Stock index futures mispricing: profit opportunities or risk premia?

Journalof ELSEVIER

Journal of Banking & Finance 18 (1994) 921-953

Stock index futures mispricing: profit opportunities or risk premia? Pradeep K. Yadav a,*, Peter F. Pope b a University ofStrathclyde, Department of Accounting and Finance, Cur-ran Building, IO0 Cathedral Street, Glasgow G4 OLN, UK b University of Lancaster, Lancaster, UK Received April 1992; final version received December

1993

Abstract This paper reports

empirical

evidence

on stock

index

futures

pricing

based

on about

four

the UK. Reported index values are based on firm quotes. Identified arbitrage opportunities are therefore actually exploitable and economically significant. The analysis controls for cash market settlement procedures. The results show that ex-ante trading rules would have generated attractive profits after transaction costs and after the risks of dividend uncertainties, marking to market and possible delays in execution are considered. The far contract and the near contract tend to be mispriced in the same direction. The mild tendency of futures to be overpriced in rising markets and underpriced in falling markets appears to be unimportant. Finally, significant positive relationships are observed between the absolute magnitude of mispricing and time to maturity and between mispricing and index option implied volatility. years

of synchronous

hourly

data

from

Keywords: Stock Index futures; Arbitrage;

Mispricing

JEL classification: G12, G13

* Corresponding author. Tel. +44-41-552 4400, Fax +44-41-552 3547, e-mail p.k.yadav @uk.ac.strath.vaxa. The authors are grateful for the helpful comments of an anonymous referee and participants at the 1992 meeting of the European Finance Association Conference in Portugal. 0378-4266/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0378-4266(94)00026-Y

922

P.K. Yadm, P.F. Pope/Journal

of Banking & Finance 18 (I 994) 921-953

1. Introduction This paper analyses stock index futures pricing using about four years of synchronous hourly data from the London markets. Existing empirical evidence based on synchronous intraday cash futures data comes from the US markets (e.g. Mackinlay and Ramaswamy, 1988; Chung, 1991) and is potentially susceptible to measurement error because cash index values based on transaction prices will, in general, not be actually tradeable values. Hence, apparently significant mispricing values may not always represent true arbitrage opportunities. On the other hand, the UK stock index analysed in this paper is based on quotes on which respective market makers are obliged to trade up to very large sizes. The stock index values thus represent actually tradeable values, synchronous with futures prices. Although differences in price adjustment delays between index stocks may still generate positive serial correlation in index returns and result in the reported index values being different from “true” values (Cohen et al., 1986, p. 114), the arbitrage opportunities identified with data based on a quote-based index are potentially exploitable and hence economically significant. The paper also aims to address several issues related to index arbitrage and futures pricing which do not appear to have been adequately examined in past research. First, contrds for cash market settlement procedures are introduced. Second, it formally tests and quantifies the economic impact of a popular view that index futures tend to be overpriced in sharply rising markets and underpriced in sharply falling markets (see e.g. Gould, 1988; Quality of Markets Quarterly, 1989). Third, in the context of the potential existence of a tax timing option, the paper investigates whether futures mispricing varies systematically with the volatility implied in index call option prices. Finally, through ex-post simulations, the paper reports estimates of the risk premium, or the increase in the width of the arbitrage window, that can be expected to exist on account of uncertainty about the magnitude and timing of dividends, the stochastic nature of daily marking to market cash flows, and potential delays in actual execution of arbitrage trades. The paper is organised as follows: Section 2 provides the forward pricing formula adjusted for cash market settlement procedures and outlines very briefly some relevant features of the UK institutional framework within which the empirical analysis is conducted; Section 3 describes the data, explains the methodology and documents the empirical results; and Section 4 presents the conclusions.

