TITLE. Efficiency Tests of the London Options Market

Transcription

1 TITLE Efficiency Tests of the London Options Market Grose Christos, University of Birmingham and University of Macedonia Dritsakis Nikolaos, University of Macedonia Keywords: efficiency, options, implied volatility, hedge, portfolio 1

2 ABSTRACT Ex ante tests of the efficiency of the London options market explain alternative hedging strategies to fund managers who seek to comprehend the opportunities in the options markets and profit by potential market inefficiencies. Over and under valued options were used to form hedge portfolios, which were mostly positive indicating potential inefficiencies in LIFFE. Therefore options appear to incorporate the role of an investment strategy on their own and not only as a hedge against positions in the underlying stocks while the Black-Scholes formula proved to be an easily computed and implemented way to make abnormal, zero risk profits. This paper also confirms the ability of a weighted implied standard deviation to explain future volatility more accurately than historical volatility by use of regression analysis. INTRODUCTION Options have existed for many centuries but in recent decades they have attracted considerable attention due to the development of organised Options Exchanges. Although options are widely used in hedging strategies the relevant literature on their various aspects remains limited probably due to the complexity of the issues involved. The purpose of this paper will be to investigate the efficiency of the London International Financial Futures and Options Exchange (LIFFE). An efficient market is a market in which no above normal, risk adusted profits can be made. Moreover, prices are unpredictable and therefore no systematic pattern in security prices can be identified. More specifically in this paper we will test the options market for the weak form efficiency whereby prices reflect all information contained in past prices. Fama (1970) established three forms of efficiency. Weak form of efficiency where prices reflect information contained in past prices, semi-

3 strong where prices reflect all relevant information known publicly and strong market efficiency where prices reflect all available information either public or private. Geweke and French identified two sub categories for Semi-Strong market efficiency: the single market and the multi market efficiency. In an efficient market prices fluctuate randomly and investors cannot normally consistently expect to achieve greater returns than those that would compensate them for the level of risk taken. It should be added though that in financial economics the concept of market efficiency is defined differently since it does not imply that equilibrium prices are always optimal. Extensive research on market efficiency provides ample ground for indepth understanding of the issues for the applied economist. Boldt et al (1984), Fama (1991), Fortune (1991), and LeRoy (1989) are amongst some of the researches testing the Efficient Market Hypothesis (EMH). Taylor (1988) attempted to investigate the possibility of abnormal profits being made irrespective of the information set. Shiller (1981), on the other hand, determines to what extent do actual prices deviate from their fundamental levels. The maority of these tests regarding weak-form market efficiency fail to support the hypothesis that abnormal profits could be made trading on information based on past prices. More recent studies, however, indicate that investors may overreact to particular information driving security prices away from their long run values (Ball, 1995). As a result, non-standard profits may be achieved by buying oversold securities and selling securities, which have reached excessively high levels. An increasing number of investment managers, realising the potential profits that could be made by identifying market inefficiencies, are engaging in the options market. Their aim is twofold: Establish solid hedges against positions taken on the underlined assets but also make profits in the options market when such opportunities arise (Korn-Trautmann, 1999). Hedgers might wish to minimise risk close to zero, under ideal conditions, but they might also want to resort to insurance to shrink future possible unfavourable outcomes. 3

4 Signals of market inefficiency could be exploited by fund managers engaging in active fund management although transaction costs often cause expected profits to be non-significant compared to the risk undertaken (Ahn, 1999; Isakov, 000). The efficiency of LIFFE is tested against a specific model, the Black- Scholes formula for the pricing of traded options. The Black-Scholes model is used to identify overvalued and undervalued option contracts on an ex post basis by using the hedging technique suggested by Black-Scholes (197). The weak form of market efficiency is tested by utilising the model s ability to identify overvalued and undervalued option contracts, forming hedged positions and constructing portfolios of hedges on a one month basis. Hence, the tests for market efficiency are ointly tests for the validity of the model and its ability in selecting the mispriced option contracts. In this way two further aspects of market efficiency are highlighted. First, the ability of a trading rule to distinguish profitable from unprofitable investments. Thus, the ability of the trading rule to explain observed prices is ascertained. The second issue is the establishment of a trading strategy based on this rule that will ensure above normal profits relative to the risk taken. In order to establish this, one can perform an ex ante test (on past data) by replicating the existing opportunities for a trader. An ex post test on such a trading model might give us false results. The lack of all relevant information might lead us into falsely determining that the market is efficient. Nonetheless, if opportunities for abnormal profits are found and seem to be persistent over time one may conclude that the market is inefficient. If, on the other hand, such opportunities are not discovered it is assumed that, with regard to the specific pricing model used, the market is efficient. However, it should be noted that if the market is efficient no model could offer a way of making above normal profits since market inefficiency phenomena are a prerequisite for the use of any option pricing models such as the B-S formula with the purpose of obtaining abnormal returns. 4

