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					 Bid-Ask Spreads and Asymmetry of Option Prices.



                                        Raisa Beygelman∗




                              This version: November 30, 2005




  ∗
      Faculty of Economics and Business Administration, Goethe University, Mertonstr. 17, D-60054 Frank-
furt am Main, Germany, e-mail: beygelman@finance.uni-frankfurt.de
         Bid-Ask Spreads and Asymmetry of Option Prices.


                                         Abstract

     In theory the prices of traded assets are unique and can be observed on the market.
     In reality, however, due to market frictions only bid and ask quotes are available
     to investors and researchers. In the existing option pricing literature one usually
     assumes that the mid quote is a good proxy for the true value of an option. This
     assumption follows the intuition that ex ante buyer and seller initiated trades are
     equally likely. In this paper we use a sample of DAX options and show that this
     assumption is not justified empirically for option markets. To estimate the true op-
                                         e
     tion price we use the model of Nord´n (2003) and adjust it for the non-synchronicity
     of the observed option prices. We show that the location of the true option price
     relatively to the bid-ask quotes changes over time and can be explained by some
     market variables. This analysis considerably contributes to the understanding of the
     price finding process on option markets.




1    Introduction and Motivation

In the existing literature different option pricing models have been fitted to the data to
test and compare their performances. One of the inputs in these studies are the prices
of options traded on the market. In the theory of option prices markets are assumed to
be frictionless, so that the equilibrium asset prices are unique and can be observed by
investors. Unfortunately, in the reality the assumption of perfect markets does not hold.
Investors buy options to higher ask prices and sell them to lower bid prices. The question
we investigate in this paper is how to detect the option prices which we would observe if
the markets were frictionless. In the following we will refer to those prices as the true or
the equilibrium option prices.

In the existing literature the problem of unobservable true option prices has by now been
solved by just using the mid quotes of option prices assuming to apply the best proxy. This
assumption follows the intuition that ex ante buy and sell transactions are equally likely,
so bid and ask quotes must then be symmetrically located around the equilibrium prices.
However, some arguments against this point of view can also be found in the literature.
For example, Chan and Chung [10] argue that mid quotes are biased proxies for the true
option prices because due to information asymmetry market makers are more averse to buy
than to sell orders. Options exhibit a limited liability property meaning that the maximal

                                              1
loss of an option buyer is just the option premium, whereas the loss of a holder of a short
positions is unrestricted. So, to ensure themselves against large losses market makers place
ask orders above the optimal values inducing that the true prices are located closer to
best bid than to best ask price. Another argument for the asymmetric price distribution
can be found in Bossaerts and Hilton [7]. They find asymmetries on the foreign exchange
markets and explain them with occasional government interventions. Bessembinder [5]
argues that asymmetric price distributions on the foreign exchange markets can be due to
inventory-control variables. Market makers use asymmetric quotations to reduce or raise
their inventory positions according to their needs. Finally, order imbalance can also be an
explanation. For example Cao, Hansch, and Wang [8] use order imbalance as a proxy for
the true value of an asset.

Several issues provide motivation for dealing with the identification of the equilibrium
option prices. First, by using mid quotes instead of the true prices to calibrate option
pricing models, it becomes much more difficult to distinguish between them. Dennis and
Mayhew [12] show that in presence of usual bid-ask spreads the power of empirical tests of
option pricing models reduces considerably. Second, it is possible to reject the true option
pricing model even if it is true. This is due to the fact that non-zero pricing errors can
arise and model parameters, which by definition are assumed to be constant, can become
time dependent if we use biased market prices in the analysis. Furthermore, Schoutens,
Simons, and Tistaert [22] show that even small errors in the estimated parameter values
reduce the quality of the pricing of exotic options considerably. Third, measurement errors
in option prices affect the estimations of the implied volatility and the calculation of hedge
ratios significantly, especially for options far away from the money (see Hentschel [15, 16]).
Forth, it is important to use the true option prices in models, which analyze the lead/lag
relationship between the option and the underlying prices. Chan, Chung, and Johnson
[9] show that using the mid quotes in place of the transaction prices reverses the leading
relationship between the assets. This indicates that the results in this area are extremely
sensitive to the choice of the proxy for the true prices. Finally, in some models, mid quotes
are taken to be the clue for the decision if a transaction is buyer or seller initiated (i.e.
see Ball and Chordia [4]). Here transaction prices are compared to the most recent bid
and ask quotes. If a transaction occurs above the mid point, it is assumed to be buyer


                                             2
initiated, otherwise it is assumed to be seller initiated. If indeed the true price is located
closer to the best bid quote and market participants realize that, a transaction can be
buyer initiated even if it takes place below the mid quote (but above the true price).

This paper contributes to the existing literature in several ways. First, we check the
assumption of independent transactions which supports the idea of using mid quotes
as proxies for the true prices. By performing an analysis of transaction directions on
the DAX option market we show that this assumption cannot hold for option markets.
We draw this conclusion after detecting a significant autocorrelation in trade directions
and dependencies of the trades on some market variables. Thus, we provide an additional
explanation for the option price asymmetry. To our knowledge we are the first who perform
that kind of analysis for option markets.

Second, we show the role of the asymmetry for the calibration of option pricing models.
Our experiments indicate that making wrong assumptions about the true option price
can lead to large biases when dealing with option pricing models with and without model
risk.

                                         e
Third, we use the model presented by Nord´n [20] to detect the true option prices and
show that non-synchronicity at the option markets can lead to biased estimation results.
Then, we extend the model by adjusting for non-synchronically observed call and put
prices and demonstrate that our model performs well.

Fourth, this paper is the first to investigate the relationship between the option price
asymmetry and the maturity of the option.

Finally, we detect that the location of the true option price relative to the bid and ask
quotes varies over time exhibiting a mean-reverting process and analyze market variables
which influence these dynamics. Our analysis contributes significantly to the understand-
ing of the price finding process on option markets.

The rest of the paper is organized as follows: in Section 2 we analyze the variables driving
the transaction direction on option markets, Section 3 shows the role of option price
                                                                      e
asymmetry for the option pricing, Section 4 presents the model of Nord´n [20], whereas
in Section 5 we perform the robustness checks and introduce the model extension. Section
6 deals with empirical results and Section 7 concludes.


                                              3
2     Transactions Directions

A trade takes place if a market order is placed at the market. If an order is placed at
the current best bid quote, then we observe a sell, otherwise it is a buy. In this section
we test if buy and sell orders in the market do arrive independently. According to the
existing literature the independence of the transaction direction is an argument for using
mid quotes as proxies for the true prices. If, in contrast, there are in contrast some factors
driving transaction direction the true prices do not necessarily correspond to mid quotes
any more. Thus, we try to provide additional motivation for using a method to estimate
the location of the true option prices. By now the analysis of the driving factors of the
transaction direction has extensively been performed for stock markets. Among others
Bessembinder [5] and Porter [21] find systematic patterns in the probability of a trade
at the ask or at the bid quote demonstrating positive autocorrelation in the direction
of the trades. Aitken, Brown, Izan, Kua, and Walter [1] analyze the influence of trade
volume, bid-ask spreads and other variables on the probability of trading at the ask
and show evidence for the influence of the variables on this probability. Whereas we
know a lot about the structure of transactions in stock markets, not much analysis has
been performed for options markets in this research area. Kalodera and Schlag [18], by
investigating the influence of stock market activity on the liquidity in option markets,
find out that the number of call or put purchases and sales can be different for rising
and falling markets. In general call purchases increase with positive and decrease with
negative returns, although the reverse cases could also be found in the data.

