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The relation between the informational content of implied volatility and arbitrage costs: Evidence from the Oslo Stock Exchange
THE RELATION INFORMATIONAL VOLATILITY EVIDENCE
BETWEEN
THE
CONTENT
AND ARBITRAGE
OF IMPLIED COSTS:
FROM THE OSLO STOCK
EXCHANGE
PARVEZ AHMED and STEVE SWIDLER
ABSTRACT This study investigates the proposition that volatility of stock returns can be predicted from the volatility implied by options on the Oslo Stock Exchange (OSE), conditional on the ability to perform arbitrage. Insights into the relation between the informational content of implied volatility and arbitrage cost can be distilled from Oslo Stock Exchange data. For Norwegian firms, options and their underlying stock trade on the Oslo Stock Exchange and have an overlapping set of market makers thereby lowering the cost of arbitrage. Other components of arbitrage trading costs, liquidity and dispersion of stock return volatility, vary widely across Norwegian firms. Moreover, restriction on the short selling of stock in Oslo allows further insight into the role of arbitrage costs in determining the informational content of implied volatility. The results yield support for the arbitrage cost hypothesis: the lower the arbitmge cost between the stock and the option, the greater the informational content of implied volatility.
The Capital College, Direct all correspondence to: Parvez Ahmed, School of Business Administration, Pennsylvania State University, Middletown, PA 17057-4898, Tel: (717) 948-6162; E-mail: pxal30psu.edu. Steve Swidler, University of Texas at Arlington. Copyright 0 1998 by JAI Press Inc. International Review of Economics and Finance, 7(4): 465-480 All rights of reproduction in any form reserved. ISSN: 1059-0560
465
PARVEZ AHMED and STEVE SWIDLER
466
I.
INTRODUCTION
This study investigates the proposition that volatility of stock returns can be predicted from the volatility implied by options on the Oslo Stock Exchange (OSE), conditional on the ability to perform arbitrage. An unbiased estimate of the future volatility of asset returns is of great interest to investors in general and option traders in particular. A model’ s option price as an estimate of the option’ s fair market value is only as good as the volatility measure being used. Thus, investors having access to superior forecasts of future volatility are likely to devise trading strategy that could generate profit by identifying m&priced options. Two approaches that have been frequently used to estimate future volatility (FV) are the realized volatility from past price data, i.e., the historical volatility (HV) and the implied volatility (IV) from option prices. Papers by Latane and Rendleman (1976), Trippi (1977), Chiras and Manaster (1978), and Beckers (198 1) show implied volatility of at-the-money options to be an unbiased predictor of future volatility of stock returns. Following the earlier works, Canina and Figlewski (1993) state that it had become almost an “article of faith” among academic researchers to treat implied volatility as the market’ s forecast of volatility. In contrast, their paper along with works by Jorion (1995) Day and Lewis (1992), and Lamoureux and Lastrapes (1993) show that implied volatility is not an unbiased forecast of future volatility. In analyzing the informational content of implied volatility, much of the empirical research assumes an underlying stock process with constant mean and variance. However, as Canina and Figlewski (1993) point out, “there is an obvious conflict in applying an approach that assumes the asset price process has a known constant volatility to a situation in which volatility must be forecast because it changes randomly over time.” They suggest that while fully rational investors should use option pricing models that explicitly account for stochastic volatility (see, e.g., Hull and White (1987), Wiggins (1987), Stein and Stein (1991) and Ball and Roma (1994)), practically speaking, they are difficult to implement.’ Thus, Canina and Figlewski argue, it is appropriate to examine the content of implied volatility from Black-Scholes or other constant variance models since they are more likely to be used by traders. In their work, Canina and Figlewski (1993) conjecture that implied volatility as a predictor of future volatility is related to the ease of performing arbitrage trades between the option and the underlying asset. At the lower end of the continuum are options that trade on the same exchange as the underlying, such as futures and option on futures traded on the Chicago Mercantile Exchange. At the other end are options such as the OEX and SPX index options where the underlying asset does not trade in the cash market. Arbitrage between an index and its option is quite costly as it requires continuous buying and selling of all the stocks that make up the index to maintain a delta neutral hedge. In between the two extremes are U.S. listed stock options which typically trade on a different exchange from the underlying asset. Thus, for U.S. stock options, arbitrage is possible but somewhat costly, since trading occurs in two different markets. Jorion (1995) tests the arbitrage cost hypothesis by using foreign exchange futures and option data from the Chicago Mercantile Exchange. Compared to Canina and Figlewski (1993), he finds evidence favoring increased informational content for implied volatility coming from at-the-money, short-term options. Nevertheless, in most instances, implied volatility is a biased forecast of foreign exchange rate volatility.
Relation Between informational
Content of Volatility and Arbitrage
467
The following analysis considers the conjecture of Canina and Figlewski (1993), and examines data from the Oslo Stock Exchange to gain further insights into the relation between arbitrage costs and the informational content of implied volatility from constant variance option pricing models. For Norwegian firms, options and their underlying stock trade on the Oslo Stock Exchange and have an overlapping set of market makers thereby lowering the cost of arbitrage. Two other components of arbitrage trading costs, liquidity and dispersion of stock return volatility, vary widely across Norwegian firms and allow additional inferences to be drawn concerning Canina and Figlewski’s (1993) arbitrage cost conjecture. Moreover, the Oslo Stock Exchange does not allow stocks to be sold short. This restriction increases the cost of arbitrage between the put option and the underlying stock. Thus, the OSE data allows for construction of several tests that examine the relations~p between arbitrage costs and the info~ation~ content of implied volatility.
Il.
MICROSTRUCTURE CONSIDERATIONS AND DATA FROM OSLO STOCK EXCHANGE
Previous studies on the forecastibility of future volatility have mostly been on Chicago Board Options Exchange (CBOE) data. The OSE has a different market microstructure compared to the U.S. securities markets. Gjerde and S&tern (1995) and Swidler, Schwartz, and Kristiansen (1994) list several differences between the Norwegian market and exchanges in U.S. First, the Norwegian markets are small compared to those in the U.S. Several markets have low liquidity, especially those for Bergesen and Hafslund Nycomed, and therefore have higher transaction costs in performing the arbitrage between the option and underlying stock. Second, Norway did not allow short selling of stock during the period of study, although Norwegian stocks listed on the London Stock Exchange could be sold short. This imposes higher transaction costs for short sales compared to the U.S. market. The short sale constraint implies that the least expensive way to express negative opinion on a stock is through buying put options or selling call options. In either case, the market maker will only fill the other side of the order at higher option premiums due to increased arbitrage costs. Thus, during times of negative market sentiment, the short sale prohibition in Norwegian stock market effectively translates into upward biased implied volatilities. Finally, for No~egi~ stocks, dividends are paid once a year (mostly around AprilMay), and the exchange lists only options with the next two quarterly expirations. This implies that many of the call options will not be exercised early and thus can be valued as European options. Moreover, for these observations, the volatility implied by the option corresponds to the expiration date. Data for the study consists of transactions prices for stock and call options traded on the Oslo Stock Exchange. This paper analyzes stock and options on Norsk Hydro (NHY), Saga Petroleum (SAG) and Bergesen (BEB) for the time period May, 1990-June, 1993. Data for Hafslund Nycomed A (HNA) covers the period December, 199 1-June, 1993 .2 The data for this study reflects the universe of options trading on the OSE for the period May 1990 to June 1993.
468
PARVEZ
AHMED
and STEVE
SWIDLER
The data set first matches a call with the underlying stock transaction price that immediately precedes the option trade. The analysis then calculates the implied volatility for each call option using the methodology described in the next section. Hence, for each date, there may be multiple implied volatilities spanning different times to expiration and different strike prices. Call options that violate the lower boundary condition, i.e., their market price is less than the current dividend adjusted stock price minus the strike price, have a negative implied volatility and are excluded from the study. The study also eliminates options with fewer than 7 days to expiration and those that are more than 20% in- or out-of-the money. Canina and Figlewski (1993), Rubinstein (1994) Derman and Kani (1994), and Dumas and Fleming (1995) find implied volatility is a U-shaped function with respect to moneyness. This is popularly referred to as the volatility “smile”. The in- and out-of-the money options have higher implied volatilities than the at-the-money options. Canina and Figlewski (1993) Shin&o (1993) and Clarke (1994) also find implied volatility to be a decreasing function of time to expiration. The revealed patterns of implied volatility suggest that investors systematically misprice certain options relative to others, although typiTable 1.
Implied Volatility
Sub-Sample
IV A.
BEBO BEBA BEBI HNAO HNAA HNAI NHYO NHYA NHYI SAG0 SAGA SAG1
on Moneyness
Average Maturio
and Maturity
Average Moneyness
Implied Volatility by Moneyness Group 0.4093 0.3890 0.3928 0.2947 0.2984 0.3321 0.2977 0.2895 0.3018 0.4361 0.4181 0.4334
B. BEBTI BEBT2 BEBT3 HNATl HNAT2 HNAT3 NHYTl NHYT2 NHYT3 SAGTl SAGT2 SAGT3
Across Subsamples
78.24 70.02 61.42 79.3 1 62.22 66.78 81.28 61.66 61.12 78.62 69.18 65.93
0.92 1.00 1.08 0.94 1.OO 1.Ol 0.93 1.oo 1.06 0.92 1.oo 1SF3
Implied Volatility by Time to Maturity Group 0.4094 0.3956 0.3916 0.3724 0.2895 0.2648 0.3 144 0.2902 0.2865 0.4430 0.4162 0.4328
26.29 65.91 118.82 26.28 64.97 120.6 1 29.92 64.96 116.46 27.19 66.33 117.01
1.00 0.98 0.97 1.00 0.98 0.98 1.OO 1.00 0.99 1.OO 0.99 0.99
This table provides estimates of average implied volatility of the various aggregated sample for all stock options. The first column refers to the sample used (e.g., BEBO implies it is the out-of-money sample for the option on BEB, BEBA implies it is the at-the-money sample for the option on BEB, BEBI implies it is the inthe-money sample for the option on BEB. BEBTl, BEBT2 and BEBT3 are for BEB subsamples with time to maturity 7-30 days, 31-90 days and greater than 90 days respectively. Column two shows the average implied volatility for the associated subsample. Column three gives the estimated mean of the days to maturity for the associated subsample. Column four gives the average moneyness for the sub-sample.
tally the ~sp~~iu~ is within arbitrage bounds. Thus, tests of average implied vola~lities are improper and give the false signal that implied volatilities from different options series are noisy observations of future volatility. In segmenting the sample by maturity, let IV< refer to the implied volatility of an option in maturity group 5, where 6 = Tl refers to options with thirty days or less to expiration, 5 = T2 are options with 31-90 days to expiration and 4 = T3 refers to options with greater than ninety days to expiration. Let M be the options moneyn~ss~ where M equals the stock price divided by the strike price (i.e. S/K). The three moneyness groups are: the out-of-the money (OTM) group with M less than 0.95; the at-the-money (ATM) group with M between 0.95 and 1.05; and the in-the-money @TM>group where M is greater than 1.05. To get a sense of the Norwegian data, estimates of average in~phed volatility appear in Table 1. For BEB, SAG and NHY options in panel A, the average implied voiatilities from at-the-money options are Iower than those from in- and out-of the-money options. For HNA, while average implied volatility of at-the-money options is lower than those from inthe-money options, they are not different from the out-of-the-money subsample. One possible reason for the difference in moneyness patterns between HNA and the rest of the sample is that HNA options traded only in the last half of the sample period.
