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). 10 (2001) 75 ± 94
1059-0560/01/$ ± see front matterD2001 Elsevier Science Inc. All rights reserved.
(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.3This 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.4The 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.
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). 1
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
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.5Investors on the RM market must meet an initial margin call representing a percentage of the total amount of their order.6Settlement 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 instantt at a priceStin 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
tiknown as ``le jour de liquidation,'' he receives (delivers) the stock and pays (receives) the priceSt.
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
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.11Since 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 mdtsdz 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
tional rates (the rates of ``report'' and ``deÂport''), then the traditional models of cost of carry apply.13In 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)):
FSebT 2
whereFis the current futures price,bis the cost of carrying the index, andTis the maturity date.14The constant proportional cost of carrying the indexbis equal to (rÿd) whereris the riskless interest rate. The constant proportional rate for the dividends, ``report'' and ``deÂport,'' corresponding to the CAC 40 isd.
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 ofdforN``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
``reports'' and ``deÂports'' using the formula for dividends in Harvey and Whaley (1992). In this context, the value of d is:
d
whereDicorresponds to the cash amounts of ``reports,'' ``deÂports'' and dividends, andtito 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
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 bÿrtN1 d1 ÿKeÿrtN1 d2
whereKstands for the strike price,t for the time left in the option's life as a proportion of a year andN1(.) is the cumulative normal distribution function.
The value of the European index put is solution to Eq. (3):
p S;t ÿSe bÿrtN1 ÿd1 KeÿrtN1 ÿd2 5
Similar equations appeared in Merton (1973) and Barone-Adesi and Whaley (1987). The value of the American index call (or put) optionC(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
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 timet, 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) byt, then the option holder should rationally exercise his implicit wildcard option if:
SÿK>c S;t :T 1ÿt
for a call and
KÿS>p S;t :T 1ÿt
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; SÿK ÿc S;t
wp;t:T1ÿtMax0; KÿS ÿp 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 byHc. The value
of the wildcard option is:
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:T1Hp;teÿrt KÿS ÿp S;t 7
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
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-the-money options in this interval. The values ÿ1, ÿ2, ÿ3, ÿ4 and ÿ5 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
Table 1
Simulations of the CAC 40 call option values with wildcard option premiums
CAC 40 index CM wc CA
1500 0.86 0.00 0.86
1525 0.86 0.00 0.86
1550 0.86 0.00 0.86
1575 0.86 0.00 0.86
1600 0.86 0.00 0.86
1625 7.76 0.01 7.77
1650 26.08 0.57 26.65
1675 50.24 1.09 51.33
1700 75.21 1.13 76.34
1725 100.20 1.15 101.35
1750 125.20 1.17 126.37
1775 150.20 1.19 151.39
1800 175.20 1.21 176.41
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
Table 2
Dividend yield each month for the CAC 40 index during the period 1994 ± 1998 in %
Month 1994 1995 1996 1997 1998
January 2.57 3.35 3.08 2.56 2.01
February 2.68 3.22 3.12 2.44 1.84
March 2.88 3.08 3.04 2.39 1.62
April 2.78 3 2.91 2.43 1.63
May 2.95 2.96 2.98 2.54 1.64
June 3.21 3.18 2.97 2.39 1.9
July 2.99 3.18 3.23 2.27 1.96
August 2.99 3.31 3.28 2.53 2.24
September 3.21 3.48 3.03 2.33 2.56
October 3.17 3.43 3.02 2.55 2.32
November 3.05 3.41 2.69 2.21 2.12
December 3.2 3.33 2.68 2.11 2.08
Year 2.97 3.24 3.15 2.40 1.99
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
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:
RvK;t
sK;t
s0;t
8
fork=ÿ5, . . ., 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 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 13.53 9.16 27.77 30.68
ÿ4 9.51 7.28 8.36 6.46
ÿ3 13.64 10.89 10.56 7.76
ÿ2 18.11 14.67 9.81 6.27
ÿ1 18.56 18.43 9.71 7.20
0 21.38 27.84 16.73 18.24
1 2.58 4.39 3.30 4.09
2 1.09 2.90 2.75 2.65
3 0.71 1.71 2.10 2.11
4 0.34 1.09 1.64 1.50
5 0.56 1.63 7.26 13.05
Total 100 100 100 100
P/C 86.87 86.46 114.11 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.