5 Lastly, the option-underlying stock relationship will be analysed. The fair value of an option contract is influenced by the price and the volatility of the underlying stock that highlights this link. The reason we want to analyse it is in order to understand the microstructure of the market. The remainder of the paper is organised as follows. In the second section the Black-Scholes model is presented and a brief history of the related literature is outlined. In chapter three there is an analysis on the various methods for the variance estimation. Then follows the methodology used and is followed throughout the rest of the paper. An analogous to the implied volatility description is done for the historical volatility to highlight differences in the two methods accuracy. In addition tests are performed in order to emphasise the superiority of the implied variance as a predictor of the future variance. In section four the data set used is represented together with a reference of the data sources. The results of the tests for the implied and historical variance are presented next. Then the hedging strategy results using both the implied variance and the historical volatility are outlined both on a monthly collective basis and for each firm separately. Finally, a brief reference of this paper s conclusions and implications are discussed in section five. OPTION MARKET EFFICIENCY: THE STATE OF THE DEBATE The Black-Scholes formula for pricing an option was presented in Their formula has been a maor contribution to financial theory and has been widely utilised by investment practitioners. The tests in parts three and four will be based on that formula. Their formula is as follows: Rt ( d1 ) EN( d e (.1) C = SN ) where 5

6 d 1 ln( S / E) + ( R + σ / ) t = (.) σ t d = d 1 σ t (.3) t is the time to expiration % of a year, R is the risk free interest rate of period t, N (.) is the cumulative normal density function calculated at d, N ( d 1 ) is the hedge ratio, σ is the annualised volatility of stock returns (the standard deviation of stock returns), ln(.) is the natural logarithm, E is the exercise price of the option, C is the fair price of the option. So as to understand more profoundly the applicability of the B-S formula for research purposes it is useful to refer to related studies that uniformly utilised both the basic formula and its adustments 1. Significant research examined the efficiency of organised options exchanges. The focus of most studies is shed on the newly established Chicago Board Options Exchange (CBOE) and their common result is that indeed evidence is inconsistent with market efficiency (Galai, 1977; Trippi, 1977; Klemkosky-Resnick, 198). However, as Jensen (1978) indicates, market efficiency implies that economic profits from trading are zero, where economic profits are risk-adusted returns net of all costs. Hence, in some of these studies after trading costs are calculated profit opportunities vanish. The issue of transaction costs is also addressed by Phillips and Smith (1980) as well as Wilmott, Hoggard and Whalley (1994). Chiras Manaster (1978) use Merton s formula, which adusts the B-S model for the inclusion of dividend payments, 1 Merton, 1976; Chiras-Manaster,

7 C D = S yt C ln( S / E) + ( R y + 1/ σ N{ σ t ) t } Ee Rt ln( S / E) + ( R y 1/ σ N{ σ t ) t} (.4) where C D is the Merton pricing formula of the call price, y is the continuously compounded dividend payment. They calculate the implied standard deviation (ISD) of each option contract by entering the current option price into the evaluation equation and using numerical solution techniques to find which price of the standard deviation equates the LHS and RHS of Merton s formula. A trading strategy is formed based on the following steps: (a). The Implied Market Value (IMV ) is calculated using the weighted average of the implied standard deviations of its stock (WISD) in the pricing formula. (b). The (IMV ) is compared to the actual market prices so as to form short and long positions on the traded options. (c). A risk-free hedge is created consisting of at least one short and long position while the amount of each option included in the hedged position depends on the hedge ratio. The hedge ratio is the reciprocal of the derivative of (.6) and is defined, C ( S D ) 1 = e yt ln( S / E) + ( R y + 1/ σ / N{ σ t ) t } (.5) THE METHODOLOGY The B-S formula requires the estimation of the historical standard deviation of stock returns. However, the computation of the volatility has always been a difficult issue in mathematical application. It is based on the 7

