Directory UMM :Data Elmu:jurnal:I:International Review of Economics And Finance:Vol10.Issue1.2001:
International Review of Economics and Finance
10 (2001) 75 ± 94
Valuation of American CAC 40 index and wildcard options
Mondher Bellalah*
Universite de Cergy, THEMA, 33 Boulevard du Port, 95 011 Cergy, France
Received 15 April 1998; revised 20 September 1999; accepted 22 October 1999
Abstract
American options traded on the CAC 40 index are cash settled. However, the stocks
underlying the CAC 40 index are traded in a particular ``forward'' market. In this market,
settlements take place periodically on a given date as in the UK. At that date, all transactions
accomplished before are settled. The settlement procedure differs from that in countries where
settlements appear as fixed number of business days after the transaction as in the US. The
characteristics of the distributions to the index underlying stocks might justify a specific model for
the valuation of these options. Besides, the organization of the Paris Bourse gives market
participants the possibility to exercise their positions during the 45 min following the close of the
stock exchange. They face each day as an exercise risk that must be accounted for. This article
applies to existing models for the valuation of CAC 40 options, and this type of risk is identified
as a wildcard option. This option is implicit in the values of index calls and puts. The magnitude
of wildcard options in a multiperiod setting is studied empirically using a new dataset by adapting
the model in Fleming and Whaley (1994) to the French market. D 2001 Elsevier Science Inc. All
rights reserved.
JEL classification: G12; G13; G14
Keywords: Wildcard options; Forward; Futures; Compound options; Volatility smile
1. Introduction
American index options are traded on the CAC 40 index. The underlying stocks are
traded in a particular market, the ``Marche aÁ ReÁglement Mensuel de la Bourse de Paris''
* E-mail address: [email protected] (M. Bellalah).
1059-0560/01/$ ± see front matter D 2001 Elsevier Science Inc. All rights reserved.
PII: S 1 0 5 9 - 0 5 6 0 ( 0 0 ) 0 0 0 7 2 - 1
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M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
(RM market).1 The RM market is a forward market providing a single monthly settlement
of all transactions at the end of the month. The liquidation day corresponds to the last
trading day of the month in which all trades are settled. Some of the features of this
market are shared by the UK system, except that the liquidation appears on any day of
the week. The cash transfers appear on the last business day of the month.
The stocks underlying the CAC 40 index give rise to 520 cash distributions a year and may
justify a specific model for the valuation of American index options. This question is of some
importance for market makers and traders operating in the Paris Bourse and for outside
investors who resort to the French market to implement some arbitrage strategies based on
index options, futures contracts and the underlying index.2 CAC 40 index options have
embedded wildcard options. In fact, the settlement procedure implies that an end-of-day
wildcard option arises in the case of the CAC 40 index options.
The wildcard option is essentially a free choice for index option holders. It results from the
fact that the option holder has 45 min to decide on exercise after the close of the Paris Bourse
while already knowing the exercise value. The exercise value of the index is set equal to the
mean index level a few seconds after 17:00 h Paris time. The choice of the mean index instead of
the closing index level is justified in order to avoid arbitrage strategies by market participants.
The decision to exercise depends on the expectations of the option holder regarding the
market trends. When the stock exchange closes, option holders have the possibility to trade the
underlying CAC 40 index options in non-organized markets. In these markets, the CAC 40
future contract is also traded at the same time. This provides market participants with more
information about the trend in the financial market. The expectations of market makers and
traders are based on the available information during the time window of 45 min after the close
of the Paris Bourse.
Wildcard options, which are implicit delivery options, exist in several financial contracts
and have been extensively studied in the literature. The wildcard option feature has been
investigated for US-listed options and futures contracts.3 This implicit option arises when the
exercise value of a derivative asset is determined before the final exercise date and when
exercise closes the underlying asset position. Even if the topic of wildcard option pricing has
been well researched for calls in several markets, little work has been done for the Paris
Bourse.4 The specific features of CAC 40 options and in particular, the ``report'' mechanism,
may have some effects on the valuation of these options and their implicit wildcard options.
1
There are roughly two settlement procedures around the world: the fixed settlement lag procedure and the
fixed settlement date. The first corresponds, for example, for the US stock transactions that are settled 5 business
days after the trade. The second procedure corresponds to countries like the UK, Italy and France, which are
examples of countries with fixed settlement date. For example, the trading year in the UK is divided into account
settlement periods of 2 weeks. This market is a forward market in which a new account period starts every other
Monday. In France, Italy and some other countries, transactions are settled once a month on a fixed date.
2
For a comprehensive discussion of the specificities of these instruments and markets and especially for the
Paris Bourse, the reader can refer to Bellalah (1990), Bellalah (1991), Bellalah and Jacquillat (1995) and Briys,
Bellalah, et al. (1998).
3
See, for example, Fleming and Whaley (1994), French and Maberly (1992), Valerio (1993) and Cohen
(1995) among others.
4
The valuation formulas for other exchanges are similar to those in Zhang (1997).
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
77
For the case of the CAC 40 index options, a potential risk exposure results from the
mechanism of cash settlement. This risk concerns both option buyers and sellers. It derives
from the possibility available to option buyers to exercise their options after the close of the
exchange. This risk has two dimensions. The first is the wildcard option that allows the
holder of an index call or put to exercise the option after the close of the exchange during
this time period (45 min). The second is the timing risk associated with the fact that the
seller of the option will not be notified of the exercise decision before the following day.
This is not the case for the option buyer who does not support this risk. Therefore, for the
option's writer, it seems that there is a greater risk in exercising before than after the close
of the stock market when the exercise value is known with certainty. In fact, if the option is
exercised before the close of the stock market, the seller is notified a few seconds after
17:00 H local time. This gives him enough time (45 min) to structure his portfolio
strategies and to reallocate the appropriate weights to his assets. If he is notified the
following day, he does not have the necessary time to reallocate his portfolio weights. This
is an important issue for institutional investors who implement asset allocation and portfolio
insurance strategies using CAC 40 index options and futures.
This paper's contribution is mainly empirical and not theoretical. In Section 2, we explain
the specific features of the French market. Some details regarding the CAC 40 options are
given. Then, two wildcard options are analyzed and identified. The first is identified in index
calls as a put option on the CAC 40 call. The analysis in this case follows the work of French
and Maberly (1992). The second is identified in index puts and is regarded as a put option on
the CAC 40 put. The empirical analysis for the Paris Bourse is new. In Section 3, we apply two
formulas for the valuation of the two wildcard options. In Section 4, we run some simulations
for index and wildcard options, and present our empirical results.
Since the formulas are valid only for a 1-day wildcard option and formulas are proposed
in the literature for the values of the wildcard call during the option's life (in a discrete
time), we provide some empirical results in this sense. A modified version of the Fleming
and Whaley's (1994) model, the FW model, is used to appreciate the value of wildcard
options in a multiperiod setting.
2. Specific features of the Paris Bourse index derivatives and wildcard options
The CAC 40 stock index is calculated using an arithmetic mean of the 40 active stocks
traded in the RM stock market. There are two separate markets for the underlying stocks
trading on the RM market. On the first RM market, stocks are traded every day from 10:00 to
17:00 h local time. On the second market known as the ``report'' market, net long and short
positions are matched once a month in order to determine for each stock a cash amount called
the ``report'' or a ``deÂport.'' This amount is established by a confrontation between all the open
positions contracted by financial institutions in the market place. Then, it is communicated to
market participants and accounted for when stock prices are quoted in the next day. Since the
assets underlying the CAC 40 index trade are in this forward market, the specificities of these
stocks must be analyzed. This allows the understanding of the specificities of the CAC 40
index options and their wildcard features.
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M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
2.1. The RM stock market and the report market
On the equity market, securities are traded either on the cash market (marche au comptant)
or on account on the monthly settlement market.5 Investors on the RM market must meet an
initial margin call representing a percentage of the total amount of their order.6 Settlement of a
trade in a security on the cash market takes place on the third business day following the date
of such trade. The monthly settlement market's account period is not a rolling settlement
period following trade as in the US but a fixed monthly calendar period for settlement of
trades that occur within the period. Thus, while transactions are firm in both price and
quantity once they have been concluded, actual cash settlement and delivery of securities take
place on the last trading day of the month.
On the account day (i.e., 6 business days before the last business day of the month,
inclusive), investors who have not closed their positions and cannot or do not wish to deliver
securities sold or present payment for securities purchased may carry their positions over to
the next account period through a special market organized once each month. This ``contago
market'' determines the cost at which buyers can obtain the cash, and sellers can buy the
securities they need to meet their obligations as buyers or sellers at the end of the month and
thus carry their positions over to the settlement day of the following month.
On the Paris Bourse, only forward prices are observed because shares are not traded on a
cash market. In the first day of the monthly settlement period or the liquidation day,
forward prices should go up by an amount that is nearly equal to 1-month interest rate. The
buyer (seller) of a stock at an instant t at a price St in the RM stock market assumes a long
(short) position at the same date. He agrees to get (to deliver) the stock on a certain
specified future date ti for a specified price St, which is the unobserved stock price at
instant t.7
On the ``jour de liquidation,'' the net long position often exceeds the net short position. The
``jour de report,'' a confrontation between net long open and short positions ``deferred'' on
each stock, and the amount of capital available on the market place determine the equilibrium
amount of ``report'' and the market share price. When the net short position exceeds the net
5
The most actively traded French and foreign shares on the official list are traded on the monthly settlement
market.
6
For an empirical analysis of the limit order book and the order flow in the Paris Bourse, see for example
Biais, Hillion, and Spatt (1995).
7
On date ti known as ``le jour de liquidation,'' he receives (delivers) the stock and pays (receives) the price St.
The day ti corresponds to the sixth business day prior to the last trading day of the civil month. It corresponds also
to the day before the last trading day in the ``RM'' month, the ``jour de report.'' The buyer (the seller) can close out
his or her position prior to date ti by an offsetting transaction on the stock. This results in a gain (loss) given by the
difference between the two values of the stock. On the nearest ti, the buyer (the seller) who wants to defer the
position (and maturity) of the initial transaction, i.e., to keep a long (short) position in the stock, sells (buys) the
stocks he bought (sold) and signs a contract. The contract commits him to buy (resell) these stocks back at the next
``jour de liquidation'' for the same price. This transaction allows the buyer (seller) to maintain his initial long
(short) position. Transactions on the last day of the settlement month could be done ``ici'' (here) or ``laÁ bas'' (there)
meaning with delivery ``this month'' (here) or ``the next month'' (there). However, investors can engage
simultaneously in two types of transactions with two different market prices.
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
79
long position, equilibrium implies a ``deÂport.'' This amount is determined in the form of an
interest rate that multiplies the share price for the holding period. Long investors get this cash
income. When the net open short and long positions delayed are exactly matched, there is no
``report'' and no ``deÂport'' on the stocks traded in the RM market.8
2.2. The CAC 40 index and wildcard options
MONEP currently trades two option contracts on the CAC 40 index: (1) the CAC 40
short-term option (PX1), which is American style and was introduced in November 1988;
and (2) the CAC 40 long-term option (PXL), which is European style and was introduced in
October 1991.