Drawing upon these results we test the following model:

                  tradet = a + b1 tradet−1 + b2 tradet−2 + b3 tradet−3

                              +b5 rt + b5 ∆midt + εt ,                                    (1)

where tradet is the trade direction for a given option at time t with tradet = 1 if the
transaction is buyer initiated and tradet = 0 if the transaction is seller initiated. The
variables tradet−1 to tradet−3 are the lagged trade directions. So, in the first place we
intend to capture the autocorrelation structure of trades. Additionaly to that, we define
the underlying return since last transaction as rt = ln(St ) − ln(St−1 ), where St is the
underlying price at time t, since it has been shown to have an impact on the trades.

                                              4
Furthermore, we control for changes in mid quotes of the option since last transaction by
adding the variable ∆midt = (midt − midt−1 )/midt−1 into the regression model. Changes
in options mid quotes between two transactions can occur if some orders are canceled
during that time period. We calculate this value in relative terms to make options with
different moneyness and maturity levels comparable. Finally, a is a constant and εt is
the error term. Note, that the time spans between the transactions and thus between the
observations of the explanatory variables can differ across the observation. We construct
our study this way to answer the question, what changes of the market situation induce
investors to perform the next transaction just at the time point we observe it? Further-
more, if a transaction occurs, is it a buy or a sell (depending on the market conditions)?
Thus, the time span between two transactions is not of interest for us, but only the fact,
that a transaction occured.

We estimate the model (1) by means of a Logit analysis using the transaction data of the
DAX options for January 1998. For every option and every transaction our data contains
the trade direction, options bid and ask quotes and the underlying price. The advantage
of the data is that we do not need to estimate the trade direction as for example has
been done by Chung, Li, and McInish [11], thus avoiding additional noise in the results.
We perform the analysis separately for call and put options and also divide the sample in
at-the-money (ATM), out-of-the-money (OTM) and in-the-money (ITM) options1 . OTM
options are the most frequently traded in our sample since these contracts are often used
for hedging and speculation purposes.

In Tables 1 and 2 the estimation results for call and put options are presented.

Our results show that there is a significant autocorrelation in trade directions for all option
groups meaning that a buy is more likely to occur given that the last transaction was also
a buy. The coefficients of the lagged transactions are in most cases significant also, but are
of much lower magnitude. Furthermore, we find a positive (negative) relationship between
the trade direction of a call (put) and the underlying return. Thus, since the value of a call
(put) rises (declines) in rising underlying markets, it seems that the market is following
a positive feedback strategy. On the other hand there is a significant inverse relationship
between the trade direction and the changes in mid quotes for both call and put options.
  1
      ITM call (put) options are those with 0.98 <= X/S < 1.02, where X is the strike price of the option.


                                                     5
This points to the contrarian feedback strategy, i.e. that investors buy if options become
cheaper. The controversial results indicate that there is no clear strategy followed by
the investors. They do rather weight up the changes in the underlying values and the
corresponding changes in the option prices and buy if options seem to be relatively cheap
given these changes. To clarify this point we extract from our sample the data where a
rise (fall) in the underlying price since last transaction accompanies with a decline of a
call (put) mid quote2 . Then we count the number of buys and sells for different option
groups. The results show that independently of the group more than 70% of transactions
were buyer initiated ones. We repeat this test for the data where a decline (rise) in the
underlying value corresponds to the rise of a call (put) value. As expected, the most of
the transactions (also about 70%) were seller initiated. To check the robustness of our
estimation we exclude both sub-samples from the original data set and estimate the model
(1) once more. We also repeat the experiment for smaller time periods. In all of cases the
results were similar to those presented in this section.

Our analysis indicates that the assumption of randomly occuring buy and sell transac-
tions is not true. Thus, we must reject the hypothesis that mid quotes are equal to the
equilibrium option prices. In the following, we investigate the location of the true option
prices relative to the bid and ask quotes.



3         Model Risk

After providing an explanation why mid quotes do not always correspond to the true op-
tion prices we ask ourselves, what drawbacks result from ignoring the issue of asymmetric
option price distribution.

One of the large fields of option research is the research on option pricing models. The most
prominent model for pricing plain vanilla options is the model of Black and Scholes (Black
and Scholes [6]) (BS), which assumes that underlying prices follow a geometric Brownian
motion. Several models that allow to consider more risk factors like stochastic volatility
or stochastic jumps have been proposed in the literature (see e.g. Bakshi, Cao, and Chen
[2] (BCC)). Therefore, the question arises, which one is doing best in terms of pricing
    2
        See Bakshi, Cao, and Chen [3] for more analysis on that topic.


                                                      6
or hedging performance. Dennis and Mayhew [12] take a closer look at the problem and
show that given the usual levels of bid-ask spreads, it is impossible to distinguish between
the Black Scholes model and the Merton’s jump diffusion model (Merton [19]) (SJ) in
many cases. Hentschel [15] investigates the effect of observation errors in option prices
on the estimation of implied volatilities and concludes that even small errors in option
prices have considerable impact on the estimation results. In another paper Hentschel
[16] analyzes the effects of observation errors on hedging parameters of the Black-Scholes
model and draw a similar conclusion.

Thus, previous studies emphasize that finding the exact location of the true option values
                                                                                e
would help improving the performance of option pricing models. Furthermore, Nord´n [20]
shows that by taking the location parameters into account, the smile is not as pronounced
as in situations where mid-prices are used.

Ignoring the problem of asymmetric option price distribution and using mid quotes to
calibrate option pricing models can lead to the following drawbacks, even if the true
model is known:


    • Estimated parameter values can be biased

    • Pricing errors can become non-zero

    • Parameters which should be constant by definition become time dependent.


To demonstrate this we assume that the true value of a European call option at time t
Ct can be expressed as follows:

                                       Ct = θBc,t + (1 − θ)Ac,t ,                                          (2)

where Bc,t and Ac,t are the bid and the ask price of the option3 . The parameter θ is
the location parameter. It determines where the true option price is located relatively to
the bid-ask quotes. If θ is equal to 1, the true value would coincide with the bid price.
Similarly, if it is equal to 0, the true value and the ask price would be equal.
   3
       For simplicity in this experiment we consider call options only. Using put options alone or additionally
to the call options would not change our results.



                                                        7
We have a cross section of 31 call options with maturity of 30 days and strike prices
ranging from 85 to 115. The volatility of the underlying asset is assumed to be 15%, the
risk free rate is 5% and the current underlying price is 100. First, we calculate the true
values of the assets. To do this we in the first run use the model of Black and Scholes and
in the second run the model of Merton. Repeating the experiment for two different models
allows us to provide deeper insights into the problem. The dynamics of the underlying
stock under the risk-neutral measure in the BS model are given by the geometric Brownian
motion (GBM) as
                                                 √
                               dSt = rSt dt +        V St dWt ,                                      (3)

whereas the dynamics in the SJ model are given by
                                                 √
                       dSt = [r − λµj ]St dt +       V St dWt + Jt dqt .                             (4)

In (3) and (4) r is the risk free rate, V is the variance of the GBM, λ is the frequency
of jumps per year, Jt is the percentage jump size, which is lognormally, identically and
independently distributed with unconditional mean µj . The standard deviation of ln[1+Jt ]
is σj . dWt are the increments of a Wiener process and qt is a Poisson jump counter with
intensity λ. For our experiment we chose the parameter values to be similar to those
attained by Bakshi, Cao, and Chen [2]: V = 0.02, λ = 0.6, µj = −0.15, σj = 0.09.

