Microstructural Biases in Empirical Tests of Option Pricing Models1
Patrick Dennis
McIntire School of Commerce
University of Virginia
Charlottesville, Virginia
Stewart Mayhew
Office of Economic Analysis
U.S. Securities and Exchange Commission
450 5th St. NW, Washington, DC 20549-1105
(202) 942-8036 [email protected]
Department of Banking and Finance
Terry College of Business
University of Georgia
Athens, GA 30602-6253
Preliminary Draft—Please do not Distribute Without Permission
Draft 0.02: November 10, 2003
1
We would like to thank seminar participants at the University of Maryland for helpful comments. The
US Securities and Exchange Commission disclaims responsibility for any private publication or statement
of any SEC employee or Commissioner. This study expresses the authors’ views and does not necessarily
reflect those of the Commission, the Commissioners, or other members of the staff.
Microstructural Biases in Empirical Tests of Option Pricing Models
Abstract
This paper examines how noise in observed option prices arising from discrete prices and other
microstructural frictions affects empirical tests of option models, and techniques for backing out
implied risk-neutral parameters from option prices. Using a conservative estimate for the amount of
noise, we demonstrate that it can be very difficult to distinguish alternative option models using a
cross-section of noisy option prices. We also find that the accuracy of traditional implied volatility
calculations, implied volatility regression forecast tests, the implied risk-neutral moment estimators
of Bakshi, Kapadia, and Madan (2003), and the univariate diffusion test proposed by Bakshi, Cao,
and Chen (2000) are all quite sensitive to the amount of noise in option prices. Our results suggest
that even in active, liquid markets such as the S&P 500 index option market, observation error
significantly reduces the power of tests, and in some cases represents an important source of bias.
The problem tends to be much more severe for less active options on lower priced stocks.
1
Introduction
Option pricing models relate option prices to parameters in hypothesized equations describing
the dynamics of the underlying asset and other state variables. Several approaches have been
suggested for empirically testing option pricing models in ways that require observed market prices
of exchange-traded options. However, well-known microstructural problems make it difficult to
observe accurate option prices. As a result, empirical studies typically focus on the most activelytraded options, the prices of which are presumed to be more directly determined by competitive
market forces, and thus more representative of the option’s “true” value. Many authors restrict
their attention to the S&P 500 or S&P 100 index option market, where microstructural problems
are believed to be relatively unimportant. Intraday data are often used to ensure synchronous
observation of prices across options of different strikes, and spread midpoints are used in place
of transaction prices in order to mitigate problems induced by bid-ask bounce. In this article,
we examine whether these safeguards are sufficient to ensure reliable results, or on the contrary
whether the lingering microstructural errors are severe enough to invalidate the conclusions of
empirical tests.
Bid-ask spreads in option markets tend to be very wide, compared to the spread in the underlying market. Contemporaneous research by Gulen and Mayhew (2003) demonstrates that option
quote midpoints are noisy estimates of option prices, as a result of microstructural features such
as the minimum price variation, or “tick size,” clustering of spread widths, and random arrival
of limit orders. Moreover, Gulen and Mayhew (2003) argue that for reasons fundamental to the
nature of options, this noise is not benign. The distribution of the observation error is likely to be
asymmetric, the quote midpoint is likely to be a biased estimate of the option’s true value, and the
degree of bias is likely to be related to the option’s strike price.
These microstructural features will interfere with empirical tests of option models, and represent
1
a source of error in techniques for backing out implied parameters or implied risk neutral densities.
We would expect this problem to be most severe for options on low-priced stocks, where the relative
tick size is large and where the interval between adjacent strike prices represents a large percentage
of the stock price. Even for higher-priced stock and index options, we would expect microstructural
biases to become severe for short-term options that are deep in or out of the money—that is, for
options with a low time value. The reason for this is that when time value is low, the differences
between the predictions of alternative models tends to be small, compared to the tick size.
Our goal is to evaluate the magnitude of this problem. A great deal of empirical research has
focused on S&P 500 and S&P 100 index options. This is a market where trading volume has
been relatively high, the tick size and strike price interval are small compared to the index level,
and microstructural frictions are thought to be relatively unimportant. Thus, we are interested in
measuring the extent to which empirical tests are likely to be influenced by microstructural biases for
actively-traded index options. If an empirical test does not have the power to distinguish between
alternative hypotheses in this context, there is little hope that it can be applied meaningfully
to individual stock options, or to other markets where microstructural biases are likely to be
significantly larger.
We examine the empirical tests using various approaches. In some cases, we are able to understand the effects of noisy option prices analytically. In other cases, we generate simulated option
prices under the null hypothesis that the Black-Scholes model is true, add plausible microstructural frictions, and then test whether the empirical technique in question can distinguish the noisy
Black-Scholes prices from prices generated under alternative models including Merton’s (1976)
jump-diffusion model. We examine various parameter values, including those calibrated to correspond to S&P 500 index options, and options on high-priced and low-priced stocks.
We find that microstructural noise in option prices is large enough to raise serious concerns as
to the validity of several empirical methods. The bias arising from the discrete tick size alone is
large enough to have a noticeable impact on empirical tests calibrated to the S&P 500 index options
2
market, and to cast serious doubt on the validity of these tests when applied to individual stock
options, particularly on lower-priced stocks. In addition, there are other potential sources of noise
and bias in option prices that may invalidate empirical results even for S&P 500 index options.
Our results suggest the need for further research on option market microstructure, so that we
may better understand how to extract useful information from observed option prices. Our findings
also suggest a shift of focus may be warranted, toward methodologies that test option models
without using option prices, for example by testing the accuracy of the assumed underlying process
against time-series data, or, where possible, by testing the performance of dynamic replicating
strategies.
The remainder of this article is organized as follows. In section 2, we summarize several categories of empirical tests of option models that rely on observed option prices. In section 3 we
give an overview of our basic methdological approach. Our main results are presented in section
4. In section 4.1 we investigate the extent to which microstructural noise makes it impossible to
distinguish between alternative option models by looking at a cross-sectional snapshot of option
prices across strikes and maturities. In section 4.2 we demonstrate that microstructural noise can
make it difficult to back out a reliable implied volatility smile. In section 4.3 we analyze the effects
of microstructural frictions on regression-based tests of whether Black-Scholes implied volatility
is a biased forecast of realized volatility. In section 4.4, we evaluate the measurement errors and
biases inherent in methods for computing implied skewness and kurtosis of the risk-neutral density,
focusing on the method proposed by Bakshi, Kapadia, and Madan (2003). In section 4.5, we examine the method of Bakshi, Cao, and Chen (2000) for testing whether options are priced according
to a diffusion model. We offer a few comments on testing models using American-style options in
section 4.6. In section 5 we conclude with a summary of our results, and a few further comments
on the implications of our findings.