Settlement takes the form of a transfer in cash equal to the difference between the strike
price of the PX1 option and the CAC 40 settlement price, taking into account the number of
contracts exercised and the trading unit. This daily index settlement price is the average of all
index values calculated and displayed between 16:40 and 17:00 h including the first index
value displayed after 17:00 H. The deadline for registering exercise instructions is set daily at
17:45 h with the exception of the expiration day when in-the-money options are automatically
exercised upon expiration. PX1 options may be exercised on any trading day from the date of
purchase up to the expiration date.9
The futures contract is based on the spot CAC 40 index. The settlement value is based on
the average of the last 40 reported CAC 40 index quotations between 15:40 and 16:00 h local
time. It is negotiated in the ``Palais de la Bourse'' between 10:00 and 17:00 h local time. On
the ``jour de liquidation,'' the buyer pays and takes the delivery of the 40 stocks, and the seller
receives the payment and delivers the stocks with their corresponding weights.10
The following example illustrates the specificities of the wildcard options. Consider a
market maker or a trader holding a call (a put) on the CAC 40 index with a strike price of
1700. Suppose that the CAC 40 index closes at 17:02 h on the business day prior to the last
day to expiration. The settlement index value is revealed to market participants immediately
after the close at 17:00 h local time. The value of this index call (put) is nearly its intrinsic
value since there is 1 day of time value left.
8
These are the specificities of observed forward prices with respect to the unobserved stock spot price given
the cost of carrying the stocks. In theory, the forward stock prices converge to the spot price that is implicit on the
last day of the account period. Then, it moves upward with respect to the spot on the first day of the account period
to account for the cost of carry. The stock price is said to be implicit because there is no organized cash market in
which the spot price can be observed. When there are no dividends, arbitrage considerations imply that the
forward price must be equal to the spot price plus the financing costs until the maturity of the forward contract.
9
Exercise of index options results in cash settlement between clearing members through the SBF following
confirmation by the SCMC of the exercise.
10
The major difference between index and equity options is that the asset underlying an index option is a
portfolio instead of an individual stock. This makes hedging between an index option and its underlying asset
more difficult. In practice, an index is duplicated by a portfolio containing few highly liquid stocks in proportions,
such that its return tracks the index's return. Therefore, index options are expected to be more severely mispriced
than stock options.
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M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
After the market closes, some bad news cause the market participants to believe that the
market will fall significantly the following day. The next day, the index call value will
probably be 0 and the index put value will be probably higher. In this case, it is better to
receive the index call intrinsic value today rather than waiting for the next day. The opposite
argument applies for the index put for which it may be worth waiting for a significant drop in
the index level.
Since market participants can exercise their positions during the 45 min following the
market close, this possibility represents the wildcard feature implicit in index calls and puts.
This feature comes into play each day from 17:00 to 17:45 h local time.11 Since the wildcard
feature represents the right given to the index call (put) holder to exercise his option, it is also
an option to sell the index call (put) at any time before 17:45 h. Immediately after 17:45 h
local time, the right to exercise is worthless. Hence, the wildcard option comes into existence
each day at the close and expires at 17:45 h. As a consequence, the option holder disposes of
a series of 45-min options, one for each day between 17:00 and 17:45 h until the option's
expiration date.12
The American-style CAC 40 call (index put) can be replicated by a portfolio of two options.
The first option is the regular option associated with the value of the underlying index. The
second is the wildcard option. The value of this privilege is positive to the buyer and negative
to the seller since the option holder decides to or not to exercise this right. Therefore, a rational
seller will demand compensation for giving the holder of an index call or put this right.
3. The valuation models
Based on the idiosyncrasies of the RM market, the specificities of CAC 40 options and
our preceding remarks consider an economy that satisfies the same assumptions as in Black
and Scholes (1973) and Merton (1973). The dynamics of the underlying index S are given
by Eq. (1):
dS=S mdt sdz
1
where m and s, respectively stand for the instantaneous expected rate of return and the
standard deviation of the CAC 40 return. The term dz is a standard Wiener process.
If the underlying index refers to a commodity contract, which is assumed to be a
``particular forward'' contract, and the distributions to this commodity are constant propor-
11
The settlement index price is established each day as the arithmetic mean of all index values quoted between
16:40 and 17:00 h local time. Each day, the index call holder (put holder) compares the payoff guaranteed by the
exercise value with the probable next day's payoff. If the call price is below the payoff guaranteed by the
settlement price, he should exercise his call (put) option. Otherwise, he should not exercise and wait until the
following day.
12
If the wildcard feature has value only after 17:00 h every day, then a jump in the option price would happen
at that time. However, the discounted value of the future wildcards is impounded into the option price prior to
17:00 h.
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
81
tional rates (the rates of ``report'' and ``deÂport''), then the traditional models of cost of carry
apply.13 In this case, the index option can be valued by applying the commodity formulas by
considering these rates as constant proportional carrying costs. If one accounts for the profile
of cash distributions and their timing, a more specific option pricing model must be used for
the valuation of American CAC 40 options. This model would be extremely difficult to write
down in a continuous time framework because of the 520 distributions to the underlying
stocks for a 1-year maturity date. However, it is possible to simplify considerably the
valuation problem by modeling the French market as a ``particular forward'' market for the
two following reasons.
First, cash receipts (or payments) corresponding to the transactions in the RM stock
market are done at the end of the month for transactions accomplished at any time. It is as if
the short (long) investor lends (borrows) the value of the portfolio duplicating the index
price in a cash account until the maturity date. In this case, the value of the index grows at
the riskless rate. Second, if one considers the rates of ``report'' and ``deÂport'' paid (received)
at the end of each month during the index option's life as constant known proportional rates,
then the familiar cost of carrying model would apply as in Merton (1973) and Barone-Adesi
and Whaley (1987). Otherwise, models with stochastic distributions to the underlying assets
must be used.
The French market can hence be seen as a particular ``forward'' market because of its
specificities. Under the above assumptions, the usual relationship between the forward price
of the index and its spot value should apply. When it is verified in the absence of costless
arbitrage opportunities, then (Eq. (2)):
F SebT
2
where F is the current futures price, b is the cost of carrying the index, and T is the maturity
date.14 The constant proportional cost of carrying the index b is equal to (r ÿ d) where r is the
riskless interest rate. The constant proportional rate for the dividends, ``report'' and ``deÂport,''
corresponding to the CAC 40 is d.
The adoption of this simplifying assumption must be clear. In fact, since option valuation
needs the construction of a hedged portfolio between the option and its underlying index,
the portfolio of 40 stocks, which duplicates the index, can be formed only when the
amounts of ``the report,'' ``deÂport'' and ``dividends'' are known. In this case, ignoring
tracking errors, a hedged portfolio aÁ la Black and Scholes (1973), can be adjusted
``continuously'' to eliminate the risk. If this assumption is relaxed, it is not an easy task
to value the CAC 40 options. It is possible to approximate the value of d for N ``dividends,''
13
French firms pay dividends once a year in the summer period. When a stock pays a dividend to a long
investor, the stock receives the shares ex-dividend at the settlement by the end of the month. Consequently, the
forward price must drop on the first day of the monthly settlement period by an amount equal to the amount of
dividends. The other forward prices are not modified by the dividend. If we ignore fiscal considerations, then the
forward price must drop by an amount equal to the discounted dividend. For more details, see Solnik (1988),
Solnik (1990), Solnik and Bousquet (1990).
14
For a comprehensive discussion of the validity of such relations in the cash and carry arbitrage in general
and for the Paris Bourse in particular, see for example Bellalah and Simon (1997a, 1997b).
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M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
``reports'' and ``deÂports'' using the formula for dividends in Harvey and Whaley (1992). In
this context, the value of d is:
N
P
r T ÿt
Di e
S
d
ln
i
i1
S
T
where Di corresponds to the cash amounts of ``reports,'' ``deÂports'' and dividends, and ti to their
timing. This approximation is easily adapted for the cash-income stream on the CAC 40 stock
index. The ``deÂport'' is treated as a dividend since it reduces the index price. The ``report'' is
assimilated to a ``negative'' dividend because the index price rises by the ``report'' amount.
When a riskless hedge between the option and the underlying index is formed, the partial
differential equation for the option price is given by:
1 2 2 @ 2 c S; t
@c S; t
@c S; t
sS
bS
ÿ rc S; t
0
2
2
@S
@S
@t
3
The American-style index call could be replicated by a portfolio consisting of a 1-day
European index call and a wildcard option, i.e., a 45-min put on the index call. Also, the
American-style index put could be replicated by a portfolio consisting of a 1-day European
index put and a wildcard option, i.e., a 45-min put on the index put.
The value of the European CAC 40 call is solution to Eq. (3):
c S; t Se bÿrt N1 d1 ÿ Keÿrt N1 d2
p
1 2
d1 ln S=K b s t =s t
2
p
d2 d1 ÿ s t
4
where K stands for the strike price, t for the time left in the option's life as a proportion of a
year and N1(.) is the cumulative normal distribution function.
The value of the European index put is solution to Eq. (3):
p S; t ÿSe bÿrt N1 ÿd1 Keÿrt N1 ÿd2
5
Similar equations appeared in Merton (1973) and Barone-Adesi and Whaley (1987). The
value of the American index call (or put) option C(S,t) [ P(S,t)] is ``nearly'' equal to the value
of the European index call (put) c(S,t) [ p(S,t)] and the value of the wildcard provision wct
(wpt). The word ``nearly'' is used because these values are not separable, and a compound
option valuation technique must be used. In a perfect market, the absence of costless arbitrage
implies the equality between the value of the American index call (put) and that of the
replicating portfolio.15
15
The valuation by duplication technique was used in different contexts by Geske (1979), Whaley (1981) and
Briys et al. (1998) among others.
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
83
Just 1 day before the American CAC 40 call's (put's) maturity date, its value is given by:
C S; t : T 1 c S; t wc;t
P S; t : T 1 p S; t wp;t
If the settlement cash flow is greater than the discounted next day's expected opening
option price, then the option holder should exercise rationally his wildcard options. At time t,
the expected following day stock index value is equal to the current index value less the
expected dividend. If we denote the time of expiration of the wildcard (about 45 min) by t,
then the option holder should rationally exercise his implicit wildcard option if:
S ÿ K > c S; t : T 1 ÿ t
for a call and
K ÿ S > p S; t : T 1 ÿ t
for a put.