8 assumption that the volatility that prevailed over the recent past will continue to hold in the future. Initially, we take a sample of returns on the stock over a single period. These returns are then converted into continuously compounded returns. Lastly, the standard deviation of the compounded returns is calculated. Lets assume that we have i continuously compounded returns, where St each return is identified as S t which equals ln( ) S t1 and t goes from 1 to i. Alternatively if there was an ex-dividend day during the interval, [( S D) S ] S = ln + where D is the dividend payment. Therefore the mean t t / t1 return and variance are as follows: S = i tσ= 1 S i t (3.1) i i ( St S) ( St ) ( St ) / i t= 1 t= 1 t= 1 σ = = i 1 i 1 (3.) i The volatility is: σ ν= (3.3) S where S is the average stock price of all S i s, S t is the weekly stock price, i is the number of observations, v is the volatility. In the above equation we divide the sum of squared deviations around the mean by i 1. Since the sample is taken from a larger population this is an 8

9 appropriate divisor, which is rendered necessary for the sample variance to be an unbiased estimate of the population variance. The returns can be daily, weekly, monthly or yearly. In the B-S model, volatility is defined as the annual standard deviation of the stock price 3. The volatility of stocks however is not constant over time. Factors such as a large stock split that increases the float of the stock may reduce the volatility. Alternatively, a company entering a more speculative line of business may result in an increase in the share price volatility. It has become apparent therefore that attempting to calculate historical volatility using annual data is not accurate because it encompasses a long period of time. A possible solution is laying greater weight on more recent closing share prices and less on older data. On the other hand, a stock s high and low prices during a particular interval could be employed. Garman and Klass (1980), Parkinson (1980), and Beckers (1983) have tested estimation methods based on this idea. Moreover, Garman and Klass suggest the possibility of utilising stocks opening-closing prices and the volume of trade. Cho and Frees (1988) make use of all recorded trades during a specific interval as well as bid and ask prices. However, such factors may introduce as much error as using the annual standard deviation. However, an implied variance technique has empirically proven to be superior to historical volatility. This procedure assumes that the market price of the option reflects its current volatility. More technically an implied variance is the value of the instantaneous variance of the stock s return which, when employed in the B-S formula, results in a model price being equal to the market price. Several techniques have been suggested: 1. Latane and Rendleman (1976) weight each option s implied standard deviation by the partial derivative of its price relative to its standard deviation c ( ) σ c where = S T N( d 1 ). σ By multiplying the variance by 50, which are the trading days in a year, or by multiplying by 50 the standard deviation one obtains the annualised standard deviation. 9

10 . Beckers (1981) concludes that the implied volatility of the option with c the highest ( ) should be chosen as that option s volatility. Furthermore, the σ implied standard deviation of the option that has an exercise price closest to the stock price proved according to his tests to be superior to the future standard deviation calculated from any weighting scheme. 3. Whaley (198) employs regression analysis in order to minimise the sum of the squared pricing errors so as to find the correct implied volatility. 4. Day and Lewis (1988) lay greater value on options that are more heavily traded. Thus, the non-synchronous trading problem is minimised. Moreover, as trading volume increases the bid-ask spread decreases and it is not as difficult to determine whether a closing price is a purchase or a sale offer. 5. Brenner and Subrahmanyam (1988) claim that the best estimates of volatility are obtained from at-the-money options, which are the most actively traded options and suffer the least from measurement errors. The formula used for the calculation of the implied standard deviation was: σ = ( C / P) (1/ t ) (3.4) where Rt C = E e σ t = P σ t (3.5) P Rt = E r (3.6) 6. Manaster and Koehler (198) attempt to confront the inability of the Newton-Raphson numerical method to find valid implied variance prices to ustify all the observed option prices. Towards this goal they present an 10