2. Index arbitrage

within the UK institutional

2.1. Settlevent-adjusted

framework

pricing of index futures contracts

Published empirical work on stock index futures pricing has implicitly assumed that cash market transactions are settled immediately. This assumption can lead to

P.K. Yadau, P.F. Pope/Journal

ofBanking & Finance 18 (1994)

921-953

923

significantly biased estimates of futures mispricing, particularly in markets like those of London and Paris, where cash market settlement takes place on a fixed future date rather than within a fixed period (see e.g. Yadav and Pope, 1992a, p. 235). The settlement-adjusted arbitrage free “fair” value F& at time t of an index futures contract maturing at time T is given by: ’

FITT = S, exp[ rt,,~,T~(T’

- t’)] -

C

d, exp[ rr,wr,Tf(T’ - w’)]

w=t+l

where S, is the spot index value at time t; t’ is the settlement date for cash market transactions made at time t; T’ is the settlement date for cash market transactions made at time T; d, is the aggregate dividend cash flows on the index associated with an ex-dividend time period W; w’ is the time at which dividend cash flow d, is actually received; and I,,,,,,~, is the forward interest rate at time t for a loan to be disbursed at time w’ for repayment at time T’. Following Mackinlay and Ramaswamy (1988), the “mispricing” of the futures contract is defined as: F, x,,,

=

T ’

Ft*T ’

(2)

4

where F, ,T is the actual futures price at t of the futures contract maturing

at T.

2.2. Risks involved in arbitrage Index arbitrage strategies are not perfectly riskless. The following risks are important in the UK context: (a) The magnitude of future dividends, their ex-dividend dates and their actual payment dates are uncertain. ’ (b) There is a risk of delayed cash market execution of arbitrage trades because execution is not through an automated system triggered by a computerised trading program as in the USA. There can also be difficulties in executing the cash leg of the arbitrage. It has been suggested that market makers have excess stock when there is an arbitrage-related sell program, and are short of stock when there is an arbitrage-related buy program. The reason given is that mispricing is significantly negative only in sharply falling markets and significantly positive only in sharply rising markets (Gould, 1988; Quality of Markets Quarterly, 1989). 3

r The arbitrage strategies underlying this valuation model are available from the authors on request. * Since dividends are typically announced at least a few weeks before the ex-dividend date, there is essentially no uncertainty arising due to stochastic dividends if the time remaining to maturity is so small that all companies going ex-dividend before futures maturity have actually declared their dividends. 3 This is relevant even in the context of obligatory market maker quotes because of two reasons. First, in their agreements with clients, market making firms offering basket trading facilities typically volatile markets. Second, arbitrageurs who have exclusion clauses protecting them in “exceptionally” arc market makers tend to use their own in-house market making arm.

924

P.K. Yadau,

P.F. Pope/Journal

ofBanking

& Finance 18 (1994)

921-953

(c) There is uncertainty about the level at which the cash leg of the arbitrage position can be closed at expiration because the settlement price at expiration is based on the average value of the cash index between 11.10 a.m. and 11.20 a.m. on the expiration day. (d) The payoffs on account of daily marking to market of the futures position are uncertain. These uncertainties lead to a risk premium and an effective increase in the width of the arbitrage band by an amount AT+ reflecting possibly the worst case scenarios. While these factors have been widely recognised in the literature (e.g. by Mackinlay and Ramaswamy, 1988; Yadav and Pope, 1990; Klemkosky and Lee, 1991), their effect has not been explicitly quantified. 4 However, an effective transaction cost discount AT- is created by the possibility of “risky” arbitrage strategies and in particular by the option of unwinding prior to expiration and the option of rollover into the next available maturity at or prior to expiration. 5 The results of Sofianos (1990) show that the profitability of index arbitrage comes essentially from the existence of these transaction cost discounts AT-. The tendency of mispricing to persist, or reverse itself, is important for arbitrageurs since it determines the magnitude of the effective transaction cost discounts resulting from the early unwinding and rollover options and determines also the implicit cost of the risk of delayed execution. It is important to highlight two other UK-specific institutional features relating to the risks involved in index arbitrage: (a) The UK cash index is based on the average of the best bid and the best ask quotes and these quotes represent prices at which competing market makers are obliged to trade for up to very large contract sizes. Thus, conditional on immediate execution, arbitrage trades triggered by reported index values are guaranteed to transact at prices identical to the ones on which the arbitrage decision was based. ’ The cash-futures mispricing computed from these prices is hence potentially economically significant. (b) There are several large market making firms making a market in all index stocks and some of these firms also offer institutional arbitrageurs index basket trading facilities at the best quotes posted on the system (across all market makers)

4 In addition to the risks involved in arbitrage strategies, institutional restrictions on short sales can inhibit the arbitrage process. In the UK, only registered market makers have special stock borrowing privileges. It is very difficult for non-market makers to undertake arbitrage transactions involving shorting stock as a matter of normal course, unless other trading books within the same institution (e.g. index funds) are already long in stock. See e.g. Puttonen and Martikainen (1991). 5 See Merrick (1989) and Yadav and Pope (1990) for the circumstances under which these options are profitable in the US and UK, respectively. 6 Studies based on US data potentially suffer from the problem of measurement error induced by the use of last transaction prices in index computation. Arbitrage trades triggered by reported index values will, in general, hit transaction prices different from the ones on which the arbitrage decision was based.