The value of the wildcard option just prior to 17:45 H should satisfy the following
conditions for the CAC 40 call and put:
wc;t:T1ÿt Max0; S ÿ K ÿ c S; t
wp;t:T 1ÿt Max0; K ÿ S ÿ p S; t
The wildcard option values at the close of the exchange, the business day prior to their
expirations, can be found using the put±call parity relationship and the compound option
formulas. Following the approach in French and Maberly (1992) for the valuation of the
wildcard in the value of index calls, let us denote the compound option price by Hc. The value
of the wildcard option is:
wc;t:T1 Hc;t eÿrt S ÿ K ÿ c S; t
6
where:
ÿbT
Hc;t Se
N2
p
p
h s t; k s T ;
r
r
t
t
ÿrT
ÿ Ke N2 h; k;
T
T
ÿ eÿrt S ÿ KN1 h
p
S
1 2
h ln r ÿ b ÿ s t =s t
S
2
p
S
1
k ln
r ÿ b ÿ s2 t =s T
K
2
and S* is given by:
p
Seÿb Tÿt N1 k s T ÿ t ÿ Keÿr Tÿt N1 k ÿ S ÿ K 0
where N2(a,b,r) is the bivariate cumulative normal distribution function with parameters a
and b and a correlation coefficient r.
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M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
Using the generalized compound option approach in Roll (1977), Zhang (1997) and
Briys et al. (1998) for the valuation of the wildcard in index puts, let us denote the
compound option price by Hp. The value of the wildcard option is:
wp;t:T 1 Hp;t eÿrt K ÿ S ÿ p S; t
7
where:
Hp;t
r
r
p
p
t
t
ÿrT
ÿ Ke N2 h; k; ÿ
Se N2 h s t; k s T ; ÿ
T
T
ÿ eÿrt S ÿ KN1 h
ÿbT
and S* is given by:
p
ÿSeÿb T ÿt N1 k s T ÿ t Keÿr T ÿt N1 k K ÿ S 0
4. Simulations and empirical results
Using the above models, wildcard option values implicit in CAC 40 index calls and
puts are simulated. The analysis concerns in a first step only options with 1 day to
expiration. This allows the appreciation of the values of one-period wildcard options.
Then, the FW model is slightly adapted and used to test the value of wildcard options in
index calls and puts during the option's life with a new dataset. This allows the study of
the magnitude of multiperiod wildcard options for the Paris Bourse and represents our
main contribution in this paper.
Simulations and empirical tests require transaction data for observed index option
prices quoted each day at 17:00 h, the history of early exercise records of the CAC 40
index calls and puts between 17:00 and 17:45 h, the closing and opening index prices
and the prices of the CAC 40 stock index futures contracts. The closing index refers to
the ``indice de compensation'' calculated using the averages of closing index prices.
Estimations of the riskless interest rates, cash dividends, the amounts of ``reports,''
``deÂports'' and volatilities of the underlying index are also required for the year 1994.
Unfortunately, many data are missing. Besides, the quoted bid and ask spreads, which
are valid only for 10 contracts, do not reflect the ``true'' market prices for the opening
and closing periods. In fact, around the market closure asset prices are affected by
actions of different market participants who change the composition of their portfolios
for hedging or some other reasons like arbitrage in some typical days. Since many data
are missing, we tried to make simulations of the wildcard option premiums as realistic
as possible.
4.1. Simulations of the models for a 1-day wildcard option
The value of the wildcard option wc,t implicit in the CAC 40 call is simulated in
Table 1 for different levels and typical parameters of the CAC 40 stock index. The
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M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
Table 1
Simulations of the CAC 40 call option values with wildcard option premiums
CAC 40 index
CM
wc
CA
1500
1525
1550
1575
1600
1625
1650
1675
1700
1725
1750
1775
1800
0.86
0.86
0.86
0.86
0.86
7.76
26.08
50.24
75.21
100.20
125.20
150.20
175.20
0.00
0.00
0.00
0.00
0.00
0.01
0.57
1.09
1.13
1.15
1.17
1.19
1.21
0.86
0.86
0.86
0.86
0.86
7.77
26.65
51.33
76.34
101.35
126.37
151.39
176.41
The following parameters are used: K = 1625, T = 1 day, t = 45 min, s = 0.229, r = 0.08, b = 0.03.
following parameters are used: K = 1625, T = 1 day, t = 45 min, s = 0.229, r = 0.08,
b = 0.03. The European option value is calculated using the modified version of
Merton's (1973) model, hereafter CM (formula (4)), and the American option value,
CA, is given as the sum of the wildcard option (formula (6)) and the European values.
Table 1 shows significant values for the wildcard option. These values are important for
in-the-money options. The value of this feature seems to be an increasing function of
the option moneyness.
4.2. Empirical tests for a multiperiod setting
Using an adapted version of the model in Fleming and Whaley (1994) to account for
the cash distributions, some simulations are run in order to appreciate the values of
wildcard options for CAC 40 options. This model allows the study of the magnitude of
wildcard options in a multiperiod setting.
Each day, option prices with strike prices within more or less 25 points of the closing
index level are calculated. The data is used from the database of the MONEP. The
historical data regarding the index PX1 and options are available on CD-ROM since
1994. For each transaction, we compare the option market price with the last reported
price of the index. The option is considered at-the-money if the difference between the
strike price and the index level is less than 25 points. We affect the value 0 to at-themoney options in this interval. The values ÿ 1, ÿ 2, ÿ 3, ÿ 4 and ÿ 5 are attributed to
in-the-money options. For out-of-the-money options, the values 1±5 are considered. At
each strike price, six expiration dates are used. It is assumed that they correspond to the
closest six Fridays to the option's maturity dates. This is because the end of each month
corresponds to a maturity date for a CAC 40 option. Generally, Thursday or Friday
corresponds to a ``jour de liquidation'' on the RM market. Hence, if valuation is done
Thursday, then the following day is the nearest expiration if it is the nearest to the end of
86
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
Table 2
Dividend yield each month for the CAC 40 index during the period 1994 ± 1998 in %
Month
1994
1995
1996
1997
1998
January
February
March
April
May
June
July
August
September
October
November
December
Year
2.57
2.68
2.88
2.78
2.95
3.21
2.99
2.99
3.21
3.17
3.05
3.2
2.97
3.35
3.22
3.08
3
2.96
3.18
3.18
3.31
3.48
3.43
3.41
3.33
3.24
3.08
3.12
3.04
2.91
2.98
2.97
3.23
3.28
3.03
3.02
2.69
2.68
3.15
2.56
2.44
2.39
2.43
2.54
2.39
2.27
2.53
2.33
2.55
2.21
2.11
2.40
2.01
1.84
1.62
1.63
1.64
1.9
1.96
2.24
2.56
2.32
2.12
2.08
1.99
the month and the remaining expiration dates are the following five Fridays corresponding
to the last Fridays in the five following months.16
Estimates of the riskless interest rates are proxied with French T-bills from the database
of ``Association FrancÎaise de Finance'' (AFFI) and the data from the SBF.17 The amounts
of ``reports,'' ``deÂports'' and dividends are taken from the database of SBF. For the
binomial method, we account for the seasonal patterns in the cash dividends, report and
deÂport in the CAC 40 index portfolio by adapting the Harvey and Whaley (1992)
methodology. We use the adapted version proposed in Briys et al. (1998). These cash
distributions are expressed in percentage for the continuous time formula. They are reported
in Table 2.
Tables 3 and 4 give some descriptive statistics regarding index calls and puts and the ratio
put/call for the years 1994±1997 according to the degree of parity.
The number of traded at-the-money index calls in % is more important in 1995 when
compared to the other years. It represents 27.84% in 1995 against 21.38% in 1994 and
16.73% in 1996. To proxy for the CAC 40 volatility rate, volatility estimates are implied from
each transaction each day using a modified lattice approach similar to that in Fleming and
Whaley (1994). The binomial model proposed by Fleming and Whaley is appropriate for the
pricing of CAC 40 options that contain a sequence of end-of-day wildcard options. Each
option expires at the end of the day. Each day, implied volatilities are aggregated with respect
to the degree of parity. Hence, each day, we obtain 11 average implied volatilities
corresponding to different degrees of parity. To get an idea about the index volatility
16
The frequent occurrence of Thursdays in the distribution of the liquidation dates is easily explained by the
fact that liquidation takes place on the seventh business day preceding the end of the calendar month. Hence, when
the month ends on Friday, Saturday or Sunday, the liquidation will appear on a Thursday. See Solnik (1990) for
more details.
17
We would like to thank Mai Huu Minh, the head of research in the SBF, for providing us with the selected
data and for implementing some algorithms.
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
87
Table 3
Descriptive statistics in % for index calls traded at the Paris Bourse according to the option moneyness for the
period 1994 ± 1997
Call
1994
1995
1996
1997
ÿ5
ÿ4
ÿ3
ÿ2
ÿ1
0
1
2
3
4
5
Total
P/C
13.53
9.51
13.64
18.11
18.56
21.38
2.58
1.09
0.71
0.34
0.56
100
86.87
9.16
7.28
10.89
14.67
18.43
27.84
4.39
2.90
1.71
1.09
1.63
100
86.46
27.77
8.36
10.56
9.81
9.71
16.73
3.30
2.75
2.10
1.64
7.26
100
114.11
30.68
6.46
7.76
6.27
7.20
18.24
4.09
2.65
2.11
1.50
13.05
100
92.47
The option is at-the-money when the difference between the strike price and the index level is less than 25
points. The sign ÿ (+) corresponds to in-the-money (out-of-the-money) options.
estimates and to check for the presence of a volatility smile, we calculate an implied ratio of
volatility (Eq. (8)). This ratio is defined as follows:
sK;t
8
RvK;t
s0;t
for k = ÿ 5, . . ., 5.
By construction, this ratio is equal to 1 for at-the-money options. Table 5 reports the
mean ratios of implied volatility according to the degree of parity. Table 5 shows clearly
Table 4
Descriptive statistics in % for index puts traded at the Paris Bourse according to the option moneyness for the
period 1994 ± 1997
Put
1994
1995
1996
1997
ÿ5
ÿ4
ÿ3
ÿ2
ÿ1
0
1
2
3
4
5
8.45
7.51
11.96
15.65
18.69
29.78
4.22
1.93
0.77
0.37
0.67
18.95
10.85
13.45
15.73
17.13
20.29
1.98
0.86
0.41
0.17
0.17
49.80
7.50
8.07
8.22
8.07
12.74
1.82
1.26
0.95
0.36
1.21
72.15
4.61
5.10
4.47
4.57
6.16
1.28
0.64
0.26
0.22
0.55
The option is at-the-money when the difference between the strike price and the index level is less than 25
points. The sign ÿ (+) corresponds to in-the-money (out-of-the-money) options.
88
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
Table 5
Estimation of the mean ratios of volatility for CAC 40 call and put options using an adapted version of the FW
model
k
C
P
ÿ5
0.957
1.097
ÿ4
0.965
1.008
ÿ3
0.969
0.986
ÿ2
0.980
0.958
ÿ1
0.988
0.932
0
1
2
3
4
5
1.021
0.918
1.056
0.912
1.133
0.959
1.202
0.995
1.303
1.028
1.576
1.129
The ratios of implied volatility are estimated with respect to the degree of parity. The following equation is
used: RvK,t = sK,t/s0,t for k = ÿ 5, . . ., 5.
the existence of a smile effect implicit in index call and put options. Since there is a smile
effect and we are interested in wildcard options, the volatility estimates implied from the atthe-money quoted option values are restricted to a 20-min window around the close of the
Paris Bourse.
