Asymmetric option price distribution and bid–ask quotes: consequences for implied volatility smiles

Asymmetric option price distribution and bid�/

ask quotes: consequences for implied volatilitysmiles�

Lars Norden *

Department of Corporate Finance, School of Business, Stockholm University, S-106 91 Stockholm, Sweden

Received 15 August 2002; accepted 7 April 2003

Abstract

This study presents a model for estimating the asymmetry of option values with respect to

option bid�/ask spreads. The model does not require knowledge of the actual option value to

evaluate the asymmetry. Using data from the Swedish equity options market, several

interesting results emerge. First, there is evidence of asymmetry in call and put values, where

values are closer to bid than to ask quotes. Second, in- and out-of-the-money calls and puts

show a higher degree of asymmetry than at-the-money options. Third, taking asymmetry into

account in the estimation of option-implied volatility, produces a less pronounced volatility

smile.

# 2003 Elsevier B.V. All rights reserved.

JEL classification: G10; G13; G14

Keywords: Option bid�/ask spread; Asymmetry; Volatility smile

�This study has benefited from comments and suggestions from Malin Engstrom, Jonathan Park-

Samson, Peter Saucrbier and an anonymous referee. I have also received valuable input from seminar

participants at Stockholm University, Uppsala University, the 2002 Annual Meeting of the German

Finance Association and the 2002 Annual Australasian Finance and Banking Conference. I am indebted

to Anders Stromberg at RPM Risk and Portfolio Management AB for his assistance.

* Tel.: �/46-8-674-7139; fax: �/46-8-674-7440.

E-mail address: [email protected] (L. Norden).

J. of Multi. Fin. Manag. 13 (2003) 423�/441

www.elsevier.com/locate/econbase

1042-444X/03/$ - see front matter # 2003 Elsevier B.V. All rights reserved.

doi:10.1016/S1042-444X(03)00019-7

mailto:[email protected]

1. Introduction

Most of the market microstructure models developed to explain the presence of

bid�/ask spreads are designed primarily for stock markets, but should of course be

relevant for options markets as well. Inventory models (see Stoll, 1978; Amihud and

Mendelson, 1980, 1982; Ho and Stoll, 1981) motivate the spread as compensation for

market makers for bearing the risk of holding undesired inventory. In asymmetric

information models (see Copeland and Galai, 1983; Glosten and Milgrom, 1985;

Easley and O’Hara, 1987; Foster and Viswanathan, 1994), market makers have an

informational disadvantage and have to quote spreads wide enough to compensate

for losses from trading with informed traders. For options markets, Cho and Engle

(1999) proposed a derivative hedge theory, according to which option bid�/ask

spreads are related to the liquidity, measured as spreads, of the underlying market. If

options markets makers are able to perfectly hedge their option positions in the

underlying market, they will not be exposed to inventory risk or informed trading in

the options markets itself. Instead, option bid�/ask spreads will arise from the

illiquidity of the underlying market, and the width of option spreads will reflect the

presence of informed trading in the underlying market.

At financial markets, asset bid�/ask spreads are, to various degrees, readily

observable*/whereas, the corresponding actual asset values are not. In addition, the

bid�/ask spread is not necessarily symmetrically positioned around the value of the

asset. In fact, several reasons for asymmetry can be found in the market

microstructure literature. For instance, in the inventory models, a market maker

changes the bid�/ask spread relative the asset value in order to attract orders that

would even out the inventory position of the market maker. Hence, if the market

maker faces an excess of buy (sell) orders, the ask (bid) commission will be in excess

of the bid (ask) commission, i.e. the asset value will be relatively closer to the bid

(ask) quote. Bossaerts and Hillion (1991) show that in foreign exchange markets,

possible government intervention gives rise to skewness in the future spot exchange

rate distribution. Hence, bid�/ask quotes at the currency forward market are not

symmetric around the forward prices (values). Furthermore, Bessembinder (1994)

reports that in foreign exchange markets, location of bid�/ask quotes relative asset

value is sensitive to dealer inventory-control variables. Another reason for

asymmetry, given by Anshuman and Kalay (1998) with reference to the stock

market, is discreteness of quotes.

Chan and Chung (1999) argue that there might be asymmetry in equity option

quotes even though the underlying stock quotes are symmetric around stock values.

Since option payoffs are asymmetric by nature, the adverse selection costs for an

options markets maker are likely to be larger for buy orders than for sell orders.

Hence, since market makers are relatively more vulnerable to incoming buy orders

than to sell orders, they would have a tendency to position the bid�/ask spread so

that the option value is closer to the bid than to the ask quote. Chan and Chung

(1999) find empirical evidence in favour of this idea; bid and ask quotes of CBOE

equity options have a tendency to be asymmetric around the option values, and

L. Norden / J. of Multi. Fin. Manag. 13 (2003) 423�/441424

indeed, values are closer to bid quotes. Also, the degree of asymmetry decreases as

the options become more in-the-money.The asymmetry of option values with respect to option bid�/ask spreads has

several important implications. First, as Chan and Chung (1999) argue, using the

midpoint of the bid�/ask spread as a proxy for the unobserved option value to test

the performance of option valuation models might produce misleading conclusions if

the option value is closer to the bid than the ask quote. Similar problems might

prevail if transaction prices are used to represent option values because of bid�/ask

bounce effects. Second, the Lee and Ready (1991) method to identify the initiator of

an option trade, by comparing the traded price with the most recent bid�/ask quotes,

might also lead to biased results if the asymmetry is not taken into account. For

instance, Easley et al. (1998) compare the traded option price at the CBOE with the

bid�/ask midpoint, and suppose that the trade is buyer (seller) initiated if the trade

price is higher (lower) than the midpoint.