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2
Empirical Tests of Option Models
Many option pricing models have been proposed over the years. Taking as a starting point the
constant volatility geometric diffusion model of Black and Scholes (1973), the field has expanded
in many directions. Given space limitations, we will mention just a few of the alternative models
that have been suggested for pricing European-style call and put options.
Alternative univariate diffusion models include the CEV model of Cox and Ross (1976), models
that incorporate the firm’s capital structure (Geske (1979), Rubinstein (1983), Toft and Prucyk (1997)),
and equilibrium models under alternative utility functions, such as those explored by Franke, Stapleton, and Subrahmanyam (1999) and Camara (2003). Merton (1973) extends the framework to
allow for time-varying but deterministic volatility. Merton (1973), Amin and Jarrow (1991) and
others have extended the framework to include stochastic interest rates. Models incorporating large
(Poisson) jumps have been suggested by Cox and Ross (1975), Merton (1976), Bates (1996), and
Duffie, Pan and Singleton (2000), among others. Stochastic volatility models have been advanced
by authors such as Hull and White (1987), Stein and Stein (1991), Heston (1993), and Jones (2003).
For general models that include stochastic interest rates, jumps, and stochastic volatility, see Scott
(1997) and Bakshi, Cao and Chen (1997). Duan (1995), Heston and Nandi (2000) and others
have examined option pricing under discrete-time GARCH processes. Several authors have priced
options under a regime-switching process, including Bollen, Gray, and Whaley (2000) and Duan,
Popova, and Ritchken (2002). Madan, Carr and Cheng (1998), Carr and Wu (2002a) and others
have developed models based on alternative processes characterized by an infinite number of small
jumps. With these and many other models to choose from, it is important for us to understand
how far we can rely on empirical testing to help distinguish between models, and which empirical
tests are most robust to microstructural frictions.
Several categories of empirical tests of option pricing models have been proposed.1 Early studies,
1
See Bates (2002) for a recent survey.
4
beginning with Black and Scholes (1972), used historical data to estimate a volatility parameter,
and then tested whether the resulting theoretical prices from their model differ systematically from
market prices. Other authors, surveyed by Figlewski (1997), have tested whether Black-Scholes
implied volatility is an accurate forecast of subsequent realized volatility. In these tests, the BlackScholes model is consistently rejected.
There are some inherent difficulties associated with this approach. For one, it does not work
well for testing models that rely on parameters that cannot easily be estimated from time-series
data, such as a market price of risk or the risk aversion of the representative investor. Certainly, one
can back out implied risk parameters from option prices and demonstrate that a particular model
is sufficiently flexible to eliminate a forecast bias in Black-Scholes, but this does not constitute an
empirical test of the model. In such a case, one can only attempt to assess whether the implied
parameter values are stable over time, or whether the magnitude of the number is economically
plausible (see, for example, Bates (2000)).
There is also a joint hypothesis problem here. For the simple tests described above, an obvious
interpretation is that the Black-Scholes model is wrong because of an omitted risk premium, or as
a result of features such as jumps, or stochastic volatility, that have been documented in studies of
the time-series properties of asset prices. However, a large portion of the apparent bias in the BlackScholes model may be attributed to measurement errors in realized volatility (Poteshman (2000)),
an error-in-variables problem that causes a bias in the regression coefficient (Christensen and Prabhala (1998)), a “peso problem” (Penttinen (2001)) and a bias resulting from Jensen’s Inequality
(Vorkink (2000)).
More generally, if a model is estimated from historical data and tested using market prices, and
the model is rejected, there are at least four potential explanations:
1. The model may be wrong,
2. The options may be mispriced,
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3. The model’s parameters may have been estimated incorrectly, or
4. The results may be influenced by error in observing option prices.
Empirical tests have been designed in an effort to distinguish from among these alternatives.
One point of this paper is to demonstrate how interference from explanation 4 makes it more
difficult to distinguish between the other three.
In terms of distinguishing between explanations 1 and 2, the typical approach implicit in many
academic studies is simply to assume a priori that markets are efficient, and rule out the possibility that the options may be mispriced. Those who believe that markets may not be efficient,
particularly those who are trying to profit from the inefficiencies, will view the distinction between
explanations 1 and 2 as crucial. One possible response is to test whether trading strategies based
on differences between theoretical and market prices are profitable. However, this is not so much a
test of the model as a test of market efficiency. If a trading strategy is found to generate positive
profits, this suggests that markets are inefficient, and that the model is exploiting at least some
degree of mispricing. But it does not rule out the possibility that the model is still wrong, or
estimated incorrectly. An accurate assessment of the profitability of such strategies would require
a method that accounts for transactions costs. As transactions costs get large, it becomes increasingly difficult to find profitable strategies, and therefore more difficult to identify a possible role
for explanation 2. Moreover, in implementing such a test, one must grapple with the thorny issue
of how to measure performance for option strategies—traditional mean-variance measures such as
Sharpe ratios are clearly inappropriate. Also, one would want to perform the test over a long time
period, to reduce the likelihood that the results are subject to a peso-problem bias.
Another way to separate explanations 1 and 2 is to test the model in ways that do not rely on
the market prices of options. For example, one can test whether the dynamics of the underlying
asset conform to the assumed process. Or, one can measure the accuracy with which a model can
dynamically replicate the payoff of an option in the presence of real-world frictions, in the spirit of
6
Bhattacharya (1980) and Figlewski (1989). The present paper does not address these tests—our
focus is on tests that are based on observed market prices of exchange-traded options.
An alternative approach to testing option models is to test the cross sectional restrictions
imposed by the model on the relative prices of the stock and options with different strikes and
maturities. For example, Macbeth and Merville (1979), Rubinstein (1985), and others have tested
the Black-Scholes model by testing whether implied volatility varies systematically as a function
of moneyness or maturity. Longstaff (1995) tests whether the stock price implied by option prices
equals the observed stock price. The advantage of this approach is that allows us to rule out
explanation 3. If the cross-sectional restrictions are systematically violated, there are no possible
parameter values that will correctly price all the options. Therefore, the rejection cannot be due
to mis-estimating parameters.