Second, we calculate the bid and the ask prices for every asset by first setting the θ to
a certain value and assuming that the difference between the both quotes divided by the
mid quote (the relative spread) for all options is 5% (high liquidity case) or 15% (low
liquidity case). Then, the ask price of a call option at time t is defined as

                              Ac,t = Ct + midc,t · sc,t · θ,                                         (5)

                                                                           Bc,t +Ac,t
where Ac,t is the ask price, Ct is the true option price, midc,t =              2
                                                                                        is the mid price
and sc,t is the relative spread. After rearranging and simplifying we get the following
formula to calculate the ask price of a call option at time t:

                                         Ct · (1 + 0.5sc,t )
                               Ac,t =                        .                                       (6)
                                        1 + sc,t · (0.5 − θ)



                                                8
The bid price is then given by

                                                      Bc,t + Ac,t
                                      Bc,t = Ac,t −               · sc,t
                                                            2
                                                        1
                                              Ac,t (1 − 2 sc,t)
                                            =                   .                          (7)
                                                 1 + 1 sc,t
                                                      2

Bid-ask quotes for put options are calculated accordingly. The differentiation between the
two liquidity cases is important since location parameters report the asymmetry of option
prices relative to the spread. In contrast to that, the observation error which is important
for calibration of option pricing models, is measured by the difference between the mid
quote and the true option value independently of the spread. Thus, for the same value of
θ, larger observation error would occur with larger spreads and vice versa.

Finally, the two option price models are calibrated for different values of the location
parameters θ. Thereby in each case we use the same model as the model we used to
calculate the true option prices, thus assuming that the true model is known to the
investor. Furthermore, mid quotes instead of the true option prices are used here, since
we assume that the true option prices are not observable.

To find the model parameters best fitting the data, we minimize the sum of squared
percentage pricing errors of all options defined as
                             31
                                   [(midn − Pn (S, r, Kn , τn ; Φ))/midn ]2 ,              (8)
                             n=1


where midn is the mid price of option n, Pn (S, r, Kn , τn ; Φ) is the model option price and Φ
is the parameter vector of one of the models we take into account. To solve the optimization
problem we use the simulated annealing algorithm4 . It searches asymptotically over the
whole parameter space and changes the direction of the searches randomly. This feature is
important, because global minima will then be found almost surely. The main drawback
of the method is its long computing time. We put up with this disadvantage to reduce
estimation risk to the minimum.

Table 3 shows the estimated parameters and Table 4 the average pricing errors in percent
for the BS and the SJ model for both the liquid (in brackets) and the illiquid (without
brackets) markets.
  4
      See i.e. Goffe, Ferrier, and Rogers [14]


                                                       9
Of course, if we set θ = 0.5 (meaning that there is no observation error), for both option
price models the estimated parameter values are very close to the true ones and the average
errors are equal to zero. This supports our claim that the estimation risk is very small if
we use the Simulating Annealing algorithm to estimate model parameters. The more θ
deviates from 0.5 the larger are errors in estimated model parameters and the larger are
the average pricing errors. In the worst case the errors exceed 5% for BS model and 2% for
SJ model in the illiquid market or 1% and 0.2% in liquid markets respectively. Again, in
liquid markets the errors are smaller, because of the smaller spreads. It is noticeable that
the estimated parameter values vary substantially with changing values of the θ, although
we use the true model to estimate them. Thus, if the location parameters are different for
options with different underlying assets, we would observe different parameter estimates,
even if both underlyings follow the same dynamics.

Our results indicate that ignoring the location parameters would lead to improper in-
ference about the true option pricing model or to a rejection of the true option pricing
model. Using wrong parameter values would also lead to hedging errors as for example
shown by Hentschel [16]. Schoutens, Simons, and Tistaert [22] exemplify that even small
biases in estimated model parameters can lead to large errors if they are used to price
exotic options.

Apart from the biases discussed above, ignoring the risk of asymmetric price distribution
can result in a problem of differentiating between two option pricing models. To clarify
this point we repeat the experiment described above by employing a model similar to the
BCC model5 . In this model the instantaneous variance of the underlying is stochastic.
The dynamics of the underlying asset and of the instantaneous variance are given by

                           dSt = [r − λµj ]St dt +      Vt St dWts + Jt dqt


                               dVt = [θv − κv Vt ]dt + σv     Vt dWtv ,                           (9)

where κv , θv /κv , σv denote the speed of adjustment, the long-run mean and the variance
of Vt respectively. dWts and dWtv are correlated increments of Wiener processes with
   5
       The only difference between the model used here and the BCC model is that we assume the interest
rate to be constant.


                                                   10
correlation coefficient ρ. The parameter values are set to V = 0.02, λ = 0.2, µj = −0.15,
σj = 0.09, κv = 1.1, θv = 0.04, σv = 0.4, ρ = −0.6. As above, for every call option bid and
ask prices are calculated according to the chosen location parameter θ and the relative
spread, which is set to 15% here. Now we assume that the true option pricing model is
not known to the investor and fit the SJ model and the model of Heston (see Heston [17])
(SV) to the mid quotes of the options. In the SV model the risk neutral dynamics are
defined as

                               dSt = rSt dt +      Vt St dWts


                           dVt = [θv − κv Vt ]dt + σv    Vt dWtv .                    (10)

We have chosen these two models, because they show similar pricing performance (see
Bakshi, Cao, and Chen [2]). Thus, it is much more difficult to distinguish between them
than for example between the SV and the BS model, especially if the observed option
prices are noisy. We fit the models by minimizing the sum of squared percentage pricing
errors as defined in (8).

The results are shown in Table 5.

In the first column the values of θ are shown. The second and third columns represent the
average percentage errors for the SJ and the SV model respectively. In the last column
the differences between the pricing errors of both models are calculated.

In cases, in which θ ≥ 0.5 the SJ model performs slightly better providing smaller average
pricing errors. In these situations we would prefer the SJ model to price derivatives. But,
if θ < 0.5, this relationship reverses. Now, the pricing errors are smaller, if we use the
SV model to fit the data. Thus, in case of option prices asymmetry it can be impossible
to distinguish between some option pricing models. Of course, our results also depend on
the parameter values of the true BCC model, but the idea remains the same.

                                                e
In the next section we present the model of Nord´n [20] which can be used to calculate
the location parameters.




