Journal of Multinational Financial Management 10 2000 461 – 479
The early exercise premium in American put option prices
Malin Engstro¨m, Lars Norde´n
Department of Corporate Finance, School of Business, Stockholm Uni6ersity, S-
106 91
Stockholm, Sweden Received 15 July 1999; accepted 18 February 2000
Abstract
This study estimates the value of the early exercise premium in American put option prices using Swedish equity options data. The value of the premium is found as the deviation of the
American put price from European put-call parity, and in addition a theoretical estimate of the premium is computed. The empirically found premium is also used in a modified version
of the control variate approach to value American puts. The results indicate a substantial value of the early exercise premium, where the premium derived from put-call parity is higher
than the theoretical premium. The premium also increases with moneyness and time left to expiration, while the effect of interest rate and volatility depends on the moneyness of the
option. The modified control variate technique works reasonably well relative to the theoretical models. In particular, for deep in-the-money options, this technique is superior.
© 2000 Elsevier Science B.V. All rights reserved.
JEL classification
:
G10; G13 Keywords
:
Early exercise premium; American put options; Modified control variate technique www.elsevier.comlocateeconbase
1. Introduction
The equity options traded on the Swedish exchange for options and other derivative securities, OM, are American options. They provide the option holder
with the right to exercise the contract any time prior to expiration. The privilege to
Corresponding author. Tel.: + 46-8-6747139; fax: + 46-8-153054. E-mail address
:
lnfek.su.se L. Norde´n. 1042-444X00 - see front matter © 2000 Elsevier Science B.V. All rights reserved.
PII: S 1 0 4 2 - 4 4 4 X 0 0 0 0 0 2 5 - 6
exercise early makes the prices of American options to include an early exercise premium compared to the prices of European options. Merton 1973a shows that
optimal early exercise never will occur for American call options written on non-dividend paying stocks. These calls could therefore be valued as if they were
European, using, e.g. the Black and Scholes 1973 formula. When American call options on dividend paying stocks are concerned, early exercise might be optimal
just before the ex-dividend instant. For an American put option on the other hand, early exercise may be optimal even if the underlying stock is not paying any
dividends. In fact, an American put option should always be exercised early if it is sufficiently in-the-money.
The possibility of early exercise complicates the valuation of the American options, and several valuation approaches, both analytical approximations and
numerical methods, have been developed. Examples of the first category are Roll 1977, Geske 1979 and Whaley 1981 for call options and Geske and Johnson
1984 and MacMillan 1986 for put options. In addition, Barone-Adesi and Whaley 1987 analyse both calls and puts. Examples of the second category are
Brennan and Schwartz 1977, Boyle 1977 and Cox et al. 1979. Blomeyer and Johnson 1988 compare the accuracy of the Geske and Johnson 1984 American
option pricing model to the Black and Scholes 1973 European model for valuing equity put options traded on the Chicago Board Options Exchange CBOE. They
find that both models undervalue options relative to market prices, but that the American model prices are much closer. They also reason that the difference
between the deviation from market prices for the two models is an estimate of the early exercise premium. On average, this early exercise premium amounts to around
5 of the actual market price.
Zivney 1991 argues that the American option pricing models do not value the early exercise premium appropriately, and suggests that the value of early exercise
ought to be established empirically. In order to measure the value of the premium, Zivney examines deviations from European put-call parity henceforth PCP of the
American SP 100 index options traded on the CBOE. A deviation from PCP can be regarded as an approximation of the value of the early exercise potential for
either the call or the otherwise identical put, depending on which one of them is in-the-money. Zivney also suggests that American options could be valued using a
modification of the control variate technique see e.g. Hull and White 1988. In the modification, an empirical early exercise premium should be added to the
theoretical value of a European option to obtain the American option value.
Zivney 1991 discusses three different ways to establish the value of the early exercise premium. First directly, as the difference between otherwise identical
American and European options, which is not possible in most markets. Secondly, as the difference between an American market price and a European value
according to some option pricing model. Unfortunately, this will be a joint test of the premium and the accuracy of the chosen pricing model. The third way is to
analyse deviations from PCP, and since this methodology does not rely on a specific option pricing model the problem of joint testing could be disregarded. However,
the deviations from PCP could be due either to the early exercise premium or
simply to inefficiencies.
1
In addition, McMurray and Yadav 1996 argue that the deviation only gives an estimate of the difference between the premiums for
corresponding calls and puts rather than the individual premium for each option. This study contributes to the existing literature in a number of aspects. First, the
value of the early exercise premium in American put option prices is determined empirically using Swedish equity option data. The premium is estimated using
deviations from PCP, and the institutional setting in Sweden facilitates the empiri- cal analysis in that dividend payments occur only once a year, which is in contrast
to four times a year in the US. The analysed period is the seven months following the dividend payments, and the call options should consequently never be exercised
early according to theory. Hence, the call option prices do not include an early exercise premium, as in e.g. Zivney 1991, and the premium is attributed entirely
to the put option. Using a dummy variable regression, the sensitivity of the early exercise premium to the pricing parameters; moneyness, time to expiration, interest
rate and volatility is examined. For comparison, a theoretical measure of the premium is also calculated as the difference between theoretical values of American
and European options. The American values are obtained using the model accord- ing to MacMillan 1986 and Barone-Adesi and Whaley 1987 henceforth the
BAW model and the European values using the Black – Scholes model.
The second contribution of the paper is to use the empirically found premium in the modified control variate approach, suggested but not used by Zivney 1991,
to value American puts. The accuracy of this pricing approach is examined by comparing the obtained option values with market prices and theoretical model
values.
2
The data set is therefore divided into two parts; an estimation period and a subsequent evaluation period. The dummy regression coefficients from the
estimation period are used to assess the early exercise premium for each put option during the out-of-sample evaluation period. This premium is then added to the
corresponding Black – Scholes value to obtain an American put option value.
The results indicate a substantial value of the early exercise premium, where the average premium is higher for the PCP measure than for the theoretical measure.
The premium also increases with the moneyness and the time left to expiration of the options, while the results concerning the effect of the interest rate and implied
volatility depends on the moneyness of the options. On the whole, the modified control variate technique works reasonably well relative e.g. the BAW model. In
particular, for the deep in-the-money options, the modified control variate tech- nique is superior to the theoretical models.
The remainder of the study is organised into five sections. The following section presents put-call parity and previous research of the early exercise premium. In
section 3, the institutional setting and data of the study is presented while section
1
However, Kamara and Miller 1995 find in their study of PCP for European options that the violations are much less frequent and smaller than in studies of American options. This could imply that
the major part of the deviations for American options could be attributed to the early exercise premium.
2
To our knowledge, no previous study uses and evaluates this method to price equity options.
4 discusses the methodology. The fifth section presents the empirical results, and section 6 contains some concluding remarks.
2. Put-call parity and the early exercise premium in American options
