Ž .
not available for it is an OTC market , for the empirical tests we used as a proxy the AbidB interest rate. We also account for costs due to the cash settlement
procedure, which distinguish index option from stock and commodities option. The only relevant difference in the results is that the trader has to pay the
additional commission cost to close on the market at maturity the position open on the underlying asset.
3. Put–call parity and transaction costs
In this section, the put–call parity conditions are derived and subjected to empirical testing on data on index options traded on the IDEM.
The data used to test the put–call parity conditions consist of infra-day prices, captured every 15 min from 10 a.m. to 5.30 p.m. from July 29, 1996 to February
18, 1997 for a total of 3642 observations. Data include ask price, bid price and Ž
. transaction price of at the money options call and put and on the MIB30 futures.
In our study, we estimated the MIB30 bid and ask prices applying to the transaction price the same spread observed for the future on the index MIB30.
5
Ž .
The put–call parity model was first developed by Stoll 1969 , and then
Ž .
extended and modified by Merton 1973 . The well-known basic put–call parity condition, when there is no dividend payment, is the following:
C s P q S y Ke
yr ŽTyt .
. 1
Ž .
Ž . The non-arbitrage conditions can be derived from Eq. 1 establishing two
portfolios, both of which result in zero pay-off at expiration. The first portfolio represents a long-hedge position, because it involves taking a long position in the
underlying share. The second portfolio represents a short-hedge position. In the absence of dividends and transaction costs, those conditions are, respectively:
C y P y S q Ke
yr ŽTyt .
F 0, 2
Ž .
P y C q S y Ke
yr ŽTyt .
F 0. 3
Ž .
Tests based on the put–call parity conditions, though weak tests of market efficiency, have been widely used in the empirical literature. Some of these studies
Ž .
basically supported the theory Nisbet, 1992; Klemkosky and Resnick, 1979 , but Ž
some inefficiency where also found to exist Stoll, 1969; Gould and Galai, 1974; .
Evnine and Rudd, 1985; Finucane, 1991 .
5
We are aware that the spread on the stocks and on the index is generally wider than that on the future. However, the error induced in the results by this approximation would be less then that induced
using, as in most empirical studies on index options, the index transaction prices.
In this section, we extend the put–call parity conditions to take into account frictions not considered in previous tests, in particular, the costs of replicating the
index and the cost of short selling. Since the option selected for the analysis are the most liquid and the spread is precisely bounded by the Italian market
regulation, we may be confident that the bid and ask prices used to account for the spread effectively corresponds to a transaction.
6
The availability of infra-day prices ensures a good synchronisation between the option prices and the underly-
ing security. Since the Italian Market trades are matched in real-time by the electronic trading system, it is reasonable to assume that the arbitrage strategy
could be implemented at the same prices prevailing when the profit opportunity were identified. This allows to overcome one of the main problems related to
put–call parity tests, that is the possibility that prices used to identify the arbitrage opportunity do not represent tradable prices. The put–call parity conditions are
Ž Ž ..
obtained as follows. If the expression representing a long hedge Eq. 2 is not
verified, a profit can be made by purchasing the index at its ask price, the put at its ask price, and selling the call at its bid price. The initial investment can be
Ž . financed at the risk free rate. In the case of a short hedge, if condition 3 is not
verified, the strategy to profit from the mispricing consists in buying the call, selling the put and short selling the portfolio which replicate the index. Accounting
for all transaction costs involved in the implementation of these strategies, the non-arbitrage conditions become, respectively, for a long and a short hedge:
C y P
y I q Ke
yr ŽTyt .
F TC q TC q TC
t
Ž .
bid ask
ask wc
bp bi
q TC T e
yr ŽTyt .
q Tk , 4
Ž . Ž .
si
P y C
q I y Ke
yr pŽTyt .
F TC q TC q TC
t
Ž .
bid ask
bid wp
bc si
q TC T e
yr ŽTyt .
q Tk , 5
Ž . Ž .
bi
where: TC and TC
are, respectively, the cost of writing a call or a put; TC
wc wp
bc
and TC are the cost of purchasing the call or the put; TC
and TC are the cost
bp bi
si
Ž of purchasingrselling the index when the position is closed at maturity T we
. calculate the present value of the commission cost and Tk are the clearing
commission on the option, usually very small and omitted from the empirical tests. Note that in the short-hedge condition the present value of K is calculated
Ž . using the repo rp rate. This allows accounting for the assumption that the cost of
the trading will be financed by the funds deposited with the lender of the securities.
6
Ž .
This contributes to overcome the objection raised by Phillips and Smith 1980 to studies based on bid and ask prices. As the authors pointed out, the average bid–ask spread generally overstates
transaction costs associated with the spread, and most trades occur inside the spread. In this case, the bid and ask quotes would not represent the effective supply and demand prices, and any tests based on
bid and ask prices would be biased in favour of the market efficiency hypothesis.
Table 1 summarises the results of the put–call parity tests under different hypothesis about transaction costs. Cases I and II represent the tests conducted
using, respectively, transaction prices and the bid and ask prices. For both cases I and II, the table reports three columns with the results obtained under three
different assumptions on the level of commission costs. In the first column, to make the results comparable with studies that ignore transaction costs, these costs
Ž .
were omitted Tc s 0 . In the second and the third column, we considered the level of transaction costs incurred, respectively, by an occasional investor and by
an arbitrageur. The position of an arbitrageur involves a lower level of transaction costs.
Referring to the average market commission, for an option contract we used a transaction cost of 10 000 ITL for the arbitrageur and 15 000 ITL for an individual
investor. Commissions on replicating the index are expressed in index points, and are 5 and 10 index points, respectively, for an arbitrageur and an individual
investor. Clearly, the larger the transaction costs, the wider the band within which prices can fluctuate without creating arbitrage opportunities.
Consistently with expectations, the number of hedges, which would have provided a profit opportunity decreases substantially when commission costs and
the bid ask spread are included. When we consider the level of transaction cost Ž
incurred by an individual investor, together with the spread bid–ask the last .
column of the table , the possibility to realise a profit from the hedge become Ž
irrelevant profit opportunities are revealed only in 2 of the simulated long and .
short hedges . However, this result is not so surprising, as this is only a weak test
Table 1 Number and average values of profitable hedges for cases I and II and different levels of transaction
costs Profitable hedges
Case I — transaction prices Case II — bra spread
Tc s 0 Tc s Tc1
Tc s Tc2 Tc s 0
Tc s Tc1 Tc s Tc2
Long hedge Number
1798 1078
546 589
213 82
of the sample 49
30 15
16 6
2 Average value
19.4 16.34
16.23 13.04
14.45 20.11
z 43.9
27.55 16.94
18.91 9.698
6.576 Short hedge
Number 1780
1083 596
519 193
70 of the sample
49 30
16 14
5 2
Average value 20.65
18.34 18.33
13.62 15.67
23.71 z
43.7 29.95
20.8 16.03
8.387 5.567
Case I: Long hedge: C y P y I q Ke
yr ŽTyt .
F Tc; Short hedge: P yC q I y Ke
yr pŽTyt .
F Tc; Case II: Long hedge: C
y P y I
q Ke
yr ŽTyt .
F Tc; Short hedge: P yC
q I y Ke
yr pŽTyt .
F
bid ask
ask bid
ask bid
Tc; where Tc is the total transaction cost. In the first column of the table, Tc s 0, in the second column Tc1 represents the lower transaction cost level incurred by an arbitrageur, in the third column Tc2 is
the transaction cost incurred by an individual investor. Average values are expressed in index points.
of market efficiency. What is interesting to note in the results is that, in 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