11 algorithm, which converges monotonically and quadratically to the unique implied variance, when it exists. 7. Johnson and Shanno (1987) develop a model that allows for a changing variance that could be helpful in testing whether the market prices options correctly since it is difficult to know which value of the variance to use in the B-S equation. The stochastic variance process is solved by the Monte Carlo method and is given by the following equation: µ st [ e 1] σ 0 σ = (3.7) µ t s where σ 0 is an initial variance value, µ s is the risk free interest rate. Chiras and Manaster (1978) emphasise that the implied standard deviations should be weighted by the price elasticity of an option with respect to its implied standard deviation. The formula used is: WISD = N = 1 ISD N = 1 C σ C σ σ C σ C (3.4) where date, N is the number of option contracts on each stock on every particular WISD is the weighted implied standard deviation for each stock on every observation date, ISD is the calculated standard deviation of each recorded option contract, 11

12 C σ σ C is the price elasticity of option relative to its implied standard deviation. The importance of WISDs is twofold. Firstly, WISDs are better indicators of future standard deviations than historical standard deviations. Secondly, they render unnecessary the collection of historical data. Nonetheless, Beckers (1981) concluded that both historical and implied volatility provide valuable information for the user of option models. For the calculation of the WISDs a numerical search routine is used that solves for σ by equating the B-S model price to the observed market price. This happens since it is impossible to solve the B-S option-pricing model for σ. Approximately 5.6 option contracts on a monthly basis were used to calculate the WISDs for each option. On some cases the market price of the call is too low to allow convergence by a model price. Lastly, throughout the sample period in only eight cases less than three standard deviation estimates were used for the calculation of the WISDs. The calculated WISDs are used as the volatility component in the B-S formula in order to calculate the option model price (IMV ). Stock margin prices are used as input in the B-S formula. The computed model price of each option contract is then compared to the market price to identify possible short or long positions. When the model price is lower (higher) than the market price the option is undervalued (overvalued) and a long (short) position is taken in the call with a corresponding short (long) position in the stock as long as their difference is not less than ten percent. In order to create a risk-free hedge at least one short and one long position are required. If for a particular stock more than one short and long position exists the ones bearing the maximum difference are chosen. A hedge ratio 4 is calculated so as to determine the amount of each option included in the hedge. 4 The ratio of the number of shares of stock that must be held in order to fully hedge against movements in the stock price is called hedge ratio. 1

13 This follows the acknowledgement that a successful option strategy involves not only a position in a mispriced option, but also an appropriate hedge in the underlying stock. The first step in the process is calculating the derivative of the B-S formula adusted for dividend payments with respect to the stock price: C yt = e S N( d1) E N( d ) e Rt 5 (3.5) C S = e yt N ( d 1 ) (3.6) Then the reciprocal of (3.10) is calculated: C S 1 = yt e N( d 1 ) (3.7) The holding period for each hedge is one-month (in the context of a five year period). The hedge position is maintained over the one-month holding period and is closed out at the opening stock and option transaction prices of the next trading day that is a month later. The pound return for every formed hedge is given by: ' ' [( C C ) + ( C C )]/( C + C ) (3.8) i i i where ' C i is the fair value of an undervalued call, C i is the market value of the undervalued call, ' C is the fair value of the overvalued call, 5 y is the dividend payment. 13

14 C is the market value of the overvalued call. The percent holding period returns are aggregated across all hedges for the twenty stocks throughout the fifty-nine holding periods. If the model is summed accurately the expected gain from each hedge should equal the difference between the market price of the option and the calculated model price. The sum of all these differences should give the total gain from the hedging strategy. In order to determine the accuracy of the hypothesis that the standard deviations deduced from option prices are a better predictor of a stock s volatility than the standard deviation inferred from historical data the following regressions are tested: SFUT, t α p + β PSHIST, t = (3.9) SFUT, t α r + β rwisd, t = (3.10) SFUT, t α s + β sshist, t + γ swisd, t = (3.11) where WISD, t is the weighted implied standard deviation inferred from option prices at time t, time SHIST, t is the historical standard deviation estimated using data from t x to t, SFUT, t is the standard deviation of option from time t to α, parameters. t + x, p, β p, α r, β r, α s, β s γ s are the estimated coefficients of the regression 14