This paper contributes to previous research in several aspects. First, the model by

Chan and Chung (1999) is extended in an important manner. In order to estimate the

degree of asymmetry of option value in relation to the option bid�/ask spread, Chan

and Chung (1999) model the bid and ask quotes for calls and puts separately. When

using this approach, the authors need to estimate the call and put option values in

terms of the underlying stock in a classical delta-hedging framework. By using the

put�/call parity relationship, this study develops a model where the asymmetry of

calls and puts can be jointly estimated. The major strength of this approach is that

there is no need for actually estimating the option values. In addition, the joint

estimation procedure enables a test of the hypothesis that calls and puts exhibit a

similar asymmetry pattern.

The second main contribution of this study recognises the possible linkage

between the asymmetry at the options markets and the so-called volatility smile. The

volatility smile refers to the graphical image of option-implied volatility as a function

of moneyness. Several previous studies have documented a higher level of implied

volatility for in- and out-of-the-money options relative to at-the-money options.1 If

different degrees of asymmetry are observed for options with different levels of

moneyness, in particular if at-the-money option values are relatively more symmetric

than in- or out-of-the-money options, this might explain the smile. Consider the

possibility that option values are closer to bid quotes than to ask quotes, and that

this asymmetry is relatively more pronounced for in- and out-of-the-money options.

Representing actual option values with bid�/ask midpoints leads to an over-

estimation of option values and of implied volatilities in general, and for in- and

out-of-the-money options in particular. In this scenario, the smile pattern is induced

by the asymmetry per se. This study is the first to account for asymmetry effects in

option values when calculating implied volatility and evaluating the smile.

1 Pena et al. (1999) provide an excellent review of the most important studies regarding the volatility

smile.

L. Norden / J. of Multi. Fin. Manag. 13 (2003) 423�/441 425

The empirical analysis of option value asymmetry and implied volatility is

conducted using data from the Swedish market for equity options.2 The results show

clear evidence of asymmetry in call and put option values, where the option values

appear to be closer to bid than to ask quotes. Furthermore, significant differences

are found with respect to option moneyness; in an out-of-the-money calls and puts

show a higher degree of asymmetry than options which are close to at-the-money.

These results imply that representing an option value with the bid�/ask midpointresults in a bias, overestimating the actual value. The estimated asymmetry

parameters for call and put values are used in- and out-of sample analysis of

option-implied volatility. Comparisons are made with the benchmark case under the

assumption of no asymmetry in option values, i.e. when the bid�/ask midpoints can

be used in the estimation of implied volatility. When the implied volatilities are

adjusted for asymmetry effects, there is considerably less evidence of a smile, relative

the benchmark case.

The rest of the study is organised as follows. Section 2 presents the methodologyof this study, first, a unified framework for investigating and testing for asymmetry

in call and put values, and second, whether the asymmetry can have consequences

for an estimation of the volatility smile. Section 3 presents some institutional features

of the Swedish market for equity options and the data. Section 4 contains the

empirical results of the study. Finally, the study ends in Section 5 with some

concluding remarks.

2. Methodology

2.1. Estimation of asymmetry

This section provides a framework for testing the degree of asymmetry of equity

option quotes relative the option values. The major problem in this analysis is thatoption values are not directly observable. Bossaerts and Hillion (1991) as well as

Bessembinder (1994) develop a model for estimating the location of asset bid�/ask

quotes relative the asset values, with reference to foreign exchange markets. Chan

and Chung (1999) extend this model for investigating possible asymmetries in option

spreads. They estimate their model individually for calls and puts, where the

estimation relies on a theoretical approximation of the option value. In this study,

the model is further developed, using the put�/call parity relationship, so that the

reliance of option value approximation can be avoided. Throughout the analysis, theunderlying stock bid�/ask spread is assumed to be symmetrically located around the

stock value. In other words, the stock value can be inferred as the average of the bid

and ask quotes, and is assumed to be observable.3

2 One reason for using Swedish equity options data, rather than e.g. data from the US markets, is that

the implied volatilities of these options exhibit a clear smile pattern, as documented by Engstrom (2002).3 The relevance of this assumption is investigated later in the empirical section (see Section 4.1).