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 8.45 18.95 49.80 72.15
ÿ4 7.51 10.85 7.50 4.61
ÿ3 11.96 13.45 8.07 5.10
ÿ2 15.65 15.73 8.22 4.47
ÿ1 18.69 17.13 8.07 4.57
0 29.78 20.29 12.74 6.16
1 4.22 1.98 1.82 1.28
2 1.93 0.86 1.26 0.64
3 0.77 0.41 0.95 0.26
4 0.37 0.17 0.36 0.22
5 0.67 0.17 1.21 0.55
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 at-the-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 wildcard-inclusive 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 wildcard-exclusive 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 wild-card-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
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 ÿ5 ÿ4 ÿ3 ÿ2 ÿ1 0 1 2 3 4 5
C 0.957 0.965 0.969 0.980 0.988 1.021 1.056 1.133 1.202 1.303 1.576 P 1.097 1.008 0.986 0.958 0.932 0.918 0.912 0.959 0.995 1.028 1.129
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 out-of-the-money as indicated by moneyness in the Intervals 3 and 1. Its value accounts for about 0.13 cent of a slightly the-money call and put option and for about 11 cents of a deep in-the-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-of-the-money options.
Table 6
Simulation of the CAC 40 call option values interest/report/dividend and wildcard early exercise premiums by option moneynessMand days to expirationTduring 1994 using the FW model
M T:1 ± 6 T: 7 ± 13 T: 14 ± 20 T: 21 ± 27 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 0.561 1082
2 0.016 0.115 0.322 0.523 0.946 1.004 1094
1 1.310 2.02 3.137 3.893 4.084 4.959 1084
0 5.194 6.521 7.230 7.986 8.302 9.089 1089
ÿ1 10.194 11.894 12.256 13.034 13.547 14.027 1084
ÿ2 17.967 18.285 18.970 19.420 19.220 21.001 1083
ÿ3 22.235 22.348 22.986 23.111 23.120 24.267 1080
No-O 1085 1085 1085 1085 1085 1085
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
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 moneynessM and days to expirationT 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).18The 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.
Table 7
Simulation of the CAC 40 put option values interest/report/dividend and wildcard early exercise premiums by option moneynessMand days to expirationTduring 1994 using the FW model
M T: 1 ± 6 T: 7 ± 13 T: 14 ± 20 T: 21 ± 27 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 0.272 1228
2 0.009 0.100 0.021 0.320 0.711 0.795 1110
1 1.120 1.84 2.917 3.524 3.948 4.572 1102
0 4.954 6.389 7.001 7.062 7.942 8.012 1089
ÿ1 10.074 11.662 12.001 12.111 12.971 13.117 1122
ÿ2 18.006 18.271 18.810 19.112 18.680 20.214 1060
ÿ3 22.128 22.013 22.527 22.714 22.975 23.173 1020
No-O 1085 1085 1085 1085 1085 1085
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.
18
The pattern seems relatively stable for all the times to maturity but not for the intervals with T:[14±20] andT:[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 exercise premium attributed to interest and cash payments. This appears in Panel A since all the values are greater than 0.5 except for some values. This tendency is not observed for CAC 40 put options in Panel B for which the results are mixed.
