The Effects of Short Sale Constraints on Derivative Prices

Transcription

1 The Effects of Short Sale Constraints on Derivative Prices Roland Nilsson Stockholm School of Economics June 24, 2003 JEL Classification: G12 Abstract This paper tests the effects of short sale constraints on derivative prices by considering deviations from put-call parity (PCP). To measure the impact of short sale constraints I focus on Sweden and a particular period during which shorting stocks was almost impossible while at the same time stock options were traded. I found that the deviations from PCP corresponding to a short position in the stock were larger in absolute magnitude during the period when shorting was restricted compared with the period after. Furthermore, the introduction of futures on individual stocks were also considered. However, futures seem not to significantly affect the degree of short constraints in the market. Finally, only for stock options that had an underlying share that couldn t be shorted abroad could a pattern consistent with a short sale constraint be observed. 1 Introduction The base case scenario in most models within financial economics is a world free of frictions, in particular, a world free of any constraints to the ability of going short. Even though constraints on shorting has been considered before, it is only during the last few years that one seriously started to investigate whether this is an innocuous assumption or not. One of the earlier papers on the subject was written by Miller (1977), in which he proposes an intuition rather than a formal model of why constraints on shorting could matter. According to Miller is shorting a way for investors that do not own the stock to get their low valuations of the asset incorporated into the equilibrium price. If shorting is restricted those beliefs will not be incorporated and thus leaving the price of the asset elevated above the equilibrium price. Part of this argument has subsequently Roland Nilsson ( roland.nilsson@hhs.se, phone: , fax: ), PhD student at Department of Finance, Stockholm School of Economics, Box 6501, SE Stockholm, Sweden. 1

2 been formalised in for instance Chen, Hong and Stein (2002). However, in order for the argument to go through one has to assume either that there is some limits to investor s rationality or some form of limits to information (i.e. investors are not aware of the fact that certain agents are short sale constrained). If one does not impose one of these restrictions the price, given the appropriate information set, will be unbiased (Diamond and Verrecchia, 1987). The implications of short constraints on equities are therefore not completely clear. For derivatives, on the other hand, since their values are determined by relying on a replicating portfolio argument where one goes long a portfolio and short an other, an absolute short constraint on the stock, in case of a stock option, would imply that the replication argument breaks down. In other words, as opposed to equity there is a very clear cut reasoning behind the effects of short sale constraints on stock option prices, which is why this will be the topic of the following paper. The paper will furthermore focus on the case of Sweden because of the very special institutional setting which prevailed there for a number of years. To be more specific, between 1980 until the end of 1991 banks and brokerages houses were forbidden by law to participate in a stock lending/borrowing transaction. They could not be a party in the transaction or bring the two parties, the lender and the borrower, together. In effect there didn t exist a market place for lending and borrowing stocks in Sweden during this period. Furthermore, in 1985 the derivatives exchange, OM, opened and started to trade stock options. Between 1985 to 1991 the rather unusual situation prevailed in Sweden where stock options were traded but the underlying stock was virtually impossible to short. At the end of 1991 the law forbidding banks and brokerages houses to involve themselves in stock lending and borrowing was abolished and several market places were established. The main purpose of this paper is therefore to investigate the effect of short sale constraints on stock option prices by comparing the period 1985 to 1991 in Sweden with the period after. Furthermore, another rather unusual feature to the Swedish derivatives market is the existence of futures on individual stocks. Whereas most countries have futures on indices few countries have it on stocks. Since the US Congress only a few years ago passed a law allowing the trade of single-stock futures the effects of the introduction of such instruments should also be of interest to the US market. The reason why any effects at all could be expected from stock-futures introduction is the fact that they allow investors to take a short position in the stock. If the cost of this transaction is smaller than the cost of going short, using the lending and borrowing market for stocks, then the constraints of shorting will be reduced. The cost of the transaction should in this context not only include the direct transaction costs but also legislational aspects as well as liquidity etc. The second purpose of this paper will therefore be to investigate the effect of the introduction of futures on individual stocks. 2