Using the notation in Chan and Chung (1999), the American call value at time t

(Ct ) is assumed to be located between the prevailing bid (Bc,t ) and ask quote (Ac,t )

according to:

Ct�uBc;t�(1�u)Ac;t (1)

where u (05/u5/l) is a constant parameter measuring how close the call value is to

the bid quote, relative the ask. The call value is closer to the bid (ask) quote if u�/ 1/2

(uB/ 1/2). For a corresponding American put, the value Pt is similarly assumed to be

located within the spread:

Pt�gBp;t�(1�g)Ap;t (2)

where Bp ,t is the bid quote and Ap ,t is the ask quote of the put at time t , and g (05/

g5/1) is a constant parameter measuring the location of the put value relative the

bid�/ask quotes. Eqs. (1) and (2) can be expressed in terms of changes in the call and

put value respectively as:

DCt�uDBc;t�(1�u)DAc;t (3)

DPt�gDBp;t�(1�g)DAp;t (4)

By denoting DSPDc ,t �/DAc ,t�/DBc ,t , as the change in the bid�/ask spread for the

call, and DSPDp ,t �/DAp ,t�/DBp ,t as the corresponding change in the put spread,

Eqs. (3) and (4) can be rearranged into:

DAc;t�uDSPDc;t�DCt (5)

DAp;t�gDSPDp;t�DPt (6)

In Eqs. (5) and (6), changes in ask quotes (DAc ,t , DAp ,t ), as well as changes in

quoted bid�/ask spreads (DSPDc ,t , DSPDp ,t), for the call and put, respectively, are

readily observable, whereas the changes in call and put option values (DCt , DPt ) are

not. Previous research, e.g. Chan and Chung (1999), argue that the expected change

in option value can be inferred from the corresponding (observable) change in the

underlying stock price (DSt). Furthermore, the idea is to replace each unobservablechange in option value in Eqs. (5) and (6) with a conditional expectation thereof,

plus an error term. In fact, using the framework of Black and Scholes (1973) and

Merton (1973), the expected change in option value can be expressed as a Taylor

expansion, and thus conditional upon the change in the underlying stock price (both

in a linear and a squared fashion), volatility, risk-free rate of interest and time left to

expiration. After properly defining the conditional expected changes in call and put

values, Eqs. (5) and (6) can be formulated as individual regression equations, which

can provide estimates of the location parameters u and g.

A more efficient solution to the problem with unobserved changes in option values

recognises that changes in call and put values are related to each other. This

relationship is of course put�/call parity, which applies for European calls and puts

with the same underlying stock, the same exercise price and the same exercise date.

To avoid arbitrage, the difference between the value of a European call (ct) and the

corresponding value of a European put (pt) must at time t be:


ct�pt�St�PVt(X ) (7)

where St is the value of the stock and PVt(X ) is the present value of the common

exercise price X . For American options, in the case when the stock provides no

dividend payments, the relationship in Eq. (7) becomes:

Ct�Pt5St�PVt(X ) (8)

where Pt ]/pt since early exercise of the American put might be optimal at any dateduring the time left to expiration, and Ct �/ct because early exercise of the American

call is never optimal in the absence of dividend payments. Following the work of

MacMillan (1986), as well as Barone-Adesi and Whaley (1987), the value of the

American put can be decomposed into the value of a corresponding European put

plus an early exercise premium according to:

Pt�pt�et (9)

Combining Eq. (9) with Eq. (7), and noting that Ct �/ct , results in:

Ct�Pt�St�PVt(X )�et (10)

This relationship must hold for changes in values as well:

DCt�DPt�DSt�DPVt(X )�Det (11)

The derived exact relationship between changes in call and put values can be

utilised by subtracting Eq. (6) from Eq. (5), and combining with Eq. (11):

DAc;t�DAp;t�uDSPDc;t�gDSPDp;t�DSt�DPVt(X )�Det (12)

In Eq. (12), the difference between the change in the call ask quote and the change

in the put ask quote can be expressed in terms of changes in five variables, whereof

only the last one*/the change in the early exercise premium of the put*/isunobservable. Note the improvement of working with Eq. (12) rather than with

Eqs. (5) and (6) individually. Instead of having to approximate the change in each

option value, the approximation now involves the change in difference between the

American put and a corresponding European put. According to the literature on

control variate techniques in option valuation, see e.g. Hull and White (1988), the

latter approximation can be made with considerably less measurement error.

Furthermore, note that if the options are European rather than American, the

term Det equals zero, and the need for approximation vanishes completely.For simplicity, the last two terms in Eq. (12) are assumed to be unrelated to stock

price changes and are together characterised as a constant term plus a residual term:

�DPVt(X )�Det�b0�ot (13)


The assumption made in Eq. (13) enables Eq. (12) to be written as a linear

regression: 4

DAc;t�DAp;t�b0�b1DSPDc;t�b2DSPDp;t�b3DSt�ot (14)

The regression coefficient b1 is an estimator for the call location parameter u

whereas �/b2 is an estimator for the put location parameter g. An individual test for

symmetry of the call (put) value with respect to the bid�/ask spread boils down to test

the null hypothesis that b1 (b2) is equal to 1/2 (�/1/2). The regression framework

developed in this study holds an advantage, compared to previous studies, as it

encompasses estimators for both call and put location parameters. Therefore, it is

possible to test the null hypothesis of equality between b1 and �/b2, i.e. whether calls

and puts exhibit the same degree of asymmetry. Finally, the coefficient b3 is expectedto be equal to one if the put�/call parity relationship in Eqs. (11) and (12) holds.