Breeden and Litzenberger (1978) noted the relationship between option prices and the riskneutral density function of the underlying stock price. Several authors have suggested methods
for backing out an implied risk-neutral density from option prices, including Rubinstein (1994)
and others surveyed by Jackwerth (1999). More recently, Bakshi, Kapadia and Madan (2003)
propose a method for estimating the moments of the risk-neutral density from the market price
of carefully constructed portfolios of options. Option models can be tested by comparing the
shape of the implied risk-neutral density to those predicted by various models, or by examining the
characteristics of the implied higher moments (Dennis and Mayhew (2002), Bakshi and Cao(2003)).
A more general approach to testing option models is to test restrictions or properties that
should hold for entire classes of models. For example, Bakshi, Cao and Chen (2000) test whether
the underlying asset price and the call option price always move in the same direction, as predicted
by all univariate diffusion models. Ait-Sahalia (2002) demonstrates that if the underlying stock
follows a diffusion process, the second partial cross-derivative of the log risk-neutral density with
respect to the stock price and strike price must be a strictly positive function, and he tests this
property on the implied risk-neutral density from option prices. Carr and Wu (2002b) introduce a
7
methodology for testing for jumps in the underlying process by examining the speed of convergence
of at-the-money and out-of-the-money option prices as they approach maturity.
3
Methodology
We now turn to our main question of how noise in observing option prices affects empirical tests
of option pricing models. Our goal is to examine the effects of microstructure-induced errors on
a variety of empirical tests. While the specific methodological details vary somewhat from test
to test, our basic approach can be roughly summarized in three steps. First, we generate a set
of option prices under the null hypothesis that the Black-Scholes model is true. Second, we add
microstructural distortions to the generated prices, calibrated to correspond to real-world frictions.
Third, we apply empirical tests to the noisy data.
In some cases, the goal is to determine whether microstructural frictions cause us to reject the
Black-Scholes model when, by construction, it is the true underlying model. In other cases, we
examine whether the empirical test can differentiate between Black-Scholes and specific alternative
models. In the interest of brevity, we confine our attention to Merton’s (1976) jump-diffusion
model as a representative example of a significant departure from the Black-Scholes framework,
and we focus on the valuation of European-style options. It would be a straightforward extension
to consider other models. In section 4.6, we offer further insights on the implications of our results
for American Style options.
In Merton’s (1976) jump diffusion model, the underlying asset is assumed to follow a process
of the form
dS
= (µ − λµj ) dt + σ dZ + dQ
S
where dQ is a Poisson jump process with jump intensity λ, dZ and dQ are uncorrelated, and the
jump return has a normal distribution with mean µj and standard deviation ξ.
Assuming that jump risk is not priced, the jump-diffusion option price is given by a weighted
average of Black-Scholes prices, with weights governed by the distribution of the number of jumps
8
over the life of the option. Specifically, let f (σ, r) denote the Black-Scholes price of a contingent
claim with constant volatility σ and risk-free rate r. Then the price of the same contract under
Merton’s jump diffusion process is
∞ −φ i
X
e φ
i!
i=1
where
f (σi , ri )
s
σi =
σ2 +
i ξ2
t
ri = r − λµj +
i ln(1 + µj )
t
and
φ ≡ λ(1 + µj ) t.
A common parameterization of Merton’s jump diffusion model is to restrict the mean jump size
to zero (µj = 0) and allow to user to specify the jump frequency parameter (λ), total volatility
(v), and the proportion of total volatility attributable to jumps (γ), where the new parameters are
related to the original ones as follows:
s
γ v2
λ
ξ=
q
σ=
v 2 − λξ 2
As γ approaches zero, options at all strike prices approach their Black-Scholes values. As γ
increases, this causes near-the-money options to have lower prices than Black-Scholes and options
with high and low strike prices to have higher prices. Also, for a given value of γ, prices tend to be
closer to Black-Scholes when there are a large number of small jumps, but differ significantly when
jumps are large but infrequent.
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4
Evaluation of Empirical Tests
4.1
Distinguishing Between Models Using Observed Option Prices
In this section, we show that the noise associated with discrete prices alone is enough to make it
difficult to distinguish between alternative pricing models using a cross-sectional snapshot of options
of different strike prices. Let us begin with the observation that it will be difficult to distinguish
between models empirically if the differences in their predictions are small compared to the amount
of noise in the data used to test them. In particular, if the difference between the option prices
from Black-Scholes and an alternative model are smaller than the tick size, this would suggest that
it is difficult to distinguish between them.
Figure 1: Difference between Jump-Diffusion and Black-Scholes prices
Difference
0.005
0
-0.005
-0.01
-0.015
1.3
1.2
0.2
1.1
0.4
Gamma
1
0.6
0.9
0.8
0.8
1
Moneyness
0.7
For example, in figure 1 we graph the difference between call option prices predicted by Merton’s (1976) Jump Diffusion model and the Black-Scholes model with the same total volatility. The
difference, expressed as a percentage of the underlying stock price, is shown as a function of the
10
option’s moneyness (K/S) and the parameter γ, which represents the proportion of total volatility
attributable to the jump component, as described in section 3. The graph in figure 1 is based on
a jump intensity parameter λ = 1, for an asset with volatility .30, maturity of 18 days, and a risk
free rate of .0113, calibrated to current market conditions on February 4, 2003.
As γ approaches zero, the Jump Diffusion prices approach Black-Scholes. As the jump size
increases, holding total volatility constant, the returns distribution becomes more fat-tailed, causing
the Jump Diffusion prices to get lower for options near the money, and higher for options in the
tails. The contour lines on the base of the graph correspond to parameter values where the two
models give the same price. In the limit as γ approaches one, the model approaches a pure jump
model.
The magnitude of the deviations between the two models depends on both the jump frequency
parameter and the jump magnitude. When λ = 1, as in this graph, and when jumps account
for less than thirty percent or less of total volatility, deviations as a percentage of the stock price
tend to be on the order of ten basis points or less. Thus, for a $100 stock, the two models would
generate prices that differ from each other by less than $0.10, or one tick. If jumps account for
eighty percent of total volatility or less, then the pricing errors may be as large as one percent of
the stock price, which represents about one tick for a low-priced stock. Holding total volatility
constant, the difference between the two models is larger when the jumps are large and infrequent.
To address this issue another way, we use the Black-Scholes model to generate a set of simulated
prices for a cross section of options with different strikes and/or maturities, at a single point in time.