15 There are 60 months (observation periods) during which our hypothesis was tested. For each observation period data from 0 individual stocks were used. The SHISTs and WISDs were tested against the SFUTs to determine which predictor explains larger percentage of the future standard deviation. The oint test of the WISDs and SHISTs tests the informational content of each parameter, i.e. whether each one contains unique information and whether one adds no additional information to the already known information from the other parameter. DATA AND EMPIRICAL RESULTS The main body of the data consists of end of month data for LIFFE listed options during the period January 1994 December 1998 written on twenty London Stock Exchange (LSE) listed options. The data were collected from monthly Financial Times issues. The quoted call prices were approximately 7,00 and for each one a corresponding fair value was calculated. The selected companies were divided into two groups of ten; one based on the January, April, July and October cycle and the other on the February, May, August and November cycle. Furthermore, the chosen listed shares have had continuously traded option contracts. The above restrictions ensured the accuracy of the ensuing results. Additionally, closing prices for the last trading day of the month were used for the share prices. During the 60 observation dates five stock splits took place and the option data were modified to accommodate the changes. The risk-free interest rate was taken from the 3-month interbank deposit rate. For the inclusion of the dividend component in the B-S formula the dividend yield for the underlying shares was used. The above data were derived from DataStream. The identification of the mispriced options required finding the implied volatility of each option contract. This was done by inserting the previous components in the B-S equation. 15

16 Historical volatility and future volatility were both calculated using weekly closing share prices from January 1989 until December 1993 and from January 1999 until July 000 respectively. In total 65 observations for the historical variance and 95 observations for the future variance were collected. Some problems that should be kept in mind when analysing our results are the following: 1. The observed call prices are not necessarily transaction prices. If the last bid is above the last transaction or the ask price below the last transaction, the bid or ask price will be the observed closing price.. Closing share prices are not transaction prices either, since the closing price is a weighted average of the last half an hour transactions. 3. Closing prices do not necessarily depict synchronisation of the transactions on the LSE and LIFFE. The regression estimates for equations 3.9, 3.10 and 3.11 are made using Ordinary Least Squares method. By observing the data from table 1 we note that a very small part of the future standard deviation is explained by the historical volatility. This feature is indicated by R which averages 14 percent. Hence, the standard deviation of the historical volatility explained only 14 percent of the deviation of the future volatility. The R results do not indicate any particular tendency towards diminishing or increasing over time. The t values recorded are significant at the five percent significance level. The higher the t values the lower is the probability that the sample used could be obtained from a distribution with an actual price of β close to zero. No negative values were recorded with mean t values approximately 4.0. Negative t values could not be accepted because they contradict the theoretical background. 16

17 Table 1: Results for the regression SFUT,t = α p + β p SHIST,t Month α(se) 0.039(0.01) 0.059(0.01) 0.053(0.00) 0.043(0.01) 0.044(0.01) 0.043(0.00) 0.048(0.00) 0.030(0.01) 0.05(0.0) 0.0(0.01) 0.05(0.01) 0.079(0.0) β(se) 0.470(0.43) -0.45(0.39) 0.(0.33) 0.538(0.5) (0.5) (0.8) -0.0(0.33) 0.66(0.56) 0.900(0.85) 0.561(0.30) 0.69(0.55) (0.69) t-value R Month α(se) 0.044(0.0) 0.089(0.0) 0.077(0.0) 0.056(0.01) 0.047(0.01) 0.050(0.01) 0.033(0.01) 0.057(0.01) 0.055(0.01) 0.041(0.00) 0.064(0.00) 0.040(0.01) β(se) 0.35(0.6) (0.93) (0.71) (0.33) 0.187(0.45) (0.57) 0.0(0.7) (0.39) (0.0) 0.18(0.19) (0.) (0.3) t-value R Month α(se) 0.036(0.00) 0.038(0.00) 0.036(0.00) 0.034(0.01) 0.048(0.01) 0.070(0.01) 0.044(0.01) 0.071(0.01) 0.014(0.01) 0.067(0.01) 0.039(0.01) 0.033(0.01)` β(se) 0.088(0.19) 0.13(0.17) 0.171(0.3) 0.550(0.49) (0.5) (0.50) 0.66(0.54) (0.60) 1.343(0.57) (0.54) 0.107(0.34) 0.511(0.41) t-value R Month α(se) 0.009(0.0) 0.047(0.00) 0.04(0.01) 0.066(0.01) 0.06(0.01) 0.054(0.00) 0.035(0.00) 0.033(0.00) 0.041(0.00) 0.044(0.01) 0.050(0.0) 0.043(0.01) β(se) 0.859(0.46) -0.04(0.1) 0.368(0.33) (0.18) (0.5) 0.14(0.8) 0.110(0.0) 0.165(0.17) 0.016(0.1) -0.03(0.31) (0.61) 0.080(0.7) t-value R Month α(se) 0.04(0.01) 0.048(0.01) 0.04(0.01) 0.098(0.01) 0.077(0.0) 0.039(0.01) 0.055(0.0) 0.058(0.01) 0.047(0.00) 0.057(0.01) 0.05(0.00) 0.048(0.01) β(se) 0.938(0.39) 0.106(0.53) 0.358(0.5) (0.60) (0.6) 0.19(0.53) (0.64) (0.3) -0.6(0.5) -0.34(0.36) (0.) 0.106(0.53) t-value R