3 To give a short preview of the results I do find that the shorting constraint increased the deviations from PCP in the direction corresponding to a short position in the stock, while no such increase could be detected for deviations from PCP where a long position in the underlying is required. The introduction of futures however did not seem to have any effect. Finally, the paper considers stocks which where shortable abroad and those that were not shortable abroad separately. The pattern mentioned above was only found for the stocks that were not shortable abroad while this was not the case for the other stocks. 1 2 Methodology 2.1 Measuring the Effects of Short Sale Constraints The effects of short sale constraints could be captured in many ways. One alternative would be to consider implied volatilities. In the spirit of Black & Scholes (1973) express the value of an European call on a non-dividend paying stock with strike price X as a fraction α t (σ, ) in the stock, S t,andafraction β t (σ, ) in a bond. Both fractions depend on the volatility of the derivative and a number of other omitted variables, c t = α t (σ, )S t β t (σ, )X If shorting the stock was impossible then obviously a situation where c< c could prevail, where c is the "true" value of the call. Since ct σ > 0 an undervalued call would imply that the implied volatility also would be too low. This would be what one should be looking for in the data. However, one has to make a stand on how to define a too low implied volatility and in the end the method chosen could always, justifiably, be criticized on different grounds. Secondly, the method is model dependent. For these reasons will I instead choose an other method by considering deviations from an arbitrage relation 1 Only after this project was started did I learn about a paper still in progress by Ofek, Richardson and Whitelaw (2002) which more or less considers the same situation as I do, namely the effect of short sale constraint on option prices by considering deviations from putcall parity. There is one major difference between our studies, however. Ofek, Richardson and Whitelaw (2002) have to rely on some measure that captures the degree of short sale constraint in the market. In their paper the measure used is the direct cost of going short which they claim also should be a proxy for how difficultyitistogoshort. Sinceashort sale constraint is a multidimensional constraint that depends on a number of variables, the direct cost only being one of them, one can always question whether all important aspects have been incorporated into the measure. This critique clearly does not apply to this study since I consider two different regimes that vastly differs in the ease by which one can go short. During one period it is almost impossible, during the second period it is easier to go short than it is currently in the US. Furthermore, I do consider the effect of introducing futures as well as how trade at different exchanges affects the short sale constraint, something they do not. 3

4 that also involves going short the stock. The relationship chosen will be the put-call-parity, henceforth PCP, which dictates that: p t c t + S t Xe r(t t) =0 The effects of an absolute short sale constraint on the stock on the PCP would be that the following relation could prevail: p t c t + S t Xe r(t t) > 0 (1a) where p t is the value of an European put on a non-dividend paying stock, r is the continuously compounded interest rate, T t is the time to maturity of both derivatives and X the strike of both derivatives. Since PCP never will hold exactly because of transaction costs, I will control for most of those to avoid the introduction of additional noise in my estimations. Consider first a situation where p t c t <Xe r(t t) S t in which case one would like to go long the left hand side and short the right hand side. If after controlling for bid and ask prices as well as other transaction costs, OT, the following holds: Xe r(t t) S ask t p ask t + c bid t OT < 0 (2) then there is no room for a profitable transaction. Note that expression (2) involves going long the stock. In the reverse scenario when p t c t >Xe r(t t) S t the following expression dictates the no arbitrage condition: p bid t c ask t Xe r(t t) + S bid t OT < 0 (3) in which case one is forced to go short the stock. If derivative prices has to adjust to any change in the price of the underlying and not the other way around, any deviation from PCP will imply that the derivatives are mispriced and not the stock. However, it will not be possible to say which one if the derivatives or whether both are mispriced. Until the end of 1991, a period exactly corresponding to the short sale constrained period considered in this study, investors were also obliged to pay a turn over tax on stocks and derivatives. This will also be controlled for, see appendix A for some further details on the adjustment procedure. Finally, commission will not be controlled for since the data is unavailable. If, on average, there are differences between the periods this could affect the results. 2.2 Normalisation To be able to aggregate across companies some form of normalisation procedure has to be applied. The first method used will be to divide the deviations from (3) and (2) by the strike price. A second version will also be used which follows the method proposed by Kamara and Miller Jr. (1995), see appendix B for further details. However, the results from the two different normalisation procedures are more or less identical and I will therefore continue to normalise by the strike price. 4

5 2.3 American Options All the stock options traded at OM are of the American type. When the underlying stock doesn t pay any dividend the value of an American call is equal to that of an European call. For the American put however, it could be optimal to exercise the option prematurely, so: P = p + EEP where P is the value of the American put and EEP is the early exercise premium. Equations (2) and (3) has to be adjusted by exchanging p for P EEP. The early exercise premium is calculated by using a programme by Cameron Rookley who uses a method proposed by Barone-Adesi and Whaley (1987) 2.It should be pointed out that this method is model dependent. However, in the end the EEP is very small so even if it were grossly miscalculated this would not change the results. Deviations from PCP has also been used to approximate the value of EEP, see Zivney (1991). Also two studies conducted on Swedish data by Engström, Nordén and Strömberg (2000) and Engström and Nordén (2000) has used the same approach. Obviously by attributing any deviation from PCP to the EEP one assumes that the markets are frictionless. In particular do not short sale constraints nor poor liquidity play any role for these deviations. This paper shows that short sale constraints do matter and that such an assumption therefore is inappropriate. 2.4 Matching Procedure PCP involves taking a position in a call and a put with the same strike price and with the same time to maturity. To match puts and calls the following matching procedure was used: Among all calls for a given company a given day, take the one with the shortest time to maturity (TTM), as long as the TTM is longer than 14 calendar days. The last 14 days have been excluded since strange price movements have been documented just prior to the expiration of the options (see Day and Lewis, 1988). By focusing on derivatives with a short TTM I will limit myself to liquid data, the importance of which will be discussed further on. From those calls with the shortest TTM choose the one which has a strike closest to the price of the underlying. Requirefurtherthatthestrikebeinthefollowingrange: 2 See < ln(x/s t ) < 0.1 (4) 5