Another suitable extension of the model is to allow for different parameter

estimates for different levels of option moneyness. At time t , moneyness is defined as

the option exercise price divided by the underlying stock value, X /St . The options are

divided into five groups with respect to moneyness, and dummy variables are formed

in order to identify the group to which a specific option belongs. Using the dummy

variables, the regression model in Eq. (14) is extended to:

DAc;t�DAp;t�X5

i�1

Dib0;i�X5

i�1

Dib1;iDSDPc;t�X5

i�1

Dib2;iDSDPp;t

�X5

i�1

Dib3;iDSt�ht (15)

where D1 is equal to one if X /St B/0.90 and zero otherwise, D2 is equal to one if

0.905/X /St B/0.98 and zero otherwise, D3 is equal to one if 0.985/X /St B/1.02 and

zero otherwise, D4 is equal to one if 1.025/X /St B/1.10 and zero otherwise, and D5 is

equal to one if X /St ]/1.10 and zero otherwise. If the dummy specification has

additional explanatory power relative the regression without dummies in Eq. (14),

the residual term ht has lower variance than ot . Using the regression framework in

Eq. (15) it is possible to test the null hypothesis that the call (put) location parameter

u (g ) is the same for different moneyness levels, simply by testing for equality amongthe b1,i (b2,i ) coefficients.

4 An alternative model specification is to rewrite Eq. (13) as �/DPVt (X )�/b ?0�/o ?t , and to alter the last

two terms on the right hand side of Eq. (14) to b ?3(DSt�/Det )�/o ?t , where the binomial model is used to

estimate the change in the early exercise premium Det . In this specification there is no assumption of

independence between DSt , and Det . On the other hand, a theoretical option valuation model has to be

used to estimate Det . Since the empirical results from the two specifications are similar*/the estimated

location parameters are identical, with the same significance levels*/the most simple model is employed in

the subsequent analysis, and the results from the alternative specification arc not reported (but are

available upon request).


2.2. Estimation of implied volatility smile

The model from Section 2.1 is estimated to investigate if Swedish call and put

equity option values are asymmetrically positioned within corresponding bid�/ask

spreads, and whether the possible asymmetry differs between calls and puts, as well

as for different levels of moneyness. The estimated regression parameters, which

describe the nature of asymmetry, are used for calculating implied volatilities. Foreach option observation, an implied volatility is calculated both with and without the

adjustment for asymmetry. The symmetric implied volatility is obtained by using the

midpoint of the option bid�/ask spread, whereas the asymmetric implied volatility

uses estimated weights uand g in order to locate each option value within the bid�/

ask spread. The analysis is out-of sample in the sense that the implied volatilities are

estimated with another data set than the one used for estimating the asymmetry

coefficients.

The volatility smile is estimated using the following regression model:

s

sATM

�a0�a1

�ln(X=S)

sATM

ffiffiffiffiffiffiffiffiffiffiffiffiffiT � t

p��a2

�ln(X=S)

sATM

ffiffiffiffiffiffiffiffiffiffiffiffiffiT � t

p�2

�j (16)

where a0, a1 and a2 are regression coefficients, and j is a residual term. Since the

analysis is performed using options on different stocks, the volatility smile is

estimated in terms of relative rather than absolute implied volatility. A volatility

ratio is formed by dividing the implied volatility (s) with the corresponding implied

volatility (sATM), which is estimated from the at-the-money option, with the same

underlying stock and expiration. The at-the-money option is defined as the option,within its class, with the moneyness measure (X /S ) closest to one. The specification

in Eq. (16) is a regression of the volatility ratio on a standardised moneyness

measure, using a linear and a quadratic term, where the log of moneyness is divided

by the at-the-money implied volatility times the square root of time to expiration

(T�/t).5

In the analysis of implied volatility smiles, calls and puts are investigated

separately. For each kind of options, calls and puts, the regression in Eq. (16) is

estimated both with and without adjustments for asymmetry of option value withrespect to the bid and ask quotes. Hence, it is possible to isolate the effects of

asymmetry on the implied volatility smile. In fact, if options exhibit different degrees

of asymmetry for different levels of moneyness, this will carry over to the estimation

of implied volatility. As a result, the regression parameters a1 and a2 are expected to

be different whether adjustments for asymmetry are made or not. Each constant

5 The definition of moneyness in volatility units is suggested by Tompkins (1994), whereas the

quadratic specification in Eq. (16) is used by Rosenberg (2000). In the literature, several implied volatility

functions for capturing the volatility smile are suggested, see e.g. Pena et al. (1999). The choice of using Eq.

(16) in this study follows from the results in Engstrom (2002), where this specification is found to have the

best explanatory power, relative several alternatives, when using equity option data from the Swedish

market.


parameter a0 is expected to be equal to one, since this represents the value of the

volatility ratio for an at-the-money option when in turn X /S is equal to one.