BE%
27,724
100
270,ooo
HNA**
136) 9,885
100
795,000
NHY
(26) 38,153
3,807.08
10,489.94
100
Looo,~
SAG
(50) 33,648
4,003.97
53,162.OO
100
5,344,4(x
WI R.
AnalystsofC$tion Li4uiciity
BE%
39.95
58.30
10
1,500
HNA**
4lLtt
99.00
10
2,x30
NHY
47.01
139.27
to
tt3,m
50.52
77.78
5
3,000
SAG (23) Notes:
*n denotes the average number of transactions per trading day rounded to the nearest whole number **HNA options began trading on December 20.1991 all others began trading on May 22,199O. t-statistics on the difference in the mean stock uansaction between BE% and NNY is 7.05 and between BEB and SAG is 2.30. t-statistics on the diffemnce in the mean option transaction between BEB and NHY is 6.29 and beteen BEB and SAG is 15.31. This table shows the summary statistics of the volume of stock and option trade. The option trading statistics include both call and put options. The column labeled N indicates the number of total transactians over the period of 1990-1993 (with the exception of HNA which did not trade in 1990). This table is designed to indicate the Iiqoidity differences of the stocks and options inctuded For analysis. Liquidity differen= is measured by both the mean v&me per tmnsactioo and the maximum votumc.
470
PARVEZ AHMED and STEVE SWIDLER
Panel B considers term structure patterns for the four options. The results reveal that the average implied volatility from short term options (subsample Tl) are higher than those from options with longer time to maturity (subsample T2 or T3). Thus, patterns from the Norwegian market are similar to U.S. results and suggest that tests on the informational content of implied volatility should be run on data stratified by time to maturity and moneyness.
111. RESEARCH DESIGN A.
Conjectures on
the Role
of Arbitrage Costs
Four conjectures test for the role of arbitrage costs in the informational content of implied volatility. First, arbitrage costs depends on trading volume. The greater the liquidity of stocks and options, the lower the arbitrage costs in maintaining a delta neutral hedge. The four Norwegian stocks and options have a wide range of liquidity as seen in Table 2. For both stock and option trading, the mean volumes of NHY and SAG are significantly greater than the mean volumes of either BEB or HNA. However, it should be noted that BEB does have the greatest number of option transactions in contrast to HNA with the lowest number of transactions per day in both the stock and the option market. As a further sign of liquidity, the largest stock and option trades for NHY and SAG are greater than the largest transactions recorded for BEB or HNA. Thus, the first hypothesis is that stocks and options with relatively high liquidity such as NHY and SAG will have lower arbitrage costs and hence have implied volatility with greater informational content. Given the large number of option trades, the informational content of BEB’s implied volatility may also be relatively high. HNA being the least liquid will have implied volatility with the lowest informational content, ceteris paribus. The actual return volatility of the underlying asset also affects arbitrage costs. Stocks with high return volatility as well as a large dispersion in volatility will be at the higher end
Table 3.
Stock BEB HNA NHY SAG
Summary
Statistics on Volatility
of Actual Stock Returns
MfXtI Volatiliry
Standard Deviation of Volatility
Minimum Volatility
Maximum Volatility
0.4226 0.4305 0.2937 0.3783
0.1888 0.3368 0.1176 0.1712
0.1668 0.1501 0.1147 0.1403
1.0109 1.5978 0.6290 0.9586
Notes: 1. t-statistics for tests of the difference in mean volatilities
are: BEB-HNA 0.551, BEB-NHY 15.711, BEB- SAG 4.711, HNA-NHY 10.393, HNA-SAG 3.742, NHY-SAG 11.042. 2. F-smistics for tests of the difference in the dispersion of volatilities are: BEB-HNA 3.182, BEBNHY 2.577, BEB-SAG 1.215, HNA-NHY 8.201, HNA-SAG 3.868, NHY-SAG 2.119. 3. The difference in mean volatilities is significant for every pair except BEB-HNA and the difference in the dispersion of volatility is significant for every pair except BEB-SAG. This table reports the summary statistics of the annualized volatility of daily returns for the four stocks being studied.Volatility is the standard deviation of stock returns over thirty trading days.
Relation Between Informational
Content of Volatility and Arbitrage
471
of the arbitrage cost continuum as the delta neutral portfolios will have to be rebalanced frequently. (See, e.g., Figlewski (1989); Jarrow & Wiggins (1989)). Table 3 presents pairwise tests on the difference in mean volatilities and dispersion of volatilities for the four stocks. The summary statistic reveal that the four stocks have a large range of volatility and dispersion in volatility. Again, HNA with its relatively high volatility and dispersion in volatility will have the highest arbitrage costs. The dispersion in volatility for BEB is between the high of HNA and the low of NHY, and is not significantly different from SAG. Given that NHY and SAG have the lowest volatility and dispersion in volatility, the second conjecture is that the informational content of implied volatility for NHY and SAG should be greater than that of BEB which, in turn, will be larger than HNA. During times of negative investor sentiment, short sale restrictions increase arbitrage costs. Given negative expectations, investors will purchase puts from market makers. However, market professionals cannot hedge their option position by shorting the underlying stock on the Oslo Stock Exchange. Faced with higher arbitrage costs of possibly selling short in London, market makers will charge a put premium. Call prices will also be higher due to put-call parity. Option prices higher than their fair market value will cause implied volatility to be biased estimators of future volatility. Thus, the third conjecture is that during periods of relatively high put trading activity, implied volatility is upward biased and therefore has lower informational content. Finally, all else being equal, arbitrage costs for markets where the option and the underlying asset have an overlapping set of market makers will be lower than costs where there is no cash market in the option’s underlying. Thus, conjecture four is that implied volatility from Norwegian stock options have higher informational content than U.S. index options.
B.
Volatility Measures
Formal tests of the of the four conjectures appear below following the derivation of implied volatility and future volatility. The Norwegian equity options are American options with known discrete dividend payments. Given the possibility of early exercise, the appropriate model to price these American options is the Cox, Ross, and Rubinstein (1979) Binomial Option Pricing Model (BOPM). In the model, the call price at time t, C,,depends on: S,
=
d
=
zd
=
r
=
2 K
= =
the the the the the the
market price for the stock, dividend paid on the underlying, time between t and the ex-dividend date, risk-free rate of interest, time to maturity where T is the option’s expiration strike price.
To account for the fall in the stock price on the ex-dividend by the present value of the dividend: S; = (St-de-rTd). For Norwegian
stocks, dividends
are paid once every year.
date and
day, the BOPM reduces S,
(1)
472
PARVEZ
AHMED
and STEVE SWIDLER
Ultimately the value of the call option depends upon whether the stock price is above or below the strike price at expiration. Thus, C,, is also a function of the underlying asset’s volatility, 6. With n periods until expiration andj the number of upticks, the value of the call option at any time I is then expressed as:
-y)
Under the assumption of risk neutrality, returns are lognormally distributed, y
“-‘max(O,S;_,l
+d+(l
Y is the probability
=
+d)n-i-K).
(2)
of an uptick. If the stock
(a-4
(3)
(U)
where, u = e0d(b), v = e-d(Ar), and a = e’ Ar. Finally, volatility, 0, is not observed directly, but can be implied from market prices using a numerical routine to solve equations 2 and 3. (For a detailed discussion see Hull (1993) and Jarrow and Rudd (1983)). The measure of implied volatility must also account for early exercise of the options, so that in solving for Q, C, can never fall below its intrinsic value, S, - K. An alternative measure of volatility can be derived from daily stock prices. Specifically, define & as the annualized standard deviation of the continuously compounded returns, {R,, RI,... RT}, where R, = In [(S, + d,) / (S,_,)]. The annualized volatility of daily returns is then given as:
0
=
s ; (R,_jj)2 “‘, 1 t=1
(4)
where 260 is the number of trading days assumed in a year and R equals the sample mean. Future volatility, FV, is then obtained using equation 4 on returns over future trading days, i.e. between time period t and the expiration of the option T.
C. Express realized volatility a zero-mean random error:
Tests of /nformationa/Content as its expected value conditional
(s = E[ol$~] +
E ,
on an information
where E [E I@]= 0.