6 By imposing this restriction only options that are quite at the money are chosen. Again, this is done in order to restrict the data to the subset which is highly liquid. Given the picked call try to find a put with matching strike and maturity. Discard any matches where either derivative have zero volume or no quoted closing price. Also, if there is no price in the underlying asset or the implied volatility of the American put could not be found (the programme didn t converge to a solution) that matched pair would not be included. If during the life time of the derivatives the underlying pays out a dividend the observation is discarded. The main reason is that the method proposed in Barone-Adesi and Whaley (1987) can handle a continuous dividend payment but is much less applicable to discrete ones. To increase the number of matched observations an adjusted matching procedure was also tried. The adjusted procedure works exactly as the above one with the only exception that instead of just taking the call which is most at the moneyandtrytofind a matching put, the next closest at the money call is also considered, and then the third next closest at the money call and so on as long as bound (4) is not violated. 3 Institutional Setting In 1980 a law prohibiting banks and brokerage houses to participate in a short selling transaction was introduced. The ban was lifted and the first market for lending and borrowing was introduced in This date will mark the end of the period during which shorting was severely restricted, denoted as period A. During the period to due to the fact that the lending of a stock was considered to be a realisation from a taxational point of view, virtually no lending took place. In this was changed and lending was no longer considered a realisation. This date will mark the beginning of the period during which shorting was no longer restricted, denoted as period C. The first future on an individual stock was introduced in and the last one, for the firms included in this study, was The period in between the introduction of futures and the abolishment of the short constraints will be denoted period B, while the period after is called D, see picture 1 for a time line. During this period not only the restriction to short was lifted but also several other restrictions imposed on the financial markets. Until the end of 1992 most shares had a restricted share which was not tradable by foreigners and a unrestricted version that for most big companies were traded abroad. For the stock options considered in this study some had an underlying which was restricted 6

7 and some had one which was unrestricted. Since the short sale constraint was a national matter, the unrestricted shares traded abroad was not subject to any constraint on the foreign exchange, something which will turn out to be important for this study. 4 Hypothesis and Confounding Factors In this study three hypothesis will be tested for: Conjecture 1: The magnitude of the deviations from relation (3) are larger during the period 1985 to 1991 compared to the subsequent period while deviations from (2) are the same across periods. Conjecture 2: The introduction of futures on individual stocks implies a reduction of the short constraint in the market and thus smaller deviations from relation (3). Again no effect on (2) should be seen. Conjecture 3: For the unrestricted shares with a significant volume traded abroad conjecture 1 should not hold. Although short sale constraints ought to be a major reason to why we observe deviations from PCP, there are several other sources for deviations as well. Suppose that new information first is incorporated into stock prices and only subsequently into derivative prices, i.e. stock prices "lead" derivative prices. Since derivative prices will adjust with a time lag, the arrival of new information to the market will cause deviations from PCP. Short sale constraints should, as in Diamond and Verrecchia (1987), make this adjustment process slower when shorting the stock is required. If the difference in adjustment speed when shorting is allowed and when it isn t is not too great, the arrival of new information will in general lead to deviations from PCP in both periods. Therefore the frequency of deviations between the periods will most likely not differ. However, this will not be the case for the magnitude. For this reason I will focus on the magnitude of the deviations from PCP rather than the frequency. Another relevant and very much related topic is liquidity. To take advantage of deviations from PCP it is crucial that all transactions are made simultaneously since any delay in any of the transactions could imply an unfavourable price change, precluding an arbitrage transaction. Firstly, variables that have been used as proxies for liquidity in previous studies will be controlled for. More specifically, in the study by Kamara & Miller Jr. (1995) they found that the volatility and the volume of the derivatives as well as the underlying were statistically significant in explaining deviations from PCP. Secondly, I will limit myself to data which is less likely to exhibit severe liquidity problems. Kamara and Miller Jr. (1995) also found that options that either are far out or in the money, as well as derivatives that have a long time to maturity are traded much 7

8 less. Therefore, in the matching procedure only options that are at the money or very close to, as well as options that have a short TTM, are included. Finally, this study only considers firms that had stock options traded at a very early stage, all of which happens to be very large companies. Large companies are more liquid than small ones. Not only the liquidity of the assets matters but also that the quotes are taken at the same point in time. Unfortunately, the derivatives exchange was open an hour and a half longer than the stock exchange until the first of April Since closing quotes are used, this will automatically imply that deviations from PCP in both directions will be higher prior to this date than compared to the period after. However, this should not cause the deviations in one direction to be more pronounced than in the other, which is the effect of short sale constraints. I.e. this should only add noise by increasing the deviations in both directions, and therefore weaken my results. Yet another important factor for explaining deviations from PCP are extreme events such as market crashes, see for instance Kleidon and Whaley (1992) that considers the effects of the 1987 market crash. One such very important extreme event was the Swedish banking crisis which prevailed during The background for the crisis was the collapse of the much inflated real estate prices during the beginning of the 90:ies. Since most debt holders used real estate as collateral the sudden drop in prices, coupled with a general economic decline, forced many debt holder to default. The crisis was so severe that several of the largest Swedish banks were on the verge of bankruptcy, forcing the government to actively intervene by supplying additional funds. One of those banks that were particularly severely hit was SEB. The effect of the crisis can easily be seen in table 1 which gives the standard deviation of the stock price in relation to the average stock price for the original 15 firms included in the sample for each year. Between 1991 and 1992 this measure increased approximately four fold for SEB and SHB, the only other bank in the sample, and three fold for the only insurance company in the sample, Skandia. No other company in the sample were even remotely affected to the same extent as these three. I argue that this is an extreme time period during which relationships between, for instance, measures of liquidity and PCP deviations might change and that therefore SEB, SHB and Skandia should be excluded 3. 5 Data The period under consideration is to Daily data from OM and the Stockholm Stock Exchange on the bid, ask and closing price as well as volume was used. 15 firms were originally included in the study, three of which subsequently were dropped for the reasons mentioned above. The 3 Only excluding for these three companies will not change the results significantly. 8