3. Institutional setting and data

In 1995, options on 27 individual stocks were listed on the Swedish options

markets. The underlying stocks were in turn listed on the Stockholm stock exchange

(StSE). The average daily trading volume for these options was around 51 000

contracts.6 Approximately 14 500 were puts whereas 36 500 were calls.7 Comparisons

with CBOE show that the average daily trading volume was around 305 000 equity

option contracts at that time, of which around 90 000 were put options.8 The trading

at the StSE and the option trading at options markets normally takes place between10:00 and 16:00 CET, Series of call and put option contracts with an initial time to

expiration of 6 months are listed every 3 months.9 The option expiration day is the

third Friday of the expiration month, but an equity option holder may exercise the

contract until 17:00 on any trading day, and until 18:00 on the expiration day.10

After receipt of exercise notices from the option holders, options markets assigns the

exercises randomly among the options writers. Options markets thus functions both

as the exchange and the clearinghouse.11

All listed derivative securities at options markets are traded within a fullycomputerised system. The trading system, which is called the ‘click trading’ system,

consists of a limit order book managed by options markets. Anytime during the

trading hours a trader can submit new market or limit orders to either buy or sell a

certain volume of a contract. If possible, the new orders are automatically matched

against the ones already present in the order book. If no matched orders can be

found to a limit order it is added to the order book. The limit order book is

complemented with an ‘upstairs market’. If the trader wishes to trade outside the

order book he/she can phone his/her order directly to options markets. An orderphoned to options markets is not added to the order book. Instead, options markets

6 One put (call) option contract gives the holder the right to sell (buy) 100 shares of the underlying

stock. A trading block in turn comprises of 10 contracts.7 During the period in question, the average daily trading volume on the StSE for the underlying;

stocks was SEK 2.6 billion for the on average 10 470 transactions per day. The market capitalisation for

the 228 listed Swedish stocks at the end of year 1995 was SEK 1179 billion (which at that time was equal to

$177 billion).8 Source: OM Factbook and Futures and Options World, February 1996, respectively.9 Long option, i.e. contracts with an initial time to expiration of 2 years, are also listed every year for

some of the underlying stocks. However, these are not included in the analysis since they are seldom

traded.10 During the analysed period the expiration months were distributed as follows: Group 1: January,

April, July and October; Group 2: February, May, August and November, and Group 3: March, June,

September and December.11 This section gives a description of the institutional setting that prevailed during the analysed period.

Since then, the trading hours and the last possible exercise times have been changed.


must try to locate a counterpart and execute the order manually. A trade can also be

conducted completely outside the exchange. Such a trade must be reported to

options markets no later than 9:45 the following trading day. All trading at options

markets is conducted via members of the exchange.12 A member is either an ordinary

dealer or a market maker. The trading environment thus constitutes a combination

of an electronic matching system and a market making system.13 The trading system

at the StSE is based on the same kind of limit order book as at options markets.

However, there are no market makers in the Swedish stock market.

The data set consists of call and put prices from the Swedish equity options

markets during the sample period July 1, 1995 through February 1, 1996. Only the 10

underlying stocks with the most frequently traded options on the market at that time

are used in the analysis.14 Daily, simultaneously recorded, closing bid�/ask quotes of

the options are obtained from options markets, whereas daily closing bid�/ask quotes

of the underlying stocks are obtained from the StSE. All option and stock quotes are

the best available in the limit order book at the closing time of options markets and

the StSE, respectively. Dividends are paid only once a year and most dividend

payments occur around May. Hence, no dividend payments are made during the

sample period. Daily rates of Swedish 1-month Treasury bills are used as proxies for

risk-free interest rates in calculating implied volatilities.

The sample period is divided into two sub-periods, where data from the first

period (July 1 to December 31, 1995) are used to estimate the degree of asymmetry,

whereas data from the second period (January 1 to February 1, 1996) are utilised for

evaluating the influence of the asymmetry on the implied volatility smile. Since both

calls and puts are analysed, only days for which non-zero quotes for calls and puts,

in an option pair, exist are included. To avoid possibly disturbing effects of trading

immediately before expiration, options with less than a week left to expiration are

omitted. After a screening procedure, observations that do not satisfy the American

put�/call parity boundary condition are excluded. In addition, observations are

omitted if either the call or the corresponding put has an option value according to

the binomial model less man SEK 0.01, the lowest tick size allowed at options

markets. The first part of the sample consists of 15 400 put�/call pairs, whereas the

second sample consists of 2256 put and 2233 call observations. In the last part of the

sample, observations are also excluded if it is not possible to obtain convergence in

the estimation of implied volatility. Each implied volatility is found using a binomial

tree with 100 time steps, following Cox et al. (1979), allowing for early exercise at

12 The options markets is the sole owner of the London Securities and Derivative Exchange (OMLX).

The two exchanges are linked to each other in real time. This means that a trader at the OMLX has access

to the same limit order book as a trader at the options markets. In 1995, 35 members were registered at the

options markets and 50 at the OMLX.13 Compare e.g. the trading system at the CBOE, which is a continuous open-outcry auction among

competitive traders; floor brokers and market makers.14 The stocks are Electrolux B, Ericsson B, MoDo B, SEB A, Trelleborg B, Astra A, SCA B, Skandia,

Volvo B and Investor B.


each node in the tree. The calculations use the midpoint of the closing bid and ask

quotes for each underlying stock.