P.K. Yadau, P.F. Pope/Journal

of Banking & Finance 18 (1994) 921-953

925

at any particular time. Therefore, there does not appear to be any need for an arbitrageur to take on “tracking error” risk by implementing index arbitrage strategies with only a subset of index stocks. This also means that the risks of delayed execution are confined to the time taken (in the absence of computerised execution) to communicate the decision to initiate or close an arbitrage basket trade. 7 There are no additional execution delays arising on account of the large number of stocks in the index. 2.3. Transaction costs Even if arbitrage is otherwise perfectly riskless, the existence of transaction costs will allow futures prices to fluctuate within a band around the “fair” price without triggering profitable arbitrage. These costs include round-trip cash and futures market trading costs, transaction taxes, and the costs of borrowing fixed interest capital and index stocks. Borrowing costs are faced only by arbitrageurs who do not have capital in treasury bills (for upper arbitrage) and index stocks (for lower arbitrage). Round-trip transaction taxes are currently 0.5% (1% before Big Bang), but market makers recycling stocks within seven days are exempt from transaction taxes for stocks in which they make a market. The spread between synchronous ask and bid futures prices in our data has varied from about 0.1% to about 0.2% of index value and round-trip futures commissions are typically less than 0.05%. Cash market-related marginal transaction costs can be close to zero for market makers and for institutional arbitrageurs who can negotiate index arbitrage trades at mid-market prices. * Different market participants have different levels of transaction costs. In this paper, we consider two levels of transaction costs, 0.25% and 0.75%, corresponding to two broad categories of potential arbitrageurs. Category A arbitrageurs are those whose marginal costs are confined essentially to transaction costs relating to the futures market. Examples of potential arbitrageurs in this category include: market makers who are not liable to pay transaction taxes; those who are otherwise committed to enter or exit the market (due to e.g. portfolio insurance or tactical price based strategies) and use the futures market only as an intermediary;

’ Our understanding is that, in their agreements with clients, market making firms offering basket trading facilities provide for execution at the prices on the system at the time the basket trading order is received unless the markets are “exceptionally” volatile. In this context, it appears reasonable to assume that execution delays of more than one hour are unlikely. s Our understanding is that market makers offer index basket trades to selected institutional clients at mid-market prices. There is no need for market makers to include adverse information costs in index arbitrage-related basket trades. If market makers can absorb arbitrage-related order flows within their normal operations, buying and selling at mid-market prices gives them half the spread on each trade. Given the estimates of Stoll (1989) for NASDAQ, this should cover order processing costs (and presumably generate other more profitable business with the institution involved).

926

P.K. YadaL>,P.F. Pope/Journal

of Banking & Finance 18 (1994) 921-953

and those with existing arbitrage positions who seek to profitably rollover their position or to profitably unwind early. Category B arbitrageurs are those who have fixed interest capital and a pool of index stocks but, besides futures market trading costs, also have to pay transaction taxes in their dealings. Examples in this category are institutions managing index funds. Arbitrageurs with higher transaction costs, for example those who have to borrow capital or stock to initiate an arbitrage position, or those who have to face significant additional cash market-related costs, 9 should ordinarily not be able to enter the arbitrage sector if there is adequate arbitrage capital available with Category A and Category B arbitrageurs. 2.4. Efect

of taxes

Investors in the futures market necessarily pay taxes in the year the capital gains arise, while investors holding the cash asset can defer the realisation of capital gains and hence the associated tax liability. This tax timing option should result in futures being underpriced relative to the fair value in Eq. (1) (Cornell and French, 1983). lo On the other hand, the marginal investor may be a tax exempt institution in which case the tax timing option will have no value, or the marginal investor may be an arbitrageur/floor trader who cannot hold the cash index indefinitely, in which case again the tax timing option will have no value. Hence, the relevance of the tax timing option for index futures pricing could be different in different markets and its impact is essentially an empirical issue. However, ceteris paribus, the value of the tax timing option should be higher when the ex-ante volatility of the underlying asset is greater. Therefore, if the tax timing option is a relevant factor for index futures pricing, mispricing should be relatively