9 firms were selected on the basis that they had both put and call options traded during the short constrained period. The interest rate used was the Swedish "stadsskuldväxlar", a zero coupon bond issued by the government and hence with very low credit risk. The interest rate was matched with the TTM of the derivatives using linear interpolation. 6 Results Table 2 gives some details concerning the first matching procedure. The total number of available observations are approximately However, for only around of those was it possible to find a match between a call and a put. Most, around 5400, could not be matched due to no volume in at least one of the derivatives. Another 4600 observations were discarded since there were no puts traded at all, which was the case in particular during the earlier years of the sample. Most of the remaining observations were lost because no match between strike or time to maturity were found, or since the calls were outside bound (4). In Table 3 the adjusted matching procedure was used in which deviations from the call closest at the money is allowed. The total number of matched observations now increase to around 4400, i.e. around more observations. One company, Ericsson, accounts for around 25% of all matched observations, while the three companies with most matched observations make up for almost 60%. The results do not change significantly between the different matching procedures, why the second with more observations will be used. Table 4 gives some descriptive statistics. The average time to maturity is 56 calendar days and the average exercise premium in relation to price of the underlying is %. In the study by Ofek, Richardson and Whitelaw (2002) where they use the method proposed by Ho, Stapleton and Subrahmanyam (1994) their reach a similar value, %. Table 5 panel i) gives some descriptive statistics for the two periods A, during which shorting was restricted, and C, the period after the ban on shorting was lifted. The interim period to has been deleted because of the tax reasons previously mentioned. In the remaining two panels, the data have been divided into positive deviations (panel ii) and negative (panel iii) ones. The observations that do not comply with PCP are not included here. In Table 6 differences between period A and C in means and medians for the negative deviations and the positive ones separately are tested for. All the observations are assumed to be iid. The null hypothesis is that there is no difference between the periods, both for the positive deviations and the negativeones. Thealternativehypothesis is that the positive deviations in period A are more positive than in period C, and that the negative deviations are more negative in period A than in period C. When testing for the difference 9

10 between two medians Wilcoxcon s test is used 4. Indeed the deviations in the A period is more negative than in the C period at virtually any significance level. This supports the hypothesis that the short sale constraint during this period led to larger deviations from PCP than in the subsequent period when shorting was allowed. However, even though there is no difference in means for the positive deviations, it is the case that the median positive deviation in the A period is statistically significantly higher than in the C period. This is partly to be expected since during more than half the sample of period C the exchanges closed at the same time while there was an hour and a half hours difference between their closing times in whole period A. This would naturally imply that there are more deviations in the A period and probably also that the magnitude of the deviations are larger. However, it could also indicate that there are liquidity reasons. To investigate the liquidity aspects a bit more closely the variables used in Kamara and Miller Jr. (1995) as proxies for the liquidity of the assets is used here. In their study they found that the standard deviation of the underlying asset in relation to the average daily volume was highly significant in explaining deviations from PCP. The rationale for their measure is that the higher the standard deviation of the asset, the more likely is an unfavourable price movement. Furthermore,thelowertheaveragevolumethemoredifficult it is to trade at a pre-specified price. In their study this was calculated both for the derivatives as well as for the underlying 5. In table 7 two regressions are carried out: t = α + β 1 D1 t + β 2 VolumeU t + β 3 VolumeD t + β 4 Std/P t + ε t (5) In the first regression t is taken to be only the positive deviations from PCP, see panel i), while the second only considers negative deviations, see panel ii). Independent variables are the daily volume of the underlying stock, VolumeU t, and the average daily volume of the put and call, VolumeD t,aswellasthe standardized standard deviation of the underlying stock price. The standard deviation is calculated over a monthly period with non-overlapping observations, normalised with the average stock price to facilitate comparisons across companies, denoted Std/P t 6. For these variables a two sided test is performed. A dummy, D1 t, is used to indicate the short sale constrained period, during which it is 1 and during the subsequent period it takes on the value 0. Only the one sided alternative hypothesis that the dummy is significantly greater/lower than zero for positive/negative deviations are considered for the dummy. Surprisingly none of the explanatory variables are significant for the positive deviations, while only Std/P t is significant for the negative ones. Furthermore, 4 In addition to Wilcoxcon s test Kolmogorov-Smirnov s goodness of fit testisalsoused with similar results. The latter test is used to detect differences between two distributions, but since it focuses on the mass of the distribution one could also think of it as a median test. 5 At OM it was possible to place an order that was contingent on that an other order was settled. Even though this will decrease the riskiness from the investors point of view, this will not decrease the likelihood of foregoing a profitable arbitrage transaction. 6 The results are rather insensitive to the choice of the time period for calculating Sdf/P t. 10