Relying on the specific features of the Paris Bourse, the adapted binomial model is used
to value European style, and wildcard-exclusive and -inclusive American-style options. The
calculations are performed each day during the simulation period. The methodology
presented in Fleming and Whaley (1994) is replicated in our paper by accounting for
the specificities of the Paris Bourse. The wildcard-exclusive American-style option is
simulated using a modified version of formula (6) in Fleming and Whaley. The wildcardinclusive American-style option is calculated using formula (7) in combination with
formula (5) in Fleming and Whaley. The adapted version of the FW model is similar to
that in Briys et al. (1998).
In the absence of the wildcard feature, the decision to early exercise CAC 40 call options
depends on the trade-off between the amounts of cash dividends, report, deÂport and the
interest income when exercise is deferred. For CAC 40 puts, the early exercise decision
represents a dilemma. The put holder compares the interest income upon immediate exercise,
and the possible index level drops as the cash dividends and report (deÂport) are paid.
Following Fleming and Whaley (1994), we refer to the difference between the wildcardexclusive American-style and the European-style option value as the interest/report/deÂport
early exercise premium. Since the wildcard feature alone may provide an incentive for early
exercise, we refer to the difference between wildcard-inclusive and -exclusive American-style
option values as the wildcard early exercise premium.
Table 6 contains simulation results of the CAC 40 call option values, interest, report,
dividend and wildcard early exercise premiums by option's moneyness and days to
expiration during the year 1994 using the FW model. Panel A shows the average wildcard-inclusive American-style option value. Panel B contains simulation results of the
average interest, report and dividend early exercise premium calculated using the difference
between the wildcard-exclusive American-style and the European-style option values. Panel
C shows simulation results of the average wildcard early exercise premium implicit in the
CAC 40 call options.
Table 7 contains the same information for CAC 40 put values. Panels B in Tables 6 and 7
indicate that the interest, report and dividend early exercise premium is significant in the
determination of the option value. This premium appears to be greater for CAC 40 puts. The
premium for a slightly in-the-money CAC 40 call is about 0.1 cent for a 1 week. It is about
89
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
Table 6
Simulation of the CAC 40 call option values interest/report/dividend and wildcard early exercise premiums by
option moneyness M and days to expiration T during 1994 using the FW model
T: 28 ± 34
T: 35 ± 41
No-O
Panel A: simulation of average wildcard-inclusive American-style option value
3
0.002
0.028
0.091
0.192
0.328
2
0.016
0.115
0.322
0.523
0.946
1
1.310
2.02
3.137
3.893
4.084
0
5.194
6.521
7.230
7.986
8.302
ÿ1
10.194
11.894
12.256
13.034
13.547
ÿ2
17.967
18.285
18.970
19.420
19.220
ÿ3
22.235
22.348
22.986
23.111
23.120
No-O
1085
1085
1085
1085
1085
0.561
1.004
4.959
9.089
14.027
21.001
24.267
1085
1082
1094
1084
1089
1084
1083
1080
M
T :1 ± 6
T: 7 ± 13
T: 14 ± 20
T: 21 ± 27
Panel B: simulation of average interest/report/dividend early exercise premium calculated using the difference
between the wildcard-exclusive American-style and the European-style option values
3
0.000
0.000
0.000
0.000
0.000
0.001
2
0.000
0.000
0.000
0.000
0.001
0.002
1
0.000
0.000
0.001
0.002
0.004
0.006
0
0.001
0.002
0.003
0.005
0.007
0.008
ÿ1
0.008
0.009
0.009
0.011
0.014
0.017
ÿ2
0.015
0.025
0.032
0.039
0.041
0.041
ÿ3
0.040
0.081
0.090
0.112
0.120
0.122
Panel C: simulation of the average wildcard early exercise premium implicit in the CAC 40 call options during
the year 1994 using the FW model
3
0.000
0.000
0.000
0.000
0.000
0.000
2
0.001
0.001
0.001
0.001
0.001
0.001
1
0.003
0.003
0.003
0.002
0.001
0.001
0
0.013
0.019
0.022
0.042
0.059
0.072
ÿ1
0.091
0.104
0.110
0.114
0.115
0.117
ÿ2
0.112
0.113
0.120
0.127
0.130
0.131
ÿ3
0.114
0.123
0.145
0.151
0.170
0.170
No-O: Number of observations.
0.4 cent for an equivalent put. This amount seems also to be an increasing function of time
and moneyness. Panel C reveals that the wildcard premium is an important part of the option
value. The value of the wildcard feature is nearly 0 when the CAC 40 index call is very outof-the-money as indicated by moneyness in the Intervals 3 and 1. Its value accounts for about
0.13 cent of a slightly in-the-money call and put option and for about 11 cents of a deep inthe-money option. The value of the mean wildcard option has a tendency to increase with
time and moneyness.
The wildcard premiums seem to be approximately equal for CAC 40 calls and puts with
the same characteristics. However, this behavior is not systematic. This premium increases
generally with time to expiration. The fact that the value of the wildcard premium increases
generally with the option's moneyness is easy to understand. This result can be explained by
the fact that in-the-money options are more frequently exercised than at-the-money or out-ofthe-money options.
90
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
Table 7
Simulation of the CAC 40 put option values interest/report/dividend and wildcard early exercise premiums by
option moneyness M and days to expiration T during 1994 using the FW model
T: 1 ± 6
T: 28 ± 34
T: 35 ± 41
No-O
Panel A: simulation of average wildcard-inclusive American-style option value
3
0.001
0.014
0.045
0.092
0.171
2
0.009
0.100
0.021
0.320
0.711
1
1.120
1.84
2.917
3.524
3.948
0
4.954
6.389
7.001
7.062
7.942
ÿ1
10.074
11.662
12.001
12.111
12.971
ÿ2
18.006
18.271
18.810
19.112
18.680
ÿ3
22.128
22.013
22.527
22.714
22.975
No-O
1085
1085
1085
1085
1085
0.272
0.795
4.572
8.012
13.117
20.214
23.173
1085
1228
1110
1102
1089
1122
1060
1020
M
T: 7 ± 13
T: 14 ± 20
T: 21 ± 27
Panel B: simulation of average interest/report/dividend early exercise premium calculated using the difference
between the wildcard-exclusive American-style and the European-style option values
3
0.000
0.000
0.001
0.003
0.006
0.0013
2
0.000
0.001
0.005
0.008
0.0011
0.021
1
0.001
0.003
0.015
0.019
0.024
0.052
0
0.004
0.054
0.048
0.0621
0.078
0.082
ÿ1
0.024
0.037
0.049
0.061
0.094
0.127
ÿ2
0.075
0.089
0.133
0.164
0.192
0.214
ÿ3
0.161
0.189
0.311
0.589
0.856
0.999
Panel C: simulation of the average wildcard early exercise premium implicit in the CAC 40 put options during
the year 1994 using the FW model
3
0.000
0.000
0.000
0.000
0.000
0.000
2
0.0006
0.0008
0.0009
0.0009
0.0009
0.001
1
0.002
0.002
0.002
0.002
0.001
0.001
0
0.013
0.020
0.021
0.039
0.054
0.069
ÿ1
0.091
0.111
0.116
0.128
0.134
0.136
ÿ2
0.112
0.111
0.118
0.131
0.142
0.143
ÿ3
0.114
0.114
0.116
0.162
0.178
0.191
No-O: number of observations.
Table 8 contains the simulation results for the value of the wildcard early exercise premium
with respect to the total early exercise premium for CAC 40 options by option moneyness M
and days to expiration T during 1994. Panel A shows the results for CAC 40 call options.
Panel B gives the results for CAC 40 put options. The average wildcard premium is expressed
as a fraction of the total early exercise premium. When the early exercise premium results
mainly from the wildcard option, a value of 1 is used. When the early exercise premium
results mainly from the interest income and report/deÂport and dividend incentives, a value of
0 is used.
Our results seem to be different from those reported in Fleming and Whaley (1994).18 The
results in Table 8 for the wildcard premium as a percentage of total exercise premiums are
not, in general, monotonic in maturity and moneyness as the results in Fleming and Whaley.
18
We thank the referee for suggesting this comparison.
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
91
Table 8
Simulation of the value of the wildcard early exercise premium with respect to the total early exercise premium for
CAC 40 options by option moneyness M and days to expiration T during 1994 using the FW model
T: 1 ± 6
T: 21 ± 27
T: 28 ± 34
T: 35 ± 41
No-O
Panel A: simulations for the CAC 40 call options
3
0.000
0.000
0.000
2
1
1
1
1
1
1
0.75
0
0.9285
0.9047
0.880
ÿ1
0.9191
0.9203
0.924
ÿ2
0.8818
0.8188
0.7894
ÿ3
0.74025
0.6029
0.617
No-O
1085
1085
1085
0.000
1
0.5
0.8986
0.910
0.7650
0.5741
1085
0.000
0.500
0.2
0.8939
0.8914
0.76023
0.5862
1085
0.000
0.3333
0.1428
0.900
0.8731
0.7616
0.5821
1085
1082
1094
1084
1089
1084
1083
1080
Panel B: simulations for the CAC 40 put options
3
0.000
0.0000
0.0000
2
1
0.8888
0.64281
1
0.6666
0.400
0.1176
0
0.7647
0.2702
0.3043
ÿ1
0.7913
0.7432
0.7030
ÿ2
0.5989
0.5500
0.4701
ÿ3
0.4145
0.3762
0.2716
No-O
1085
1085
1085
0.000
0.5294
0.09524
0.3857
0.6772
0.426
0.2157
1085
0.000
0.45
0.04
0.4090
0.5877
0.4251
0.1721
1085
0.000
0.0455
0.088
0.4569
0.5171
0.4005
0.1605
1085
1228
1110
1102
1089
1122
1060
1020
M
T: 7 ± 13
T: 14 ± 20
No-O: number of observations.
The pattern seems relatively stable for all the times to maturity but not for the intervals with
T:[14±20] and T:[21±27]. This last period corresponds, in general, for ``report'' and ``deport''
payments on the underlying assets of the CAC 40 index. These cash distributions can affect
the option values and the value of the exercise premium upward or downward.
For at-the-money or in-the-money calls and puts, the wildcard premium is increasing, in
general (with some exceptions), in both maturity and moneyness mainly because of the
effects of the CAC 40 cash distributions that affect early exercise of American options. In
fact, the exercise decision of options traded in the MONEP is based on an arbitrage
relationship between calls and puts. Using that relationship, market makers view calls and
puts as perfect substitutes. This might explain the observed behavior of wildcard premiums.19
Besides, our wildcard premium as a percentage of the call values is much smaller than those
reported in Fleming and Whaley (1994) because CAC 40 calls are not frequently exercised
before the maturity date even if the wildcard period is of 45 min.