Also assume that the forecast of future volatility, F(cP), uses information (1966) suggests the following regression to test rationality of the forecast: FV=a+
PF(@)+q.
set @ plus
(5) set @. Theil
(6)
If F(Q) is, on average, the true future value, FV, then a and p will take the values 0.0 and 1.O respectively. Deviation from these values is evidence of bias and inefficiency in forecasts. The estimation of equation 6 uses implied volatility as the conditional forecast mea-
Relation
Between
lnforrnationa/
Content
of Volatility
and Arbitrage
473
sure, and the analysis tests for informational efficiency on all samples stratified on time to maturity and moneyness. As long as successive disturbances are uncorrelated in equation 6 the least square estimates of a and j3 will be unbiased and consistent. Canina and Figlewski (1993) note that for daily data, the disturbance term, q, is serially correlated for any pair of options whose time to expiration overlap. The presence of serial correlation does not bias the regression parameters, but the familiar inference procedures based on F and t distributions are no longer valid. While Canina and Figlewski (1993) use generalized method of moments (GMM) to account for serial correlation, this paper bootstraps the data to address the problem.3 Bootstrapping is more intuitively appealing compared to GMM. Also, GMM has statistical properties that are known only asymptotically. Generally it is consistent, but its performance is unknown in “small” samples. Bootstrapping is the more appropriate method to address the serial correlation problem since many of the subsamples from the OSE data can be classified as “small”.
Bootstrapping is a type of Monte Carlo experiment that uses a single available data set to design an experiment in which the data approximate the distribution of the error terms or other random quantities (see Efron (1979, 1982) ). To outline the bootstrapping technique, consider an N x L matrix, where N is the number of observations and L the total number of variables. For the purpose of exposition, assume a regression is run using the two variables in equation 6. In the2first run, estimate and store the coefficients 8, and 81 and coefficient of determination R, . The data is then resampled with replacement N times, and the second run estimates and stores regression coefficients h2 , p2 , and Rz . Repeating the process B times yields the mean values: 4 B i=l
G’b) i=l
(7c) i=l
Finally, based on the distribution of each estimated coefficient, statistics to test if a = 0 and p = 1 in equation 6.5
E.
Tests for the
it is possible to calculate f-
four Conjectures
Tests of conjectures one and two analyze the regression coefficients from equation 6. If liquidity and dispersion in volatility affect arbitrage costs, then implied volatility from short term, at-the-money options on NHY and SAG will have higher explanatory power
474
PAUVEZ AHMED and STEVE SWIDLER
than those on BEB and HNA. Moreover, HNA with the lowest liquidity, highest volatility and dispersion in volatility will have the least explanato~ power. Thus, tests of the two hypotheses examine the regression results from equation 6 on the different maturity and moneyness subsamples for each of the four stocks. The analysis tests the explanatory power of implied volatility across subsamples by comparing the regression parameters from equation 6. For im lied volatility to be unbiased a must equal 0 and p equal 1, Further, comparing R P across subsamples within each time to maturity group allows for additional inferences on the informational content of implied volatility. Most of the trading occurs for options that are at- or slightly out-of-the money. Subsample ATM will be most liquid followed by OTM and ITM. Higher liquidity implies greater info~ation~ content for such subsamples. Thus, R2 from the info~ational content regression in equation 6 on subsamples ATM will be higher than those from OTM and ITM. The third conjecture tests far the role of the short selling constraint on the info~a~onai content of implied volatility. The test considers only the at-the-money, shot-t term options, and the analysis classifies the options according to relative put-to-call trading volume for the previous three trading days. The average put-to-call trading volume proxies as a short term price expectation for the underlying stock. A high ratio is indicative of relatively high put trading and is a signal of negative expected price performance for the underlying stock. For each firm, the test creates three subsamples from the put-call trading volume distribution: PC75 has observations with put-call trading volume greater than the seventy fifth percentile, PC50 includes options with put-call trading volume between the twenty fifth and seventy fifth percentile and PC25 is the group in the lowest quartile for put-call volume. If high put-call volume signals negative investor sentiment, then the short selling constraint on the OSE implies that arbitrage costs are greatest for the PC75 subsample. A regression based on equation 6 is run on each of the put-call trading volume subsamples. Like conjectures one and two, the analysis tests unbiasedness (a = 0 and p = 1). Comp~ng R2 across subsamples which pass the test of unbiasedness allow for additional inferences on the informational content of implied volatility. If short sale constraints increase arbitrage costs, then subsample PC75 will have biased implied volatilities with the lowest informational content. A test of the final conjecture, implied volatilities for QSE stock options have higher informational content than those on CBOE index options, compares the average CX,l3, and R2 from equation 6 to those from a corresponding equation in Canina and Figlewski (1993). If arbitrage costs do have a role in the info~ational content of implied volatility, then the average R2 from the GSE options will be greater, the average a closer to zero and the average p closer to one than those reported in table 3 of Canina and Figlewski (1993).
IV,
EMPIRICAL RESULTS
Table 4 presents the informational content regression results for stratified moneyness and time to maturity subsamples. The results in table 4 fail to reject the null hypothesis of a = 0 and p = 1 for short term, at-the-money options on NHY and SAG. For al1 other sub-Samples, the results reject the nulI hypothesis. NHY and SAi; are the most liquid and least volatile of the four stocks, witb short term, at-the-money options being the most liquid of all subsamples. Thus, arbitrage costs between stocks on NHY and SAG and their short term,
Relation Between informational Content of Vo~ati/jtyand Arbitrage T~~Ie 4.
OTM
Info~ation~ Content Regressions on Moneyness and Time to Maturity Subsampl~s
ATM
Q t-statistics B t-statistics R2 N
BEB / TI .” _-..--I_X--. 0.048 0.050 1.279 1.649 0.544 0.561 -5.635* -5.709* 0.318 0.277 122 144
a f-statistics P t-statistics R2 N
0.479 3.222* -0.093 -X772* 0.009 70
ci t-statistics P r-statistics R= N
~-
0.088 1.463 0.545 -2.691* 0.084 107
HNA I TI __..“__ 0.376 2.704” 0.055 -3.114” 0.008 73
ITM
OTM
0.250 4.260’ 0.401 -5.202* 0.096 145
“0.08 1 -1.094 1.153 0.78 1 0.219 154
KTM
OTM
0.138 4.1!%* 0.307 -8.27V 0.201 138 0.462 6.719’ -0.190 -15.47% 0.025 69
1.150 7.446” -2.265 -6.320* 0.166 111
0.156 2.924* 0.310 -3.907* 0.056 144
0.261 7.235* 0.051 -8.194* 0.003 237
0.275 .--9.247” -0.005 -9.320” 0.003 187
o.sofi7.976* -0.068 -8.18 I* 0.008 125
-0.147 2.327* 0.055 -7.633* 0.003 255
0.076
0.325 7.762* 0.085 -9.870* 0.004 229
SAG/z? ---_~ 0.222 7.834* 0.294 -10.63* 0.046 183
0.370 10.13” 0.024 - 12.56* 0.0@4 172
0.043 0.46 1 0.379 -7.823* 0.082 256
1.065 0.720 -2x522* 0.185 159
ATM
ITM
BEB / T3 0.249 6.149* 0.178 -8.160* 0.045 126
0.392 13.05* -0.027 -14.61* 0.003 232
0.328 8.098* 0.096 -9.778” 0.016 138
HNA/Iz .._“_.-~ 1.334 0.522 5.881* 3.543* -2.771 -0.463 -5.543* -3.552” 0.221 0.017 54 48
1.325 4.917* -3.822 -10.62* 0.366 124
HNA / T3 ~1.388 6.8x? -3.440 -6.362* 0.387 41 NHY/T3
NHY/T2
NHY/TI -0.047 -O.&r4 1.016 0.087 0.205 198
ATM
BEB/Tz -.------0.178 0.213 5.597* 7.249* 0.421 0.275 -6.O!??* -10.12” 0.113 0.129 227 165
SAGlTi a f-statistics P t-statistics R2 N
475
0.325 7.303* -0.207 -7.944” 0.019 122
_-_-0.438 10.33* -0.087 -10.92” 0.014 108 0.771 5.1&3*
-1.185 -4.237” 0.101 52
-“~“--. 0.315 6.060* -0.147 -6.728* 0.015 86
SAG/T3 0.149 3.850* 0.493 -5:787* 0.143 154
0.32 I --.-. 7.075* 0‘147 -7.938” 0.022 121
iVotes: Realized volatility over the remaining tife of the option regressed on implied volatility: Equation 6, FV = CYc pF(+) + vj F{$) = TV. Regression coefficients and their associated r-sratistics are presented. An informationally efficient forecast requires cz= 0 and p = 1. The 9 sub-samples are defined by intrinsic value and maturity. N is the number of observation in the sub-sample. ATM = At-TV-money options; ITM = In-the-money options; OTM = Out-of-the-money options. T1 = Near-term options with 30 days or less to maturity; T2 = Near-term options with 3 l-90 days to mato~ty and T3 I Longterm options with days to maturity greater than 91 days. The R* for the Bootstrap method is the average R’ of all one thousand regressions. *Sign&ant at 5%
at-the-money options is low. Lower arbitrage costs between such stocks and their options make implied volatility au unbiased estimate of future volatility. For short term BEB options, while implied volatility is a biased estimate, informational content as measured by the regression’s R2 compares favorably to NHY and SAG. With the highest arbitrage costs of the four stocks, short term HNA implied volatility has the lowest info~ational content. AH estimated coefficients are statistically significant in HNA/Tl and the R2 coefficients are small. Thus, these results support conjectures one and two, A comp~son across time to Marty subsamples provides further support of conjecture one. In general, option liquidity is greatest for at-the-money or slightly out-of-the-money options, i.e. options in the ATM subsample. If liquidity lowers arbitrage costs, then ATM
476
PARVEZ Table 5.