11 Std/P t has a negative sign which means that the higher the standardised volatility the larger are the deviations, which is the expected result. Of course, since Kamara and Miller Jr.(1995) used the standard deviation in relation to volume thefactthatonlystd/p t is significant for the negative deviations is not contradictory to their results. The variable of interest, however, is the dummy which is not significant for the positive side while it is significant on the negative one at the 1% level, which is in accordance with previous results. Taken together these evidence does not support the hypothesis that liquidity is a driving force behind the results. Turning to the effects of the introduction of futures, table 8 gives some descriptive statistics. In table 9 the same things are tested for as in table 6, but in addition also differences between the periods A and B and B and D are considered. B being the period during which shorting was allowed but no futures were traded and D the period during which both shorting and futures were available. There seems not to be any difference between the A and B period on the positive side, just as expected. On the negative side the average is significantly higher at the 1% level, while the median is significant at the 5% level. Over all this supports the view that the abolishment of the shorting constraint has a significant impact on deviations from PCP. When comparing period B with period D only the mean positive deviations and the median negative deviations are significant, the latter only at the 5% level. Even though this indicates that the introduction of futures did not have any significant effect one should keep in mind that the period prior to the introduction of the futures roughly corresponds to currency crisis, during which Sweden ceased to uphold a fixed exchange rate. Comparing period A with D, i.e. do the joint effect of the introduction of futures as well as the abolishment of the short sale constraint have a significant effect on deviations from PCP. The answer is definitely yes since all deviations are higher intheaperiodthaninthedperiod. However,sincethepositivedeviations also are significantly higher these results are not solely explainable by short sale constraint arguments. To again investigate whether liquidity could explain the significant positive deviations, the same regressions as before are performed in table 10, adding a new dummy, D2 t, which is 1 during period B and 0 otherwise: t = α + β 1 D1 t + β 2 D2 t + β 3 VolumeU t + β 4 VolumeD t + β 5 Std/P t + ε t (6) Only Std/P t is significant among the liquidity variables. However, in contrast to the previous regression it is significant both when considering positive as well negative deviations. The sign is again the one predicted, which is to say a positive sign for positive deviations and a negative sign for the negative deviations. The D1 t dummy is significant at the 5% level for both types of deviations, while D2 t is highly significant at the 0,1% level only for positive 11

12 deviations. Furthermore, panel iii) shows that the negative deviations are significantly lower at the 0,1% level in period A than in period B, while they are not significantly higher for the positive deviations. In summary the regression analysis confirms the previous results. The introduction of futures does not seem have any effect, while the abolishment of the short sale constraint does. As for liquidity, among the variables tested for only the volatility of the underlying has a significant impact by increasing the size of the deviations. 7 Foreign Trade As mentioned before, until the end 1992 certain shares were so called restricted, implying that they were not allowed to be traded by foreigners. One way to get around a short sale constraint would be to short the share abroad where no such restriction is imposed, which only could be done with the unrestricted shares. Since certain stock options had unrestricted underlying and others had a restricted underlying, one would expect, to the extent that shorting was carried out abroad, that conjecture 1 only holds for the restricted shares. One could argue that if the volume of shorting abroad was small then it wouldn t matter. However, because of the turnover tax, which was imposed during a period which exactly corresponds to the short constraint period, much of the trade emigrated abroad to London and NASDAQ. In table 11, second column, Stockholm s turn over in relation to the total turn over at London, NASDAQ and Stockholm is given for each company for the period (roughly corresponding to the short sale constrained period). For eight out of the 12 companies more than half of total turn over took place in London and NASDAQ. Obviously foreign trade was very important during this period for the companies considered here. Furthermore, since the supply of shares and the number of shorted shares should be highly correlated, it is reasonable to suspect that the volume of shorted shares abroad also was high for almost all theunrestrictedsharesinthesample.inthethirdcolumnintable12thetype of underlying share is given, RE denoting the restricted shares, UN being the unrestricted ones. A denotes vote-strong shares and B are the shares with less voting power. To investigate the possible effects of foreign trade each company is analysed separately to see which do support my working hypothesis and which do not. Table12givesthedifference between the average deviation (either the positive ones or the negative ones) between the periods on company basis. The second column gives the average negative deviation for company x in period A minus the average negative deviation for company x in period C. In a similar manner the positive deviations are considered in column 3 and in columns 4 and 5 the same thing is done for medians. For companies that had no deviation in a 12