It is interesting to note that for call options, the wildcard premium is generally greater than
the early
10 (2001) 75 ± 94
Valuation of American CAC 40 index and wildcard options
Mondher Bellalah*
Universite de Cergy, THEMA, 33 Boulevard du Port, 95 011 Cergy, France
Received 15 April 1998; revised 20 September 1999; accepted 22 October 1999
Abstract
American options traded on the CAC 40 index are cash settled. However, the stocks
underlying the CAC 40 index are traded in a particular ``forward'' market. In this market,
settlements take place periodically on a given date as in the UK. At that date, all transactions
accomplished before are settled. The settlement procedure differs from that in countries where
settlements appear as fixed number of business days after the transaction as in the US. The
characteristics of the distributions to the index underlying stocks might justify a specific model for
the valuation of these options. Besides, the organization of the Paris Bourse gives market
participants the possibility to exercise their positions during the 45 min following the close of the
stock exchange. They face each day as an exercise risk that must be accounted for. This article
applies to existing models for the valuation of CAC 40 options, and this type of risk is identified
as a wildcard option. This option is implicit in the values of index calls and puts. The magnitude
of wildcard options in a multiperiod setting is studied empirically using a new dataset by adapting
the model in Fleming and Whaley (1994) to the French market. D 2001 Elsevier Science Inc. All
rights reserved.
JEL classification: G12; G13; G14
Keywords: Wildcard options; Forward; Futures; Compound options; Volatility smile
1. Introduction
American index options are traded on the CAC 40 index. The underlying stocks are
traded in a particular market, the ``Marche aÁ ReÁglement Mensuel de la Bourse de Paris''
* E-mail address: [email protected] (M. Bellalah).
1059-0560/01/$ ± see front matter D 2001 Elsevier Science Inc. All rights reserved.
PII: S 1 0 5 9 - 0 5 6 0 ( 0 0 ) 0 0 0 7 2 - 1
76
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
(RM market).1 The RM market is a forward market providing a single monthly settlement
of all transactions at the end of the month. The liquidation day corresponds to the last
trading day of the month in which all trades are settled. Some of the features of this
market are shared by the UK system, except that the liquidation appears on any day of
the week. The cash transfers appear on the last business day of the month.
The stocks underlying the CAC 40 index give rise to 520 cash distributions a year and may
justify a specific model for the valuation of American index options. This question is of some
importance for market makers and traders operating in the Paris Bourse and for outside
investors who resort to the French market to implement some arbitrage strategies based on
index options, futures contracts and the underlying index.2 CAC 40 index options have
embedded wildcard options. In fact, the settlement procedure implies that an end-of-day
wildcard option arises in the case of the CAC 40 index options.
The wildcard option is essentially a free choice for index option holders. It results from the
fact that the option holder has 45 min to decide on exercise after the close of the Paris Bourse
while already knowing the exercise value. The exercise value of the index is set equal to the
mean index level a few seconds after 17:00 h Paris time. The choice of the mean index instead of
the closing index level is justified in order to avoid arbitrage strategies by market participants.
The decision to exercise depends on the expectations of the option holder regarding the
market trends. When the stock exchange closes, option holders have the possibility to trade the
underlying CAC 40 index options in non-organized markets. In these markets, the CAC 40
future contract is also traded at the same time. This provides market participants with more
information about the trend in the financial market. The expectations of market makers and
traders are based on the available information during the time window of 45 min after the close
of the Paris Bourse.
Wildcard options, which are implicit delivery options, exist in several financial contracts
and have been extensively studied in the literature. The wildcard option feature has been
investigated for US-listed options and futures contracts.3 This implicit option arises when the
exercise value of a derivative asset is determined before the final exercise date and when
exercise closes the underlying asset position. Even if the topic of wildcard option pricing has
been well researched for calls in several markets, little work has been done for the Paris
Bourse.4 The specific features of CAC 40 options and in particular, the ``report'' mechanism,
may have some effects on the valuation of these options and their implicit wildcard options.
1
There are roughly two settlement procedures around the world: the fixed settlement lag procedure and the
fixed settlement date. The first corresponds, for example, for the US stock transactions that are settled 5 business
days after the trade. The second procedure corresponds to countries like the UK, Italy and France, which are
examples of countries with fixed settlement date. For example, the trading year in the UK is divided into account
settlement periods of 2 weeks. This market is a forward market in which a new account period starts every other
Monday. In France, Italy and some other countries, transactions are settled once a month on a fixed date.
2
For a comprehensive discussion of the specificities of these instruments and markets and especially for the
Paris Bourse, the reader can refer to Bellalah (1990), Bellalah (1991), Bellalah and Jacquillat (1995) and Briys,
Bellalah, et al. (1998).
3
See, for example, Fleming and Whaley (1994), French and Maberly (1992), Valerio (1993) and Cohen
(1995) among others.
4
The valuation formulas for other exchanges are similar to those in Zhang (1997).
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
77
For the case of the CAC 40 index options, a potential risk exposure results from the
mechanism of cash settlement. This risk concerns both option buyers and sellers. It derives
from the possibility available to option buyers to exercise their options after the close of the
exchange. This risk has two dimensions. The first is the wildcard option that allows the
holder of an index call or put to exercise the option after the close of the exchange during
this time period (45 min). The second is the timing risk associated with the fact that the
seller of the option will not be notified of the exercise decision before the following day.
This is not the case for the option buyer who does not support this risk. Therefore, for the
option's writer, it seems that there is a greater risk in exercising before than after the close
of the stock market when the exercise value is known with certainty. In fact, if the option is
exercised before the close of the stock market, the seller is notified a few seconds after
17:00 H local time. This gives him enough time (45 min) to structure his portfolio
strategies and to reallocate the appropriate weights to his assets. If he is notified the
following day, he does not have the necessary time to reallocate his portfolio weights. This
is an important issue for institutional investors who implement asset allocation and portfolio
insurance strategies using CAC 40 index options and futures.
This paper's contribution is mainly empirical and not theoretical. In Section 2, we explain
the specific features of the French market. Some details regarding the CAC 40 options are
given. Then, two wildcard options are analyzed and identified. The first is identified in index
calls as a put option on the CAC 40 call. The analysis in this case follows the work of French
and Maberly (1992). The second is identified in index puts and is regarded as a put option on
the CAC 40 put. The empirical analysis for the Paris Bourse is new. In Section 3, we apply two
formulas for the valuation of the two wildcard options. In Section 4, we run some simulations
for index and wildcard options, and present our empirical results.
Since the formulas are valid only for a 1-day wildcard option and formulas are proposed
in the literature for the values of the wildcard call during the option's life (in a discrete
time), we provide some empirical results in this sense. A modified version of the Fleming
and Whaley's (1994) model, the FW model, is used to appreciate the value of wildcard
options in a multiperiod setting.
2. Specific features of the Paris Bourse index derivatives and wildcard options
The CAC 40 stock index is calculated using an arithmetic mean of the 40 active stocks
traded in the RM stock market. There are two separate markets for the underlying stocks
trading on the RM market. On the first RM market, stocks are traded every day from 10:00 to
17:00 h local time. On the second market known as the ``report'' market, net long and short
positions are matched once a month in order to determine for each stock a cash amount called
the ``report'' or a ``deÂport.'' This amount is established by a confrontation between all the open
positions contracted by financial institutions in the market place. Then, it is communicated to
market participants and accounted for when stock prices are quoted in the next day. Since the
assets underlying the CAC 40 index trade are in this forward market, the specificities of these
stocks must be analyzed. This allows the understanding of the specificities of the CAC 40
index options and their wildcard features.
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M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
2.1. The RM stock market and the report market
On the equity market, securities are traded either on the cash market (marche au comptant)
or on account on the monthly settlement market.5 Investors on the RM market must meet an
initial margin call representing a percentage of the total amount of their order.6 Settlement of a
trade in a security on the cash market takes place on the third business day following the date
of such trade. The monthly settlement market's account period is not a rolling settlement
period following trade as in the US but a fixed monthly calendar period for settlement of
trades that occur within the period. Thus, while transactions are firm in both price and
quantity once they have been concluded, actual cash settlement and delivery of securities take
place on the last trading day of the month.
On the account day (i.e., 6 business days before the last business day of the month,
inclusive), investors who have not closed their positions and cannot or do not wish to deliver
securities sold or present payment for securities purchased may carry their positions over to
the next account period through a special market organized once each month. This ``contago
market'' determines the cost at which buyers can obtain the cash, and sellers can buy the
securities they need to meet their obligations as buyers or sellers at the end of the month and
thus carry their positions over to the settlement day of the following month.
On the Paris Bourse, only forward prices are observed because shares are not traded on a
cash market. In the first day of the monthly settlement period or the liquidation day,
forward prices should go up by an amount that is nearly equal to 1-month interest rate. The
buyer (seller) of a stock at an instant t at a price St in the RM stock market assumes a long
(short) position at the same date. He agrees to get (to deliver) the stock on a certain
specified future date ti for a specified price St, which is the unobserved stock price at
instant t.7
On the ``jour de liquidation,'' the net long position often exceeds the net short position. The
``jour de report,'' a confrontation between net long open and short positions ``deferred'' on
each stock, and the amount of capital available on the market place determine the equilibrium
amount of ``report'' and the market share price. When the net short position exceeds the net
5
The most actively traded French and foreign shares on the official list are traded on the monthly settlement
market.
6
For an empirical analysis of the limit order book and the order flow in the Paris Bourse, see for example
Biais, Hillion, and Spatt (1995).
7
On date ti known as ``le jour de liquidation,'' he receives (delivers) the stock and pays (receives) the price St.
The day ti corresponds to the sixth business day prior to the last trading day of the civil month. It corresponds also
to the day before the last trading day in the ``RM'' month, the ``jour de report.'' The buyer (the seller) can close out
his or her position prior to date ti by an offsetting transaction on the stock. This results in a gain (loss) given by the
difference between the two values of the stock. On the nearest ti, the buyer (the seller) who wants to defer the
position (and maturity) of the initial transaction, i.e., to keep a long (short) position in the stock, sells (buys) the
stocks he bought (sold) and signs a contract. The contract commits him to buy (resell) these stocks back at the next
``jour de liquidation'' for the same price. This transaction allows the buyer (seller) to maintain his initial long
(short) position. Transactions on the last day of the settlement month could be done ``ici'' (here) or ``laÁ bas'' (there)
meaning with delivery ``this month'' (here) or ``the next month'' (there). However, investors can engage
simultaneously in two types of transactions with two different market prices.
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
79
long position, equilibrium implies a ``deÂport.'' This amount is determined in the form of an
interest rate that multiplies the share price for the holding period. Long investors get this cash
income. When the net open short and long positions delayed are exactly matched, there is no
``report'' and no ``deÂport'' on the stocks traded in the RM market.8
2.2. The CAC 40 index and wildcard options
MONEP currently trades two option contracts on the CAC 40 index: (1) the CAC 40
short-term option (PX1), which is American style and was introduced in November 1988;
and (2) the CAC 40 long-term option (PXL), which is European style and was introduced in
October 1991.