4. Empirical results

4.1. Asymmetry results

Table 1 reports simple S.D.s of daily changes in bid and ask quotes, respectively.

The results are presented separately for calls and puts, and for different levels ofmoneyness. Overall, the S.D. of option bid quote changes is lower than the

corresponding S.D. of ask quote changes. For all observations, the S.D. ratio for ask

and bid quote changes equals 1.08 for calls and 1.12 for puts. These results indicate

that option bid quotes change less than the corresponding ask quotes. Furthermore,

two distinct patterns can be discerned with respect to moneyness. First, the S.D. of

changes in bid as well as ask quotes is decreasing in moneyness for calls and

increasing for puts. In other words, the more out-of-the-money an option is the less

variable its quotes. Second, the S.D. ratio is not the same for different moneynesslevels. In particular, for options, which are deep out-of-the-money, there is a large

diversity between daily bid and ask quote changes.

Table 1 also reports S.D.s for changes in stock bid and ask quotes. Evidently, the

S.D. ratio between bid and ask quote changes is much closer to one for the

underlying stocks than for the corresponding calls and puts. This notion supports the

assumption made in the theoretical analysis that the stock bid�/ask spread is

symmetric around its value, whereas the option spreads are not.

One reason, recognised by Chan and Chung (1999), for option bid quote changesto be smaller, and less variable, than the ask quote changes is that call (put) bid

quotes are not as frequently revised downwards when the underlying stock market

midpoint quote decreases (increases). In particular, this scenario is likely when the

option is deep out-of-the-money, and the option price is so low that it makes no

sense to lower the bid quote any further. If this is the case, one would expect to see

smaller call (put) bid quote changes when the underlying stock market is going down

(up), than when it is going up (down). To investigate if the results in Table 1 are

driven by this potential effect, the S.D.s of option quote changes are stratified withrespect to up and down market conditions, i.e. when the underlying stock midpoint

quote change is positive or negative, in Table 2.15 Interestingly, the stratified results

in Table 2 point in the opposite direction than indicated. For calls, and using all

observations, the S.D. ratio is slightly higher during an up-market than during a

down-market condition. Furthermore, the ratio for deep out-of-the-money calls

equals 1.61 (1.30) when the underlying market increases (decreases), indicating a

presence of asymmetry in bid and ask quote changes during both kinds of market

conditions. For puts, the results are similar; out-of-the-money puts are showing a

15 The cases when the underlying stock midpoint quote is unchanged are not shown in Table 2.


Table 1

S.D. of bid and ask quote changes for options and stocks

Calls Puts Stocks

Number of obser-

vations

Bid quote

changes

Ask quote

changes

Ratio Bid quote

changes

Ask quote

changes

Ratio Bid quote

changes

Ask quote

changes

Ratio

X /S B/0.90 3360 3.3475 3.6684 1.0959 0.3888 0.5147 1.3239 3.4600 3.4503 0.9972

0.905/X /

S B/0.98

3432 2.7549 2.8305 1.0275 0.9942 1.1247 1.1312 3.5719 3.5669 0.9986

0.985/X /

S B/1.02

1923 2.0430 2.2083 1.0809 1.7026 1.9861 1.1665 3.4571 3.4140 0.9875

1.025/X /

SB/ 1.10

3307 1.5360 1.7489 1.1386 2.3740 2.6511 1.1167 3.5067 3.4928 0.9960

X /S ]/1.10 3378 0.7675 1.0319 1.3446 3.2871 3.6260 1.1031 3.5385 3.5477 1.0026

Total 15 400 2.3087 2.5029 1.0841 2.0710 2.3104 1.1156 3.5366 3.5274 0.9974

This table contains estimated S.D.s for changes in daily bid and ask quote changes during the period July 1 to December 31, 1995. Results are presented for

calls, puts and the corresponding underlying stocks, as well as for different levels of moneyness X /S . ‘Ratio’ is defined as the ratio between S.D. of ask quote

changes and S.D. of bid quote changes.

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Table 2

S.D. of option bid and ask quote changes, up and down market

Calls Puts

Number of

observations

Bid quote changes Ask quote changes Ratio Bid quote changes Ask quote changes Ratio

Panel A: Up market condition (DS�/0)

X /SB/ 0.90 1623 2.8341 3.1930 1.1266 0.4327 0.5288 1.2221

0.905/X /SB/ 0.98 1558 2.1811 2.3396 1.0727 0.9624 1.0406 1.0813

0.985/X /S B/1.02 834 1.5919 1.8566 1.1663 1.6348 1.8276 1.1179

l.025/X /S B/1.10 1359 1.1115 1.4055 1.2646 1.8751 2.1217 1.1315

X /S ]/1.10 1194 0.6241 1.0056 1.6112 2.2347 2.5566 1.1440

Total 6568 2.1233 2.3804 1.1211 1.6650 1.8560 1.1147

Panel B: Down market condition (DSB/0)

