Volatility trading and market efficiency

of market efficiency. What is interesting to note in the results is that, in contrast Ž . 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 number of profitable short hedges than profitable long hedges. Different elements can Ž . contribute to interpret this result: i in contrast with other markets, in the Italian Ž Market 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 call parity, including the ones mentioned above, have not explicitly allowed for transaction costs associated with the securities lending market when the strategy involve a short position in the underlying security. Another explanation is that market makers and options dealers prefer trading on the future contract on the MIB30 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 is that tests of market efficiency critically depend on the treatment of transaction costs. This is particularly true for tests based on an option-pricing model, which rely for their validity on continuous portfolio rebalancing. This problem will be investigated further in Section 4, where a stronger test of market efficiency is implemented.

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 exploit deviations between actual option prices and theoretical prices. More precisely, a volatility trading consists of formulating trading strategies on the basis of one’s own anticipation of the future volatility of the security underlying the option Ž . contract. To expect a future volatility higher lower than that currently registered Ž . on the market, corresponds to foreseeing a rise lowering in the price of the Ž . options. The related strategy consists in purchasing selling the options. The first studies aimed at verifying the efficiency of the options market based on an analysis of the deviations between theoretical and effective prices were Ž . Ž . conducted by Black and Scholes 1972 , Galai 1977 , and MacBeth and Merville Ž . 1979 . These authors examined the possibility of realising above-normal returns Ž by 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 the realisation 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 1993 analyses the efficiency of the European Option Market, developing a dynamic hedging 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 the informational efficiency of currency options. 4.2. Data and Õolatility estimates A sample consisting of daily data was used for the present analysis. This sample differs from the one used to test the put–call parity condition, not only for the period and the frequency of the data, but also because in this sample, apart from Aat-the-moneyB options, also AinB and Aout-of-the-moneyB options are available. The period surveyed is from December 1996 to September 1997. Synchronisation is essential to obtain an accurate estimate of the implied volatility and to avoid distortion of the results of the hedging simulations. This is the reason why this sample has been constructed through a very accurate and time-consuming procedure. From the data on the index, available minute by minute, we selected the price at the same time every day. On a daily basis, from the set of prices of all put and call options contracts effectively concluded, continuously quoted, we Ž selected 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 for Ž . 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 trading days. 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 partly mitigates 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 second observation every day to use to calculate implied volatilities. Calculating implied volatilities using call and put prices on several strikes, in turn calculated as the average of two daily values, we believe that we 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 is Ž . 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 the stock exchange is closed, and that, in a certain sense, the trading itself seems to cause some of the volatility. Implied volatility estimates are obtained as a weighted average of single Ž . implied volatilities. Following Chiras and Manaster 1978 , the weights used are the elasticity of the individual prices to volatility: n EW s j j ISD Ý j Es W j j js1 WISD s , n EW s j j Ý Es W j j js1 where WISD is the AWeighted Implied Standard DeviationB for the index Mib30 on the observation date, ISD the AImplied Standard DeviationB of option j, W is j Ž .Ž . the option price and dW rds s rW is the price elasticity of option j with j j j j Ž . respect to the index standard deviation s . We calculated three measures of Ž . implied volatility: the first WISDc is constructed using the three call option Ž . 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 weighted average of the first two. Some descriptive evidence on the time series properties of the implied volatilities is provided in Table 2. All implied volatilities evidence persistence in the level of volatility, all the series 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. 1987 Ž . 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 table shows that the average implied volatilities of the calls are always significantly lower 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 implied Ž . 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 volatilities from 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 a sensibly lower level of autocorrelation, which tend to disappear at lag 3. A possible explanation is that in the money call options are more likely to be affected by measurement problems, as we will precise in discussing volatility trading strategy results. 