’ The average inner market spread for heavily traded UK stocks, as reported by the Stock Exchange Quarterly from time to time, has averaged about 0.8%, except for a few quarters after Black Monday. Considering that the spread for index arbitrage should not include adverse information costs (estimated by Stoll, 1989 as 43% for NASDAQ), cash market transaction costs should not have exceeded about 0.5% on average. lo There is another tax-related factor indirectly relevant to index arbitrage. UK fund management institutions have been somewhat reluctant to trade stock index futures because of the difficulty of being sure whether the profit from the transaction would be classified as income or capital. This distinction is important for institutions like pension funds and unit trusts since in their case where an item of profit is regarded as capital, there is an exemption from tax. Ordinarily, this obscurity should be completely irrelevant for arbitrageurs. Because cash and futures trades are directly linked, they are taxed on a similar basis and the gross profits from arbitrage will be the same for all arbitrageurs irrespective of their tax status. These profits are also unambiguously taxable as income. This should pose no difficulty even for tax exempt institutions since, as market participants have pointed out to us, some after-tax income on this account is better than no income at all. However, we have been given to understand that tax exempt institutions generally prefer to bypass arbitrage opportunities rather than have a taxable component in their income and face the resultant detailed scrutiny of the Inland Revenue. This results in a relatively lesser commitment of arbitrage capital from certain categories of potential arbitrageurs than is otherwise possible.

P.K. Yadau, P.F. Pope/Journal

ofBanking & Finance 18 (1994) 921-953

927

more negative when the ex-ante volatility of the index is greater. l1 In this paper, we investigate empirically the dependence of mispricing on the ex-ante estimate of market volatility implied in index call option prices. l2

3. Empirical

evidence

3.1. Data The empirical evidence is based on Fl’SElOO stock index futures contracts. This is an arithmetic average, market value weighted index of one hundred (highest capitalisation) stocks. The contracts are traded on the London International Financial Futures Exchange (LIFFE), and expire four times a year in March, June, September and December on the last business day of the month. Though three maturities are traded at any particular time, only the two earlier maturities have significant trading volume. Our analysis is hence confined to the two earliest maturity contracts. The contract nearest to maturity at any time is labelled as the “near” contract and the next maturing contract is labelled as the “far” contract. Expiration day observations are not included. Details of the data collection procedures for the cash and futures prices and dividend flows are described in Yadav and Pope (1992a). The results are based on hourly data for the period April 28, 1986 to March 23, 1990. The hourly frequency is determined by the availability of cash index data over the sample period. The analysis is based only on futures prices that are synchronous with hourly cash prices. Opening cash prices are not synchronous with opening futures prices and therefore a maximum of 7 hours per day, or 6930 hours over the sample period, are available for analysis. However, we use conservative synchronicity criteria and define the futures prices as synchronous with the cash price only if it is time stamped less than 60 seconds before the cash price. r3 This leaves a total of 3862 near contract and 1172 far contract mispricing observations. Stock index option contracts based on the FTSElOO index are traded on the London Traded Options Market and expire on the last business day of every month. Over the sample period, four maturities (corresponding to the four

rt The general equilibrium model of Hemler and Longstaff (1991) also suggests that futures mispricing should vary systematically with the ex-ante forecast volatility of the cash market. I2 There is conflicting evidence on the information content of implied volatility as an ex-ante predictor of future volatility. Canina and Figlewski (1991) find that implied volatility is a very poor predictor of future volatility, but Day and Lewis (1992) find that implied volatility adds substantial predictive power. In this paper, we use implied call volatility as the proxy for ex-ante market volatility. r3 Strickland and Xu (1992) also examine settlement-adjusted index futures mispricing using hourly data for 1988 and 1989. However, they do not impose the 60 seconds synchronicity restriction and hence their results can be potentially biased by asynchronicity.