Name
Informational
of
Content Regressions
P-C Volume Less Than 25th Percentile
Stock
PC25
BEB
HNA
NHY
a Is N a
N a
N SAG
a
Fc: N
AHMED
and STEVE
SWIDLER
on Put-Call Volume Subsamples
P-C Volume Greater Than 25th but Less Than 75th Percentile PC50
P-C Volume Greater Than 75th Percentile PC75
Coefficient
r-statistics
Coefficient
t-stafistics
CoefJicient
t-stafistics
-0.015 0.969 0.538 36 0.440 -0.087 0.082 18 -0.157 1.253 0.314 50 -0.132 1.329 0.205 37
-1.550 -0.163
0.079 0.502 0.292 69 0.483 -0.154 0.026 35 -0.120 1.452 0.379 98 -0.250 1.460 0.380 71
1.869
0.117 0.458 0.159 39 0.359 0.738 0.516 20 0.284 0.277 0.029 50 0.393 0.106 0.025 40
1.708 -2.956*
1.357 -2.52*
-1.261
0.430
-0.911 0.808
-5.209*
1.989*
-1.935*
-1.741 1.867
-2.012* 1.611
0.35 1 -0.840
3.528* -3.224*
4.862* -4.970*
Realized volatility over the remaining life of tbe option regressed on implied volatility; Equation 6, FV = a + PF(@) + qF($) = IV. Regression coefficients and their associated t-statistics are presented. An informationally efficient forecast requires a = 0 and p = 1. Short-rerm, at-the-money options are stratified on the 3.dayput-call volume ratio. PC75 has observations with Put-Call Volume ratios in the 75th percentile, PC50 has observations with Put-Call Volume ratios between the 20th and 75th percentile, PC25 has observations with Put-Call Volume ratios less than the 25th percentile. The R* for the Bootstrap method is the average R* of all five hundred regressions. *Significant at 5%
options should have greater informational content than the other moneyness groups. The results in table 4 are consistent with the liquidity hypothesis as the R2 for ATM options exceed the R2 for OTM and ITM options in 9 of 12 cases (75%). Using a binomial distribution, this outcome is significant at the 0.0193 level. If Norway’s short sale restriction has an effect on arbitrage costs, then the informational content of implied volatility will differ between subsamples segregated by put-call trading volume. With arbitrage cost a factor, the expectation is that the informational content of IV will be poorer for subsample PC75 compared to PC50 and PC25. Table 5 presents the Bootstrapping results on put-call trading volume subsamples based on equation 6. For BEB, NHY and SAG, implied volatility for subsample PC75 is a biased estimate of future volatility as the results in table 5 reject the null hypothesis of cx = 0 and p = 1. In contrast, unbiasedness cannot be rejected for BEB/PC25, SAG/PC25, NHY/PC25 and NHY/PCSO. Moreover, for all three stocks, the R2 for PC75 is always lower than the R2 for PC50 and PC25. Both tests of unbiasedness and R2 suggest that short selling constraints increase arbitrage costs and lower informational content of implied volatility. The results for HNA appear somewhat anomalous. However, it should be noted that there are few observations for HNA and that most of the observations for PC75 are clustered around two time periods. Finally, the analysis examines the conjecture four comparison between OSE and OEX options. The sampling technique for this study is similar to Canina and Figlewski (1993) on OEX data and facilitates a meaningful comparison of the informational content regres-
Relation
Between
Informational
Content
of Volatility
and Arbitrage
477
Table 6. Comparison of Regression Parameters from the Informational Content Regressions on Norwegian Equity and U.S. Index Options OSE
BEB
HNA
Average p
0.226 0.252
0.867 1.574
0.202 0.172
0.186 0.410
0.370 -0.185
0.204 0.278
0.134 0.050
Average R2
0.124
0.144
0.043
0.089
0.100
0.085
0.039
Max R2
0.318
0.387
0.205
0.219
0.387
0.318
0.197
Min R2
0.003
0.008
0.003
0.004
0.003
0.003
-0.002
9
9
9
36
27
A. Average
a
N
9
B. Average
NHY
ALL OSE without ALL OSE HNA
SAG
OEX
For all subsamples
For Short Term, At-The-Money
32
Subsamples
0.050
0.376
-0.047
-0.08 1
0.0745
-0.025
0.098
Average p
0.56 1
0.055
1.016
1.153
0.6962
0.910
0.013
Average R2
0.277
0.008
0.205
0.219
0.1112
0.233
0.067
Max R2
0.277
0.008
0.205
0.219
0.277
0.277
0.067
Min R2
0.277
0.008
0.205
0.219
0.008
0.205
0.067
4
3
N
a
1
1
1
1
1
This table shows the average a. p, and R* from the informational content regressions for the OSE and OEX data. The results for the OEX data are from Canina and Figlewski (1993). The &statisric for the difference of means between R2 from regressions on OEX and OSE data is 3.041 and is significant at 5% In 7 cases the OSE has Rz greater than the highest value of 0.1975 for the OEX data. For 19 subsamples the R* for the OSE data exceeds the mean from the OEX samples. The f- statistic for the difference of means between R* from regressions on OEX and OSE data wirhout HNA is 2.36 and is significant at 5%. The pairwise difference in mean R* between each of the OSE stock and the OEX data is significant at 5% except for the NHY-OEX pair.
sions between the two studies. Both studies run the informational content regressions on data stratified by time to maturity and moneyness. While this study has thirty six subsamples from its four stocks, the Canina and Figlewski (1993) study has thirty two subsamples for the OEX options. Table 6 compares the average a, p, and R2 from the informational content regressions on these subsamples. Panel A of table 6 reveals that the average p is closer to one for BEB, NHY, and SAG compared to the OEX sample. The average p is not closer to one for the overall OSE sample. This is due to the large negative p coeffcients for HNA. Average p for the overall OSE sample without HNA is closer to one and significantly higher than the OEX estimates. Also, the average R2 for the Norwegian data is greater than OEX index options and the difference significant at 5%. Moreover, in seven of the thirty six cases (19.5%), R2 from the OSE data is greater than the highest R2 of 0.197 obtained from the OEX index options. Panel B of table 6, compares average a, p, and R2 for only the short term, at-the-money options, the subsamples with the lowest arbitrage costs. Again with the exception of HNA, the results show that the c1 is closer to zero, p closer to one and R2 higher for OSE than the corresponding estimates from the OEX subsample. Thus, the results in table 6 appear to support the conjecture that all else being equal, informational content of implied volatility where the option and the underlying asset trade side-by-side is higher than in markets where the underlying does not trade at all.
PARVEZ AHMED and STEVE SWIDLER
478
V.
CONCLUSIONS
The Norwegian data allows for tests of the arbitrage cost hypothesis in determining the informational content of implied volatility. Using liquidity and volatility criteria, NHY and SAG appear to have lower arbitrage costs than BEB and HNA. Support for the arbitrage cost hypothesis includes findings that IV is an unbiased predictor of FV for short term, atthe-money NHY and SAG options. Due to the short sale restrictions for OSE stocks, arbitrage costs increase during periods of negative market sentiment. If put-call volume proxies for market sentiment, then relatively high put trading activity should produce implied volatility that is biased. The results support this conjecture as implied volatility tends to have lower informational content when put-call trading volume is relatively high. Finally, arbitrage costs in Norway are lower than those on U.S. exchanges because of an overlapping set of stock and option market makers. With the exception of lightly traded HNA, implied volatility regressions have higher R* when compared to the OEX index option results of Canina and Figlewski (1993). This result agrees with Jorion (1995) that side-by-side trading is a major determinant of low arbitrage costs. Taken as a whole, the above results suggest that arbitrage cost is an important factor in determining the informational content of implied volatility. The results have implications for market participants and researchers. Implied volatility can be an unbiased predictor of future volatility when conditioned upon the ability to perform low cost arbitrages between the option and its underlying asset.
ACKNOWLEDGEMENTS We would like to thank the two anonymous reviewers and the editor, Carl Chen, for their helpful suggestions and encouragement in developing the manuscript. Any remaining errors are the responsibility of the authors.
NOTES 1. Jarrow and Wiggins (1989) also argue that alternative stock process models are difficult to employ because of additional parameters and costly computational procedures. They further contend that the Black-Scholes model using implied volatility offers an attractive choice compared to the more complex models. 2. The OSE delisted options on Hafslund Nycomed B after seven months due to low trading activity. The OSE also listed Den Norsk Bank (DNB) options during the initial part of the three year window, but the options ceased trading after bankruptcy proceedings. The OSE introduced options on Kvzmer A (KVA) April 29, 1993 and they are not examined due to their short trading horizon. 3. Tests were also run using the GMM outlined in Canina and Figlewski (1993). The GMM methodology yields a similar set of conclusions, and results are available upon request.
Relation Between informational
Content of Volatility and Arbitrage
479
4. The size of each bootstrap sample depends on the data set used to estimate a and p. It ranges from 20 to 256. The resampling with replacement process is repeated B = 10,000 times. 5. An alternative bootstrapping method is to resample the error term, q. However, the residuals in equation 6 are not independent and identically distributed and it is better to use the method presented in the paper. Davidson and Mackinnon (1993) notes that “this form of bootstrapping yields results that are often very similar to those from using heteroskedasticity-consistent covariance matrix estimators.”
REFERENCES Ball, C., & Roma, A. (1994). Stochastic volatility option pricing, Journal of Financial and Quantitative Analysis, 29, 589-607. Beckers, S. (1981). Standard deviation implied in option prices as predictors of future stock price variability, Journal of Banking and Finance, 5, 363-38 1. Canina, L., & Figlewski, S. (1993). The information content of implied volatility, The Review of Financial Studies, 6, 659-68 1. Chiras, D.P., & Manaster, S. (1978). The information content of option prices and a test of market efficiency, Journal of Financial Economics, 6, 2 13-234. Clarke, R.G. (1994). Estimating and using volatility: Part 2, Derivatives Quarterly, 35-40. Cox, J.C., Ross, S.A., & Rubinstein, M. (1979). Option pricing: A simplified approach, Journal of Financial Economics, 7,229-263. Davidson, J., & Mackinnon, J. (1993). Estimation and Inference in Econometrics, New York: Oxford University Press. Day, T.E, & Lewis, C.M. (1992). Stock market volatility and the information content of stock index options, Journal of Econometrics, 52, 267-287. Derman, E., & Kani, I. (1994). Riding on a smile, Risk 7(4), 2-9. Dumas, B., Fleming, J., & Whaley, R.E. (I 995). Implied volatility smiles: Empirical tests, Duke University Working Paper. Efron, B. (1982). The jackknife, the bootstrap and other resampling plans, Philadelphia, Society of Industrial and Applied Mathematics. Efron, B. (1979). Bootstrapping methods: Another look at jackknife, Annals of Statistics 7, l-26. Figlewski, S. (1989) Options arbitrage in imperfect markets, Journal of Finance, 44, 12891312. Gjerde, Id., & Sazttem, F. (1995) Option initiation and underlying market behavior: Evidence from Norway, The Journal of Futures Market, 15(8), 88 l-899. Hull, J.C. (1993). Options, Futures and other Derivative Securities, Second Edition, New Jersey, Prentice Hall. Hull, J., & White, A. (1987). The pricing of options on assets stochastic, Journal of Finance, 42,28 I-300. Jarrow, R.A., & Rudd, A. (1983). Option Pricing, Richard D. Irwin, Illinois. Jarrow, R., & Wiggins, J. (1989). Option pricing and implicit volatilities, Journal of Economic Surveys, 3,59-8 1.