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 moneynessMand days to expirationTduring 1994 using the FW model
M T: 1 ± 6 T: 7 ± 13 T: 14 ± 20 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 0.000 0.000 0.000 1082
2 1 1 1 1 0.500 0.3333 1094
1 1 1 0.75 0.5 0.2 0.1428 1084
0 0.9285 0.9047 0.880 0.8986 0.8939 0.900 1089
ÿ1 0.9191 0.9203 0.924 0.910 0.8914 0.8731 1084
ÿ2 0.8818 0.8188 0.7894 0.7650 0.76023 0.7616 1083
ÿ3 0.74025 0.6029 0.617 0.5741 0.5862 0.5821 1080
No-O 1085 1085 1085 1085 1085 1085
Panel B: simulations for the CAC 40 put options
3 0.000 0.0000 0.0000 0.000 0.000 0.000 1228
2 1 0.8888 0.64281 0.5294 0.45 0.0455 1110
1 0.6666 0.400 0.1176 0.09524 0.04 0.088 1102
0 0.7647 0.2702 0.3043 0.3857 0.4090 0.4569 1089
ÿ1 0.7913 0.7432 0.7030 0.6772 0.5877 0.5171 1122
ÿ2 0.5989 0.5500 0.4701 0.426 0.4251 0.4005 1060
ÿ3 0.4145 0.3762 0.2716 0.2157 0.1721 0.1605 1020
No-O 1085 1085 1085 1085 1085 1085
No-O: number of observations.
19
Our results are different from those reported from the SP 100 options because of the specific elements of CAC 40 options. In fact, the presence of several cash distributions (report, deÂport and dividends) affect the values of CAC 40 options in different ways. These distributions can increase or decrease option values depending on the existence of a report or a deÂport in the market place. Since the values of these American options can increase or decrease in time according to the effects of cash distributions, it is natural to expect different patters in the evolution of the wildcard options.
Market data is used to examine the number of contracts exercised early as a percentage of the total number of contracts exercised for 1±80 days to expiration. It is found that nearly 5% of total exercises take place on the last Thursday before expiration and especially immediately after the futures contract's maturity date. The number of contracts exercised generally decreases as time to expiration increases, since less than 0.1% occurs between 30 and 80 days to expiration.
These results may be explained by several reasons. If the option is not exercised just before the maturity date, it would be worth the greater of its intrinsic value and 0. Second, the distributions to the underlying index may influence the decision of the option holder regarding early exercise through a given relationship between the dividend, ``report,'' ``deÂport'' and the level of the riskless interest rate. A final plausible explanation is associated with arbitrage strategies. This might explain why the wildcard premium is particularly important just 1 day before the option's expiration and especially the Thursday or Friday before the option's maturity date.
5. Conclusion
The purpose of this article is to study empirically the French CAC 40 index options and the wildcard features embedded in these contracts. First, the characteristics of the French-based index markets are explained. It turns out that the French market is a particular ``forward'' market. Then, the specificities of the CAC 40 index, index options and arbitrage in these markets are analyzed. The exposition of the factual details is based on the fact that very little published work on the French option markets appeared in scholarly journals. Second, some existing models are applied for the valuation of the CAC 40 options and their embedded wildcard options. The simple monoperiodic model for the valuation of wildcard options implicit in the CAC 40 options parallels that in French and Maberly (1992). The formula proposed for the wildcard option embedded in the index put is an adapted version of the models in Zhang (1997) and Briys et al. (1998). It seems that the values of wildcard options implicit in index puts are almost of the same magnitude as those in index calls.
based on the CAC 40 index. This work is also of interest to researchers and can be extended in several ways. In fact, there are other possible modeling of the cash distributions to the underlying CAC 40 index. As a consequence, specific models may report different prices when the exact timing and the stochastic character of these distributions are accounted for. Also, the model adopted here for a one-period wildcard options can be extended to a multiperiod setting by defining the upper bounds for the wildcard options as in Cohen (1995). It is also possible to extend the empirical work using the observed intra-day bid and ask spreads around market closure.
Acknowledgments
The author thanks the editor and an anonymous reviewer for their helpful comments that contribute significantly to the results of this version of the paper. We thank also Bertrand Jacquillat, Jacques Hamon, Darell Duffie, Bruno Solnik, Yves Simon, Jean Mathis and all the participants at the seminar conferences at the University of Paris-Dauphine, HEC and ESSEC for their helpful comments on earlier drafts of this paper.
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