13 particular direction in both periods no difference between the periods is given. Note that a negative sign implies that the company has a deviation that goes in the direction prescribed by my working hypothesis. Total and Total+ gives the total number of negative and positive deviations in all periods per company. For convenience the type of underlying is given again in the last column. Let s first consider the negative deviations. All companies, except for Ericsson, that had unrestricted shares as underlying, and therefore were allowed to be traded abroad, has deviations that do not support my working hypothesis. While all companies that have restricted shares as underlying have deviations that goes in the right direction. This pattern also holds perfectly for positive deviations. In table 13 the firms has been subdivided into two groups, one containing only the unrestricted shares (panel i) and one with the restricted shares (panel ii) and some descriptive statistics are given. In table 14 differences between period A and C are tested for in the same manner as before; but now for the two groups separately. In panel i) the unrestricted group is considered. Only the median positive deviation in period A is significantly higher than in the C period and that only at the 5% level. I.e. for this group there is virtually no difference between periods, as expected. In panel ii) the restricted group is considered. Here the negative deviations are significantly higher in the A period compared to the C period at the 0,1% level both when considering means and medians, while the positive median is weakly higher at the 5% level in period A than in C. Again, this is supports my original hypothesis and is in line with what is to be expected. In summary, these results only strengthens the previous findings. Short sale constraints do matter since only the firms where shorting abroad wasn t allowed confirm the working hypothesis, while for firms that has an underlying asset that can be shorted abroad no such pattern can be observed. 8 Conclusion This paper tests the effects of short sale constraints on derivative prices, by considering deviations from put-call parity in the Swedish market between the beginning of 1988 to the end of I focus on Sweden and this particular period because lending and borrowing of stocks between 1980 and 1991 was virtually impossible while stock options were introduced already In other words, during 1985 to 1991 the very particular setting in which stock options were traded but shorting the underlying stock was almost impossible prevailed in Sweden. I found that the deviations from PCP corresponding to a short position in the stock were larger in absolute magnitude prior to 1992 compared with the period after, indicating that short sale constraints on stocks do matter for the pricing of stock options. Controlling for several proxies for liquidity did not alter these results. 13

14 Furthermore, the introduction of futures on individual stocks were also considered. If futures is a cheaper or easier way to replicate a short position than using the lending and borrowing market for stocks then deviations from PCP in the direction corresponding to a short position in the stock should decrease. This was however not found to be the case. Finally, systematic differences between companies where the underlying of the stock option was traded abroad and therefore also shortable abroad was found. Only for companies with an underlying that could not be shorted abroad did I find a pattern consistent with a short constraint, for companies that could be shorted abroad that pattern disappeared. References [1] Barone-Adesi G. and R. Whaley (1987), "Efficient Analytic Approximation of American Option Values", Journal of Finance, Vol. 42, pp [2] Black F. and Scholes M. (1973), The Pricing of Options and Corporate Liabilities, Journal of Political Economy, Vol. 81, No. 3, pp [3] Chen J., H. Hong and J. Stein (2002), "Breadth of Ownership and Stock Return", Journal of Financial Economics, Vol. 66, No. 2-3, pp [4] Day T. and C. Lewis (1988), "The Behavior of the Volatility Implicit in the Prices of Stock Index Options", Journal of Financial Economics, Vol. 22, pp [5] Diamond D. and R. Verrecchia (1987), "Constraints on Short-Selling and Asset Price Adjustment to Private Information", Journal of Financial Economics, Vol. 18, pp [6] Engstöm M. and L. Nordén (2000), "The Early Exercise Premium in American Put Option Prices", Journal of Multinational Financial Management, Vol. 10, pp [7] Engstöm M., L. Nordén and A. Strömberg (2000), "Early Exercise of American Put Options: Investor Rationality on the Swedish Equity Options Market", The Journal of Futures Markets, Vol. 20, No. 2, pp [8] Ho T.S., R. C. Stapleton and M. G. Subrahmanyam (1994), "A Simple Technique for the Valuation and Hedging of American Options", The Journal of Derivatives, Vol. 1 pp [9] Kamara A. and T. Miller Jr. (1995), "Daily and Intraday Tests of European Put-Call Parity", Journal of Financial and Quantitative Analysis, Vol. 30, No. 4, pp

15 [10] Kleidon, A. W. and R. E. Whaley (1992), "One Market? Stocks, Futures and Options During 1987", Journal of Finance, Vol. 47, pp [11] Ofek E., M. Richardson and R. Whitelaw (2002), "Limited Arbitrage and Short Sale Restrictions: Evidence from Options Markets", working paper at NYU, Stern School of Business [12] Miller E. (1977), "Risk, Uncertainty, and Divergence of Opinion", Journal of Finance, Vol. 32, No. 4, pp [13] Zivney T. L. (1991), "The Value of Early Exercise in Option Prices: An Empirical Investigation", The Journal of Financial and Quantitative Analysis, Vol. 26, pp Appendix A The turnover tax on stocks were paid when the stock were sold or bough; half of the tax at each transaction. It varied over the period from 2% of the value of the stock in to 1% in As for the derivative the turnover tax was paid when it is exercised on the value of the underlying stock. Since a PCP transaction involves buying both a call and a put, even though the price at maturity of the stock is unknown, one of the derivatives will be exercised and tax paid. I will assume that the expected price of the stock at maturity equals the price today. A more appropriate assumption would be to take the drift of the underlying into account. However, since the average time to maturity is only around 55 days such an adjustment would not make any difference. The tax is then the value of the stock today discounted, on average, 55 days. 10 Appendix B Here follows a brief summary of the normalisation procedure proposed in Kamara and Miller Jr. (1995). Consider first the portfolio St ask + p ask t c bid t. At time T this portfolio will always yield the strike price X no matter what happens. At time t, however, the funds to buy the portfolio has to be borrowed. The rate r B is the borrowing rate at which the transaction breaks even: X (S ask t + p ask t c bid t )(1 + r B )=0 In a similar fashion will r L be the lending rate that is involved in taking the opposite position at date t: (c ask t S bid t p bid t )(1 + r L ) X =0 15