480
PARVEZ
AHMED
and STEVE
SWIDLER
Jorion, P. (1995). Predicting volatility in the foreign exchange market, Journal ofFinance, 50(2), 507-528. Lamoureux, C.G., & Lastrapes, W.D. (1993). Forecasting stock return variance: Toward an understanding of stochastic implied volatility, The Review of Financial Studies, 6, 293-326. Latane, H., & Rendleman, R. (1976). Standard deviation of stock price ratios implied in option prices, Journal of Finance, 31,369-389. Rubinstein, M. (1994). Implied Binomial Tree, Journal of Finance, 49 (3), 77 1-818. Shimko, D. (1993). Bounds of probability, Risk, 6(4), 33-37. Stein, E., & Stein, J. (1991). Stock price distributions with stochastic volatility: An analytic approach, Review of Financial Studies, 4, 727-752. Swidler, S., Schwartz, L., & Kristiansen, R. (1994). Option expiration day effects in small markets: Evidence from the Oslo stock exchange, The Journal of Financial Engineering, 3(2), 177- 195. Theil, H. (1966). Applied economic forecasting, North Holland, Amsterdam, Chapter 2. Trippi, R.R. (1977). A test of option market efficiency using random walk valuation models, Journal of Economics and Business, 29, 93-98. Wiggins, J. (1987). Option values under stochastic volatility: Theory and empirical estimates, Journal of Financial Economics, 44, 101 l-1023.
BETWEEN
THE
CONTENT
AND ARBITRAGE
OF IMPLIED COSTS:
FROM THE OSLO STOCK
EXCHANGE
PARVEZ AHMED and STEVE SWIDLER
ABSTRACT This study investigates the proposition that volatility of stock returns can be predicted from the volatility implied by options on the Oslo Stock Exchange (OSE), conditional on the ability to perform arbitrage. Insights into the relation between the informational content of implied volatility and arbitrage cost can be distilled from Oslo Stock Exchange data. For Norwegian firms, options and their underlying stock trade on the Oslo Stock Exchange and have an overlapping set of market makers thereby lowering the cost of arbitrage. Other components of arbitrage trading costs, liquidity and dispersion of stock return volatility, vary widely across Norwegian firms. Moreover, restriction on the short selling of stock in Oslo allows further insight into the role of arbitrage costs in determining the informational content of implied volatility. The results yield support for the arbitrage cost hypothesis: the lower the arbitmge cost between the stock and the option, the greater the informational content of implied volatility.
The Capital College, Direct all correspondence to: Parvez Ahmed, School of Business Administration, Pennsylvania State University, Middletown, PA 17057-4898, Tel: (717) 948-6162; E-mail: pxal30psu.edu. Steve Swidler, University of Texas at Arlington. Copyright 0 1998 by JAI Press Inc. International Review of Economics and Finance, 7(4): 465-480 All rights of reproduction in any form reserved. ISSN: 1059-0560
465
PARVEZ AHMED and STEVE SWIDLER
466
I.
INTRODUCTION
This study investigates the proposition that volatility of stock returns can be predicted from the volatility implied by options on the Oslo Stock Exchange (OSE), conditional on the ability to perform arbitrage. An unbiased estimate of the future volatility of asset returns is of great interest to investors in general and option traders in particular. A model’ s option price as an estimate of the option’ s fair market value is only as good as the volatility measure being used. Thus, investors having access to superior forecasts of future volatility are likely to devise trading strategy that could generate profit by identifying m&priced options. Two approaches that have been frequently used to estimate future volatility (FV) are the realized volatility from past price data, i.e., the historical volatility (HV) and the implied volatility (IV) from option prices. Papers by Latane and Rendleman (1976), Trippi (1977), Chiras and Manaster (1978), and Beckers (198 1) show implied volatility of at-the-money options to be an unbiased predictor of future volatility of stock returns. Following the earlier works, Canina and Figlewski (1993) state that it had become almost an “article of faith” among academic researchers to treat implied volatility as the market’ s forecast of volatility. In contrast, their paper along with works by Jorion (1995) Day and Lewis (1992), and Lamoureux and Lastrapes (1993) show that implied volatility is not an unbiased forecast of future volatility. In analyzing the informational content of implied volatility, much of the empirical research assumes an underlying stock process with constant mean and variance. However, as Canina and Figlewski (1993) point out, “there is an obvious conflict in applying an approach that assumes the asset price process has a known constant volatility to a situation in which volatility must be forecast because it changes randomly over time.” They suggest that while fully rational investors should use option pricing models that explicitly account for stochastic volatility (see, e.g., Hull and White (1987), Wiggins (1987), Stein and Stein (1991) and Ball and Roma (1994)), practically speaking, they are difficult to implement.’ Thus, Canina and Figlewski argue, it is appropriate to examine the content of implied volatility from Black-Scholes or other constant variance models since they are more likely to be used by traders. In their work, Canina and Figlewski (1993) conjecture that implied volatility as a predictor of future volatility is related to the ease of performing arbitrage trades between the option and the underlying asset. At the lower end of the continuum are options that trade on the same exchange as the underlying, such as futures and option on futures traded on the Chicago Mercantile Exchange. At the other end are options such as the OEX and SPX index options where the underlying asset does not trade in the cash market. Arbitrage between an index and its option is quite costly as it requires continuous buying and selling of all the stocks that make up the index to maintain a delta neutral hedge. In between the two extremes are U.S. listed stock options which typically trade on a different exchange from the underlying asset. Thus, for U.S. stock options, arbitrage is possible but somewhat costly, since trading occurs in two different markets. Jorion (1995) tests the arbitrage cost hypothesis by using foreign exchange futures and option data from the Chicago Mercantile Exchange. Compared to Canina and Figlewski (1993), he finds evidence favoring increased informational content for implied volatility coming from at-the-money, short-term options. Nevertheless, in most instances, implied volatility is a biased forecast of foreign exchange rate volatility.
Relation Between informational
Content of Volatility and Arbitrage
467
The following analysis considers the conjecture of Canina and Figlewski (1993), and examines data from the Oslo Stock Exchange to gain further insights into the relation between arbitrage costs and the informational content of implied volatility from constant variance option pricing models. For Norwegian firms, options and their underlying stock trade on the Oslo Stock Exchange and have an overlapping set of market makers thereby lowering the cost of arbitrage. Two other components of arbitrage trading costs, liquidity and dispersion of stock return volatility, vary widely across Norwegian firms and allow additional inferences to be drawn concerning Canina and Figlewski’s (1993) arbitrage cost conjecture. Moreover, the Oslo Stock Exchange does not allow stocks to be sold short. This restriction increases the cost of arbitrage between the put option and the underlying stock. Thus, the OSE data allows for construction of several tests that examine the relations~p between arbitrage costs and the info~ation~ content of implied volatility.
Il.
MICROSTRUCTURE CONSIDERATIONS AND DATA FROM OSLO STOCK EXCHANGE
Previous studies on the forecastibility of future volatility have mostly been on Chicago Board Options Exchange (CBOE) data. The OSE has a different market microstructure compared to the U.S. securities markets. Gjerde and S&tern (1995) and Swidler, Schwartz, and Kristiansen (1994) list several differences between the Norwegian market and exchanges in U.S. First, the Norwegian markets are small compared to those in the U.S. Several markets have low liquidity, especially those for Bergesen and Hafslund Nycomed, and therefore have higher transaction costs in performing the arbitrage between the option and underlying stock. Second, Norway did not allow short selling of stock during the period of study, although Norwegian stocks listed on the London Stock Exchange could be sold short. This imposes higher transaction costs for short sales compared to the U.S. market. The short sale constraint implies that the least expensive way to express negative opinion on a stock is through buying put options or selling call options. In either case, the market maker will only fill the other side of the order at higher option premiums due to increased arbitrage costs. Thus, during times of negative market sentiment, the short sale prohibition in Norwegian stock market effectively translates into upward biased implied volatilities. Finally, for No~egi~ stocks, dividends are paid once a year (mostly around AprilMay), and the exchange lists only options with the next two quarterly expirations. This implies that many of the call options will not be exercised early and thus can be valued as European options. Moreover, for these observations, the volatility implied by the option corresponds to the expiration date. Data for the study consists of transactions prices for stock and call options traded on the Oslo Stock Exchange. This paper analyzes stock and options on Norsk Hydro (NHY), Saga Petroleum (SAG) and Bergesen (BEB) for the time period May, 1990-June, 1993. Data for Hafslund Nycomed A (HNA) covers the period December, 199 1-June, 1993 .2 The data for this study reflects the universe of options trading on the OSE for the period May 1990 to June 1993.
468
PARVEZ
AHMED
and STEVE
SWIDLER
The data set first matches a call with the underlying stock transaction price that immediately precedes the option trade. The analysis then calculates the implied volatility for each call option using the methodology described in the next section. Hence, for each date, there may be multiple implied volatilities spanning different times to expiration and different strike prices. Call options that violate the lower boundary condition, i.e., their market price is less than the current dividend adjusted stock price minus the strike price, have a negative implied volatility and are excluded from the study. The study also eliminates options with fewer than 7 days to expiration and those that are more than 20% in- or out-of-the money. Canina and Figlewski (1993), Rubinstein (1994) Derman and Kani (1994), and Dumas and Fleming (1995) find implied volatility is a U-shaped function with respect to moneyness. This is popularly referred to as the volatility “smile”. The in- and out-of-the money options have higher implied volatilities than the at-the-money options. Canina and Figlewski (1993) Shin&o (1993) and Clarke (1994) also find implied volatility to be a decreasing function of time to expiration. The revealed patterns of implied volatility suggest that investors systematically misprice certain options relative to others, although typiTable 1.