Next, using the Merton (1976) model, we generate simulated prices for the same set of options.
We generate prices for each point on a fine grid of possible parameter values, that includes the
Black-Scholes model as a limiting case. Then, we search this grid to identify the set of parameters
that price all the options to within one-half of a tick of the Black-Scholes price. If the parameters
of the alternative model fall within this set, it will be difficult or impossible for an observer to
distinguish empirically between that model and Black-Scholes by looking only at option prices.
11
Given the large magnitude of the microstructural biases discussed above, the criterion of onehalf tick is generous—in practice, we would expect typical microstructural errors to be considerably
larger than this. However, as we will see even this small error is enough to make it difficult to
distinguish between alternative models.
In order to convey the essence of our results more effectively, we wish to depict the set of
indistinguishable parameters graphically. Accordingly, we vary only two parameters at a time, and
select the other parameters to ensure that the model converges to Black-Scholes at the appropriate
limits. Allowing additional parameters to vary can only expand the range of potential option prices
spanned by the model. Our results, therefore, will be biased in a way that make the models appear
easier to distinguish from Black-Scholes than they actually are.
Figure 2: Jump Diffusion Parameters Indistinguishable from Black-Scholes
50
45
SPX
40
35
IBM
30
Lambda
25
20
CSCO
15
10
5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Gamma
Figure 2 depicts the set of jump parameters that yield prices that are observationally indistinguishable from Black-Scholes. Specifically, the graph shows the different combinations of the jump
12
frequency parameter (λ) and the total proportion of volatility due to jumps (γ) that price a string
of call options with a single maturity but different strike prices, all to within half of one tick2 of
their respective Black-Scholes prices. Parameter values to the left of the border are indistinguishable from Black-Scholes. Three different regions are depicted, calibrated to roughly correspond to
S&P 500 (SPX) index options, IBM options, and Cisco options, each with 22 trading days (about
one month) to maturity.3
As indicated in the figure, there is a region of parameter values for which the jump diffusion
model is observationally indistinguishable from Black-Scholes. The extent of this region is directly
related to the size of the tick size as a percentage of the option prices. As indicated in the graph, the
region is much larger for IBM options than for SPX options, and is even larger for CSCO options.
We have examined the differences between the Black-Scholes and Jump Diffusion prices from
two angles, reflected in figures 1 and 2. In both cases, we have held constant the total volatility
across the two models. Thus, the magnitude of the differences between the two models, relative
to the tick size, tells us something about how difficult it would be for an empiricist to distinguish
between the models, assuming the true volatility is known. In practice, the true volatility is not
known, and this makes it even more difficult to distinguish between the models.
This point is illustrated in figure 3. Consider a world where the true prices are generated by
Black-Scholes with σ = .30, and the econometrician is testing Black-Scholes against the alternative hypothesis of Merton’s Jump-Diffusion model with parameters λ = 1 and γ = .8—that is,
eighty percent of total variance is generated by a jump component with one jump per year. If the
econometrician knows the true total volatility to be .30, then the difference between theoretical
prices predicted by the two models is shown in the lower line in figure 3 (which is simply a slice
out of figure 1). The absolute difference can be as large as ten basis points. Now suppose that the
2
Half of a tick represents .025 for options priced under $3.00 and .05 for options priced above $3.00.
For the SPX options, the index level was set to 848.20, and option prices were generated for strike prices every
five dollars between 800 and 900. For IBM, the stock price was set to 77.09, with strikes every five dollars from 55
to 90. For Cisco, the stock price was 13.20, with strikes every $2.50 from $2.50 to $17.50.
3
13
econometrician does not know the true volatility but selects the volatility parameter by minimizing
the maximum pricing error across options, in the spirit of Rubinstein’s (1985) minimax statistic.
In this case, the econometrician would select a total volatility of .4162. As shown in the upper line
in figure 3, the two models never differ by more than four basis points.
Figure 3: Distinguishing Between Models when True Volatility is Unknown
This Graph shows the difference between Merton’s Jump Diffusion prices and Black-Scholes prices
when the true volatility parameter is known (v = .30) and when the parameters of the Jump
Diffusion Model are calibrated to fit the option prices (v = .4162).
0.006
v=.30
v=.4162
0.004
0.002
Difference
0
-0.002
-0.004
-0.006
-0.008
-0.01
-0.012
0.7
0.8
0.9
1
Moneyness
14
1.1
1.2
1.3
4.2
Implied Volatility Smiles
The sensitivity of an option’s price with respect to volatility is a function of how far the option
is in or out of the money, and of the option’s time to maturity. In particular, long-term options
and options that are near the money are the most sensitive to volatility. For short-term options
that are deep in or out of the money, large changes in volatility correspond to small changes in the
option price. As a result, a small amount of error in observing the option price translates into a
large error in observing the implied volatility. As demonstrated by Hentschell (2002), this noise
can lead to systematic bias in the estimation of the volatility “smile.”
Figure 4: Implied Volatility Bid-Ask Spread Under Black-Scholes
0.7
Bid
Midpoint
Ask
0.6
Implied Volatility
0.5
0.4
0.3
0.2
0.1
0
650
700
750
800
850
Strike Price
900
950
1000
1050
To illustrate, consider figure 4, which is based on the following exercise. First, we generate call
option prices using the Black-Scholes formula, using parameters calibrated to represent the S&P 500
market. Next, we add a “typical” bid-ask spread around the Black-Scholes price, assuming the true
price to be the quote midpoint, and taking guidance from the spreads reported for S&P 500 index
15
options by Gulen and Mayhew (2003). Then, we calculate the Black-Scholes implied volatility of
these hypothetical bid and ask prices. The resulting implied volatility spread will bracket the true
parameter used to generate the option prices. Figure 4 depicts bid and asked implied volatility as a
function of strike price, using parameters calibrated to the February S&P 500 options on February
4, 2003. For comparison purposes, figure 5 shows the implied volatility of actual quoted prices at
10:00 AM on that day.
Figure 5: Implied Volatility Bid-Ask Spread for S&P 500 Index Options
0.6
Bid
Midpoint
Ask
0.5
Implied Volatility
0.4
0.3
0.2
0.1
0
650
700
750
800
850
Strike Price
900
950
1000
1050
For options that are at-the-money, the bid-ask spread, in implied volatility terms, is relatively
narrow (for the 850 call, the spread is .278–.295). As we move away from the money, the volatility
spread tends to get wider. Once we get sufficiently far in or out of the money, the bid price
moves below the lower arbitrage bound on the option’s price, at which point, implied volatility is
undefined—any positive value for the volatility parameter will yield a Black-Scholes price higher
than the bid. Around the same point, the implied volatility of the ask price begins to diverge
16
rapidly from the true parameter value.
