

035

06.8

y74

4.2

024

18.6

4y

377.

20

4873

.22

012

71.6

1210

948.

9W

ISD

pc0

0y

2116

.70

1451

.35

y37

7.2

872.

970

012

3.14

49y

46.4

Out

ofth

em

oney

Hsd

y13

921.

129

80.5

y11

048.

311

972.

466

87.1

472

58.4

910

672.

755

7.40

5y

12.3

9y

3577

.946

1156

8.8

WIS

Dp

y52

305

y69

.620

95.5

2115

.23

1056

.89

089

2.89

125

87.7

310

86.2

y11

61.1

733

73.2

WIS

Dpc

y77

83.2

1314

.111

26.3

7231

.43

1266

.72

012

34.9

9y

545.

6110

86.2

y90

9.50

240

21.6

Inth

em

oney

Hsd

0y

1970

.28

y18

83.4

1762

.36

2645

.53

0y

2417

2y

3914

.8y

1346

.5y

2824

.17

y31

703.

3W

ISD

p0

5496

.858

33.7

1415

.59

4144

.57

00

00

016

890.

8W

ISD

pc0

4158

.755

2626

.893

1415

.59

2746

.42

00

00

010

947.

7

Thi

sta

ble

repo

rts

the

resu

lts

ofth

est

rate

gyob

tain

edus

ing

diff

eren

tes

tim

ates

ofth

ein

dex

vola

tili

ty.

Hsd

isth

eH

isto

rica

lst

anda

rdde

viat

ion;

WIS

Dc,

WIS

Dp

are,

resp

ecti

vely

,th

ew

eigh

ted

Ž.

impl

ied

vola

tili

tyca

lcul

ated

usin

gth

ree

call

opti

onpr

ices

atdi

ffer

ent

stri

kepr

ices

and

thre

epu

top

tion

pric

es.

WIS

Dpc

isth

ew

eigh

ted

impl

ied

vola

tili

tyob

tain

edus

ing

the

thre

eca

llpr

ice

and

the

thre

epu

tpr

ices

.


efficiency of the MIBO30 market. Results seem also to be consistent with thefinding of many studies, asserting that the implied volatilities are better predictors

Žof future volatility than those obtained from historic price data Canina and.Figlewski, 1993 . Despite that this finding needs further and more direct investiga-

Ž . Ž .tion, we can observe that profits losses are generally higher lower when thevolatility is predicted using the implied rather than the historical estimate.

5. Conclusions

The purpose of this study was to test the efficiency of the recently introducedItalian Index Option contract. The results obtained show that the market is on thewhole efficient.

Ž .1 The put–call parity test, once the bid–ask spread and the transaction costsare taken into account, supports the hypothesis of market efficiency. In fact, thepossibility of profitable hedges was encountered in only 2% of the cases withretail operators and, respectively, in 5% and 6% of the cases with institutionaloperators dealing in long or short hedges. Moreover, the presence of a very similarnumber of cases of profitable short and long hedges implicitly upholds the

Žhypothesis of efficiency of the market for securities lending at least for the 30.securities which constitute the MIB30 index . Alternatively, it may support the

hypothesis that the market makers and dealers in securities do not make the hedgethrough the short-sale of the portfolio that replicates the index, but use predomi-nantly the future contract on the MIB30 index as a substitute.

Ž .2 Tests conducted on the Black and Scholes model and on simulated tradingenforce the finding that the MIBO30 market is efficient even if with some pointsof divergence. Comparison between the theoretical prices calculated according tothe Black and Scholes model and the actual prices revealed the presence ofsignificant mispricings. This does not necessarily imply market inefficiency,because mispricings could derive from inaccuracies in the theoretical modelreferred to. A strategy of volatility trading was therefore simulated in order tocheck the effective possibility of obtaining profits by taking advantage of themispricings. The simulation experimented overcomes some of the model’s limita-tions, such as the absence of transaction costs and the assumption of constantvolatility. The results of the simulation show that it is not possible to makesystematic profits on the basis of singling out potential mispricings. Furthermore,implied volatility estimates seem to have higher predictive power in identifyingmispricings that ex-post demonstrated to be profitably exploitable. The resultsobtained support, on the one hand, the hypothesis of efficiency of the MIBO30market and, on the other, are consistent with the most recent and authoritativeliterature that affirms that implied volatility constitutes a better predictor ofeffective volatility.


Acknowledgements

The authors thank participants to the A11th Annual European Futures ResearchŽ .SymposiumB organised by the CBOT for useful insights. We also gratefully

acknowledge the comments and suggestions of M. Bagella, E. Barone, Don M.Chance, A. Cybo Ottone, N. Di Noia, L. Mastroeni, W. Perraudin, D. Sabatini, C.Wolff and an anonymous referee. The usual disclaimer applies. The opinionsexpressed in this work do not necessarily reflect those of the Citibank and of thePrime Minister’s Office. We also thank the CONSOB for the data. Although thiswork was the result of the authors’ joint effort, Sections 1, 2.2, 3, 4.2 and 4.4 werewritten by Laura Cavallo and Sections 2.1, 4.1, 4.3 and 5 by Paolo Mammola. Alltests were programmed in gauss by Laura Cavallo and are available upon request.

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