Implied Volatility
Sub-Sample
IV A.
BEBO BEBA BEBI HNAO HNAA HNAI NHYO NHYA NHYI SAG0 SAGA SAG1
on Moneyness
Average Maturio
and Maturity
Average Moneyness
Implied Volatility by Moneyness Group 0.4093 0.3890 0.3928 0.2947 0.2984 0.3321 0.2977 0.2895 0.3018 0.4361 0.4181 0.4334
B. BEBTI BEBT2 BEBT3 HNATl HNAT2 HNAT3 NHYTl NHYT2 NHYT3 SAGTl SAGT2 SAGT3
Across Subsamples
78.24 70.02 61.42 79.3 1 62.22 66.78 81.28 61.66 61.12 78.62 69.18 65.93
0.92 1.00 1.08 0.94 1.OO 1.Ol 0.93 1.oo 1.06 0.92 1.oo 1SF3
Implied Volatility by Time to Maturity Group 0.4094 0.3956 0.3916 0.3724 0.2895 0.2648 0.3 144 0.2902 0.2865 0.4430 0.4162 0.4328
26.29 65.91 118.82 26.28 64.97 120.6 1 29.92 64.96 116.46 27.19 66.33 117.01
1.00 0.98 0.97 1.00 0.98 0.98 1.OO 1.00 0.99 1.OO 0.99 0.99
This table provides estimates of average implied volatility of the various aggregated sample for all stock options. The first column refers to the sample used (e.g., BEBO implies it is the out-of-money sample for the option on BEB, BEBA implies it is the at-the-money sample for the option on BEB, BEBI implies it is the inthe-money sample for the option on BEB. BEBTl, BEBT2 and BEBT3 are for BEB subsamples with time to maturity 7-30 days, 31-90 days and greater than 90 days respectively. Column two shows the average implied volatility for the associated subsample. Column three gives the estimated mean of the days to maturity for the associated subsample. Column four gives the average moneyness for the sub-sample.
tally the ~sp~~iu~ is within arbitrage bounds. Thus, tests of average implied vola~lities are improper and give the false signal that implied volatilities from different options series are noisy observations of future volatility. In segmenting the sample by maturity, let IV< refer to the implied volatility of an option in maturity group 5, where 6 = Tl refers to options with thirty days or less to expiration, 5 = T2 are options with 31-90 days to expiration and 4 = T3 refers to options with greater than ninety days to expiration. Let M be the options moneyn~ss~ where M equals the stock price divided by the strike price (i.e. S/K). The three moneyness groups are: the out-of-the money (OTM) group with M less than 0.95; the at-the-money (ATM) group with M between 0.95 and 1.05; and the in-the-money @TM>group where M is greater than 1.05. To get a sense of the Norwegian data, estimates of average in~phed volatility appear in Table 1. For BEB, SAG and NHY options in panel A, the average implied voiatilities from at-the-money options are Iower than those from in- and out-of the-money options. For HNA, while average implied volatility of at-the-money options is lower than those from inthe-money options, they are not different from the out-of-the-money subsample. One possible reason for the difference in moneyness patterns between HNA and the rest of the sample is that HNA options traded only in the last half of the sample period.
BE%
27,724
100
270,ooo
HNA**
136) 9,885
100
795,000
NHY
(26) 38,153
3,807.08
10,489.94
100
Looo,~
SAG
(50) 33,648
4,003.97
53,162.OO
100
5,344,4(x
WI R.
AnalystsofC$tion Li4uiciity
BE%
39.95
58.30
10
1,500
HNA**
4lLtt
99.00
10
2,x30
NHY
47.01
139.27
to
tt3,m
50.52
77.78
5
3,000
SAG (23) Notes:
*n denotes the average number of transactions per trading day rounded to the nearest whole number **HNA options began trading on December 20.1991 all others began trading on May 22,199O. t-statistics on the difference in the mean stock uansaction between BE% and NNY is 7.05 and between BEB and SAG is 2.30. t-statistics on the diffemnce in the mean option transaction between BEB and NHY is 6.29 and beteen BEB and SAG is 15.31. This table shows the summary statistics of the volume of stock and option trade. The option trading statistics include both call and put options. The column labeled N indicates the number of total transactians over the period of 1990-1993 (with the exception of HNA which did not trade in 1990). This table is designed to indicate the Iiqoidity differences of the stocks and options inctuded For analysis. Liquidity differen= is measured by both the mean v&me per tmnsactioo and the maximum votumc.
470
PARVEZ AHMED and STEVE SWIDLER
Panel B considers term structure patterns for the four options. The results reveal that the average implied volatility from short term options (subsample Tl) are higher than those from options with longer time to maturity (subsample T2 or T3). Thus, patterns from the Norwegian market are similar to U.S. results and suggest that tests on the informational content of implied volatility should be run on data stratified by time to maturity and moneyness.
111. RESEARCH DESIGN A.
Conjectures on
the Role
of Arbitrage Costs
Four conjectures test for the role of arbitrage costs in the informational content of implied volatility. First, arbitrage costs depends on trading volume. The greater the liquidity of stocks and options, the lower the arbitrage costs in maintaining a delta neutral hedge. The four Norwegian stocks and options have a wide range of liquidity as seen in Table 2. For both stock and option trading, the mean volumes of NHY and SAG are significantly greater than the mean volumes of either BEB or HNA. However, it should be noted that BEB does have the greatest number of option transactions in contrast to HNA with the lowest number of transactions per day in both the stock and the option market. As a further sign of liquidity, the largest stock and option trades for NHY and SAG are greater than the largest transactions recorded for BEB or HNA. Thus, the first hypothesis is that stocks and options with relatively high liquidity such as NHY and SAG will have lower arbitrage costs and hence have implied volatility with greater informational content. Given the large number of option trades, the informational content of BEB’s implied volatility may also be relatively high. HNA being the least liquid will have implied volatility with the lowest informational content, ceteris paribus. The actual return volatility of the underlying asset also affects arbitrage costs. Stocks with high return volatility as well as a large dispersion in volatility will be at the higher end
Table 3.
Stock BEB HNA NHY SAG
Summary
Statistics on Volatility
of Actual Stock Returns
MfXtI Volatiliry
Standard Deviation of Volatility
Minimum Volatility
Maximum Volatility
0.4226 0.4305 0.2937 0.3783
0.1888 0.3368 0.1176 0.1712
0.1668 0.1501 0.1147 0.1403
1.0109 1.5978 0.6290 0.9586
Notes: 1. t-statistics for tests of the difference in mean volatilities
are: BEB-HNA 0.551, BEB-NHY 15.711, BEB- SAG 4.711, HNA-NHY 10.393, HNA-SAG 3.742, NHY-SAG 11.042. 2. F-smistics for tests of the difference in the dispersion of volatilities are: BEB-HNA 3.182, BEBNHY 2.577, BEB-SAG 1.215, HNA-NHY 8.201, HNA-SAG 3.868, NHY-SAG 2.119. 3. The difference in mean volatilities is significant for every pair except BEB-HNA and the difference in the dispersion of volatility is significant for every pair except BEB-SAG. This table reports the summary statistics of the annualized volatility of daily returns for the four stocks being studied.Volatility is the standard deviation of stock returns over thirty trading days.
Relation Between Informational
Content of Volatility and Arbitrage
471
of the arbitrage cost continuum as the delta neutral portfolios will have to be rebalanced frequently. (See, e.g., Figlewski (1989); Jarrow & Wiggins (1989)). Table 3 presents pairwise tests on the difference in mean volatilities and dispersion of volatilities for the four stocks. The summary statistic reveal that the four stocks have a large range of volatility and dispersion in volatility. Again, HNA with its relatively high volatility and dispersion in volatility will have the highest arbitrage costs. The dispersion in volatility for BEB is between the high of HNA and the low of NHY, and is not significantly different from SAG. Given that NHY and SAG have the lowest volatility and dispersion in volatility, the second conjecture is that the informational content of implied volatility for NHY and SAG should be greater than that of BEB which, in turn, will be larger than HNA. During times of negative investor sentiment, short sale restrictions increase arbitrage costs. Given negative expectations, investors will purchase puts from market makers. However, market professionals cannot hedge their option position by shorting the underlying stock on the Oslo Stock Exchange. Faced with higher arbitrage costs of possibly selling short in London, market makers will charge a put premium. Call prices will also be higher due to put-call parity. Option prices higher than their fair market value will cause implied volatility to be biased estimators of future volatility. Thus, the third conjecture is that during periods of relatively high put trading activity, implied volatility is upward biased and therefore has lower informational content. Finally, all else being equal, arbitrage costs for markets where the option and the underlying asset have an overlapping set of market makers will be lower than costs where there is no cash market in the option’s underlying. Thus, conjecture four is that implied volatility from Norwegian stock options have higher informational content than U.S. index options.
B.
Volatility Measures
Formal tests of the of the four conjectures appear below following the derivation of implied volatility and future volatility. The Norwegian equity options are American options with known discrete dividend payments. Given the possibility of early exercise, the appropriate model to price these American options is the Cox, Ross, and Rubinstein (1979) Binomial Option Pricing Model (BOPM). In the model, the call price at time t, C,,depends on: S,
=
d
=
zd
=
r
=
2 K
= =
the the the the the the
market price for the stock, dividend paid on the underlying, time between t and the ex-dividend date, risk-free rate of interest, time to maturity where T is the option’s expiration strike price.
To account for the fall in the stock price on the ex-dividend by the present value of the dividend: S; = (St-de-rTd). For Norwegian
stocks, dividends
are paid once every year.
date and
day, the BOPM reduces S,
(1)
472
PARVEZ
AHMED
and STEVE SWIDLER
Ultimately the value of the call option depends upon whether the stock price is above or below the strike price at expiration. Thus, C,, is also a function of the underlying asset’s volatility, 6. With n periods until expiration andj the number of upticks, the value of the call option at any time I is then expressed as:
-y)
Under the assumption of risk neutrality, returns are lognormally distributed, y
“-‘max(O,S;_,l
+d+(l
Y is the probability
=
+d)n-i-K).