TZ'I

ST‘0

8l'Z-

90'11-

8I'P-

OI'P-

6Z'O-

28'8

28'2

kx-

ZO'E-

01'9-

PS’Z -

EP'O

86'0

96'0

28'0

19'0

6P'O

PE'O

EO'O-

28'0

01'1

t6'0

EZ'O

ZP'O

LO'0

S8'0

8Z'I-

IZ'O-

88'0-

Z8'0-

Z8'E-

E8'0-

16'0-

EI'I-

EZ'I-

81'0-

8Z.O-

PI'I-

19'1-

98'0-

LI'I-

SL'O

ES.0

26'0

99'0

6p'l

SS'O

9L'O

ZS'O

18'0

OS'0

6p'O

OS'0

so.1

6E'O

98'0

00'0

99'0

6S'0

LE'O

tz.0

PO'0

00'0

ZE'O-

8E'O

Ol’OIS.0 PL’O

EO'O-

LI'O-

ZE’O

PL'O-

ZI'O

EE'O-

6Z'O-

8Z'I-

OS'O-

9L'O-

P8'0-

PP'O-

PZ'O

ZO'O

09'0-

60'1-

9s.o-

PS’O-

S8'0

SE'1

ZP'T

6E.I

00'1

28'0

OL'O

9Z'O

ZI'I

89'1

62'1

Is'0

IL'0

EZ'O

E6'1

ES'I-

OP'O-

OI'l-

EE'T-

08'01-

II'I-

IZ'I-

PO'L-

lb'1-

ZP'O-

8p'O-

OS.-

LZ'Z-

Sl'l-

08'OIF

OE'O-

IV0

9Z'O

OO'O-

OE'O-

9Z'O-

9p'o-

LS'O-

IO'0

Es'0

OE'O

9E'O-

EL'O-

6E'O-

60'0-

zso

9E'O

65'0

IS'0

SS'I

6E'O

WO

WI

19'0

6E'O

SE'0

6E'O

P9'0

8Z'O

LL'O

SE'O-

6E'O

PI'0

ZO'O

9L'O-

ZZ'O-

LE'O-

ZL'O-

LO'O-

OS'0

62'0

8E'O-

Z9'0-

8E'O-

ST’O-

O'SL

L'S1

E'8E

8'0s

L'P9

2'69

1'SL

L'96

Z'6P

8'11

P'EZ

6.18

S'9L

L'68

9'9s

8Pt

SEZ

WZ

P9Z

ZLZ

IZZ

'C61

IPZ

OPZ

EIE

S6Z

182

OIE

ZIE

Z98E

98

L8JW

L8W

18das

~8

88'wI

88U"f

88 3aa 88das

68W

68das

68

06'W

LO'O-

88'0

99.0-

60'0

EL'0

E6'1

LZ'I-

Z8'0-

VE'O-

PP.0

SP'O

9S'O

SE'O-

OS'0

SO8

o'oz

El1

091

98W

98d'Js

3aa

Jaa

68Jwl

3aa

IW

SE'E-

te.1

65'0

%uu!dxa

)X,J”O3

VI alqu

KWJIIODreaN

.ON

30

.sqo

08'0

sanp2a ,dsyru

aa-

30 %

aagdgsap-L?upuds!r

ueaK

73

EO'I-

‘PlS

‘asp

OE'O-

wpaN

PS'I

‘“!IN

zp'o

:OH lW-J JaMoI

‘cl

% Iaddn

sa1g.m~

xeJ.q

IE'S

(‘d-=d)

6P'Z0~7,

!Pe ‘1JO3

IVaS

sys~~e~s

No. of obs.

37 62 42 79 90 62 65 86 87 78 75 105 73 102 84 45 1172

Contract expiring

Jun 86 Sep 86 Dee 86 Mar 87 Jun 87 Sep 87 Dee 87 Mar 88 Jun 88 Sep 88 Dee 88 Mar 89 Jun 89 Sep 89 Dee 89 Mar 90 Ah

32.4 25.8 45.2 8.9 1.1 48.4 10.8 67.4 88.5 100.0 100.0 8.6 63.0 100.0 96.4 100.0 56.6

% of -ve misp. values

0.24 0.66 -0.01 0.65 1.31 0.02 0.92 -0.16 -0.27 - 1.53 -0.50 0.82 -0.19 -0.86 -0.89 -1.10 -0.05

CL

Mean

0.44 0.92 0.55 0.51 0.51 0.52 0.79 0.44 0.26 1.03 0.29 0.46 0.43 0.30 0.68 0.28 0.98

Std. dev.

