本文主要从对OPTION PRICES AND MARKET EFFICIENCY分析，由代写留学生论文中心策划组提供。是英文语种、Financ代写留学生论文ial management研究方向、不需要数据处理的硕士课程论文 Master Assignment，不需要盲审（博士或硕士生有这个需要），如有需求请联系本站论文中心或者提交相关文章的代写需求。
Black and &holes (1973) have derived an option pricing formula which,
given their assumptions, can be used to calculate the equilibrium price of stock
purchase options. One of the assumptions employed by Black and Scholes is
that the stock pays no dividend. However, dividends on some stocks may be
substantial and can have a significant effect on the valuation of options whose
stocks make such payments during the life of the options. Merton (1973) adjusts
the Black-Scholes mode1 for a specific dividend policy. Dividends are assumed
to be paid continuously such that the yield is constant.
This assumption does not conform to actual dividend policies of firms. To
apply Merton’s model it is necessary to convert discrete payments to an equivalent
continuous rate. Although payments are not made at a constant rate this
assumption appears more reasonable than ignoring dividends entirely. At
present there is no known closed form solution to the option pricing equation
for a discrete dividend payment.’
According to Merton’s formula, equilibrium option prices are a function of
the stock price, maturity date of the option, the exercise price, the risk free rate
of interest, the dividend yield on the stock, and the standard deviation of the
stock’s rate of return. The stock price, maturity date and exercise price are
easily observed; the risk free rate of interest may be closely approximated by
the rate of return on short term government securities, and the stock’s dividend
yield over a specified time period can easily be calculated. The standard deviation
of the stock’s rate of return cannot be directly observed. However, Merton’s
formula may be employed to calculate ‘implied standard deviations’. An implied
standard deviation is the value of a stock’s standard deviation of returns which,
when employed in the option pricing formula, will equate an observed option
price with the price calculated from the option formula. In this study we find
that prices observed for different options on the same stock result in different
calculated implied standard deviations. If the option model is correct and the
true standard deviation of a stock’s return is unique, the presence of the multiple
implied standard deviations implies that the observed option prices are not in
equilibrium. This hypothesis is tested by attempting to identify ‘overpriced’
and ‘underpriced’ options. It is concluded that, contrary to the efficient market
hypothesis, ‘underpriced’ and ‘overpriced’ options can be identified from
currently available information. A trading rule which, in theory, is ‘risk free’
but earns returns substantially greater than the risk free rate is presented and
tested empirically.
This paper is organized as follows: in section 2 the model is presented and
previous related empirical tests are discussed; in section 3 a method for calculating
implied standard deviations is presented and tested; in section 4 a trading
rule, which exploits the informational content of the implied standard deviations,
is presented and tested; and in section 5 the results from the above
sections are summarized and the implications are discussed.
2. The model and previous tests
The assumptions underlying the original Black-Scholes model are as follows
(1) There are no penalties for short sales.
(2) Transactions costs and taxes are zero.
(3) The market operates continuously.
(4) The risk-free interest rate is constant.
(5) The stock price is continuous.
(6) The stock pays no dividends.
(7) The option can only be exercised at maturity.
The option formula of Black and Scholes is
W = XlV(Z,) - Ce-"N(Z2),
where
z _ In(X/C)+(r++u’)t
1- vt’12
t
2, = z1 -VP,
W = the current option price for a single share of stock,
x = the current stock price,
c = the exercise price of the option,
e = the base of natural logarithms,
t = the time remaining until expiration of the option,
r = the continuous risk-free rate of interest for the period t,
V = the standard deviation of returns on the stock during the period t
(assumed to be constant),
IV(*) = the cumulative normal density function of (a).
Merton generalized the Black-Scholes model by relaxing the assumption
regarding dividends (assumption 6). Merton allows for a constant known
continuous dividend yield (y) on the stock.
The form of his evaluation equation follows:
where
W = e-Y’XN(dl) - e-”CN(d,),
dl =
ln(X/C)+(r-y++u2)t
vt It2
,
(2)
d2 = dl -vtli2,
and y = the continuous dividend yield on the stock.
It should be noted that when y = 0, eq. (2) reduces to the original Black-
Scholes model. We will derive our results using the more general Merton
model.
Empirical testing of the option pricing models have generally been confined
to tests of eq. (1). Black and &holes compared option prices calculated from
(1) with actual market prices. They observe that ‘the model tends to overestimate
the value of an option on a high variance security, market traders
tend to underestimate the value, and similarly while the model tends to underestimate
the value of an option on a low variance security market traders tend
to overestimate the value’ [Black and Scholes (1972, pp. 416-417)]. Part of the
deviation of observed option prices from the prices predicted by eq. (1) is
thought, by Black and Scholes, to be due to errors in measurement of the
variance of stock returns. The measure of stock return variance used in their
study was simply the sample variance of historic stock returns. In spite of
employing this imprecise estimate of u in eq. (1) Black and Scholes found that a
strategy of selling options which are ‘overvalued’, relative to eq. (I), and
buying ‘undervalued options’ could yield substantial profits. However, after
accounting for transactions costs it was found that there is no opportunity for
traders to take advantage of price discrepancies.
Galai (1977) in a similar study, finds some evidence of profit opportunities,
even after allowing for transactions costs. His study also has the weaknesses
associated with using historical estimates of standard deviation (0).
In order to avoid the shortcomings associated with the use of estimation of
variance from past data, implied standard deviations (ISDs) are used as an
estimate of variance in this study. The usefulness of ISDs have been previously
investigated by Latane and Rendleman (1976) and by Trippi (1977).
Latant and Rendleman calculated ISDs using eq. (1). Since stocks generally
have many traded options, several different ISDs were calculated for each stock.
To obtain a single estimate of v, the ISDs associated with each stock were
combined into a single weighted average standard deviation.’
Trippi calculated an arithmetic average of the ISDs to obtain a ‘collective
assessment of volatility’ for each stock. LatanC and Rendleman had previously
labeled the use of such an arithmetic average as ‘unreasonable’. The average
ISD obtained was used as the value of v in eq. (1) in’order to calculate estimates
of equilibrium prices. Options whose observed prices were 15 y0 below or above
their estimated equilibrium prices were bought or sold respectively. Trippi
concluded that his trading rule could have allowed one to realize large profits
after commission costs during the period covered by his study.
This study differs from the previous studies in the following ways:
(1) An estimate of each stock’s return variance is derived from current market
conditions instead of from historical data. Tests indicate that the market derived
estimates of variance are superior to those based only on historical data.
(2) A method is introduced for calculating a collective assessment of volatility
at any point in time which more properly accounts for the informational value
of each ISD.
(3) The changes in the predictive characteristics of option prices over time is
examined.
(4) The use of Merton’s adjustment allows inclusion of dividend data.