18 The results for equation 3.10 (table ) support the hypothesis that WISDs provide better estimates of the future volatility than SHISTs. R average value is 6 percent almost double the SHISTs value which can be interpreted as WISDs having double the predictive ability of SHISTs. Nonetheless, there does not seem to exist any particular trend on these results either, a fact which would indicate a change in the predictive ability throughout the sample. The maority of t values are significant at the five percent significance level with five values bearing significance at the 0.01 level. Since t values are still particularly high averaging 3.86 it could be stated that the null hypothesis of β equalling zero is reected for both regressions. The coefficient standard errors appear to be higher for β ' s rather than for the constant in both regressions. From the results in table 1 the calculated mean value for a' s is and for while the mean β is β ' s The mean constant value is 0.034, In table 3 the results from the oint regression are shown. The figures partially support the conclusions drawn by the previous regressions. The t values are significantly higher for the WISDs rather than for the SHISTs even though in both negative values appear. The WISDs mean t value is 0.93 whereas the SHISTs mean value is This notable difference in their predictive power is not fully supported by the R results that are slightly higher than regression (3.11) R results. The mean of 0.3 (compared to 0.8 in regression 3.11) signals the existence of at least some informational content in SHISTs that is not fully contained in the WISDs. In spite of this result WISDs still appear to be better predictors of future standard deviations and to contain the multitude of information required to make an accurate prediction of SFUTs. 18

19 Table : Results for the regression SFUT,t = α r + β r WISD,t Month α(se) 0.05(0.01) 0.06(0.01) 0.036(0.01) 0.076(0.01) 0.034(0.01) 0.055(0.01) 0.038(0.01) 0.05(0.01) 0.049(0.01) 0.06(0.01) 0.055(0.0) 0.09(0.0) β(se) 0.003(0.05) (0.05) 0.037(0.04) (0.03) 0.017(0.05) -0.05(0.04) 0.015(0.03) -0.08(0.04) 0.004(0.07) (0.04) 0.015(0.07) 0.083(0.07) t-value R Month α(se) 0.060(0.03) 0.041(0.05) 0.05(0.04) 0.047(0.03) 0.083(0.04) (0.04) -0.09(0.0) 0.101(0.03) 0.054(0.03) 0.018(0.0) 0.047(0.0) 0.001(0.0) β(se) -0.05(0.15) 0.063(0.0) 0.166(0.19) (0.14) -0.13(0.16) 0.58(0.19) 0.308(0.1) -0.11(0.16) (0.1) 0.1(0.1) 0.043(0.08) 0.16(0.11) t-value R Month α(se) 0.06(0.01) 0.030(0.01) 0.034(0.01) (0.0) 0.063(0.0) 0.046(0.03) 0.0(0.0) 0.073(0.0) 0.030(0.03) 0.051(0.01) 0.08(0.0) 0.034(0.0) β(se) 0.057(0.06) 0.057(0.08) 0.041(0.07) 0.77(0.1) (0.09) 0.056(0.14) 0.16( (0.11) 0.118(0.13) 0.061(0.07) 0.063(0.09) 0.083(0.10) t-value R Month α(se) 0.08(0.0) 0.043(0.01) 0.048(0.0) 0.047(0.01) 0.033(0.01) 0.058(0.00) 0.018(0.01) 0.06(0.01) 0.035(0.01) 0.048(0.01) 0.048(0.0) 0.038(0.01) β(se) 0.094(0.0) -0.01(0.05) 0.05(0.09) 0.001(0.08) 0.065(0.07) (0.00) 0.100(0.08) 0.068(0.07) 0.031(0.06) -0.05(0.07) 0.008(0.11) 0.04(0.09) t-value R Month α(se) 0.061(0.03) (0.03) 0.057(0.04) 0.01(0.03) 0.00(0.04) 0.04(0.03) 0.071(0.03) 0.01(0.03) 0.019(0.0) 0.015(0.0) 0.050(0.0) (0.03) β(se) (0.1) 0.3(0.1) (0.13) 0.11(0.1) 0.09(0.13) 0.065(0.10) (0.1) 0.11(0.1) 0.071(0.06) 0.109(0.09) -0.01(0.08) 0.3(0.1) t-value R