                                              11
4                          e
          The Model of Nord´n

The model we use in our paper was first presented by Chan and Chung [10] and was
                     e
then extended by Nord´n [20]. As mentioned above we assume that the true value of a
European call option C at time t can be expressed as

                                       Ct = θBc,t + (1 − θ)Ac,t .                                      (11)

Taking first differences leads to

                                    ∆Ct = θ∆Bc,t + (1 − θ)∆Ac,t ,                                      (12)

where ∆Ct , ∆Bc,t and ∆Ac,t denote changes in the true value, the bid and the ask price,
respectively. After defining ∆SP Dc,t = ∆Ac,t − ∆Bc,t as the change in the bid-ask spread
and rearranging equation (12) we get the formula for changes in ask price as being

                                      ∆Ac,t = θ∆SP Dc,t + ∆Ct .                                        (13)

Analogous to equation (13), the formula for a European put option is

                                     ∆Ap,t = γ∆SP Dp,t + ∆Pt ,                                         (14)

where ∆Pt denotes the change in the true put option values and γ is the location parameter
of the put option6 .

In equations (13) and (14) the changes in the true option values ∆Ct and ∆Pt are the only
                                                        e
parameters, which cannot be observed on the market. Nord´n [20] solves this problem by
using the put-call parity. It says that the following relationship must hold:

                                        Ct − Pt = St − Ke−r·T ,                                        (15)

where St denotes the value of the underlying, r is the risk free rate and T is the time to
maturity of the options. Again, expressing the relationship in difference notation leads to
the following relationship:

                                  ∆Ct − ∆Pt = ∆St − ∆(Ke−r·T ).                                        (16)
    6
        Since the locations of the call and put options need not be the same, we use different variable names
here.


                                                      12
After combining the equations (13), (14) and (16) we get the relationship between the
changes in ask prices of the call and the put values and other parameters:

            ∆Ac,t − ∆Ap,t = θ∆SP Dc,t − γ∆SP Dp,t + ∆St − ∆(Ke−r·T )                  (17)

To estimate θ and γ we can now perform a regression according to equation (17) of the
following form:

           ∆Ac,t − ∆Ap,t = β0 + β1 ∆SP Dc,t − β2 ∆SP Dp,t + β3 ∆St + εt ,             (18)

where εt is the error term. Since ∆(Ke−r·T ) is a constant, it is assumed to be represented
through the constant term β0 . Estimation results for regression coefficients β1 and β2
correspond to the estimates of θ and γ respectively. If the assumption that the true
option values correspond to mid prices holds, estimates should result in 0.5 for both of
them.

There are several points concerning the methodology above to mention. First, all input
parameters needed to estimate the values of location parameters are observable on the
market. Thus, we do not have to choose any proxy. Second, we do not need to know the
true dynamics of the underlying and of state variables, i.e. we do not have to know the
true option pricing model and thus we are not affected by model risk.

Note that the above model is valid for European options only, since we used the corre-
sponding put-call parity to derive it. If we wish to estimate the location parameters of
American style options, we have to modify our result by adding an early exercise premium
as the additional independent variable into the estimation regression (18). Since the early
exercise premium is not observable and has to be estimated first, which is not an easy
exercise, one could drop this variable, by assuming it to be included in the error term ε
                           e
as it has been done by Nord´n [20].

                                                  e
In the following section we test the model of Nord´n [20], show, what problems can arise
by implementing it and suggest some extensions to improve its performance.



5    Implementation Issues

By implementing the method described above one can face some problems due to the
available data. For example, by now we have assumed that the true value of the underlying

                                            13
asset is known. In the reality, this is in general not the case. Like in option markets,
in stock markets we cannot observe the true market value of an asset, but instead we
observe bid and ask quotes. Although stocks exhibit much smaller spreads than options,
the estimated option’s location parameters could be biased, if we just use the mid prices
of the underlying asset for the regression. The bias might be more pronounced if the true
value of the asset is instead located very close to the bid or to the ask price.

Another problem one could face is the problem of non-synchronicity of the observed
option prices. For the estimations described above one observation vector consists of the
ask price of a call option, the ask price of the corresponding put option, the spreads of
these contracts and the underlying price. In illiquid markets however, options are traded
very infrequently, so that often call and put prices cannot be observed at the same point in
time. The problem of non-synchronicity called attention of many researches. For example,
in Dennis [13] the author uses options observed within a 5 minute interval only. Chan
and Chung [10] restricts his data to call and put prices with less then 30 minutes of time
difference between the observations.

In this section we construct a simulation study to investigate whether observation errors
in the underlying price and the non-synchronicity of option prices can lead to biases in
estimates of the location parameters.



5.1    Observation Error in the Underlying Prices

Besides the true values of options, the true values of the underlying assets are in general not
                                                                                       e
observable as well. Instead only bid and ask prices can be observed on the market. Nord´n
[20] assumes that the true value of the underlying assets corresponds to the mid point
of the bid-ask spread. To justify this assumption, he analyzes bid and ask quote changes
of 27 stocks, which were listed on the Stockholm stock exchange in 1995. Differences in
the sum of bid and ask changes can be one possible explanation for asymmetric location
                                                                    e
of the true prices within the spread (see Chan and Chung [10]). Nord´n [20] finds that
in contrast to options, where variations in the bid are usually lower than those in the
ask, the changes for underlying assets were approximately equal. He uses this result as an
explanation for taking mid prices of the underlying in his study.


                                              14
Despite this finding, we check if the incorrect assumption about the true value of the
underlying asset does affect the estimation results in (18) by constructing the following
simulation study. Analogous to the definitions for the options we define the true value of
the underlying asset to be located between the respective bid and ask price at time t as
follows:

                                St = φBs,t + (1 − φ)As,t,                                  (19)

where φ is the location parameter, Bs,t is the bid price and As,t ist the ask price of the asset
at time t. Furthermore, we consider a call and a put option each with strike price equal
to 100 and time to expiration equal to 30 days. We assume that the current underlying
stock price is 100, the risk free interest rate is 5% and the volatility of the underlying
process is 15%.

First, we simulate a path of the underlying price for the time period of one month with
10 minutes intervals between every observation using the geometric Brownian motion.
Second, given the underlying price at every point in time we calculate the BS price of the
call and the put option. Third, we determine the bid and ask prices for every observed
option price as follows. To be more flexible, we allow the relative spreads at each time
point to change within defined bounds. Thus, for each asset we define a relative spread
lower bound (SLB) and a relative spread upper bound (SUB), and then determine the
relative spread for each time point by drawing a number from the uniform distribution
between SLB and SUB. In this analysis we chose the following values: SLBs = 0.01,
SUBs = 0.02, SLBc = SLBp = 0.02, SUBc = SUBp = 0.08 (the subscripts p, c and s
distinguish the respective quantities for puts, calls and the underlying stock). Since the
results of this study are sensitive to the set up, it is important to chose the parameter
values carefully. Thus, we analyze the dynamics of the relative spread of the at-the-money
DAX options and detected similar dynamics as assumed here. The spread bound for the
underlying asset is chosen to be smaller, since underlying markets are in general more
liquid than option markets.

In our example we set θ and γ values to 0.2 and 0.7 respectively. Armed with dynamics
of the relative spread and the values of the location parameters we are able to calculate
the bid and ask quotes for the call and the put option at every time point as shown in (6)
and (7).

                                              15
Finally, we perform the regression (18) for different values of φ and use the mid quotes of
the underlying stock instead of the true value St . For every calibration we compare the θ
and the γ estimates with the true ones. By doing so, we can show how large the estimation
biases in θ and γ are depending on the observation error in the underlying asset. All other
parameters are left unchanged for every regression. Table 6 shows the results.