16 The conditions consistent with no arbitrage are then: r B > e rt 1 r L < e rt 1 In other words, a too high borrowing rate and a too low lending rate are consistent with no arbitrage. Deviations from PCP are measured as the difference between the right hand side and the left for the borrowing rate, and as the left hand minus the right hand side for the lending rate. 16

17 Picture 1 Time Line A B D Stock Lending Introduced Taxational Change Introduction of Futures C In the above picture a time line over the important events in the study. Period A refers to the period to during which shorting wasn t allowed while period C is the period after ending when shorting wasn t restricted. Period B is the period when shorting was allowed but no futures on individual stocks were traded. The futures were introduced for the different companies during the period to Period D is the period during which both shorting was allowed and futures were traded.

18 Table 1 Standard deviation of underlying stock price in relation to average stock price Asea 0,088 0,165 0,158 0,106 0,072 0,115 0,065 Astra 0,070 0,216 0,121 0,148 0,108 0,107 0,082 Atlas Copco 0,165 0,090 0,247 0,127 0,107 0,116 0,058 Avesta 0,208 0,169 0,179 0,123 0,172 0,222 0,191 Electrolux 0,088 0,096 0,233 0,141 0,156 0,110 0,066 Ericsson 0,189 0,291 0,161 0,180 0,155 0,244 0,080 Pharmacia 0,089 0,092 0,090 0,119 0,109 0,082 0,077 SCA 0,086 0,093 0,147 0,077 0,153 0,056 0,102 SEB 0,087 0,087 0,138 0,129 0,520 0,678 0,152 SHB 0,132 0,066 0,117 0,113 0,433 0,348 0,128 Skandia 0,083 0,091 0,165 0,102 0,314 0,212 0,166 SKF 0,172 0,147 0,273 0,108 0,181 0,200 0,065 Skanska 0,098 0,082 0,200 0,134 0,278 0,264 0,144 Trelleborg 0,128 0,114 0,176 0,143 0,255 0,189 0,089 Volvo 0,063 0,066 0,211 0,154 0,189 0,091 0,061 Average 0,116 0,124 0,174 0,127 0,213 0,202 0,102 Table gives the standard deviation of the underlying stock price in relation to the average stock price, for each of the years included in the study.

19 Table 2 Matching Statistics No Deviations from At the Money Total # Days Matches Not ATM No Puts Maturity Strike No Volume Dividend Volatility Other Asea Astra Atlas Copco Avesta Electrolux Ericsson Pharmacia SCA SKF Skanska Trelleborg Volvo Sum The table describes the results from the first matching procedure in which no deviations from the most at the money call was allowed. First column contains the 12 companies under consideration. The second, the total number of days available for each company. Out of these observations the number of matched observations are given in the next column. The number of matched observations refers to the number of matched pairs of puts and calls. The remaining column describes where the observations were lost in the matching process. Not at the money (ATM) refers to the fact that no derivative with a strike price in the region 0.1 < ln(x/s) < 0.1 were found. No Puts refers to the fact that no puts were traded at all. Maturity implies that no put with a matching maturity were found, and strike to the fact that no put with appropriate strike price were found. No Volume means that either put or the call had zero volume traded. If during the life time of the options the underlying stock paid a dividend, that observation was excluded (column Dividend). Volatility refers to the fact that the implied volatility of the American put couldn t be calculated, and then a few other observations were lost due to no price in some asset (either bid, ask or close price of derivatives or underlying).

20 Table 3 Matching Statistics Deviations from At the Money Total # Days Matches Not ATM No Puts Maturity Strike No Volume Dividend Volatility Other Asea Astra Atlas Copco Avesta Electrolux Ericsson Pharmacia SCA SKF Skanska Trelleborg Volvo Sum As table 2 except that it describes the second matching procedure in which deviations from the stock options most at the money is allowed.