Settlement takes the form of a transfer in cash equal to the difference between the strike
price of the PX1 option and the CAC 40 settlement price, taking into account the number of
contracts exercised and the trading unit. This daily index settlement price is the average of all
index values calculated and displayed between 16:40 and 17:00 h including the first index
value displayed after 17:00 H. The deadline for registering exercise instructions is set daily at
17:45 h with the exception of the expiration day when in-the-money options are automatically
exercised upon expiration. PX1 options may be exercised on any trading day from the date of
purchase up to the expiration date.9
The futures contract is based on the spot CAC 40 index. The settlement value is based on
the average of the last 40 reported CAC 40 index quotations between 15:40 and 16:00 h local
time. It is negotiated in the ``Palais de la Bourse'' between 10:00 and 17:00 h local time. On
the ``jour de liquidation,'' the buyer pays and takes the delivery of the 40 stocks, and the seller
receives the payment and delivers the stocks with their corresponding weights.10
The following example illustrates the specificities of the wildcard options. Consider a
market maker or a trader holding a call (a put) on the CAC 40 index with a strike price of
1700. Suppose that the CAC 40 index closes at 17:02 h on the business day prior to the last
day to expiration. The settlement index value is revealed to market participants immediately
after the close at 17:00 h local time. The value of this index call (put) is nearly its intrinsic
value since there is 1 day of time value left.
8
These are the specificities of observed forward prices with respect to the unobserved stock spot price given
the cost of carrying the stocks. In theory, the forward stock prices converge to the spot price that is implicit on the
last day of the account period. Then, it moves upward with respect to the spot on the first day of the account period
to account for the cost of carry. The stock price is said to be implicit because there is no organized cash market in
which the spot price can be observed. When there are no dividends, arbitrage considerations imply that the
forward price must be equal to the spot price plus the financing costs until the maturity of the forward contract.
9
Exercise of index options results in cash settlement between clearing members through the SBF following
confirmation by the SCMC of the exercise.
10
The major difference between index and equity options is that the asset underlying an index option is a
portfolio instead of an individual stock. This makes hedging between an index option and its underlying asset
more difficult. In practice, an index is duplicated by a portfolio containing few highly liquid stocks in proportions,
such that its return tracks the index's return. Therefore, index options are expected to be more severely mispriced
than stock options.
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M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
After the market closes, some bad news cause the market participants to believe that the
market will fall significantly the following day. The next day, the index call value will
probably be 0 and the index put value will be probably higher. In this case, it is better to
receive the index call intrinsic value today rather than waiting for the next day. The opposite
argument applies for the index put for which it may be worth waiting for a significant drop in
the index level.
Since market participants can exercise their positions during the 45 min following the
market close, this possibility represents the wildcard feature implicit in index calls and puts.
This feature comes into play each day from 17:00 to 17:45 h local time.11 Since the wildcard
feature represents the right given to the index call (put) holder to exercise his option, it is also
an option to sell the index call (put) at any time before 17:45 h. Immediately after 17:45 h
local time, the right to exercise is worthless. Hence, the wildcard option comes into existence
each day at the close and expires at 17:45 h. As a consequence, the option holder disposes of
a series of 45-min options, one for each day between 17:00 and 17:45 h until the option's
expiration date.12
The American-style CAC 40 call (index put) can be replicated by a portfolio of two options.
The first option is the regular option associated with the value of the underlying index. The
second is the wildcard option. The value of this privilege is positive to the buyer and negative
to the seller since the option holder decides to or not to exercise this right. Therefore, a rational
seller will demand compensation for giving the holder of an index call or put this right.
3. The valuation models
Based on the idiosyncrasies of the RM market, the specificities of CAC 40 options and
our preceding remarks consider an economy that satisfies the same assumptions as in Black
and Scholes (1973) and Merton (1973). The dynamics of the underlying index S are given
by Eq. (1):
dS=S mdt sdz
1
where m and s, respectively stand for the instantaneous expected rate of return and the
standard deviation of the CAC 40 return. The term dz is a standard Wiener process.
If the underlying index refers to a commodity contract, which is assumed to be a
``particular forward'' contract, and the distributions to this commodity are constant propor-
11
The settlement index price is established each day as the arithmetic mean of all index values quoted between
16:40 and 17:00 h local time. Each day, the index call holder (put holder) compares the payoff guaranteed by the
exercise value with the probable next day's payoff. If the call price is below the payoff guaranteed by the
settlement price, he should exercise his call (put) option. Otherwise, he should not exercise and wait until the
following day.
12
If the wildcard feature has value only after 17:00 h every day, then a jump in the option price would happen
at that time. However, the discounted value of the future wildcards is impounded into the option price prior to
17:00 h.
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
81
tional rates (the rates of ``report'' and ``deÂport''), then the traditional models of cost of carry
apply.13 In this case, the index option can be valued by applying the commodity formulas by
considering these rates as constant proportional carrying costs. If one accounts for the profile
of cash distributions and their timing, a more specific option pricing model must be used for
the valuation of American CAC 40 options. This model would be extremely difficult to write
down in a continuous time framework because of the 520 distributions to the underlying
stocks for a 1-year maturity date. However, it is possible to simplify considerably the
valuation problem by modeling the French market as a ``particular forward'' market for the
two following reasons.
First, cash receipts (or payments) corresponding to the transactions in the RM stock
market are done at the end of the month for transactions accomplished at any time. It is as if
the short (long) investor lends (borrows) the value of the portfolio duplicating the index
price in a cash account until the maturity date. In this case, the value of the index grows at
the riskless rate. Second, if one considers the rates of ``report'' and ``deÂport'' paid (received)
at the end of each month during the index option's life as constant known proportional rates,
then the familiar cost of carrying model would apply as in Merton (1973) and Barone-Adesi
and Whaley (1987). Otherwise, models with stochastic distributions to the underlying assets
must be used.
The French market can hence be seen as a particular ``forward'' market because of its
specificities. Under the above assumptions, the usual relationship between the forward price
of the index and its spot value should apply. When it is verified in the absence of costless
arbitrage opportunities, then (Eq. (2)):
F SebT
2
where F is the current futures price, b is the cost of carrying the index, and T is the maturity
date.14 The constant proportional cost of carrying the index b is equal to (r ÿ d) where r is the
riskless interest rate. The constant proportional rate for the dividends, ``report'' and ``deÂport,''
corresponding to the CAC 40 is d.
The adoption of this simplifying assumption must be clear. In fact, since option valuation
needs the construction of a hedged portfolio between the option and its underlying index,
the portfolio of 40 stocks, which duplicates the index, can be formed only when the
amounts of ``the report,'' ``deÂport'' and ``dividends'' are known. In this case, ignoring
tracking errors, a hedged portfolio aÁ la Black and Scholes (1973), can be adjusted
``continuously'' to eliminate the risk. If this assumption is relaxed, it is not an easy task
to value the CAC 40 options. It is possible to approximate the value of d for N ``dividends,''
13
French firms pay dividends once a year in the summer period. When a stock pays a dividend to a long
investor, the stock receives the shares ex-dividend at the settlement by the end of the month. Consequently, the
forward price must drop on the first day of the monthly settlement period by an amount equal to the amount of
dividends. The other forward prices are not modified by the dividend. If we ignore fiscal considerations, then the
forward price must drop by an amount equal to the discounted dividend. For more details, see Solnik (1988),
Solnik (1990), Solnik and Bousquet (1990).
14
For a comprehensive discussion of the validity of such relations in the cash and carry arbitrage in general
and for the Paris Bourse in particular, see for example Bellalah and Simon (1997a, 1997b).
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M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
``reports'' and ``deÂports'' using the formula for dividends in Harvey and Whaley (1992). In
this context, the value of d is:
N
P
r T ÿt
Di e
S
d
ln
i
i1
S
T
where Di corresponds to the cash amounts of ``reports,'' ``deÂports'' and dividends, and ti to their
timing. This approximation is easily adapted for the cash-income stream on the CAC 40 stock
index. The ``deÂport'' is treated as a dividend since it reduces the index price. The ``report'' is
assimilated to a ``negative'' dividend because the index price rises by the ``report'' amount.
When a riskless hedge between the option and the underlying index is formed, the partial
differential equation for the option price is given by:
1 2 2 @ 2 c S; t
@c S; t
@c S; t
sS
bS
ÿ rc S; t
0
2
2
@S
@S
@t
3
The American-style index call could be replicated by a portfolio consisting of a 1-day
European index call and a wildcard option, i.e., a 45-min put on the index call. Also, the
American-style index put could be replicated by a portfolio consisting of a 1-day European
index put and a wildcard option, i.e., a 45-min put on the index put.
The value of the European CAC 40 call is solution to Eq. (3):
c S; t Se bÿrt N1 d1 ÿ Keÿrt N1 d2
p
1 2
d1 ln S=K b s t =s t
2
p
d2 d1 ÿ s t
4
where K stands for the strike price, t for the time left in the option's life as a proportion of a
year and N1(.) is the cumulative normal distribution function.
The value of the European index put is solution to Eq. (3):
p S; t ÿSe bÿrt N1 ÿd1 Keÿrt N1 ÿd2
5
Similar equations appeared in Merton (1973) and Barone-Adesi and Whaley (1987). The
value of the American index call (or put) option C(S,t) [ P(S,t)] is ``nearly'' equal to the value
of the European index call (put) c(S,t) [ p(S,t)] and the value of the wildcard provision wct
(wpt). The word ``nearly'' is used because these values are not separable, and a compound
option valuation technique must be used. In a perfect market, the absence of costless arbitrage
implies the equality between the value of the American index call (put) and that of the
replicating portfolio.15
15
The valuation by duplication technique was used in different contexts by Geske (1979), Whaley (1981) and
Briys et al. (1998) among others.
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
83
Just 1 day before the American CAC 40 call's (put's) maturity date, its value is given by:
C S; t : T 1 c S; t wc;t
P S; t : T 1 p S; t wp;t
If the settlement cash flow is greater than the discounted next day's expected opening
option price, then the option holder should exercise rationally his wildcard options. At time t,
the expected following day stock index value is equal to the current index value less the
expected dividend. If we denote the time of expiration of the wildcard (about 45 min) by t,
then the option holder should rationally exercise his implicit wildcard option if:
S ÿ K > c S; t : T 1 ÿ t
for a call and
K ÿ S > p S; t : T 1 ÿ t
for a put.