If we interpret the bid-ask spread as a kind of informal confidence interval around the option’s
true value, we might conclude that in this example, option prices become uninformative about
implied volatility as we move beyond ten to twenty percent away from the money.
To what extent is it possible to differentiate between alternative pricing models by observing
their prices projected through the Black-Scholes model into implied volatility space? This depends
on the relative tick size and the interval between adjacent strike prices. For S&P 500 index options,
both the tick size and the interval between strike prices are small, as a percentage of the level of
the underlying asset. The resulting implied volatility smile is at least somewhat informative. As
illustrated in figure 5, the smile appears to deviate from the flat line predicted by Black-Scholes,
even after explicitly allowing for bid-ask spreads.
Figure 6: Implied Volatility Bid-Ask Spread for CSCO Options
4
Bid
Midpoint
Ask
3.5
Implied Volatility
3
2.5
2
1.5
1
0.5
0
2.5
5
7.5
10
Strike Price
12.5
15
17.5
However, microstructural problems can be more severe in cases where the tick size is large
17
compared to the option price—that is, for low-priced stocks, low-volatility stocks, and shorter-term
options. For an example of a lower-priced stock, see figure 6, which shows the implied volatility
smile for Cisco February call options at 10:00 AM on February 4, 2003. The figure is based on
NBBO bid and ask prices across the five exchanges, which, at this moment, happened to be only one
tick wide at all strikes. And yet, even the strike prices nearest to the money have reasonably wide
implied volatility spreads (.689 to .735 for the 12.50, .587 to .651 for the 15.00), and beyond this,
the spreads are so wide as to be virtually meaningless. There is no guarantee that there will be any
strike prices that can be used to obtain a meaningful estimate of implied volatility. Consider the
hypothetical extreme case depicted in figure 7. This example is based on Black-Scholes prices, as
is figure 4, but volatility is lower (.20) the options are shorter-lived (one week), and the underlying
stock price is low ($11.25). The bid-ask spread is assumed to be only one tick wide, centered around
the true value. For these parameters, there would be essentially no useful information about implied
volatility at any strike price.
Figure 7: Implied Volatility Spread Under Black-Scholes, Severe Case
1.2
Bid
Midpoint
Ask
1
Implied Volatility
0.8
0.6
0.4
0.2
0
10
12.5
Strike Price
18
4.3
Regression Forecast Tests
Several authors, including Canina and Figlewski (1993), Jorion (1995), Figlewski (1997) and others
cited therein, use a regression framework to test whether Black-Scholes implied volatility is an
unbiased forecast of realized volatility. The basic premise underlying this test is that if X is an
unbiased forecast of Y, then in the regression equation
Y = α + βX + ²
we should expect to find α = 0 and β = 1. In this case, Y represents realized volatility over
the life of the option, and X represents implied volatility. This equation can be estimated using
non-overlapping time periods, or using a technique that explicitly accounts for the overlapping
horizons. Generally, those authors who have performed this test have found that α > 0 and β < 1,
suggesting that Black-Scholes implied volatility is a biased forecast of realized volatility. This has
been interpreted as evidence of an omitted priced risk factor, such as volatility or jump risk.
Let us mention two important problems with this regression test. First, the classical regression
framework assumes that the independent variables are observed without noise. But due to microstructural frictions, implied volatility is observed with noise. Therefore, the regression is subject
to an errors-in-variables problem, and the OLS estimate of β will be biased toward zero.
A common perception is that measurement errors are not important for actively-traded, atthe-money options. For example, in their comprehensive survey article on volatility forecasting,
Granger and Poon (2003) state the following, with respect to the error-in-variables problem in
implied volatility forecast regressions:
It has been suggested to us that implied biasedness could not have been caused by
model misspecification or measurement errors because this has relatively small effects
for ATM options, which is used in most of the studies that report implied biasness.
In their study of the S&P 100 market, however, Christensen and Prabhala (1998) verify that the
19
errors-in-variables problem can be severe enough to alter the conclusions of the regression forecast
test. They confirm the results of previous studies that the beta coefficient is significantly lower than
one under OLS. But when lagged implied volatility is used as an instrumental variable, they find
the coefficient to be higher, and they can no longer reject the null hypothesis that Black-Scholes
implied volatility is an unbiased forecast.
Second, in a world where the Black-Scholes model is strictly true, the volatility would not change
over time, so the market’s volatility forecast should not change over time. Without variation in the
explanatory variable, the relation between it and the dependent variable cannot be estimated. It
seems somewhat odd to test a model using a technique that is not valid under the null hypothesis.
In practice, this is not a fatal problem, as it is widely recognized that volatility changes over time.
If volatility is stochastic but volatility risk is not priced, as in Hull and White (1987), the BlackScholes implied volatility for an at-the-money option is a good approximation to expected volatility.
Thus, the question of whether Black-Scholes implied volatility is a biased forecast remains valid,
even if the model in its strictest form is violated.
Even in a world where volatility is changing, the errors-in-variables problem will be exacerbated
if there is only a small amount of variation in volatility within the sample period. When the
independent variable is observed with noise:
X̃ = X + u
the OLS estimate b in a univariate regression of Y on X̃ converges to:
Ã
plim b = β
σx2
σx2 + σu2
!
where σx2 measures the variation of the true value of X in the sample, and σu2 is variance of the
measurement error. Thus, in a regression-based forecast test of Black-Scholes, the attenuation bias
depends not only on the amount of noise with which implied volatility is observed, but also on the
extent to which true volatility changes over the sample. In the limit as the true volatility never
20
Table 1: Extent of Attenuation Bias
σu
.001
.001
.001
.005
.005
.005
.01
.01
.01
σx
.10
.05
.01
.10
.05
.01
.10
.05
.01
plim b
.9999
.9996
.9901
.9975
.9901
.8000
.9901
.9615
.5000
changes, the OLS estimate will converge to zero, even if implied volatility is observed with a small
amount of noise. To illustrate, table 1 shows the expected OLS coefficient for differing levels of
noise in observing implied volatility, and for differing amounts of variation in true volatility.