Table 1B Far contract rnispricing-descriptive statistics

0.23 0.41 0.12 0.70 1.41 0.07 1.10 -0.18 -0.28 - 1.38 - 0.47 0.89 -0.13 - 0.84 - 0.67 - 1.10 -0.13

Median

-0.51 - 0.95 - 1.01 - 0.73 - 0.07 - 1.07 - 1.79 - 0.96 -0.79 - 7.84 - 1.31 - 0.77 - 1.51 - 1.74 - 2.73 - 1.84 - 7.84

Min.

1.28 2.68 0.88 1.65 2.53 0.93 2.90 0.95 0.59 - 0.79 - 0.02 1.57 0.60 -0.19 0.13 -0.44 2.90

Max.

Quartiles

-0.15 - 0.08 - 0.57 0.35 1.00 - 0.39 0.54 - 0.50 - 0.47 - 1.54 - 0.67 0.72 - 0.43 - 1.04 - 1.49 - 1.31 - 0.68

QI

lower

0.55 1.42 0.40 0.98 1.67 0.45 1.40 0.13 -0.10 -1.18 -0.27 1.08 0.11 -0.62 -0.34 -0.89 0.70

upper Q3

0.70 1.50 0.97 0.63 0.67 0.85 0.86 0.62 0.38 0.36 0.40 0.36 0.54 0.42 1.14 0.42 1.37

(a,-Q,)

-

-

0.50 0.73 0.87 0.27 0.10 0.86 0.77 0.85 0.66 1.98 1.10 0.30 1.10 1.45 2.10 1.61 1.48

PS

0.93 2.26 0.78 1.47 2.04 0.85 1.75 0.64 0.14 -0.98 -0.12 1.36 0.33 - 0.44 -0.07 -0.71 1.55

p95

95th

Percentiles 5th

1.44 2.99 1.65 1.74 1.94 1.71 2.52 1.49 0.81 0.99 0.97 1.66 1.43 1.01 2.02 0.90 3.02

(P,,-P,)

1.57 2.35 - 0.04 6.30 15.32 0.13 6.29 - 1.69 - 14.92 - 10.68 - 6.29 8.82 - 1.82 -8.84 - 3.41 - 21.96 - 0.21

Serial corr. adj T-stat Ha: /L=O

930

P.K. Yadau, P.F. Pope/Journal

of Banking & Finance 18 (1994) 921-953

immediately following month-end dates) and several exercise prices were available at any one time. At-the-money and relatively near maturity options tend to have the greatest liquidity, but there is also evidence of instability close to expiration. Hence, the implied volatility estimate used in this study is calculated as follows: l4 for each day in the sample period, daily closing index call option prices, and the corresponding synchronous cash index value, were collected from the Financial Times, and the implied volatility was calculated l5 for the two call option series which were closest to being at the money and which were expiring more than one month but less than two months later. The implied volatility estimate for the day was taken to be the arithmetic mean of the two implied volatilities so calculated. 3.2. it&pricing Mispricing of the futures contract was calculated on the basis of the following assumptions: (a> forecast dividends to maturity for each date are identical to the actual ex-post daily cash dividend inflow for the FTSElOO basket; (b) the forward interest rate at time t for a loan made at time w to be redeemed at time T, is identical to the (future) spot interest rate at time w on a Treasury Bill maturing at time T; (c) the value of day t of one- and three-month maturity Treasury Bill interest rates can be used to estimate a linear term structure from which the implied forward interest rate for the period t’ to T’ (in Eq. (1)) can be calculated; and (d) actual payment of dividends is made 53 calendar days after the ex-dividend date, this being the average ex-post delay between the ex-dividend date and the actual dividend payment date for index stocks over the sample period. Given estimates of transaction costs, an examination of the mispricing time series can provide indications of the categories of arbitrageurs that are active in the market. Alternatively, mispricing in excess of marginal transactions costs can be interpreted as evidence about the profitability of arbitrage-related trading strategies for different categories of potential arbitrageurs. Tables 1A and 1B report descriptive statistics of r’:icentage mispricing for the near and far contracts, respectively. They also report the t-statistic for the hypothesis that mean mispricing is zero, allowing for the autocorrelation structure of the mispricing. l6 These tables have several interesting features. It would appear to be inappropriate to conclude that the forward pricing formula is, on average, a biased estimate of the actual futures price. Mean mispricing is signifi-

I4 The authors are grateful to Oystein Johnsen for providing these implied volatility estimates. l5 The European style dividend adjusted Black&holes formula was used. It is acknowledged that this can potentially introduce errors as highlighted by Harvey and Whaley (1992). However, the errors are not expected to be systematic in relation to futures mispricing. I6 Standard error of the mean calculated as standard deviation divided by fi (where N is the number of observations) is inappropriate in view of autocorrelation in mispricing.