10 Ž . Harvey and Whaley 1992 underline that negative serial correlation is only a weak evidence against the hypothesis that volatility changes are unpredictable. In fact, it may be spuriously induced by asynchronous 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. L. Ca Õ allo, P. Mammola r Journal of Empirical Finance 7 2000 173 – 193 183 Table 2 Summary 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 Autocorrelations deviation r r r r r r r r 1 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 y 13.260 14.010 y 0.39 0.037 0.027 y 0.012 y 0.087 y 0.039 y 0.087 0.136 In 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 y 0.476 0.056 0.073 y 0.208 y 0.051 y 0.118 y 0.071 0.028 Out 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 y 8.930 9.740 y 0.241 y 0.024 0.059 0.02 y 0.096 y 0.004 y 0.067 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 y 8.690 9.980 y 0.449 0.045 y 0.009 0.039 y 0.037 y 0.036 0.065 y 0.102 In 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 y 8.280 7.710 y 0.383 y 0.121 0.213 y 0.106 y 0.119 0.135 y 0.028 y 0.108 Out 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 y 6.550 7.350 y 0.271 y 0.199 0.095 0.065 y 0.093 0.008 0.008 y 0.066 Weighted implied Õolatilities WISDc 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 y 11.890 12.830 y 0.378 y 0.07 0.155 y 0.131 0.033 y 0.077 y 0.035 0.037 WISDpc 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 y 6.720 6.900 y 0.313 y 0.112 0.014 0.078 y 0.096 0.082 0.012 y 0.105 WISDp 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 y 0.319 y 0.111 0.249 y 0.185 0.075 y 0.095 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 the Ž . 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 implied volatilities for call and put options is upward sloping. Empirical evidence indicates that upward sloping term structure is a characteristic of markets with lower Ž . Ž . volatility as in the United States , while high volatility markets notably Japan usually present downward-sloping term structures. 13 4.3. Methodology Ž First 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 t y 1 as a forecast of volatility at time t. Once an option appears to be undervalued–overvalued by the market, a Adelta hedgingB strategy is simulated 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 hedging Ž . 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 increasing both 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 a function 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 . The positions thus built up are maintained until maturity of the options. If it appears profitable within maturity to open more than one position, the equal sign positions are accumulated and opposite sign ones counterbalance one another to get the net position. The MIB30 index positions are then closed at maturity, at the settlement price of the contract, fixed, on the basis of a CONSOB deliberations, as equal to the value of the index calculated on the opening prices of its securities. 14 It is assumed that the purchase of the option or of the index is financed by borrowing at the risk-free rate, and therefore involves the payment of interest. As evidenced in the previous analysis, the short sale of the option or of the index involves a cost deriving from the difference between the market interest rate and the repo rate at which the sale proceeds can be invested. We used as a proxy of this interest rate the bid rate. Apart from the cost due to payment of interest, the commission costs on the index and the options must also be considered. We recall from Section 3 that commission varies according to whether a trader of a certain importance or an individual investor is involved. It is assumed that, if the simulated strategies were actually implemented by professional arbitrageurs, the tariffs would be 10 000 ITL for 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 price and at maturity allows to eliminate the so called Adivergency riskB represented by a difference in the settlement price of the call and the price of trading in the index. In fact, the settlement price of the call is calculated at the value of the index calculated at the opening price of the assets constituting the index. Table 3 Number of mispricings at various thresholds using different volatility estimates d 5 10 15 20 25 30 Ž . a Call options Call at the money Hsd 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 money Hsd 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 money Hsd 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 money Hsd 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 money Hsd 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 money Hsd 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 2 23 7 5 2 1 1 The Table reports the number of mispricings both in absolute value and as a percentage of the total number 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 and three put option prices. WISDpc is the weighted implied volatilit y obtained using the three call price and the three put prices. Running a trading simulation as described, we tested the MIBO30 market efficiency, verifying the possibility of achieve profits reducing the effects of the MIB30 index variations on the results of the strategy. The simulation undertaken permitted 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 used in 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 for Ž . differences attributable to any inefficiency of the model. ii The dynamic hedging allows to account for eventual variations in the volatility of the index returns Ž . 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 MIB30 indices were taken into account. 15 The simulation also permitted to have some indication 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 . . OTP y OP rOP 5, 10, 15, 20, 25, 30 . It can be noted that, for both call and put options, the highest number of arbitrage opportunities is discovered for options out of the money and the lowest number for options in the money. Moreover, the number of cases in which the mispricing exceeds 15 is quite restricted. 