Very roughly speaking, the order of magnitude of the noise introduced by the discrete tick size
alone, expressed in terms of standard deviation of the implied volatility estimate, is something
like σu = .001 for at-the-money SPX index options, and σu = .01 for individual stock options.
Additional noise may be introduced by the other factors discussed above, such as randomly arriving
limit orders, or from stale trading in the index components (see Jorion (1995)). As for the variation
in the true volatility, estimating this properly would require a time-series decomposition. Based
on the time series of realized volatility over the past twenty years since the introduction of index
options, we can roughly approximate that a standard deviation of σx = .01 corresponds to a
relatively short sample or a period of unusually stable volatility, while a standard deviation of
σx = .10 represents a sample carefully chosen to include a large structural change in volatility.
4.4
Implied Risk-Neutral Moments
A substantial body of literature has addressed the question of how to extract from observed option
prices an estimate of the risk-neutral density of the underlying asset returns (see Jackwerth (1999)
for a survey). Other authors, such as Bakshi, Kapadia, and Madan (2003) (BKM) have suggested
21
methods for estimating the higher moments of the risk-neutral density, without having to estimate
the entire density. Dennis and Mayhew (2002) have used the BKM technique to examine the cross
sectional determinants of risk-neutral skewness, and Bakshi and Cao (2003) have used the approach
to further explore risk-neutral kurtosis, for individual stock options.
In their paper, BKM derive a relation between the moments of the risk-neutral density and the
price of securities with quadratic, cubic, and quartic payoff functions, which can be replicated using
a portfolio of out-of-the money calls and puts with a continuum of strike prices. Specifically, they
demonstrate that the risk-neutral skewness and kurtosis of the continuously-compounded return
distribution at horizon τ are given by
SKEW (t, τ ) =
KU RT (t, τ ) =
erτ W (t, τ ) − 3µ(t, τ )erτ V (t, τ ) + 2µ(t, τ )3
[erτ V (t, τ ) − µ(t, τ )2 ]3/2
erτ X(t, τ ) − 4µ(t, τ )erτ W (t, τ ) + 6µ(t, τ )2 V (t, τ ) − 3µ(t, τ )4
[erτ V (t, τ ) − µ(t, τ )2 ]2
where the prices of the expected return, volatility, cubic, and quartic payoffs are given by:
erτ
erτ
erτ
V (t, τ ) −
W (t, τ ) −
X(t, τ ),
24
³ 2 h
i´ 6
K
2 1 − ln S(t)
C(t, τ ; K)dK
K2
µ(t, τ ) ≈ erτ − 1 −
V (t, τ ) =
Z ∞
S(t)
+
W (t, τ ) =
0
K2
h
i
³
P (t, τ ; K)dK,
h
Z ∞ 6 ln K − 3 ln K
S(t)
S(t)
C(t, τ ; K)dK
h
i
³ h
i´2
Z S(t) 6 ln S(t) + 3 ln S(t)
K
K
0
³
h
Z ∞ 12 ln K
S(t)
S(t)
+
i´2
K2
S(t)
−
X(t, τ ) =
i´
³
h
Z S(t) 2 1 + ln S(t)
K
Z S(t) 12
³
i´2
h
ln
K2
³
− 4 ln
K2
S(t)
K
i´2
22
K
S(t)
³
+ 4 ln
K2
0
h
P (t, τ ; K), dK
i´3
h
C(t, τ ; K)dK
S(t)
K
i´3
P (t, τ ; K)dK.
The prices of these polynomial contracts represent weighted sums of prices of out-of-the-money
calls and puts, across a continuum of strike prices. Thus, the risk-neutral moments can be estimated
directly from the prices of traded options. The accuracy of the resulting estimates depends on the
accuracy with which the traded option prices are observed. Even if we can observe option prices
without error, however, there can still be error in the risk-neutral moment estimation from two
sources - the fact that the interval between adjacent strike prices is not infinitely small and the fact
that we do not observe all strike prices from 0 to ∞.
Since options do not trade with a continuum of strike prices, the interval between strike prices
matters since we have to approximate the integral with a sum. Namely, due to contract specifications in the index and stock options market, ∆K is usually $2.50 or $5. To examine the impact
of a discrete strike price interval, assume that we are in a Black-Scholes world and can observe
option prices with no noise. Furthermore, assume that there is a stock that has a price of 50, a
return volatility of 20% per year, the riskless rate is 5% per year, and we can observe option prices
that have strikes from 30 to 70. If we set ∆K to some small number, say $0.10, and compute the
risk-neutral skewness, we recover the correct value of 0. If we increase ∆K from this value and
repeat the experiment, the risk-neutral skewness gradually diverges from 0 in an oscillatory fashion
shown in Figure 8.
Furthermore, the economic effect is large. Since we are starting with the Black-Scholes model,
the skewness should be zero, yet for strike price intervals in the range of $5 the error in skewness
can be plus or minus 0.4, causing the researcher to reject the null of zero skewness even though it
is true.
Examining Figure 8, we see that the skewness looks like it is zero at strike price interval of 5, in
which case the BKM metric is unbiased at the strike price interval that we observe most frequently
in the market. However, the value of the skewness at this strike price interval is actually -0.026;
it appears to be zero is simply due to the scale of the graph. More importantly, the skewness and
kurtosis metrics are homogeneous of degree 0 with respect to K, ∆K and S. If we compute the
23
0.4
0.3
0.2
Skewness
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
0.5
1
1.5
2
2.5
3
Strike Price Interval
3.5
4
4.5
5
Figure 8: Skewness vs. Strike Price Interval
This figure contains a plot of the risk-neutral skewness computed by the method of Bakshi, Kapadia
and Madan vs. the strike price interval. Strike prices are generated at intervals of ∆K from a low
strike of 30 to a high strike of 70. The strike price interval is initially set to ∆K = 0.1 and then
increased to ∆K = 5 in increments of 0.1. For each set of strike prices, corresponding option prices
are generated from the Black-Scholes model with a time to maturity of one month, a riskfree rate
of 5%, a stock price of $50, and a volatility of 20% per year. The BKM method is then used to
compute the risk-neutral skewness from these prices.
24
skewness metric at αK, α∆K, and αS for some positive constant α, the values of the integrand
of the volatility, cubic, and quartic payoffs will be the same as if it were evaluated at K, ∆K and
S, and, as a result, we will get the same skewness and kurtosis. This means that evaluating the
skewness at a stock price of $50 and a strike price interval of $4.75 will be the same as evaluating
³
the skewness at a stock price of ($50)
5
4.75
´
= $52.63 and a strike price interval of $5. In either
case we would (incorrectly) measure the skewness as -0.45.