(2)
of an uptick. If the stock
(a-4
(3)
(U)
where, u = e0d(b), v = e-d(Ar), and a = e’ Ar. Finally, volatility, 0, is not observed directly, but can be implied from market prices using a numerical routine to solve equations 2 and 3. (For a detailed discussion see Hull (1993) and Jarrow and Rudd (1983)). The measure of implied volatility must also account for early exercise of the options, so that in solving for Q, C, can never fall below its intrinsic value, S, - K. An alternative measure of volatility can be derived from daily stock prices. Specifically, define & as the annualized standard deviation of the continuously compounded returns, {R,, RI,... RT}, where R, = In [(S, + d,) / (S,_,)]. The annualized volatility of daily returns is then given as:
0
=
s ; (R,_jj)2 “‘, 1 t=1
(4)
where 260 is the number of trading days assumed in a year and R equals the sample mean. Future volatility, FV, is then obtained using equation 4 on returns over future trading days, i.e. between time period t and the expiration of the option T.
C. Express realized volatility a zero-mean random error:
Tests of /nformationa/Content as its expected value conditional
(s = E[ol$~] +
E ,
on an information
where E [E I@]= 0.
Also assume that the forecast of future volatility, F(cP), uses information (1966) suggests the following regression to test rationality of the forecast: FV=a+
PF(@)+q.
set @ plus
(5) set @. Theil
(6)
If F(Q) is, on average, the true future value, FV, then a and p will take the values 0.0 and 1.O respectively. Deviation from these values is evidence of bias and inefficiency in forecasts. The estimation of equation 6 uses implied volatility as the conditional forecast mea-
Relation
Between
lnforrnationa/
Content
of Volatility
and Arbitrage
473
sure, and the analysis tests for informational efficiency on all samples stratified on time to maturity and moneyness. As long as successive disturbances are uncorrelated in equation 6 the least square estimates of a and j3 will be unbiased and consistent. Canina and Figlewski (1993) note that for daily data, the disturbance term, q, is serially correlated for any pair of options whose time to expiration overlap. The presence of serial correlation does not bias the regression parameters, but the familiar inference procedures based on F and t distributions are no longer valid. While Canina and Figlewski (1993) use generalized method of moments (GMM) to account for serial correlation, this paper bootstraps the data to address the problem.3 Bootstrapping is more intuitively appealing compared to GMM. Also, GMM has statistical properties that are known only asymptotically. Generally it is consistent, but its performance is unknown in “small” samples. Bootstrapping is the more appropriate method to address the serial correlation problem since many of the subsamples from the OSE data can be classified as “small”.
Bootstrapping is a type of Monte Carlo experiment that uses a single available data set to design an experiment in which the data approximate the distribution of the error terms or other random quantities (see Efron (1979, 1982) ). To outline the bootstrapping technique, consider an N x L matrix, where N is the number of observations and L the total number of variables. For the purpose of exposition, assume a regression is run using the two variables in equation 6. In the2first run, estimate and store the coefficients 8, and 81 and coefficient of determination R, . The data is then resampled with replacement N times, and the second run estimates and stores regression coefficients h2 , p2 , and Rz . Repeating the process B times yields the mean values: 4 B i=l
G’b) i=l
(7c) i=l
Finally, based on the distribution of each estimated coefficient, statistics to test if a = 0 and p = 1 in equation 6.5
E.
Tests for the
it is possible to calculate f-
four Conjectures
Tests of conjectures one and two analyze the regression coefficients from equation 6. If liquidity and dispersion in volatility affect arbitrage costs, then implied volatility from short term, at-the-money options on NHY and SAG will have higher explanatory power
474
PAUVEZ AHMED and STEVE SWIDLER
than those on BEB and HNA. Moreover, HNA with the lowest liquidity, highest volatility and dispersion in volatility will have the least explanato~ power. Thus, tests of the two hypotheses examine the regression results from equation 6 on the different maturity and moneyness subsamples for each of the four stocks. The analysis tests the explanatory power of implied volatility across subsamples by comparing the regression parameters from equation 6. For im lied volatility to be unbiased a must equal 0 and p equal 1, Further, comparing R P across subsamples within each time to maturity group allows for additional inferences on the informational content of implied volatility. Most of the trading occurs for options that are at- or slightly out-of-the money. Subsample ATM will be most liquid followed by OTM and ITM. Higher liquidity implies greater info~ation~ content for such subsamples. Thus, R2 from the info~ational content regression in equation 6 on subsamples ATM will be higher than those from OTM and ITM. The third conjecture tests far the role of the short selling constraint on the info~a~onai content of implied volatility. The test considers only the at-the-money, shot-t term options, and the analysis classifies the options according to relative put-to-call trading volume for the previous three trading days. The average put-to-call trading volume proxies as a short term price expectation for the underlying stock. A high ratio is indicative of relatively high put trading and is a signal of negative expected price performance for the underlying stock. For each firm, the test creates three subsamples from the put-call trading volume distribution: PC75 has observations with put-call trading volume greater than the seventy fifth percentile, PC50 includes options with put-call trading volume between the twenty fifth and seventy fifth percentile and PC25 is the group in the lowest quartile for put-call volume. If high put-call volume signals negative investor sentiment, then the short selling constraint on the OSE implies that arbitrage costs are greatest for the PC75 subsample. A regression based on equation 6 is run on each of the put-call trading volume subsamples. Like conjectures one and two, the analysis tests unbiasedness (a = 0 and p = 1). Comp~ng R2 across subsamples which pass the test of unbiasedness allow for additional inferences on the informational content of implied volatility. If short sale constraints increase arbitrage costs, then subsample PC75 will have biased implied volatilities with the lowest informational content. A test of the final conjecture, implied volatilities for QSE stock options have higher informational content than those on CBOE index options, compares the average CX,l3, and R2 from equation 6 to those from a corresponding equation in Canina and Figlewski (1993). If arbitrage costs do have a role in the info~ational content of implied volatility, then the average R2 from the GSE options will be greater, the average a closer to zero and the average p closer to one than those reported in table 3 of Canina and Figlewski (1993).
IV,
EMPIRICAL RESULTS
Table 4 presents the informational content regression results for stratified moneyness and time to maturity subsamples. The results in table 4 fail to reject the null hypothesis of a = 0 and p = 1 for short term, at-the-money options on NHY and SAG. For al1 other sub-Samples, the results reject the nulI hypothesis. NHY and SAi; are the most liquid and least volatile of the four stocks, witb short term, at-the-money options being the most liquid of all subsamples. Thus, arbitrage costs between stocks on NHY and SAG and their short term,
Relation Between informational Content of Vo~ati/jtyand Arbitrage T~~Ie 4.
OTM
Info~ation~ Content Regressions on Moneyness and Time to Maturity Subsampl~s
ATM
Q t-statistics B t-statistics R2 N
BEB / TI .” _-..--I_X--. 0.048 0.050 1.279 1.649 0.544 0.561 -5.635* -5.709* 0.318 0.277 122 144
a f-statistics P t-statistics R2 N
0.479 3.222* -0.093 -X772* 0.009 70
ci t-statistics P r-statistics R= N
~-
0.088 1.463 0.545 -2.691* 0.084 107
HNA I TI __..“__ 0.376 2.704” 0.055 -3.114” 0.008 73
ITM
OTM
0.250 4.260’ 0.401 -5.202* 0.096 145
“0.08 1 -1.094 1.153 0.78 1 0.219 154
KTM
OTM
0.138 4.1!%* 0.307 -8.27V 0.201 138 0.462 6.719’ -0.190 -15.47% 0.025 69
1.150 7.446” -2.265 -6.320* 0.166 111
0.156 2.924* 0.310 -3.907* 0.056 144
0.261 7.235* 0.051 -8.194* 0.003 237
0.275 .--9.247” -0.005 -9.320” 0.003 187
o.sofi7.976* -0.068 -8.18 I* 0.008 125
-0.147 2.327* 0.055 -7.633* 0.003 255
0.076
0.325 7.762* 0.085 -9.870* 0.004 229
SAG/z? ---_~ 0.222 7.834* 0.294 -10.63* 0.046 183
0.370 10.13” 0.024 - 12.56* 0.0@4 172
0.043 0.46 1 0.379 -7.823* 0.082 256
1.065 0.720 -2x522* 0.185 159
ATM
ITM
BEB / T3 0.249 6.149* 0.178 -8.160* 0.045 126
0.392 13.05* -0.027 -14.61* 0.003 232
0.328 8.098* 0.096 -9.778” 0.016 138
HNA/Iz .._“_.-~ 1.334 0.522 5.881* 3.543* -2.771 -0.463 -5.543* -3.552” 0.221 0.017 54 48
1.325 4.917* -3.822 -10.62* 0.366 124
HNA / T3 ~1.388 6.8x? -3.440 -6.362* 0.387 41 NHY/T3
NHY/T2
NHY/TI -0.047 -O.&r4 1.016 0.087 0.205 198
ATM
BEB/Tz -.------0.178 0.213 5.597* 7.249* 0.421 0.275 -6.O!??* -10.12” 0.113 0.129 227 165
SAGlTi a f-statistics P t-statistics R2 N
475
0.325 7.303* -0.207 -7.944” 0.019 122
_-_-0.438 10.33* -0.087 -10.92” 0.014 108 0.771 5.1&3*
-1.185 -4.237” 0.101 52
-“~“--. 0.315 6.060* -0.147 -6.728* 0.015 86
SAG/T3 0.149 3.850* 0.493 -5:787* 0.143 154
0.32 I --.-. 7.075* 0‘147 -7.938” 0.022 121
iVotes: Realized volatility over the remaining tife of the option regressed on implied volatility: Equation 6, FV = CYc pF(+) + vj F{$) = TV. Regression coefficients and their associated r-sratistics are presented. An informationally efficient forecast requires cz= 0 and p = 1. The 9 sub-samples are defined by intrinsic value and maturity. N is the number of observation in the sub-sample. ATM = At-TV-money options; ITM = In-the-money options; OTM = Out-of-the-money options. T1 = Near-term options with 30 days or less to maturity; T2 = Near-term options with 3 l-90 days to mato~ty and T3 I Longterm options with days to maturity greater than 91 days. The R* for the Bootstrap method is the average R’ of all one thousand regressions. *Sign&ant at 5%
at-the-money options is low. Lower arbitrage costs between such stocks and their options make implied volatility au unbiased estimate of future volatility. For short term BEB options, while implied volatility is a biased estimate, informational content as measured by the regression’s R2 compares favorably to NHY and SAG. With the highest arbitrage costs of the four stocks, short term HNA implied volatility has the lowest info~ational content. AH estimated coefficients are statistically significant in HNA/Tl and the R2 coefficients are small. Thus, these results support conjectures one and two, A comp~son across time to Marty subsamples provides further support of conjecture one. In general, option liquidity is greatest for at-the-money or slightly out-of-the-money options, i.e. options in the ATM subsample. If liquidity lowers arbitrage costs, then ATM
476
PARVEZ Table 5.