As expected, the best result is achieved for the case where the true value of the stock
corresponds to the mid price (φ = 0.5). Here, the estimations of θ and γ are closest to
the true values of 0.2 for θ and 0.7 for γ apart from a small estimation error. Also, the
adjusted R2 is close to one as one would expect. The further away from the mid point the
true value of the underlying is located, the more biased are the estimates for regression
parameters. In the worst cases (where φ = 0 and φ = 1) we get an estimate for θ of around
0.15 and for γ of around 0.68. The values of the adjusted R2 go down to approximately
0.3 indicating only a moderate fit of the model. On the one hand, these errors are not
that pronounced. On the other hand, however, as has been shown in previous sections, the
deviations from the true values of the location parameters of that magnitude can in some
cases lead to biases in estimated parameters of option pricing models and to non-zero
pricing errors of the models.

We conclude that in some cases using the right underlying value for estimation of location
                                                                         e
parameters becomes an important issue. Both, Chan and Chung [10] and Nord´n [20]
use mid prices for the underlying price, so their results are probably biased. To avoid
the problem one can for example use approximative values for the true underlying asset
prices as supposed in Cao, Hansch, and Wang [8]. Another possibility is to use the model
presented in Bessembinder [5]. Here, the location parameter of the underlying asset can
be estimated by using the following simple regression model:

                           ∆Bs,t = α0 + α1 ∆SP Ds,t + ηt ,

where ∆Bs,t are the changes in the bid price of the underlying, ∆SP Ds,t are the changes in
the spreads and ηt is the error term. Estimation results for α1 correspond to the estimated
value of φ. To detect the performance of the suggested correction methods more tests are
needed. We leave this issue for the further research.



                                            16
5.2        Non-Synchronicity of Option Prices

To implement the model described above one needs observation vectors, each containing
an ask price of a call option, an ask price of the corresponding put option and an underlying
price observed at the same time. However, in illiquid markets options are traded at a lower
frequency. A quote is assumed to be valid as long as no new quote arrives. If the price
of the underlying however changes in the meanwhile, the true value of an option must
also change, because otherwise the put-call parity (which were used to derive the model)
would be violated. To account for this liquidity effect, we adjust the put-call parity, derive
an adjusted regression model and test whether we are able to eliminate the problem of
non-synchronicity.

Consider four points in time: t1 , t1 , t2 , and t2 with t1 < t1 < t2 < t2 . Assume that in t1
                               c    p    c        p       c    p    c    p                   c

and in t2 we observe the bid and the ask price of the call option of interest Bc,t1 , Ac,t1 ,
        c                                                                         c       c

Bc,t2 and Ac,t2 but no quotes of the corresponding put option. In t1 and t2 we observe
    c         c                                                    p      p

the put values Bp,t1 , Ap,t1 , Bp,t2 and Ap,t2 , but no call quotes are available. We assume
                   p       p       p         p

that the underlying asset is traded frequently enough to find the true underlying asset
price at any point in time. Since stock markets are in general more liquid than option
markets, this is a realistic assumption. We now have to build regression vectors at time
t1 and at time t2 out of the quotes described above. Since the call and the corresponding
 c              c

put prices are observable at different points in time, we must assign to the observed call
price a ”wrong” put price. To correct for that problem we suppose an adjusted put-call
parity. At time t1 it can be written as
                 c


                                Ct1 − (Pt1 + ω(t1 )) = St1 − Ke−r·T ,
                                  c      p      c        c
                                                                                                        (20)
                                             Pt1
                                               c


where St1 is the underlying price observed at time t1 , and ω(t1 ) is the error corresponding
        c                                           c          c

to the problem of non-synchronicity. The term Pt1 + ω(t1 ) is the unobserved true price
                                                p      c

Pt1 at time t1 . Similarly, the put-call parity at time t2 can be defined as7
  c          c                                           c


                                Ct2 − (Pt2 + ω(t2 )) = St2 − Ke−r·T .
                                  c      p      c        c
                                                                                                        (21)
                                             Pt2
                                               c

   7
       Note that the term T in (20) is smaller than the term T in (21) by (t2 − t1 ). Since this difference is
                                                                            c    c

negligible small relatively to the magnitude of the values T , we ignore this fact.


                                                      17
The error terms ω(t1 ) and ω(t2 ) cannot be observed, but they can be approximated by
                   c          c

using the Taylor expansion as
                           dPt1                                 dPt2
                ω(t1 ) =
                   c
                              p
                                   (St1 − St1 ),     ω(t2 ) =
                                                        c
                                                                   p
                                                                       (St2 − St2 ).                            (22)
                           dSt1
                              p
                                      c     p
                                                                dSt2
                                                                   p
                                                                          c     p




For simplicity, we denote the terms (St1 − St1 ) and (St2 − St2 ) by Std1 and Std2 respectively
                                       c     p          c     p

in the following. Then, we proceed as in Section 3 by additionally inserting equations (22)
in (20) and (21) to get the following regression relation at time t2 :
                                                                   c

                                                                dPt2                dPt1
∆Ac,t2 − ∆Ap,t2 = θ∆SP Dp,t2 − γ∆SP Dc,t2 + ∆St2 +                 p
                                                                       Std2 −          p
                                                                                            Std1 −∆(Ke−r·T ).(23)
     c        c            c            c      c
                                                                dSt2
                                                                   p
                                                                                    dSt1
                                                                                       p

                                                                            E

Compared to the equation (17), we have the additional term E. It represents the adjust-
                                                                            dPt2               dPt1
ment for option prices observed non-synchronically. The terms               dSt2
                                                                                    p
                                                                                        and    dSt1
                                                                                                    p
                                                                                                        represent the
                                                                                    p               p

deltas of the put option at times t2 and t1 respectively. If we assume that the moneyness
                                   p      p
                                                                           dPt2             dPt1
of the option does not change too much over time, we can set               dSt2
                                                                                p
                                                                                        =      p
                                                                                            dSt1
                                                                                                   . Then, the term
                                                                                p              p
                           dPt2
E can be simplified to         p
                           dSt2
                                  (Std2 − Std1 ), leading to the ”new” regression equation (24):
                              p


                 ns   ns            ns            ns       ns        d
∆Ac,t − ∆Ap,t = β0 + β1 ∆SP Dp,t − β2 ∆SP Dc,t + β3 ∆St + β4 (Std − St−1 ) + εt , (24)

       ns
where β4 represents the estimated delta of the put options as described above. The
subscript ”ns” emphasizes that the parameters were estimated after the adjustment for
the non-synchronicity.

To check whether our approximation method is useful, we conduct an experiment similar
to the one described in Section 5.1. As above, we simulate a path of the underlying prices
using the GBB with equal time intervals between the observations. However, in contrast
to the previous study, we calculate call and put prices (using the BS model) at every
second point in time. Additionally, we alternate call and put price calculations, so that at
one point in time either a call or a put price is available (but never both of them at the
same time point). The larger the time difference between the time points, the less liquid
the market, and the larger is the problem of the non-synchronicity.