20 Table 3: Results for the regression SFUT,t = α s + β s SHIST,t +γ s WISD Month β γ t-value(β) t-value(γ) R Month β γ t-value(β) t-value(γ) R Month β γ t-value(β) t-value(γ) R Month β γ t-value(β) t-value(γ) R Month β γ t-value(β) t-value(γ) R

21 The results from pooling all the available data into a single regression for equations can be seen in table 4. They are first calculated uncorrected using Ordinary Least Squares estimate. They are then obtained after being corrected for the presence of heteroskedasticity. In order to obtain the corrected results the White Heteroskedasticity test with cross terms was used. After examining the obtained results it is evident that the corrected model improved the t values and R of our estimates. The historical volatility t value rose from 1.43 to 1.93, a 30 percent increase. The WISDs t value rose from 4.38 to 5.90 and for the generalised regression by 50 percent to 6.39 for β and by 30 percent to 7.3 for α. Furthermore, WISDs t value is notably higher that SHISTs values supporting therefore the original conclusions that WISDs are providing a more accurate prediction of the future volatility than SHISTs. Our previous conclusions of some predictive power in the SHIST is also supported by the R results of the corrected model. Even though the transformed WISD model s R is close to the transformed generalised model s R it is still rather smaller (0.158 for WISD and 0.07 for the generalised model). This means that SHISTs contain some small informational content that is not included in the WISDs alone. Table 4: Ordinary Least Squares results from pooled results Dependent Variable SFUTAVER Variable Coefficient Std. Error t-statistic Prob. C HISTAVER R-squared Mean dependent var Adusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) White Heteroskedasticity Test: F-statistic Probability Obs*R-squared Probability

22 Dependent Variable SFUTAVER Variable Coefficient Std. Error t-statistic Prob. C HISTAVER R-squared Mean dependent var 7.1E-05 Adusted R-squared S.D. dependent var 9.70E-05 S.E. of regression 9.84E-05 Akaike info criterion Sum squared resid 1.45E-07 Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Dependent Variable SFUTAVER Variable Coefficient Std. Error t-statistic Prob. C WISDAVER R-squared Mean dependent var Adusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) White Heteroskedasticity Test: F-statistic Probability Obs*R-squared Probability Dependent Variable SFUTAVER Variable Coefficient Std. Error t-statistic Prob. C WISDAVER R-squared Mean dependent var 6.45E-05 Adusted R-squared S.D. dependent var 7.15E-05 S.E. of regression 7.7E-05 Akaike info criterion Sum squared resid 8.46E-08 Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Dependent Variable SFUTAVER Variable Coefficient Std. Error t-statistic Prob. C HISTAVER WISDAVER R-squared Mean dependent var Adusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) White Heteroskedasticity Test: F-statistic Probability Obs*R-squared Probability