Name
Informational
of
Content Regressions
P-C Volume Less Than 25th Percentile
Stock
PC25
BEB
HNA
NHY
a Is N a
N a
N SAG
a
Fc: N
AHMED
and STEVE
SWIDLER
on Put-Call Volume Subsamples
P-C Volume Greater Than 25th but Less Than 75th Percentile PC50
P-C Volume Greater Than 75th Percentile PC75
Coefficient
r-statistics
Coefficient
t-stafistics
CoefJicient
t-stafistics
-0.015 0.969 0.538 36 0.440 -0.087 0.082 18 -0.157 1.253 0.314 50 -0.132 1.329 0.205 37
-1.550 -0.163
0.079 0.502 0.292 69 0.483 -0.154 0.026 35 -0.120 1.452 0.379 98 -0.250 1.460 0.380 71
1.869
0.117 0.458 0.159 39 0.359 0.738 0.516 20 0.284 0.277 0.029 50 0.393 0.106 0.025 40
1.708 -2.956*
1.357 -2.52*
-1.261
0.430
-0.911 0.808
-5.209*
1.989*
-1.935*
-1.741 1.867
-2.012* 1.611
0.35 1 -0.840
3.528* -3.224*
4.862* -4.970*
Realized volatility over the remaining life of tbe option regressed on implied volatility; Equation 6, FV = a + PF(@) + qF($) = IV. Regression coefficients and their associated t-statistics are presented. An informationally efficient forecast requires a = 0 and p = 1. Short-rerm, at-the-money options are stratified on the 3.dayput-call volume ratio. PC75 has observations with Put-Call Volume ratios in the 75th percentile, PC50 has observations with Put-Call Volume ratios between the 20th and 75th percentile, PC25 has observations with Put-Call Volume ratios less than the 25th percentile. The R* for the Bootstrap method is the average R* of all five hundred regressions. *Significant at 5%
options should have greater informational content than the other moneyness groups. The results in table 4 are consistent with the liquidity hypothesis as the R2 for ATM options exceed the R2 for OTM and ITM options in 9 of 12 cases (75%). Using a binomial distribution, this outcome is significant at the 0.0193 level. If Norway’s short sale restriction has an effect on arbitrage costs, then the informational content of implied volatility will differ between subsamples segregated by put-call trading volume. With arbitrage cost a factor, the expectation is that the informational content of IV will be poorer for subsample PC75 compared to PC50 and PC25. Table 5 presents the Bootstrapping results on put-call trading volume subsamples based on equation 6. For BEB, NHY and SAG, implied volatility for subsample PC75 is a biased estimate of future volatility as the results in table 5 reject the null hypothesis of cx = 0 and p = 1. In contrast, unbiasedness cannot be rejected for BEB/PC25, SAG/PC25, NHY/PC25 and NHY/PCSO. Moreover, for all three stocks, the R2 for PC75 is always lower than the R2 for PC50 and PC25. Both tests of unbiasedness and R2 suggest that short selling constraints increase arbitrage costs and lower informational content of implied volatility. The results for HNA appear somewhat anomalous. However, it should be noted that there are few observations for HNA and that most of the observations for PC75 are clustered around two time periods. Finally, the analysis examines the conjecture four comparison between OSE and OEX options. The sampling technique for this study is similar to Canina and Figlewski (1993) on OEX data and facilitates a meaningful comparison of the informational content regres-
Relation
Between
Informational
Content
of Volatility
and Arbitrage
477
Table 6. Comparison of Regression Parameters from the Informational Content Regressions on Norwegian Equity and U.S. Index Options OSE
BEB
HNA
Average p
0.226 0.252
0.867 1.574
0.202 0.172
0.186 0.410
0.370 -0.185
0.204 0.278
0.134 0.050
Average R2
0.124
0.144
0.043
0.089
0.100
0.085
0.039
Max R2
0.318
0.387
0.205
0.219
0.387
0.318
0.197
Min R2
0.003
0.008
0.003
0.004
0.003
0.003
-0.002
9
9
9
36
27
A. Average
a
N
9
B. Average
NHY
ALL OSE without ALL OSE HNA
SAG
OEX
For all subsamples
For Short Term, At-The-Money
32
Subsamples
0.050
0.376
-0.047
-0.08 1
0.0745
-0.025
0.098
Average p
0.56 1
0.055
1.016
1.153
0.6962
0.910
0.013
Average R2
0.277
0.008
0.205
0.219
0.1112
0.233
0.067
Max R2
0.277
0.008
0.205
0.219
0.277
0.277
0.067
Min R2
0.277
0.008
0.205
0.219
0.008
0.205
0.067
4
3
N
a
1
1
1
1
1
This table shows the average a. p, and R* from the informational content regressions for the OSE and OEX data. The results for the OEX data are from Canina and Figlewski (1993). The &statisric for the difference of means between R2 from regressions on OEX and OSE data is 3.041 and is significant at 5% In 7 cases the OSE has Rz greater than the highest value of 0.1975 for the OEX data. For 19 subsamples the R* for the OSE data exceeds the mean from the OEX samples. The f- statistic for the difference of means between R* from regressions on OEX and OSE data wirhout HNA is 2.36 and is significant at 5%. The pairwise difference in mean R* between each of the OSE stock and the OEX data is significant at 5% except for the NHY-OEX pair.
sions between the two studies. Both studies run the informational content regressions on data stratified by time to maturity and moneyness. While this study has thirty six subsamples from its four stocks, the Canina and Figlewski (1993) study has thirty two subsamples for the OEX options. Table 6 compares the average a, p, and R2 from the informational content regressions on these subsamples. Panel A of table 6 reveals that the average p is closer to one for BEB, NHY, and SAG compared to the OEX sample. The average p is not closer to one for the overall OSE sample. This is due to the large negative p coeffcients for HNA. Average p for the overall OSE sample without HNA is closer to one and significantly higher than the OEX estimates. Also, the average R2 for the Norwegian data is greater than OEX index options and the difference significant at 5%. Moreover, in seven of the thirty six cases (19.5%), R2 from the OSE data is greater than the highest R2 of 0.197 obtained from the OEX index options. Panel B of table 6, compares average a, p, and R2 for only the short term, at-the-money options, the subsamples with the lowest arbitrage costs. Again with the exception of HNA, the results show that the c1 is closer to zero, p closer to one and R2 higher for OSE than the corresponding estimates from the OEX subsample. Thus, the results in table 6 appear to support the conjecture that all else being equal, informational content of implied volatility where the option and the underlying asset trade side-by-side is higher than in markets where the underlying does not trade at all.
PARVEZ AHMED and STEVE SWIDLER
478
V.
CONCLUSIONS
The Norwegian data allows for tests of the arbitrage cost hypothesis in determining the informational content of implied volatility. Using liquidity and volatility criteria, NHY and SAG appear to have lower arbitrage costs than BEB and HNA. Support for the arbitrage cost hypothesis includes findings that IV is an unbiased predictor of FV for short term, atthe-money NHY and SAG options. Due to the short sale restrictions for OSE stocks, arbitrage costs increase during periods of negative market sentiment. If put-call volume proxies for market sentiment, then relatively high put trading activity should produce implied volatility that is biased. The results support this conjecture as implied volatility tends to have lower informational content when put-call trading volume is relatively high. Finally, arbitrage costs in Norway are lower than those on U.S. exchanges because of an overlapping set of stock and option market makers. With the exception of lightly traded HNA, implied volatility regressions have higher R* when compared to the OEX index option results of Canina and Figlewski (1993). This result agrees with Jorion (1995) that side-by-side trading is a major determinant of low arbitrage costs. Taken as a whole, the above results suggest that arbitrage cost is an important factor in determining the informational content of implied volatility. The results have implications for market participants and researchers. Implied volatility can be an unbiased predictor of future volatility when conditioned upon the ability to perform low cost arbitrages between the option and its underlying asset.
ACKNOWLEDGEMENTS We would like to thank the two anonymous reviewers and the editor, Carl Chen, for their helpful suggestions and encouragement in developing the manuscript. Any remaining errors are the responsibility of the authors.
NOTES 1. Jarrow and Wiggins (1989) also argue that alternative stock process models are difficult to employ because of additional parameters and costly computational procedures. They further contend that the Black-Scholes model using implied volatility offers an attractive choice compared to the more complex models. 2. The OSE delisted options on Hafslund Nycomed B after seven months due to low trading activity. The OSE also listed Den Norsk Bank (DNB) options during the initial part of the three year window, but the options ceased trading after bankruptcy proceedings. The OSE introduced options on Kvzmer A (KVA) April 29, 1993 and they are not examined due to their short trading horizon. 3. Tests were also run using the GMM outlined in Canina and Figlewski (1993). The GMM methodology yields a similar set of conclusions, and results are available upon request.
Relation Between informational
Content of Volatility and Arbitrage
479
4. The size of each bootstrap sample depends on the data set used to estimate a and p. It ranges from 20 to 256. The resampling with replacement process is repeated B = 10,000 times. 5. An alternative bootstrapping method is to resample the error term, q. However, the residuals in equation 6 are not independent and identically distributed and it is better to use the method presented in the paper. Davidson and Mackinnon (1993) notes that “this form of bootstrapping yields results that are often very similar to those from using heteroskedasticity-consistent covariance matrix estimators.”
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480
PARVEZ
AHMED
and STEVE
SWIDLER
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