The value of the wildcard option just prior to 17:45 H should satisfy the following
conditions for the CAC 40 call and put:
wc;t:T1ÿt Max0; S ÿ K ÿ c S; t
wp;t:T 1ÿt Max0; K ÿ S ÿ p S; t
The wildcard option values at the close of the exchange, the business day prior to their
expirations, can be found using the put±call parity relationship and the compound option
formulas. Following the approach in French and Maberly (1992) for the valuation of the
wildcard in the value of index calls, let us denote the compound option price by Hc. The value
of the wildcard option is:
wc;t:T1 Hc;t eÿrt S ÿ K ÿ c S; t
6
where:
ÿbT
Hc;t Se
N2
p
p
h s t; k s T ;
r
r
t
t
ÿrT
ÿ Ke N2 h; k;
T
T
ÿ eÿrt S ÿ KN1 h
p
S
1 2
h ln r ÿ b ÿ s t =s t
S
2
p
S
1
k ln
r ÿ b ÿ s2 t =s T
K
2
and S* is given by:
p
Seÿb Tÿt N1 k s T ÿ t ÿ Keÿr Tÿt N1 k ÿ S ÿ K 0
where N2(a,b,r) is the bivariate cumulative normal distribution function with parameters a
and b and a correlation coefficient r.
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M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
Using the generalized compound option approach in Roll (1977), Zhang (1997) and
Briys et al. (1998) for the valuation of the wildcard in index puts, let us denote the
compound option price by Hp. The value of the wildcard option is:
wp;t:T 1 Hp;t eÿrt K ÿ S ÿ p S; t
7
where:
Hp;t
r
r
p
p
t
t
ÿrT
ÿ Ke N2 h; k; ÿ
Se N2 h s t; k s T ; ÿ
T
T
ÿ eÿrt S ÿ KN1 h
ÿbT
and S* is given by:
p
ÿSeÿb T ÿt N1 k s T ÿ t Keÿr T ÿt N1 k K ÿ S 0
4. Simulations and empirical results
Using the above models, wildcard option values implicit in CAC 40 index calls and
puts are simulated. The analysis concerns in a first step only options with 1 day to
expiration. This allows the appreciation of the values of one-period wildcard options.
Then, the FW model is slightly adapted and used to test the value of wildcard options in
index calls and puts during the option's life with a new dataset. This allows the study of
the magnitude of multiperiod wildcard options for the Paris Bourse and represents our
main contribution in this paper.
Simulations and empirical tests require transaction data for observed index option
prices quoted each day at 17:00 h, the history of early exercise records of the CAC 40
index calls and puts between 17:00 and 17:45 h, the closing and opening index prices
and the prices of the CAC 40 stock index futures contracts. The closing index refers to
the ``indice de compensation'' calculated using the averages of closing index prices.
Estimations of the riskless interest rates, cash dividends, the amounts of ``reports,''
``deÂports'' and volatilities of the underlying index are also required for the year 1994.
Unfortunately, many data are missing. Besides, the quoted bid and ask spreads, which
are valid only for 10 contracts, do not reflect the ``true'' market prices for the opening
and closing periods. In fact, around the market closure asset prices are affected by
actions of different market participants who change the composition of their portfolios
for hedging or some other reasons like arbitrage in some typical days. Since many data
are missing, we tried to make simulations of the wildcard option premiums as realistic
as possible.
4.1. Simulations of the models for a 1-day wildcard option
The value of the wildcard option wc,t implicit in the CAC 40 call is simulated in
Table 1 for different levels and typical parameters of the CAC 40 stock index. The
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M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
Table 1
Simulations of the CAC 40 call option values with wildcard option premiums
CAC 40 index
CM
wc
CA
1500
1525
1550
1575
1600
1625
1650
1675
1700
1725
1750
1775
1800
0.86
0.86
0.86
0.86
0.86
7.76
26.08
50.24
75.21
100.20
125.20
150.20
175.20
0.00
0.00
0.00
0.00
0.00
0.01
0.57
1.09
1.13
1.15
1.17
1.19
1.21
0.86
0.86
0.86
0.86
0.86
7.77
26.65
51.33
76.34
101.35
126.37
151.39
176.41
The following parameters are used: K = 1625, T = 1 day, t = 45 min, s = 0.229, r = 0.08, b = 0.03.
following parameters are used: K = 1625, T = 1 day, t = 45 min, s = 0.229, r = 0.08,
b = 0.03. The European option value is calculated using the modified version of
Merton's (1973) model, hereafter CM (formula (4)), and the American option value,
CA, is given as the sum of the wildcard option (formula (6)) and the European values.
Table 1 shows significant values for the wildcard option. These values are important for
in-the-money options. The value of this feature seems to be an increasing function of
the option moneyness.
4.2. Empirical tests for a multiperiod setting
Using an adapted version of the model in Fleming and Whaley (1994) to account for
the cash distributions, some simulations are run in order to appreciate the values of
wildcard options for CAC 40 options. This model allows the study of the magnitude of
wildcard options in a multiperiod setting.
Each day, option prices with strike prices within more or less 25 points of the closing
index level are calculated. The data is used from the database of the MONEP. The
historical data regarding the index PX1 and options are available on CD-ROM since
1994. For each transaction, we compare the option market price with the last reported
price of the index. The option is considered at-the-money if the difference between the
strike price and the index level is less than 25 points. We affect the value 0 to at-themoney options in this interval. The values ÿ 1, ÿ 2, ÿ 3, ÿ 4 and ÿ 5 are attributed to
in-the-money options. For out-of-the-money options, the values 1±5 are considered. At
each strike price, six expiration dates are used. It is assumed that they correspond to the
closest six Fridays to the option's maturity dates. This is because the end of each month
corresponds to a maturity date for a CAC 40 option. Generally, Thursday or Friday
corresponds to a ``jour de liquidation'' on the RM market. Hence, if valuation is done
Thursday, then the following day is the nearest expiration if it is the nearest to the end of
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M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
Table 2
Dividend yield each month for the CAC 40 index during the period 1994 ± 1998 in %
Month
1994
1995
1996
1997
1998
January
February
March
April
May
June
July
August
September
October
November
December
Year
2.57
2.68
2.88
2.78
2.95
3.21
2.99
2.99
3.21
3.17
3.05
3.2
2.97
3.35
3.22
3.08
3
2.96
3.18
3.18
3.31
3.48
3.43
3.41
3.33
3.24
3.08
3.12
3.04
2.91
2.98
2.97
3.23
3.28
3.03
3.02
2.69
2.68
3.15
2.56
2.44
2.39
2.43
2.54
2.39
2.27
2.53
2.33
2.55
2.21
2.11
2.40
2.01
1.84
1.62
1.63
1.64
1.9
1.96
2.24
2.56
2.32
2.12
2.08
1.99
the month and the remaining expiration dates are the following five Fridays corresponding
to the last Fridays in the five following months.16
Estimates of the riskless interest rates are proxied with French T-bills from the database
of ``Association FrancÎaise de Finance'' (AFFI) and the data from the SBF.17 The amounts
of ``reports,'' ``deÂports'' and dividends are taken from the database of SBF. For the
binomial method, we account for the seasonal patterns in the cash dividends, report and
deÂport in the CAC 40 index portfolio by adapting the Harvey and Whaley (1992)
methodology. We use the adapted version proposed in Briys et al. (1998). These cash
distributions are expressed in percentage for the continuous time formula. They are reported
in Table 2.
Tables 3 and 4 give some descriptive statistics regarding index calls and puts and the ratio
put/call for the years 1994±1997 according to the degree of parity.
The number of traded at-the-money index calls in % is more important in 1995 when
compared to the other years. It represents 27.84% in 1995 against 21.38% in 1994 and
16.73% in 1996. To proxy for the CAC 40 volatility rate, volatility estimates are implied from
each transaction each day using a modified lattice approach similar to that in Fleming and
Whaley (1994). The binomial model proposed by Fleming and Whaley is appropriate for the
pricing of CAC 40 options that contain a sequence of end-of-day wildcard options. Each
option expires at the end of the day. Each day, implied volatilities are aggregated with respect
to the degree of parity. Hence, each day, we obtain 11 average implied volatilities
corresponding to different degrees of parity. To get an idea about the index volatility
16
The frequent occurrence of Thursdays in the distribution of the liquidation dates is easily explained by the
fact that liquidation takes place on the seventh business day preceding the end of the calendar month. Hence, when
the month ends on Friday, Saturday or Sunday, the liquidation will appear on a Thursday. See Solnik (1990) for
more details.
17
We would like to thank Mai Huu Minh, the head of research in the SBF, for providing us with the selected
data and for implementing some algorithms.
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
87
Table 3
Descriptive statistics in % for index calls traded at the Paris Bourse according to the option moneyness for the
period 1994 ± 1997
Call
1994
1995
1996
1997
ÿ5
ÿ4
ÿ3
ÿ2
ÿ1
0
1
2
3
4
5
Total
P/C
13.53
9.51
13.64
18.11
18.56
21.38
2.58
1.09
0.71
0.34
0.56
100
86.87
9.16
7.28
10.89
14.67
18.43
27.84
4.39
2.90
1.71
1.09
1.63
100
86.46
27.77
8.36
10.56
9.81
9.71
16.73
3.30
2.75
2.10
1.64
7.26
100
114.11
30.68
6.46
7.76
6.27
7.20
18.24
4.09
2.65
2.11
1.50
13.05
100
92.47
The option is at-the-money when the difference between the strike price and the index level is less than 25
points. The sign ÿ (+) corresponds to in-the-money (out-of-the-money) options.
estimates and to check for the presence of a volatility smile, we calculate an implied ratio of
volatility (Eq. (8)). This ratio is defined as follows:
sK;t
8
RvK;t
s0;t
for k = ÿ 5, . . ., 5.
By construction, this ratio is equal to 1 for at-the-money options. Table 5 reports the
mean ratios of implied volatility according to the degree of parity. Table 5 shows clearly
Table 4
Descriptive statistics in % for index puts traded at the Paris Bourse according to the option moneyness for the
period 1994 ± 1997
Put
1994
1995
1996
1997
ÿ5
ÿ4
ÿ3
ÿ2
ÿ1
0
1
2
3
4
5
8.45
7.51
11.96
15.65
18.69
29.78
4.22
1.93
0.77
0.37
0.67
18.95
10.85
13.45
15.73
17.13
20.29
1.98
0.86
0.41
0.17
0.17
49.80
7.50
8.07
8.22
8.07
12.74
1.82
1.26
0.95
0.36
1.21
72.15
4.61
5.10
4.47
4.57
6.16
1.28
0.64
0.26
0.22
0.55
The option is at-the-money when the difference between the strike price and the index level is less than 25
points. The sign ÿ (+) corresponds to in-the-money (out-of-the-money) options.
88
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
Table 5
Estimation of the mean ratios of volatility for CAC 40 call and put options using an adapted version of the FW
model
k
C
P
ÿ5
0.957
1.097
ÿ4
0.965
1.008
ÿ3
0.969
0.986
ÿ2
0.980
0.958
ÿ1
0.988
0.932
0
1
2
3
4
5
1.021
0.918
1.056
0.912
1.133
0.959
1.202
0.995
1.303
1.028
1.576
1.129
The ratios of implied volatility are estimated with respect to the degree of parity. The following equation is
used: RvK,t = sK,t/s0,t for k = ÿ 5, . . ., 5.
the existence of a smile effect implicit in index call and put options. Since there is a smile
effect and we are interested in wildcard options, the volatility estimates implied from the atthe-money quoted option values are restricted to a 20-min window around the close of the
Paris Bourse.