21 Table 4 Descriptive Statistics (Averages) Stock Price TTM Calendar TTM Trading Moneyness EEP EEP/StockP (%) EEP/PutP (%) Asea ,000 0,64 0,143 3,799 Astra ,006 0,66 0,142 4,068 Atlas Copco ,015 0,27 0,112 3,131 Avesta ,001 0,05 0,085 1,254 Electrolux ,004 0,37 0,152 3,338 Ericsson ,005 0,50 0,144 3,133 Pharmacia ,003 0,19 0,066 2,993 SCA ,008 0,35 0,176 3,727 SKF ,002 0,16 0,145 2,825 Skanska ,000 0,16 0,104 1,679 Trelleborg ,002 0,16 0,133 2,577 Volvo ,007 0,50 0,143 3,387 Average ,003 0,334 0,129 2,99 Call Spread Call Volume OI Call IV Call Put Price Spread Put Volume Put OI Put IV Put Price (%) Call (%) Asea 21,0 7, ,22 15,5 9, ,24 Astra 22,1 3, ,27 13,5 7, ,28 Atlas Copco 15,8 12, ,28 8,6 14, ,32 Avesta 3,8 16, ,41 3,4 23, ,43 Electrolux 14,8 5, ,29 10,7 9, ,31 Ericsson 23,1 2, ,35 15,5 7, ,35 Pharmacia 8,5 8, ,35 6,6 19, ,31 SCA 8,4 8, ,27 8,6 15, ,29 SKF 6,6 7, ,33 5,1 11, ,35 Skanska 10,8 7, ,35 9,1 11, ,37 Trelleborg 7,4 6, ,43 5,9 12, ,44 Volvo 22,2 4, ,29 14,7 9, ,30 Average 13,7 7, ,32 9,8 12, ,33 The tables gives the average values of each of the companies included in the study. The top table gives the average stock price, time to maturity (TTM) in calendar days and in trading days, the moneyness defined as ln(s/k). The early exercise price (EEP) and EEP in relation to the stock price as well as the put price in %. The next table gives the average statistic for both calls and puts. The average price, the average mid spread of the derivative prices in %, volume of derivative, open interest (OI) and finally, the average implied volatility (IV).

22 Table 5 Descriptive Statistics Shorting Abolishment Panel i): Whole Sample First Observ. Last Observ. Min Max Average Std # A : No Shorting ,0322 0,0237-0,0007 0, C : No Restriction ,0452 0,0731-0,0004 0, Panel ii): Long Stock Pos. Dev. Average Median Std # A + 0,0070 0,0059 0, C + 0,0051 0,0019 0, Panel iii): Short Stock Neg. Dev. Average Median Std # A -0,0053-0,0037 0, C -0,0041-0,0022 0, In the table the data has been divided into two periods A and C, the first corresponding to the short constrained period and the second to the period without a constraint. The interim period to has been deleted since, even though it was allowed, virtually no one lended and borrowed during this period for tax reasons. Furthermore, positive deviations from PCP, corresponding to the case when one goes long the stock, and negative deviations from PCP, corresponding to the case when one goes long the stock, has been separated in panel ii) and iii). The data has been aggregated over all firms where deviations from PCP have been normalised by the strike price.

23 Table 6 Testing Shorting Abolishment A + > C + A < C Differences in means 0,69 (0,244) 2,68 (0,003)** Differences in medians 2,72 (0,003)** 3,95 (0)*** Test the differences between the period A, the short restricted period, and C, during which no restriction on shorting existed. The data are aggregated across firms and normalised by the strike price. Positive and negative deviations from PCP are considered separately, while observations that comply with PCP are not included. Finally, the interim period to is excluded for tax reasons. The tests concerns the mean and median. The null hypothesis is that there is no difference between the periods. The alternative one sided hypothesis is that the positive deviations in period A is larger than those in period C and that the negative deviations in period A are more negative than those in period C. Differences in medians are tested using Wilcoxon s test, means using a t-test (since the degrees of freedom here is very large it is assumed that the normal distribution could be used instead of the t distribution). The numbers in the tables are the test statistics and the numbers within parenthesis the p-values. */**/*** indicates significance levels at 5%/1% and 0,1% level. Panel i) Table 7 Testing Shorting Abolishment - Regression Independent Coefficient Standard Deviation P-value Variable VolumeU 1,19E-11 7,39E-11 0,872 VolumeD 4,81E-07 1,01E-06 0,636 Std/P 0,0707 0,0482 0,146 Intercept 0,0019 0,0021 0,346 Dummy 0,0024 0,0028 0,1894 Panel ii) Independent Coefficient Standard Deviation P-value Variable VolumeU 1,57E-11 1,99E-11 0,43 VolumeD -2,31E-08 4,42E-07 0,958 Std/P -0,0296 0, ,0010** Intercept -0,0030 0, *** Dummy -0,0012 0, ,0045** In the above table the following regression has been performed: t = α + β1d1t + β 2VolumeU t + β 3VolumeDt + β 4Std / Pt + ε t where denotes the deviations from put call parity (PCP). In panel i) only positive deviations from PCP are used as dependent variable and in panel ii) only negative deviations are considered. D denotes a dummy that takes on the value 1 during the short constrained period and 0 when no restriction apply. VolumeU is the daily volume of the underlying stock and VolumeD is the average daily volume of the put and the call. Std/P is the standard deviation of the underlying stock calculated on a monthly basis with non-overlapping observations. Furthermore, the measure has been standardised with the average underlying stock price over the same period. For all variables but the dummy the two tailed p-value is given. For the dummy the one sided p-value for the alternative hypothesis that it is significantly greater than zero in panel i) and significantly lower than zero in panel ii).