4.4. Empirical eÕidence The simulation of the arbitrage strategies are based on estimates of historical and weighted implied volatilities. Table 4a and b gives the number of options, respectively, call and put, on which hedging was applied. The strategy consists in assuming a longrshort position in the option if it results to be undervalued or overvalued by the market. 15 Ž . Cavallo 1999 compares the results obtained using different option pricing models which explicity account for transaction costs on a similar trading strategy. Table 4 Number of profitable long and short hedges Position Dec Jan Feb Mar Apr May June July Aug Sept Ž . a Call options At the money Hsd Long 3 2 Short 4 2 7 2 5 14 4 1 WISDc Long 1 1 Short 1 WISDpc Long 2 Short 1 Out of the money Hsd Long 7 1 6 Short 5 6 14 5 9 19 6 3 2 WISDc Long 1 1 2 1 1 1 1 1 1 Short 1 1 1 1 1 1 WISDpc Long 3 7 4 4 2 1 4 3 Short 1 1 1 In the money Hsd Long 1 9 1 Short 4 1 WISDc Long 1 8 1 1 1 Short 4 WISDpc Long 1 8 1 1 Short 4 Ž . b Put options At the money Hsd Long 4 1 2 Short 4 9 1 9 1 3 20 10 2 2 WISDp Long 2 2 2 1 1 1 Short 1 1 1 WISDpc Long 3 1 1 Short 1 2 1 Out of the money Hsd Long 6 1 2 Short 7 13 1 17 12 14 22 19 7 5 WISDp Long 2 2 1 1 Short 3 3 5 1 4 4 2 2 1 WISDpc Long 3 2 Short 4 7 3 4 4 6 4 2 1 In the money Hsd Long 1 1 1 Short 1 1 15 4 1 1 WISDp Long 2 1 1 2 Short 1 1 Ž . Table 4 continued Position Dec Jan Feb Mar Apr May June July Aug Sept Ž . b Put options WISDpc Long 2 3 1 2 Short 1 Hsd is the Historical standard deviation; WISDc, WISDp are, respectively, the weighted implied Ž . 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 put prices. Overall results of the strategies are presented synthetically in Table 5a and b. Despite being quite controversial, these results do not indicate the possibility of realising systematic profits using a delta hedging strategy, whatever is the estimate used 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 period considered, monthly results present a high variability, with positive and negative signs. Moreover, results obtained for in the money call option, positive for almost all the sub-periods, have to be taken with caution. We can observe that the majority of the profits are concentrated in the month of February, when the index registered a strong and unexpected rise. This evidence indicates the sensibility of the portfolio to movements in the index. It should be kept in mind that the delta represents only a rough estimate of the price variation of the option as compared to the variation of the underlying security. For in the money options, the hedging error is particularly high. Therefore, these options are subjected to a wider distortion resulting from the movements of the MIB30 index. To this we must add that, although the risk of a lack of synchrony between the option price and the index price is minimised using a high-quality data set, this risk is higher when in the money options are involved. In fact, in the money options are the least liquid, with effective exchanges and related prices taking place intermittently. For these options, therefore, it is possible that the time lapse between the moment of disclosure of the index and that of the price of the option may be sufficient to determine an error in the calculation of the theoretical price. The risk of error in the case of in the money calls is such that it compensates for the profits obtainable by the arbitrage. In fact, even if in this case the error was favourable and led to profits, 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 synchrony seem 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. L. Ca Õ allo, P. Mammola r Journal of Empirical Finance 7 2000 173 – 193 190 Table 5 Ž . Overall results of the volatility trading strategy values expressed in index points Dec Jan Feb Mar April May June July Aug Sept Total Ž . a Call options At the money Hsd y9902.4 793.112 y5268.4 1752.15 3502.15 3752.93 y155.41 y738.05 206.497 y6057.424 WISDc y2600 2639.98 39.939 WISDpc 2071.51 2071.512 Out of the money Hsd y4363.4 y15234.1 y11120.1 4191.86 6380.47 11838.8 y31319 y8499.5 y2350 3457.123 y47017.3 WISDc 26.8 y5156.5 994.1 273.979 1842.2 y1026.6 y1082.1 3174.47 28.883 y1336.00 y2260.8 WISDpc 1669.9 21309.6 y2449.1 y395.26 867.905 y1026.6 5361.67 13565.2 28.883 38932.3 In the money Hsd 3807.8 33225.1 317.014 1549.2 38899.18 WISDc 3517.8 32510.7 583.71 y1182.6 1549.2 36978.96 WISDpc 3517.8 32294.3 934.979 1549.2 38296.35 Ž . b Put options At the money Hsd y10336 y3466.4 y6383.7 2892.91 2915.36 3312.59 y851.85 154.413 y85.54 1371.263 y10476.9 WISDp 3506.8 y744.2 2418.64 y377.2 4873.22 1271.612 10948.9 WISDpc y2116.7 1451.35 y377.2 872.97 123.1449 y46.4 Out of the money Hsd y13921.1 2980.5 y11048.3 11972.4 6687.14 7258.49 10672.7 557.405 y12.39 y3577.946 11568.8 WISDp y5230 5 y69.6 2095.5 2115.23 1056.89 892.891 2587.73 1086.2 y1161.17 3373.2 WISDpc y7783.2 1314.1 1126.3 7231.43 1266.72 1234.99 y545.61 1086.2 y909.502 4021.6 In the money Hsd y1970.28 y1883.4 1762.36 2645.53 y24172 y3914.8 y1346.5 y2824.17 y31703.3 WISDp 5496.8 5833.7 1415.59 4144.57 16890.8 WISDpc 4158.755 2626.893 1415.59 2746.42 10947.7 This table reports the results of the strategy obtained using different estimates of the index volatility. Hsd is the Historical standard deviation; WISDc, WISDp are, respectively, the weighted Ž . implied volatility 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 put prices. efficiency of the MIBO30 market. Results seem also to be consistent with the finding of many studies, asserting that the implied volatilities are better predictors Ž of 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 the volatility is predicted using the implied rather than the historical estimate.

5. Conclusions