P.K. Yadav, P.F. Pope/Journal

ofBanking & Finance 18 (1994) 921-953

931

cantly negative and the proportion of negative mispricing values is significantly greater than 50% for 9 (6) near (far) contracts. Mean mispricing is also significantly positive and the proportion of positive mispricing values is significantly greater than 50% for 4 (5) near (far> contracts. Mean mispricing is not different from zero for only 3 (5) near (far) contracts. There is a tendency for mispricing to be either predominantly negative or predominantly positive. Whether mispricing is predominantly negative or predominantly positive varies from contract to contract and often from the far phase to the near phase of the same contract. The overall inference about mean mispricing for the aggregate sample depends on the choice of sample period. For example, for a three year sample period starting from the third quarter of 1986 to the second quarter of 1989, mispricing is predominantly negative for 5 (3), and predominantly positive for 5 (5) near (far> contracts; and mean mispricing is positive! This finding is consistent with US evidence. l7 The first and third quartiles, and the 5th and 95th percentiles, of the mispricing variable, also vary substantially from contract to contract. If the forward pricing formula is an unbiased estimator of the fair futures price, the first and third quartiles, or the 5th and 95th percentiles, could be regarded as proxies for the lower and upper boundaries of the arbitrage windows for different categories of arbitrageurs. Similarly, the interquartile range (a,-Q,) or the difference (P,, - P5) could be regarded as a proxy for the corresponding overall width of the arbitrage window. In this context, the boundaries of the arbitrage window appear to be very volatile. For 8 out of 16 near contracts, both quartiles are of the same sign and for the September 88 near contract, even the 5th and 95th percentiles have the same sign. The interquartile range (a,--a,>, or the difference (P,, - P,), appears relatively more stable. However, there appears to be no obvious relationship between the variation in the transaction cost estimates in different quarters and the variation in mean/median mispricing or variation in Q,/Q,/P,/P,, in Tables 1A and lB, or even changes in the overall width of the arbitrage window proxied by, say, (a,-Q,) or (P,, - P,). This again suggests changing levels of ex-post systematic biases in the forward pricing formula estimate of the futures price, and persistence in mispricing, rather than rapidly changing boundaries of the arbitrage window. The minimum and maximum values, the 5th and the 95th percentile, and, in several cases, even the lower and upper quartiles suggest that the absolute magnitude of mispricing often exceeds the estimated transaction costs of Category A and Category B arbitrageurs. Furthermore, there does not appear to have been

I7 For example, Cornell and French (1983), conclude that average mispricing is negative, (1991) and Chug (1991) conclude that average recognise that mispricing exhibited systematic and predominantly negative in others.

Modest and Sundereshan (1983) and Figlewski (1984) while Bhatt and Cakici (1990), Klemkosky and Lee mispricing is positive. Klemkosky and Lee (1991) also trends, being predominantly positive in some periods

F2.9, (1.72 *)

lqi.2) (-1.80*)

***)

**I

< X,,,, Q 0.5%)

i6.8) t-2.24**)

tl7.6) (-3.09 ***> 289 (239.3) (3.75 ***j 28 (46.3) C-2.76***)

LOI (-2.12

i -0.5%

pz.2) (- 1.49)

F14r.9, (2.37 * * 1

fi7.2, (0.45)

;12.1) (-2&l***)

k.5) (- 0.82)

(0.5% < x,,,,

Q 1.0%)

***)

***t

**)

;:S, (8.49 ***I

(6:9.0, (11.13 ***)

tz8.3) (-4.84

F4.5, (-2.13 3 (15.4) (-3.19

{LO% < A&.,,}

X near denotes mispricing of the near contract. X,,, denotes mispricing of the far contract. Only these 899 cases are considered for which (both) near and far contract futures prices were posted within 60 seconds of the cash index quote. In each cell of each of the two matrices above, the main observation is the number of cases which fall into the joint classification corresponding to that cell. The first parentheses in every cell contain the number of cases expected if there was no relationship between near and far contract mispricing. The second parentheses in every cell contain the z-statistic for the hypothesis that the the difference between the actuai number of cases and the expected number of cases coutd have arisen purely by chance. * Significant at the 10% level; ** significant at the 5% ievel; *** significant at the 1% level.