Relying on the specific features of the Paris Bourse, the adapted binomial model is used
to value European style, and wildcard-exclusive and -inclusive American-style options. The
calculations are performed each day during the simulation period. The methodology
presented in Fleming and Whaley (1994) is replicated in our paper by accounting for
the specificities of the Paris Bourse. The wildcard-exclusive American-style option is
simulated using a modified version of formula (6) in Fleming and Whaley. The wildcardinclusive American-style option is calculated using formula (7) in combination with
formula (5) in Fleming and Whaley. The adapted version of the FW model is similar to
that in Briys et al. (1998).
In the absence of the wildcard feature, the decision to early exercise CAC 40 call options
depends on the trade-off between the amounts of cash dividends, report, deÂport and the
interest income when exercise is deferred. For CAC 40 puts, the early exercise decision
represents a dilemma. The put holder compares the interest income upon immediate exercise,
and the possible index level drops as the cash dividends and report (deÂport) are paid.
Following Fleming and Whaley (1994), we refer to the difference between the wildcardexclusive American-style and the European-style option value as the interest/report/deÂport
early exercise premium. Since the wildcard feature alone may provide an incentive for early
exercise, we refer to the difference between wildcard-inclusive and -exclusive American-style
option values as the wildcard early exercise premium.
Table 6 contains simulation results of the CAC 40 call option values, interest, report,
dividend and wildcard early exercise premiums by option's moneyness and days to
expiration during the year 1994 using the FW model. Panel A shows the average wildcard-inclusive American-style option value. Panel B contains simulation results of the
average interest, report and dividend early exercise premium calculated using the difference
between the wildcard-exclusive American-style and the European-style option values. Panel
C shows simulation results of the average wildcard early exercise premium implicit in the
CAC 40 call options.
Table 7 contains the same information for CAC 40 put values. Panels B in Tables 6 and 7
indicate that the interest, report and dividend early exercise premium is significant in the
determination of the option value. This premium appears to be greater for CAC 40 puts. The
premium for a slightly in-the-money CAC 40 call is about 0.1 cent for a 1 week. It is about
89
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
Table 6
Simulation of the CAC 40 call option values interest/report/dividend and wildcard early exercise premiums by
option moneyness M and days to expiration T during 1994 using the FW model
T: 28 ± 34
T: 35 ± 41
No-O
Panel A: simulation of average wildcard-inclusive American-style option value
3
0.002
0.028
0.091
0.192
0.328
2
0.016
0.115
0.322
0.523
0.946
1
1.310
2.02
3.137
3.893
4.084
0
5.194
6.521
7.230
7.986
8.302
ÿ1
10.194
11.894
12.256
13.034
13.547
ÿ2
17.967
18.285
18.970
19.420
19.220
ÿ3
22.235
22.348
22.986
23.111
23.120
No-O
1085
1085
1085
1085
1085
0.561
1.004
4.959
9.089
14.027
21.001
24.267
1085
1082
1094
1084
1089
1084
1083
1080
M
T :1 ± 6
T: 7 ± 13
T: 14 ± 20
T: 21 ± 27
Panel B: simulation of average interest/report/dividend early exercise premium calculated using the difference
between the wildcard-exclusive American-style and the European-style option values
3
0.000
0.000
0.000
0.000
0.000
0.001
2
0.000
0.000
0.000
0.000
0.001
0.002
1
0.000
0.000
0.001
0.002
0.004
0.006
0
0.001
0.002
0.003
0.005
0.007
0.008
ÿ1
0.008
0.009
0.009
0.011
0.014
0.017
ÿ2
0.015
0.025
0.032
0.039
0.041
0.041
ÿ3
0.040
0.081
0.090
0.112
0.120
0.122
Panel C: simulation of the average wildcard early exercise premium implicit in the CAC 40 call options during
the year 1994 using the FW model
3
0.000
0.000
0.000
0.000
0.000
0.000
2
0.001
0.001
0.001
0.001
0.001
0.001
1
0.003
0.003
0.003
0.002
0.001
0.001
0
0.013
0.019
0.022
0.042
0.059
0.072
ÿ1
0.091
0.104
0.110
0.114
0.115
0.117
ÿ2
0.112
0.113
0.120
0.127
0.130
0.131
ÿ3
0.114
0.123
0.145
0.151
0.170
0.170
No-O: Number of observations.
0.4 cent for an equivalent put. This amount seems also to be an increasing function of time
and moneyness. Panel C reveals that the wildcard premium is an important part of the option
value. The value of the wildcard feature is nearly 0 when the CAC 40 index call is very outof-the-money as indicated by moneyness in the Intervals 3 and 1. Its value accounts for about
0.13 cent of a slightly in-the-money call and put option and for about 11 cents of a deep inthe-money option. The value of the mean wildcard option has a tendency to increase with
time and moneyness.
The wildcard premiums seem to be approximately equal for CAC 40 calls and puts with
the same characteristics. However, this behavior is not systematic. This premium increases
generally with time to expiration. The fact that the value of the wildcard premium increases
generally with the option's moneyness is easy to understand. This result can be explained by
the fact that in-the-money options are more frequently exercised than at-the-money or out-ofthe-money options.
90
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
Table 7
Simulation of the CAC 40 put option values interest/report/dividend and wildcard early exercise premiums by
option moneyness M and days to expiration T during 1994 using the FW model
T: 1 ± 6
T: 28 ± 34
T: 35 ± 41
No-O
Panel A: simulation of average wildcard-inclusive American-style option value
3
0.001
0.014
0.045
0.092
0.171
2
0.009
0.100
0.021
0.320
0.711
1
1.120
1.84
2.917
3.524
3.948
0
4.954
6.389
7.001
7.062
7.942
ÿ1
10.074
11.662
12.001
12.111
12.971
ÿ2
18.006
18.271
18.810
19.112
18.680
ÿ3
22.128
22.013
22.527
22.714
22.975
No-O
1085
1085
1085
1085
1085
0.272
0.795
4.572
8.012
13.117
20.214
23.173
1085
1228
1110
1102
1089
1122
1060
1020
M
T: 7 ± 13
T: 14 ± 20
T: 21 ± 27
Panel B: simulation of average interest/report/dividend early exercise premium calculated using the difference
between the wildcard-exclusive American-style and the European-style option values
3
0.000
0.000
0.001
0.003
0.006
0.0013
2
0.000
0.001
0.005
0.008
0.0011
0.021
1
0.001
0.003
0.015
0.019
0.024
0.052
0
0.004
0.054
0.048
0.0621
0.078
0.082
ÿ1
0.024
0.037
0.049
0.061
0.094
0.127
ÿ2
0.075
0.089
0.133
0.164
0.192
0.214
ÿ3
0.161
0.189
0.311
0.589
0.856
0.999
Panel C: simulation of the average wildcard early exercise premium implicit in the CAC 40 put options during
the year 1994 using the FW model
3
0.000
0.000
0.000
0.000
0.000
0.000
2
0.0006
0.0008
0.0009
0.0009
0.0009
0.001
1
0.002
0.002
0.002
0.002
0.001
0.001
0
0.013
0.020
0.021
0.039
0.054
0.069
ÿ1
0.091
0.111
0.116
0.128
0.134
0.136
ÿ2
0.112
0.111
0.118
0.131
0.142
0.143
ÿ3
0.114
0.114
0.116
0.162
0.178
0.191
No-O: number of observations.
Table 8 contains the simulation results for the value of the wildcard early exercise premium
with respect to the total early exercise premium for CAC 40 options by option moneyness M
and days to expiration T during 1994. Panel A shows the results for CAC 40 call options.
Panel B gives the results for CAC 40 put options. The average wildcard premium is expressed
as a fraction of the total early exercise premium. When the early exercise premium results
mainly from the wildcard option, a value of 1 is used. When the early exercise premium
results mainly from the interest income and report/deÂport and dividend incentives, a value of
0 is used.
Our results seem to be different from those reported in Fleming and Whaley (1994).18 The
results in Table 8 for the wildcard premium as a percentage of total exercise premiums are
not, in general, monotonic in maturity and moneyness as the results in Fleming and Whaley.
18
We thank the referee for suggesting this comparison.
M. Bellalah / International Review of Economics and Finance 10 (2001) 75±94
91
Table 8
Simulation of the value of the wildcard early exercise premium with respect to the total early exercise premium for
CAC 40 options by option moneyness M and days to expiration T during 1994 using the FW model
T: 1 ± 6
T: 21 ± 27
T: 28 ± 34
T: 35 ± 41
No-O
Panel A: simulations for the CAC 40 call options
3
0.000
0.000
0.000
2
1
1
1
1
1
1
0.75
0
0.9285
0.9047
0.880
ÿ1
0.9191
0.9203
0.924
ÿ2
0.8818
0.8188
0.7894
ÿ3
0.74025
0.6029
0.617
No-O
1085
1085
1085
0.000
1
0.5
0.8986
0.910
0.7650
0.5741
1085
0.000
0.500
0.2
0.8939
0.8914
0.76023
0.5862
1085
0.000
0.3333
0.1428
0.900
0.8731
0.7616
0.5821
1085
1082
1094
1084
1089
1084
1083
1080
Panel B: simulations for the CAC 40 put options
3
0.000
0.0000
0.0000
2
1
0.8888
0.64281
1
0.6666
0.400
0.1176
0
0.7647
0.2702
0.3043
ÿ1
0.7913
0.7432
0.7030
ÿ2
0.5989
0.5500
0.4701
ÿ3
0.4145
0.3762
0.2716
No-O
1085
1085
1085
0.000
0.5294
0.09524
0.3857
0.6772
0.426
0.2157
1085
0.000
0.45
0.04
0.4090
0.5877
0.4251
0.1721
1085
0.000
0.0455
0.088
0.4569
0.5171
0.4005
0.1605
1085
1228
1110
1102
1089
1122
1060
1020
M
T: 7 ± 13
T: 14 ± 20
No-O: number of observations.
The pattern seems relatively stable for all the times to maturity but not for the intervals with
T:[14±20] and T:[21±27]. This last period corresponds, in general, for ``report'' and ``deport''
payments on the underlying assets of the CAC 40 index. These cash distributions can affect
the option values and the value of the exercise premium upward or downward.
For at-the-money or in-the-money calls and puts, the wildcard premium is increasing, in
general (with some exceptions), in both maturity and moneyness mainly because of the
effects of the CAC 40 cash distributions that affect early exercise of American options. In
fact, the exercise decision of options traded in the MONEP is based on an arbitrage
relationship between calls and puts. Using that relationship, market makers view calls and
puts as perfect substitutes. This might explain the observed behavior of wildcard premiums.19
Besides, our wildcard premium as a percentage of the call values is much smaller than those
reported in Fleming and Whaley (1994) because CAC 40 calls are not frequently exercised
before the maturity date even if the wildcard period is of 45 min.
It is interesting to note that for call options, the wildcard premium is generally greater than
the early