23 Dependent Variable SFUTAVER Variable Coefficient Std. Error t-statistic Prob. C HISTAVER WISDAVER R-squared Mean dependent var 5.81E-05 Adusted R-squared S.D. dependent var 5.74E-05 S.E. of regression 5.59E-05 Akaike info criterion Sum squared resid 3.75E-08 Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) After establishing the potential short and long positions hedges are formed. There are 59 holding periods one less than the observation dates since each hedge is held for one month. Hedge positions were taken for all months averaging 6.35 analogous positions every holding period. During each of periods 3, 8 and 57 one hedge was held. Moreover, for periods, 7 and 51 two hedges were selected. For the rest of the holding periods more than three hedges were selected. The maximum held positions were in period 7 when 11 pairs of short and long positions were assumed. During the first and last part of the sample period the number of option contracts was smaller and profits through hedging relatively low. However, in the middle of the examination period some significant profit opportunities arose with profits from hedging reaching the 0-5 percent margin. This phenomenon might be due to higher volatility during that period. Among the 59 holding periods in six cases the hedges resulted in losses that did not exceed the six percent boundary. Nevertheless, in the maority of cases the observed inefficiencies in the options market resulted in profits being made. The average profit for a trader during the holding period would have been approximately 6.9 percent. The highest overall profit made on one month s hedge positions was percent on the 4 th month. On the contrary, the highest overall loss made on a single month s hedge was 5.74 percent on the 58 th month. 3

24 In addition of the 375 hedges 31 were profitable which constitutes 83 percent of the total number of hedges. The gain from the short and long positions was 1 and minus 18 of the anticipated returns respectively. Furthermore, 89 percent of the constructed portfolios were profitable, a fact that verifies the correctness of the followed strategy. It is noteworthy that it was not tested what percent of the noted profits is attainable. It is useful to refer to Trippi s work (1977), which simulated the potential profit opportunities by purchasing options at the opening price on the day following the establishment of a hedge. 37 out of the 71 options selected had opening prices that were considered favourable. Both the favourable and unfavourable options categories indicated weekly profits of about 10 percent. Hence it was proven that most of the observed gain is attainable. A similar strategy was followed using the historical volatility for underlying stocks using weekly return data for the period January 1989 December By examining the effects using historical volatility rather than implied volatility it was attempted to identify discrepancies in our results. The use of the historical volatility altered significantly our results without nonetheless enabling us to identify whether it undervalued or overvalued option contracts. Hence no definite conclusions can be made as to the nature that historical volatility affects our results. However, it is certain, and it has been verified by the work of Chiras-Manaster (1978), Johnson-Shanno (1987) etc., that the implied volatility provides better estimates of the volatility of a traded stock. The results of the monthly hedge returns show that the overall profits made by using the monthly hedges were 8. percent. During hedge periods 7, 16, 9 and 45 only one hedge was selected while the maximum number of hedges was 16 in periods 3 and 5. Furthermore, 9 percent of the formed hedges were profitable while only three monthly portfolios were negative. Out of the total number of 395 hedges 33 were profitable while only one negative 4

25 hedge exceeded the 10 percent margin. The other three negative monthly portfolios were minus 3.3, 4.89 and 1.03 percent respectively. The maximum percentage gain from a monthly portfolio was 6.09 on the 31 st holding period. In addition, the gain on the short positions is 141 percent of the anticipated returns while the loss on the long positions was 9 percent of the anticipated returns. Lastly, it should be emphasised that the pattern of higher returns in the tails of the returns distribution that was seen using the implied volatility is not observable in the historical volatility returns. CONCLUSION This paper has examined the ability of the Black-Scholes option pricing formula adusted for dividend payments to identify mispricing in the options market for option contracts traded in the London Financial Futures and Options Exchange. The model proved successful in identifying over and under valued call options on an ex-post basis. Towards this goal a weighted implied standard deviation is calculated for each underlying stock for the most accurate calculation of each stock s standard deviation. Regression results proved that the implied standard deviation provides superior to historical volatility estimates. These results suggested that only small part of the future volatility was explained by the calculated standard deviation based on past data. On the contrary, improved estimates of the future volatility are obtained when the implied standard deviation was used. Furthermore, the model displayed the previously observed pricing bias by undervaluing, relative to the market price, out-of-the-money call options and pricing fairly at and in-the-money call options. The results of ex-ante performance tests did not support option market efficiency. Therefore, a trading strategy in the options market would offer above normal profits for an investor. In order to determine this result a hedging strategy was followed whereby above a specified level undervalued option 5

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