Again, our underlying price process starts at 100, the volatility of the price process is 15%,
the risk free rate is 5% and we consider a call option with strike price 100 and an initial
maturity of 30 days. In contrast to the last experiment we do not define the length of the

                                                   18
whole observation period, since this would lead to less data points for larger time spans
between the observations and could influence the results. We instead generate the data
for 1000 time points. If an option expires before the simulation is over, we introduce a new
option with 30 days to expiration. To avoid additional biases, we set the bid-ask spread
of the underlying equal to 0. The bounds of the relative bid-ask spreads of the call and
put options are set to the following values: SLBc = SLBp = 0.02, SUBc = SUBp = 0.08.
The true values of the location parameters are left unchanged at 0.2 for the θ and 0.7
for the γ. As above, we are able to calculate the bid and the ask quotes using the true
option prices, the relative spreads and the values of θ and γ. We perform the simulation
for different time spans between the observations and apply the regressions (18) and (24)
for each of the data sets. Finally, we compare the estimation results of the two models.

Table 7 shows the results of the study. The first column represents the differences between
the corresponding call and put observations in minutes. The next three columns show the
                                                    e
results obtained by using the model proposed in Nord´n [20], i.e. the regression equation
(18). In the last three columns the results of the adjusted regression equation (24) are
presented.

If we compare the results for θ and for θns , we do not find much difference, although in
general the results of the adjusted model are slightly better than those of the unadjusted
one. The reason for such a good performance is that by construction always the ”right” call
prices are used for estimation. In contrast to that, the put prices are observed with delay,
and so we can observe large differences between the results for the location parameters of
the put γ and γ ns . In our scenario 10 minutes of time span between call and put values
are sufficient to obtain a difference between the estimated value and the true value of
more than 0.3, whereas the adjusted model still performs very well. For the largest time
difference of 60 minutes, the unadjusted model provides us with extremely wrong values.
In contrast to that the results of the adjusted model are still reliable.

We have checked the results using the SV model instead of the BS model and the results
were similar to those presented here. One possible problem of using the approximation in
(22) is the changing moneyness of the options, since moneyness is one of the factors, which
affects the delta of an option. Putting options with different delta values in one regression
leads to biases. This problem can be avoided by using only options with similar moneyness

                                             19
in one regression. As will be shown later, the subdivision of the data sample into different
moneyness categories is one important part of dealing with location parameters anyway,
since the location parameters take different values for different moneyness levels. Thus, we
do not have to make additional assumptions to eliminate the problem of different deltas.

At this point we can conclude that the problem of non-synchronicity of the observations
is an important issue by dealing with location parameters. In very illiquid markets one
should use the adjusted model to estimate the values of the location parameters, because
otherwise in some cases wrong results can be the consequence. Chan and Chung [10]
impose the restriction that the call and the corresponding put values have to be observed
within a thirty-minute interval. According to our example allowing for this time span
could lead to biased results. However, to be able to appraise the magnitude of the error,
an extensive analysis of the data structure is needed. In particular one has to relate the
number of observations with large time spans to the whole number of the observations.
As shown in this subsection one should use the extended model introduced in this section
to be sure that errors are minimized.



6     Empirical Results

6.1    Data

To investigate the location parameters empirically we collect a time series of best bid and
best ask quotes of call and put options on the German blue chip index (DAX) traded
in 1998 on the electronic trading platform. The index cobrains the 30 biggest and most
actively traded German shares. The quote data for the underlying is obtained from Xetra.
The choice of our data set has some advantages compared to other alternatives. First, the
DAX is a performance index, where dividends are reinvested. Thus, we do not have to
adjust our model for dividend payments. Second, options written on the DAX are all of
European style, so we avoid any biases due to an early exercise premium. Third, we avoid
the problem of observation errors in the underlying. Finally, these options are very liquid
making the liquidity bias relatively small. This also ensures that we have enough data to
                                                                   e
make a detailed analysis using options on one underlying only. Nord´n [20], for example,


                                            20
uses options on many underlying assets which can lead to biases. We use one, two, three
and sixth month Fibor rates as a proxy for the risk free interest rate.

We applied several filters to the data set. First, we eliminate observations which do not
satisfy the put-call parity. Second, observations with a relative spread larger then 50%
are deleted from the sample. Finally, options with less than 7 days prior to maturity are
eliminated.

Observation vectors are constructed by first searching for put-call pairs with the smallest
time difference between the corresponding observations. Then, pairs with a time difference
of more than 10 minutes were deleted from the sample (according to the results in Section
3.2., this should provide us with reliable results, if we use the adjusted regression model
(24)).



6.2       Location Parameters and Time to Maturity

Some previous studies show that the values of the location parameters vary with the
moneyness of the options8 . In this subsection, we test if θ and γ also depend on option’s
time to maturity. First, we define four maturity groups: T <= 2 weeks (Group I), 2 weeks
<= T <= one month (Group II), one month <= T <= 2 months (Group III), and T >=
2 months (Group IV), where T denotes the maturity of the option. Then we perform the
regression model (24) by additionally introducing a dummy variables Di for every of the
maturity groups:
                              4                    4                            4
                                       ns                   ns                           ns
          ∆Ac,t − ∆Ap,t =          Di β0,i   +          Di β1,i ∆SP Dp,t   −         Di β2,i ∆SP Dc,t
                             i=1                  i=1                          i=1
                                              4                     4
                                                       ns
                                       +           Di β3,i ∆St +             ns          d
                                                                         Di β4,i (Std − St−1 ) + εt ,   (25)
                                             i=1                   i=1

where D1 is equal to one if maturity of the option belongs to the Group I and zero
otherwise, D2 is equal to one if the maturity belongs to the Group II and zero otherwise
                                         e
and so on. As already pointed out by Nord´n [20] the introduction of dummy variables
allows to test the null hypothesis that the values of location parameters for puts and calls
are equal. Furthermore, we test whether the location parameters are equal for different
  8
                   e
      see i.e. Nord´n [20]


                                                          21
maturity groups. To answer this question we additionally divide our data sample into three
moneyness9 categories: 0.9 <= X/St < 0.98, 0.98 <= X/St < 1.02, 1.02 <= X/St < 1.1,
and perform the estimation (25) separately for every moneyness category.

Figure 1 presents the location parameter as a function of the time to maturity for different
levels of moneyness. The x-axis shows the maturity groups in months, whereas the value
for the last moneyness category is set to 3. The y-axis represents the results for θ (solid
line) and γ (dashed line)10 .

Consistently with previous results the location parameters of the OTM DAX calls and
puts are in general higher than θ and γ of the ITM options and are larger than 0.5. This
implies that the true values of these options are on average closer to the bid quotes, and
thus the mid quotes are a biased proxy for the true option prices. In fact, OTM options
(especially the OTM puts) are sometimes found to be too expensive. Our results seem to
support this finding.

Furthermore, we can observe pronounced patterns of θ and γ depending on the time to
maturity for ITM and OTM options. While the prices of the short term maturity calls
and puts exhibit a considerable asymmetry, the true values of the longer term options are
located around the mid quote which consequently seem to be a good proxy for the true
prices in this case. For ATM options no significant relationship can be found. We could
not reject the hypothesis that the θ parameters are different for different maturity groups
at 5% significance level.