We repeated the above experiment for kurtosis and found similar results, shown in Figure 9.
At small strike price intervals, the kurtosis is 3.0, but as the strike price interval is increased to $5,
the error introduced into the kurtosis is on the order of plus or minus 2.0.
25
6
5.5
5
4.5
Kurtosis
4
3.5
3
2.5
2
1.5
1
0
0.5
1
1.5
2
2.5
3
Strike Price Interval
3.5
4
4.5
5
Figure 9: Kurtosis vs. Strike Price Interval
This figure contains a plot of the risk-neutral kurtosis computed by the method of Bakshi, Kapadia
and Madan vs. the strike price interval. Strike prices are generated at intervals of ∆K from a low
strike of 30 to a high strike of 70. The strike price interval is initially set to ∆K = 0.1 and then
increased to ∆K = 5 in increments of 0.1. For each set of strike prices, corresponding option prices
are generated from the Black-Scholes model with a time to maturity of one month, a riskfree rate
of 5%, a stock price of $50, and a volatility of 20% per year. The BKM method is then used to
compute the risk-neutral kurtosis from these prices.
Above we assumed that we could measure the option price with no error and had a large set of
strike prices (from 30 to 70) available to us. The only error was introduced by the discrete strike
price interval. In option markets, however, often we do not have a full set of strike prices available
to us, either because a contract does not exists at a certain strike price, or a contract exists but is
relatively illiquid. Next we examine the impact of limiting the number of observed strike prices on
the risk-neutral moments. Our method is essentially the same as before, except we fix the strike
price interval at a very small value of $0.10 and gradually decrease the number of strike prices
available. The domain of available strikes is from 50 − w to 50 + w. Initially we set the domain
26
0
-0.05
Skewness
-0.1
-0.15
-0.2
-0.25
-0.3
0
2
4
6
8
10
12
Domain Half-Width
14
16
18
20
Figure 10: Skewness vs. Domain Half-Width
This figure contains a plot of the risk-neutral skewness computed by the method of Bakshi, Kapadia
and Madan vs. the half-domain width (w). Strike prices are generated at intervals of 0.1 from a
low strike of 50 − w to a high strike of 50 + w. The domain half-width is initially set to 20, and
then decreased in steps of 0.1 to a value of 1.0. For each set of strike prices, from 50 − w to 50 + w
corresponding option prices are generated from the Black-Scholes model with a time to maturity of
one month, a riskfree rate of 5%, a stock price of $50, and a volatility of 20% per year. The BKM
method is then used to compute the risk-neutral skewness from these prices.
half-width, w at 20 and then gradually decrease the domain half-width from 20 down to 1. The
results for skewness are shown in Figure 10 and the results for kurtosis are shown in Figure 11. At
a large domain half-width, the skewness is 0 and kurtosis is 3. However, as the number of available
strikes drops, both skewness and kurtosis are biased downward.
It is important to note that the only error that we have introduced has been in the form of a
coarse strike price interval and a limited number of strike prices. We have assumed that prices come
from a Black-Scholes world and are observed without error. In reality, of course, error in option
prices is present and can come from many sources. This additional noise makes it more difficult to
27
3
2.5
Kurtosis
2
1.5
1
0.5
0
2
4
6
8
10
12
Domain Half-Width
14
16
18
20
Figure 11: Kurtosis vs. Domain Half-Width
This figure contains a plot of the risk-neutral kurtosis computed by the method of Bakshi, Kapadia
and Madan vs. the half-domain width (w). Strike prices are generated at intervals of 0.1 from a
low strike of 50 − w to a high strike of 50 + w. The domain half-width is initially set to 20, and
then decreased in steps of 0.1 to a value of 1.0. For each set of strike prices, from 50 − w to 50 + w
corresponding option prices are generated from the Black-Scholes model with a time to maturity of
one month, a riskfree rate of 5%, a stock price of $50, and a volatility of 20% per year. The BKM
method is then used to compute the risk-neutral kurtosis from these prices.
28
distinguish between option pricing models on the basis of the moments of the risk-neutral density.
4.5
Tests of Diffusion Properties
Bakshi, Cao, and Chen (2000) observe that all univariate diffusion option pricing models have the
property that the stock price and the call option price, after correcting for time decay, should
always move in the same direction. They proceed to test this property by examining the frequency
with which observed call option and underlying prices move in the same direction. They find that
this frequency to be significantly different from zero, and interpret this as evidence that the prices
are not being generated by a univariate diffusion model.
Given that option prices are observed with noise, it is possible that the observed option price
may move in the opposite direction as the underlying, even if the true price is moving in the same
direction.
Suppose that the underlying asset price can be observed without noise, and let St represent the
asset price at time t. And suppose that the true call option price is given by the Black-Scholes
formula, or some other univariate diffusion model, in which the call price is monotonically increasing
with respect to the underlying price:
Ct = bs(St , Φ),
where the vector Φ includes the other parameters of the model.
Further, suppose that due to market microstructure effects, option prices are observed with
noise:
Ĉt = Ct + ²t .
Under the conservative assumption that the observed option price is always equal to the true
price rounded off to the nearest tick, we would expect the unconditional distribution of ²t to be
uniform over the range [−D/2, D/2], where D is the tick size.
Consider two points in time, designated t0 and t1 . We are able to observe S0 , S1 , Cˆ0 , and
Cˆ1 . Between time zero and one, the stock price and the observed call price can move in opposite
29
directions if the change in the noise term more than offsets the change in the true option price. For
example, the probability that the stock price increases and the observed call price decreases is:
P R(S1 > S0 ; Ĉ1 < Ĉ0 ) = P R(S1 > S0 ; C1 + ²1 < C0 + ²0 )
= P R(S1 > S0 ; bs(S1 ; Φ) − bs(S0 ; Φ) < −∆²
=
Z ∞
S0
P R[bs(X; Φ) − bs(S0 ; Φ) < −∆²]f (X)dX
where ∆² ≡ ²1 − ²0 and f ( ) represents the probability density function of S1 , conditional on
S0 . If ²0 and ²1 are independent, a uniform distribution for ²t implies that ∆² has a triangular
distribution with a range of [−D, D], and the expression simplifies to:
P R(S1 > S0 ; Ĉ1 < Ĉ0 ) =
Z S∗
(D + bs(S0 ; Φ) − bs(X; Φ))2
2D2
S0
f (X)dX
where S ∗ satisfies the equation
bs(S ∗ ; Φ) = bs(S0 ) + D.