X /SB/ 0.90 1518 2.1289 2.4911 1.1701 0.2982 0.4526 1.5177

0.905/X /SB/ 0.98 1611 2.0109 1.9992 0.9942 0.7030 0.9015 1.2824

0.985/X /S B/1.02 935 1.6383 1.7436 1.0643 1.1768 1.5269 1.2975

l.025/X /S B/1.10 1698 1.4227 1.5882 1.1163 1.7542 2.0553 1.1716

X /S ]/1.10 1932 0.7783 1.0131 1.3018 2.7466 3.0845 1.1230

Total 7694 1.7582 1.9216 1.0929 1.9181 2.1741 1.1335

This table contains estimated S.D.s for changes in daily bid and ask quote changes during the period July 1 to December 31, 1995. Results are presented for

calls and puts as well as for different levels of moneyness, X /S . ‘Ratio’ is defined as the ratio between S.D. of bid quote changes and S.D. of ask quote changes.

Results are stratified with respect to up market condition, where the underlying stock price changes are positive (DS �/0), and down market condition, where

the underlying stock price changes are negative (DS B/0).

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relatively larger diversity between bid and ask quote changes when the underlying

stock market is going down. This contradicts the idea from Chan and Chung (1999),

and suggests that asymmetry is present during all underlying market conditions.

Table 3 contains the results from the estimation of the regression model according

to Eq. (14). Using the model without dummies, the difference between call and put

ask quote changes is regressed on call and put spread changes as well as stock price

changes. The results give an estimate of the call location parameter u�/b1 equal to

0.80, and a corresponding estimate of the put parameter g�/�/b2 equal to 0.76. The

individual null hypotheses that each parameter is equal to 1/2 can be strongly

rejected, with P -values B/0.0001. There is also some evidence of difference between

the parameters for calls and puts: the hypothesis of equality between u and g can be

rejected at the 10%-level of significance (P -value�/0.0901). These results support the

idea of asymmetry; option values are closer to bid quotes than to ask quotes, for

both calls and puts.

Table 3

Regression results, estimated location parameters

Regression coefficients

Constant DSPDc DSPDp DS R2

Without dummies �/0.0611 0.8009 �/0.7562 0.9913 0.9317

(0.0001) (0.0001) (0.0001) (0.0249)

With dummies X /SB/ 0.90 �/0.0468 0.8751 �/0.7304 0.9850

(0.0082) (0.0001) (0.0001) (0.0702)

0.905/X /SB/ 0.98 �/0.0728 0.6557 �/0.5411 0.9692

(0.0001) (0.0001) (0.2927) (0.0001)

0.985/X /S B/1.02 �/0.0393 0.7603 �/0.6587 0.9672

(0.1023) (0.0001) (0.0055) (0.0111)

l.025/X /S B/1.10 �/0.0397 0.7526 �/0.7117 1.0040

(0.0320) (0.0001) (0.0001) (0.6776)

X /S ]/1.10 �/0.0534 0.8405 �/0.8793 1.0223 0.9332

(0.0010) (0.0001) (0.0001) (0.0033)

LR test 2.345 83.23 151.4 79.00

(0.7994) (0.0001) (0.0001) (0.0001)

This table contains results from the estimation of the regression model in Eq. (14), regression without

dummies, and Eq. (15), regression with dummies. Both regressions are run using data from the period July

1 to December 31, 1995. Results are presented for calls and puts as well as for different levels of moneyness

X /S. The dependent variable is the difference between call ask quote changes and put ask quote changes.

The explanatory variables are the change in call spread DSPDc , the change in put spread DSPDp and the

change in underlying stock midpoint quote DS . Figures in parentheses below coefficients are P -values. A

P -values below a Constant coefficient test the null hypothesis that the coefficient is zero, a P -value below

a DSPDc (DSPDp ) coefficient test the null hypothesis that the coefficient is equal to 0,5 (�/0.5), and a P -

value below a DS coefficient test the null hypothesis that the coefficient is equal to 1. The last row displays

LR tests, each x2-distributed with five degrees of freedom under the null hypothesis of equality across

moneyness for the coefficients belonging to each explanatory variable. Correction for heteroskedasticity in

the residuals is made according to White (1980).


Table 3 also presents the results from the estimation of the extended regression

model in Eq. (15), where different location parameters are allowed for different levels

of moneyness (regression with dummies). Starting with the calls, the location

parameter u is significantly different from 1/2 for each of the five levels of

moneyness. Furthermore, a likelihood ratio (LR) test strongly rejects the hypothesis

of equality between the call location parameters for different moneyness levels. There

appears to be a relatively higher degree of asymmetry for deep in- and deep out-of-the-money calls than for the calls in between. The puts display a similar pattern. The

deep in- and deep out-of-the-money options show a more severe asymmetry, with

put values closer to bid quotes, than the options with moneyness around at-the-

money. In fact, puts that are slightly out-of-the-money, and belong to the group

where 0.905/X /St B/0.98, are the only options for which the individual null

hypothesis of symmetry cannot be rejected.