6.3        Time Series of Location Parameters

In the previous subsection we assumed the location parameters to be constant in time.
However this assumption is only an approximation to the reality, since empirically we can
observe a significant variation of the location parameters over time. To clarify this issue
we re-estimate the location parameters at intervals of two hours. The results for ATM call
options with time to maturity between 2 weeks and 2 months are represented in Figure 2.
Obviously, the dynamics of θ oscillate significantly. However, the time series is stationary
   9
       Moneyness at time t is defined as X/St .
  10
       All of the parameter values are highly significant with relatively small standard errors indicating
stable results.


                                                    22
and exhibits a strong mean reversion. Interestingly, we rarely observe that the location
parameter leaves the 0–1–bound. This implies that the true value does not lie within the
bid-ask spread. However, those periods are only short-lived and the market participants
seem to adjust their quotes immediately.

Next, we analyze the relative error that may result if mid quotes are used instead of true
                                                                                midt −Pt
prices. We define the relative observation error of an asset at time t as ψt =    midt
                                                                                         ,   where
Pt is the true price of the asset at time t. Using our data sample we estimate ψ. Figure 3
represents the median of the estimated ψ for call options as a function of time of day11 .
One can observe that the relative observation error is more pronounced immediately after
the opening of the exchange and before the closing of the market. The accumulation of
the information overnight and the absence of an opportunity to trade may imply a higher
price uncertainty in the morning. Since the return distribution overnight is different from
those over continuous trading intervals, investors may be reluctant to open new positions
during the last hours of trading. Thus, liquidity decreases which may result in higher price
uncertainty. Given these observations, if option price data is needed at a daily frequency,
it seems advisable to collect the time series around noon and not to use opening or closing
prices.



6.4        Location Parameters and Market Variables

In this subsection we analyze the impact of important market variables on the dynamics
of the location parameters. For this purpose we estimate the following regression

                                      d            d            d
                         θ(γ)t = α + β1 θ(γ)t−1 + β2 θ(γ)t−2 + β3 θ(γ)t−3
                                        d         d       d
                                      +β4 rt−1 + β5 vt + β6 T It + εd ,
                                                                    h                         (26)

where θ(γ)t is the location parameter of call (put) options calculated at intervals of two
hours and θ(γ)t−1 to θ(γ)t−3 are the lagged location parameters. rt−1 is the underlying
asset return calculated over the last interval and vt is the volatility of the underlying
estimated for the current interval. T It (trade imbalance) is the difference between the
number of buyer and seller initiated trades. The results are represented in Tables 8 and 9
  11
       The results for put options are similar.



                                                    23
for call and put options respectively. The location parameters are positively autocorrelated
up to the second lag. This seems to be consistent with the results presented in Figure 2.
The return of the underlying has a negative (positive) impact on the location parameters
of call (put) options. This seems to be intuitive. Imagine a situation where the price of the
underlying is going up. Then, the true value of a call option increases as well. However, if
market participants do not adjust the bid-ask spread, the parameter θ decreases. Market
participants may be reluctant to adjust the bid-ask spread of the option immediately as
the relative tick size of options is usually much higher than those of the underlyings. A
symmetric explanation can be derived for put options.

If market participants recognize that the change of the underlying price was significant
they adjust their order submission strategy and thus the bid-ask quotes. Again, imagine
a call option. If the underlying price increases significantly the true value of the option
increases as well and market participants submit more market buy orders. Thus, the trade
imbalance increases and the true value returns to the center of the bid-ask spread. This
intuition is supported by the sign of the regression coefficient of the trade imbalance.

The volatility of the underlying has a positive impact on the location parameter of both
call and put options. If volatility shocks occur market participants seem to buy aggressively
options by submitting market orders and thus consuming liquidity on the ask side of the
order book. As a consequence, the ask price rises and the true value appears to be closer
to the bid quote, so that the location parameter increases.



7     Conclusions

This paper deals with the question of how to identify the true option prices which cannot
be observed due to a number of market frictions. We argue that mid quotes are in general
a biased proxy for the true option prices and provide an explanation for that by analyzing
a sequence of transactions of call and put options. We detect a significant autocorrelation
in transaction directions which contradicts the assumption of independently arriving buy
and sell orders.

An additional motivation for dealing with the topic is provided by showing that consider-
able drawbacks can arise if one uses biased option prices when dealing with option pricing

                                             24
models.

                                                    e
To identify the true prices we use the model of Nord´n [20]. However, we show that if call
and put prices are observed non-synchronically, it maylead to biased results. To correct
for that we extend the model by adjusting it for the case of non-synchronicity using Taylor
expansion.

We then use the DAX option data to analyze the location parameters for the year 1998.
In line with the existing literature the true option prices are on average closer to bid then
to ask quotes indicating that mid quotes are indeed biased proxies for the true option
prices. By additionally dividing the data sample into time to maturity groups we are able
to get new insights into the structure of the location parameters.

In the last subsection we show that the location of the true option prices does not remain
constant but changes over time exhibiting dynamics similar to a mean-reverting process.
The observation error is the highest in the morning and at the evening, thus it seems
advisable to collect daily time series of option prices around noon and not to use opening
or closing prices. Furthermore, we find that the location parameters are autocorrelated
and are partly driven by the underlying return and the underlying volatility. Finally, we
find an evidence that market participants adjust their trading strategies in times of high
option price asymmetry. Our results help to understand the price finding process on option
markets.




                                             25
                   ITM      S. Error         ATM      S. Error         OTM       S. Error
    Intercept      -0.642    0.116     ***   -0.732   0.0527     ***   -0.831    0.0472     ***
     Lag (dt )     0.650     0.124     ***   1.157    0.0618     ***   1.447     0.0531     ***
    Lag2 (dt )     0.387     0.116     ***   0.278    0.0567     ***   0.354     0.0465     ***
    Lag3 (dt )     0.433     0.116     ***   0.112    0.0563     **    0.211     0.0463     ***
  Underl. return   72.00    12.811     ***   166.00   15.003     ***   207.00    11.273     ***
   Mid change      -4.687    0.579     ***   -7.803    0.427     ***   -11.736    0.406     ***
    pseudo R2      0.135                     0.169                     0.236
  Observations     1334                      5963                       9544


Table 1: Estimated parameters of the model
tradet = a + b1 tradet−1 + b2 tradet−2 + b3 tradet−3 + c rt + d ∆midt + εt
for call options, where tradet is the trade direction for a given option at time t with tradet
= 1 if the transaction is buyer initiated and tradet = 0 if the transaction is seller initiated.
The variables tradet−1 to tradet−3 are the lagged trade directions and rt , ∆midt and εt
are the underlying asset return, change of the option mid quote since the last transaction
and the error variable at time t, respectively. The table presents the results for ITM,
ATM und OTM options. ***, ** and * correspond to 1%, 5% and 10% significance level,
respectively.