An analogous expression can be derived for the probability that the stock price declines while
the observed option price increases. Under Black-Scholes, the distribution f ( ) is lognormal, but
the expression may be applied to univariate diffusion option models in general.
This equation tells us that the probability that the stock price and call price move in opposite
directions depends on the tick size and the distribution of the true stock price change over the
period. In particular, it depends on the probability that the true option price will change by less
than one tick. The magnitude of the typical option price change will be larger if the observation
horizon is longer, or if the options are more in the money. Thus, in a world where the BlackScholes model is true but option prices are noisy, we would expect to see the proportion of spurious
30
violations of the Bakshi, Cao, and Chen (2000) condition to be decreasing with time horizon, and
increasing with moneyness. That these are the exact findings reported by those authors, suggests
that their results are likely to be influenced by microstructural biases. We would also expect the
proportion of spurious violations to be an increasing function of the relative tick size, suggesting
that the Bakshi, Cao, and Chen (2000) technique should be more biased for options on lower-priced
stocks.
4.6
A Note on American Options
In this paper, we have focused on models for pricing European-style options. Of course, individual
stock options, the S&P 100 index option, and the most popular ETF options, such as the QQQ
options, are American-style. Generally, the conclusions drawn from our simulation experiments
would apply equally well to American options. The model might be slightly different, but the
fundamental relation between measurement error and the difficulty in testing models remains the
same.
One could argue on theoretical grounds that it should be harder to test a model using American
option prices than European options. The basic argument would be that when American options
get sufficiently far in the money, it becomes optimal to exercise them early, so we should not expect
to see any trading activity beyond this boundary. This reduces the number of usable strike prices in
an empirical test. Even if prices were available at these in-the-money strikes, they would not help
distinguish between models. Different models make different predictions as to the precise location
of the early exercise boundary, but all arbitrage-free models agree that beyond the boundary, the
option price should equal its intrinsic value.
On the bright side, there is hope that American options can offer additional types of empirical
tests. For example, it may be possible to make inferences about the option model by observing
early exercise behavior.4 In cases where European and American options trade side-by-side, we
4
Existing research on option exercise indicates apparently irrational behavior. For example, see the recent paper
by Poteshman and Serbin (2003).
31
can refine our empirical tests by incorporating our observation the early exercise premium. For
example, Dueker and Miller (2002) have looked briefly at the short period from April to June of
1986 when European and American options on the S&P 500 index traded side by side. Others,
such as McMurray and Yadav (2000), have looked at the FT-SE 100 market, where European and
American options trade with staggered strike prices. Another promising data source for such a test
is in the S&P 100 index options market, where American and European options have been trading
side by side since July 2001.
5
Comments and Conclusions
Bid-ask spreads in option markets tend to be large, compared to the differences between theoretical
predictions of alternative models. For various microstructural reasons, spread midpoints are not
only noisy, but are likely to be biased estimates of an option’s value. Microstructural features such
as discrete prices, the random arrival of limit orders, and a wide interval between adjacent strike
prices make it very difficult to perform reliable empirical tests of option pricing models that rely
on observed option prices, especially options on low-priced stocks.
The theoretical literature provides us a wide variety of potential option models. Many of these
models are characterized by a large number of parameters, and the models differ from each other
in fairly subtle ways. Rather than focusing our attention on the latest, most sophisticated models,
we have tried to address the question of testability on a more fundamental level. We have shown
that it it is often very difficult to distinguish between simple models that differ in basic ways.
Disentangling models with more subtle differences must present an even bigger challenge. If we
cannot differentiate between a constant volatility model with and without jumps, how can we hope
to distinguish a stochastic volatility jump model with stochastic jump intensity from a stochastic
volatility jump model with constant jump intensity?
In the body of this paper, we have pointed out a number potential problems that may arise in
empirical tests of option pricing models. Here, we will offer some additional insights, and summarize
32
some of our main points, trying along the way to offer constructive suggestions as to how future
research may be improved in light of the potential problems.
A recurring theme in this paper is that noise in observing option prices can severely hamper
the power of empirical techniques. This problem has been recognized in the past—the response has
been correct but insufficient. It makes sense to focus our attention on only the most actively-traded
options, use midpoints instead of trade prices, and avoid using closing prices. Beyond this, we can
attempt to reduce the noise in observed prices through careful use of intraday data. For example,
we can attempt to come up with a more accurate estimate of current option prices by applying
time-series techniques to high-frequency intraday data.
Researchers studying individual stock options should be especially careful to recognize the potential biases associated with the tick size and wide interval between adjacent strikes, and should
consider using only higher-priced stocks. Bid ask spreads on individual stock have became considerably tighter since 1999, as a result of competitive forces in the industry. Also, quotes are updated
more frequently than they were in the mid 1990s and before. This suggests in may be a good
idea to use current data rather than the Berkeley Options Data Base. However, due to the recent
market decline, the average stock price is considerably lower than before. Some of the most actively
traded options in recent years have been on stocks with prices below $20.00, where microstructural
problems can be severe.
Let us now comment on the idea of testing an option model by testing the performance of a
trading strategy based on the model. Even in the most liquid exchange-traded option markets, bidask spreads tend to be sufficiently wide that the round-trip trading cost of a strategy implemented
through market orders would obliterate the trading profits of all but the most extremely profitable
strategies. This is particularly true for strategies involving out-of-the-money options. Thus, a key
question is the degree to which such trading strategies could be implemented at lower cost by option
trading firms with a seat on an exchange, or by investors through the use of limit order trading.
We believe that a careful study of limit order execution quality in the option market would be
33
particularly valuable.
Finally, we would like to emphasize that all of the potential problems we discuss in this paper
result from difficulty in observing option prices, not underlying stock prices. While there may also
be errors associated with observing underlying stock prices, we submit that these are likely to be
much less important, perhaps an order of magnitude smaller. Bid-ask spreads for actively-traded
stocks have become very tight in recent years, often they are down to one penny. As transaction
costs have plummeted, it has become much cheaper to implement dynamic trading strategies.
Therefore, we suggest that a productive direction for future research would be to focus on directly
testing option models by testing the accuracy of the assumed equation describing the dynamics of
the underlying asset, and, in models where it would be appropriate to do so, testing the performance
of dynamic replicating strategies.
34
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