4.2. Volatility smile results

The asymmetry results from Table 3 are consistent with the presence of a volatility

smile for both calls and puts. If indeed the midpoints between option bid and ask

quotes are used to estimate implied volatilities, there will be a tendency for

exaggerating volatility for in- and out-of-the-money options relative at-the-money

options. Table 4 presents the results from running the regression in Eq. (16), which is

intended to produce an estimated volatility smile, one for calls and one for puts. The

Table 4

Regression results, volatility smile functions

Regression coefficients

Constant ln(X /S )/(sATM�T�/t ) ln(X /S )/(sATM�T�/t )2 R2

Calls Symmetric 0.9826 �/0.0934 0.1304 0.6188

(0.0001) (0.0001) (0.0001)

Asymmetric 0.9449 �/0.0448 0.0733 0.3993

(0.0001) (0.0001) (0.0001)

Puts Symmetric 0.9770 0.1342 0.1781 0.7904

(0.0001) (0.0001) (0.0001)

Asymmetric 0.9479 0.0464 0.1117 0.6158

(0.0001) (0.0001) (0.0001)

This table contains the results from the estimation of the regression model in Eq. (16). Separate

regressions are run for calls and puts using data from the period July 1 to December 31, 1995. The

dependent variable is the option-implied volatility divided by corresponding at-the-money implied

volatility. The implied volatility is estimated using midpoint option quotes (symmetric) as well as

asymmetry-adjusted quotes, using the estimated location parameters from Table 3. The explanatory

variables are linear and quadratic terms of a standardised moneyness measure, where X /S is option

exercise price divided by the underlying stock price, sATM is the-money implied volatility, and T�/t is the

time left to expiration of the option. Figures in parentheses below coefficients are P -values, used to test the

null hypothesis that each coefficient is zero. Correction for heteroskedasticity in the residuals is made

according to White (1980).


dependent variables, the implied volatilities, are estimated under the symmetric

assumption, using option midpoints, as well as under the asymmetric alternative,

where the estimated location parameters from Table 3 are used to infer option

values. The regression model performs better when the asymmetry is not taken into

consideration. The R2 for the call (put) volatility smile regression equals 0.62 (0.79)

when the option midpoints are used as proxies for option values, whereas the

corresponding figure is 0.40 (0.62) when adjustments are made for asymmetry. Theregression coefficients for the linear and quadratic standardised moneyness measure

take on relatively smaller numbers in the asymmetric regression equations for calls as

well as for puts. Still, the coefficients for the linear and quadratic moneyness measure

are significantly different from zero both for the symmetric and asymmetric

specifications.

Fig. 1a and b plots the ‘actual’ and predicted symmetric (asymmetric) implied

volatility ratio, using the regression in Eq. (16), as a function of the linear

explanatory variable*/the standardised moneyness measure. Evidently, the esti-mated volatility smile, as a function of moneyness, exhibit relatively less ‘happiness’

when the asymmetry of call option value with respect to the bid�/ask spread, is taken

into account. A similar pattern can be observed if the estimated put volatility smile

functions in Fig. 2a and b are compared. However, even if the option value

asymmetry is accounted for, there is still evidence of a volatility smile pattern.

Nevertheless, the asymmetry might explain some*/although not all*/of the

volatility smile.

5. Concluding remarks

A model is developed in order to estimate the asymmetry of option values with

respect to option bid�/ask spreads. The model provides an extension to the model by

Chan and Chung (1999), since it does not require knowledge of the actual option

value to evaluate the asymmetry. This extension is important as the asymmetry

pattern can be investigated without assuming a particular option valuation frame-work. Using data from the Swedish market for equity options, clear empirical

evidence of asymmetry in call and put option values is found; the option values tend

to be closer to bid than to ask quotes. Significant differences are found with respect

to option moneyness; in- and out-of-the-money calls and puts show a higher degree

of asymmetry than options which are close to at-the-money. These results imply that

representing an option value with the bid�/ask midpoint results in a bias,

overestimating the value. The bias also appears to be larger for in- and out-of-the-

money options relative at-the-money options.The potential relation between the asymmetry at the options markets and the so-

called volatility smile is recognised. The estimated asymmetry parameters for call

and put option values are used in an out-of sample analysis of option-implied

volatility. Comparisons are made with the benchmark case under the assumption of

no asymmetry in option values, i.e. when the option bid�/ask midpoints are used in


the estimation of implied volatility. When the implied volatilities are adjusted for

asymmetry, there is less evidence of a smile, relative the benchmark case.

This study is the first to highlight the linkage between asymmetry in option values

with respect to bid�/ask spreads and the implied volatility smile. The results

presented here have several implications for future research. One is that the search

Fig. 1. (a) Implied call volatility ratio using midpoint of bid�/ask spread. (b) Implied call volatility ratio

adjusted for asymmetry.


for an explanation of the volatility smile in essence might be a search for an

explanation of the asymmetry in options markets. Consequently, if the asymmetry

story is true, the chain of evidence ought to be that e.g. stochastic volatility or fat

tails in the underlying distribution leads to a higher degree of asymmetry in the

options markets, which in turn leads to an observed volatility smile.

Fig. 2. (a) Implied put volatility ratio using midpoint of bid�/ask spread. (b) Implied put volatility ratio

adjusted for asymmetry.


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Documents

Asymmetric option price distribution and bid–ask quotes: consequences for implied volatility smiles