                                               26
                   ITM      S. Error         ATM       S. Error         OTM       S. Error
   Intercept      -0.552     0.142     ***   -0.985    0.0555     ***   -1.0688    0.044     ***
    Lag (dt )      1.239     0.153     ***   1.293     0.0689     ***   1.435     0.0538     ***
   Lag2 (dt )      0.099     0.143           0.248     0.0611     ***   0.377     0.0478     ***
   Lag3 (dt )      0.209     0.142           0.231     0.0607     ***   0.316     0.0475     ***
 Underl. return   -119.70   17.750     ***   -195.90   15.501     ***   -220.50   11.998     ***
  Mid change      -9.608     1.195     ***   -12.219    0.614     ***   -11.596    0.433     ***
   pseudo R2       0.206                     0.200                      0.236
  Observations      996                       5250                       8946


Table 2: Estimated parameters of the model
tradet = a + b1 tradet−1 + b2 tradet−2 + b3 tradet−3 + c rt + d ∆midt + εt
for put options, where tradet is the trade direction for a given option at time t with tradet
= 1 if the transaction is buyer initiated and tradet = 0 if the transaction is seller initiated.
The variables tradet−1 to tradet−3 are the lagged trade directions and rt , ∆midt and εt
are the underlying asset return, change of the option mid quote since the last transaction
and the error variable at time t, respectively. The table presents the results for ITM,
ATM und OTM options. ***, ** and * correspond to 1%, 5% and 10% significance level,
respectively.




                                              27
          θ          V                  λ                    µj               σj
   BS     0   0.0203 (0.0201)
         0.3 0.0201 (0.0201)
         0.5 0.0200 (0.0200)
         0.7 0.0199 (0.0200)
          1   0.0197 (0.0199)
    SJ    0   0.0197 (0.0203)     0.1116 (0.1453)     -0.9914 (-0.5759) 1.9534 (0.3412)
         0.3 0.0203 (0.0202)      0.1032 (0.3701)     -0.8093 (-0.2324) 0.6310 (0.1284)
         0.5 0.0200 (0.0200)      0.6007 (0.6024)     -0.1499 (-0.1496) 0.0900 (0.0898)
         0.7 0.0185 (0.0197)      3.3333 (1.0032)     -0.0412 (-0.0979) 0.0385 (0.0663)
          1   0.0072 (0.0188) 99.9964 (2.5250) -0.0050 (-0.0627) 0.0125 (0.0430)


Table 3: Estimated parameter values of the BS model and the SJ model in case of a liquid
(in brackets) and an illiquid (without brackets) market without model risk for different
values of the location parameter θ.


                         θ   Avg. error in % BS     Avg. error in % SJ
                         0    0.0558 (0.0194)        0.0168 (0.0018)
                      0.3     0.0223 (0.0078)        0.0022 (0.0008)
                      0.5     0.0000 (0.0000)        0.0000 (0.0000)
                      0.7     0.0222 (0.0078)        0.0064 (0.0014)
                         1    0.0553 (0.0194)        0.0258 (0.0052)


Table 4: Estimated average percentage pricing errors of the BS model and the SJ model
in case of a liquid (in brackets) and an illiquid (without brackets) market without model
risk for different values of the location parameter θ.




                                            28
                      θ    Avg. error SJ Avg. error SV         Difference
                      0       0.0475              0.0442            -0.0033
                     0.3      0.0236              0.0187            -0.0049
                     0.5      0.0071              0.0093            0.0022
                     0.7      0.0119              0.0217            0.0098
                      1       0.0374              0.0477            0.0104


Table 5: Estimated average percentage pricing errors of the SJ and the SV models for
different values of the location parameter θ. The true option pricing model is the BCC
model. In last column the differences between the errors are calculated.

                              φ     θ (0.2) γ (0.7) Adj. R2
                              0     0.1553   0.6721        0.3405
                              0.3   0.1652   0.6662        0.4405
                              0.5   0.2081   0.7005        0.9946
                              0.7   0.1572   0.7003        0.4198
                              1     0.1514   0.6880        0.3290


Table 6: Estimated location parameters of call and put options for different location
parameters of the underlying asset. In parenthesis the true values of θ and γ are presented.



          Minutes    θ (0.2) γ (0.7) Adj. R2         θns (0.2) γ ns (0.7)     Adj. R2
             0.1     0.2016    0.7044   0.9675         0.2            0.7        1
              1      0.2056    0.7263   0.7604        0.2012         0.7023   0.9986
              5      0.2012    0.8034   0.4411        0.1938         0.6912   0.9665
             10      0.2129    0.9153   0.3678        0.2060         0.6861   0.9609
             30      0.1934    1.3744   0.3138        0.2045         0.6493   0.9466
             60      0.2305    2.0098   0.3221        0.2458         0.5253   0.9478


                                                e
Table 7: Estimation results of the model of Nord´n [20] and the adjusted regression (24)
(columns 5 to 7) for different time intervals between the call and the corresponding put
prices. In parenthesis the true values of θ and γ are presented.

                                             29
                                                θ           Std. Error
                         Intercept         0.27520   ***     0.02547
                          lag (θ)          0.34416   ***     0.03304
                         lag2 (θ)          0.17234   ***     0.03460
                         lag3 (θ)         -0.00233           0.03310
                  Lagged underl. return -4.11384 ***         1.32646
                     Underl. volatility    63.1121   **     29.19256
                     Trade imbalance      -0.00249 ***      0.000642
                          adj. R2          0.2336
                       Observations            932


Table 8: Estimation results of the regression model (26) for call options. ***, ** and *
correspond to 1%, 5% and 10% significance level, respectively.




                                               γ            Std. Error
                        Intercept          0.35705    ***     0.02973
                         lag (θ)           0.28110    ***     0.03149
                         lag2 (θ)          0.05955     *      0.03226
                         lag3 (θ)          0.02959            0.03096
                  Lagged underl. return    6.11364    ***     1.36277
                    Underl. volatility    136.37045 ***      46.63772
                    Trade imbalance       -0.00478    ***    0.000726
                         adj. R2           0.1825
                      Observations             986


Table 9: Estimation results of the regression model (26) for put options. ***, ** and *
correspond to 1%, 5% and 10% significance level, respectively.




                                          30
                          0.9 <= X/S < 0.98                                               0.98 <= X/S < 1.02                                       1.02 <= X/S < 1.1

                 1                                                               1                                                        1




                                                                theta / gamma




                                                                                                                         theta / gamma
theta / gamma




                0.8                                                             0.8                                                      0.8
                0.6                                                             0.6                                                      0.6
                0.4                                                             0.4                                                      0.4
                0.2                                                             0.2                                                      0.2
                 0                                                               0                                                        0
                      0   1          2           3          4                         0   1           2          3   4                         0   1          2           3   4
                              Time to Maturity                                                Time to Maturity                                         Time to Maturity


Figure 1: Estimations of the location parameter θ (solid line) and γ (dashed line) for
different maturity and moneyness levels. The results on the left graph were generated
using options with 0.9 <= X/S < 0.98, the middle graph using options with 0.98 <=
X/S < 1.02, the right graph using options with 1.02 <= X/S < 1.1.




                                                         1.5
                                                           1
                                                         0.5
                                                 theta




                                                           0
                                                         -0.5
                                                          -1
                                                                                                   Time



Figure 2: Location parameter θ for call options with 1.02 <= X/St < 1.1 and time to
maturity between 2 weeks and one month for a half year period. Estimation frequency:
120 minutes.




                                                                                              31
                         0.015
                   psi




                         0.005



                                  8   10   12      14    16   18
                         -0.005
                                           Time of Day




Figure 3: Median observation error for call options as a function of time of day.




                                           32
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                                            34

				
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