(1.0% < A?,,}

F21.8) (-4.08

Fl9.9) (-4.52

(0.5% < x,,

***)

;:03.2) (-2.11**1

{ - 0.5% < X,, < O.S%}

< 1.0%)

(2f7.7, (1.501 12s (113.0) (1.21)

:p6.2) (8.48 ***I

< -OS%}

T5.21 (1.68 *I

{ - 1.0% < A?,,,, < - 0.5%)

{- 1.0% < x,,

d - 1.0%)

(4.7) (4.26 ***)

14

&e,,

of near and far contracts

(X,, sl - 1.0%)

Table 2 Relative mispricing

P.K. Yadau, P.F. Pope/Journal

ofBanking & Finance 18 (1994) 921-953

933

any systematic reduction with the passage of time in the magnitude of mean mispricing, or the standard deviation of the mispricing variable. The mean absolute magnitude of mispricing, the standard deviation of the mispricing variable, the interquartile range (a,-Q,), and the difference (P,, - Ps) are all larger for the far contract than for the near contract. Since each of these measures can be viewed as proxies for the overall width of the arbitrage window, this is consistent with index arbitrage being regarded as more risky when the time to maturity is longer, because of greater dividend and interest rate uncertainty (Mackinlay and Ramaswamy, 1988; Yadav and Pope, 1990). Out of the nine quarters in which mean mispricing is significantly different from zero for both the near and the far contract, there are eight quarters in which the direction of the significant mean mispricing in the near contract is the same as the direction of the significant mean mispricing for the far contract. It is only for the quarter ending December 87 (the somewhat atypical contract period spanning Black Monday) that the direction of the significant mean mispricing for the near contract is opposite to the direction of the significant mean mispricing for the far contract. Hence, at any particular point of time the far contract has tended to be mispriced in the same direction as the near contract, suggesting that time period specific factors (like market sentiment and market volatility) could have been important determinants of mispricing. On the other hand, there are 3 contracts in which mean mispricing is significant in one direction in the far phase and also significant in the opposite direction in the near phase of the same contract, while there are 4 contracts in which mean mispricing is significant in one direction in the far phase and also significant in the same direction in the near phase of that contract. Hence, for a particular contract, the systematic bias in mispricing has tended to persist from the far phase to the near phase (about) as often as it has tended to reverse itself. This suggests that contract-specific factors (like misspecification of dividends) may not have been important determinants of systematic bias in futures pricing. The relative direction of mispricing in the near and the far contracts is examined also by analysing the subset of data observations corresponding to only those hours for which values of both the near contract mispricing and the far contract mispricing were simultaneously available. This confines the subsamples to those hours for which both near and far futures prices were posted during the last 60 seconds of an hourly interval. There are 899 such cases. In 640 (71.2%) of these cases, the near contract is mispriced in the same direction as the far contract -significantly greater than the proportion expected in the absence of any association between the direction of near and far contract mispricing (p-value < 0.0001). Furthermore, of the remaining 259 cases, as many as 120 cases correspond to the account period spanning near futures maturity, where the behaviour of near contract mispricing could potentially be of a qualitatively different nature because of the complete irrelevance of short selling constraints for index arbitrage with respect to the near contract during that period (Pope and Yadav, 1994). Table 2

264 272 221 193

241 240

313 295

281 310 312

Sep 87 Dee 87 Mar 88 Jun 88

Sep 88 Dee 88

Mar 89 Jun 89

Sep 89 Dee 89 Mar 90

3862

113

160 148 235 264

Sep 86 Dee 86 Mar 87 Jun 87

All

first order serial correlation mispricing

16

466

17 20 34

21 37

12 26

43 40 42 25

33 33 34 25

29.2 5.4 19.7 3.9 12.6 10.5 14.2 14.4 24.8 67.6 16.0 4.7 5.4 40.8 32.2 21.2 19.3

7.1

10.2 7.7 6.5 8.2

2.3 16.5

34.8 17.8

9.1 10.0 2.4

14.8

-ve mispricing

21.3 6.7 21.2 20.2

+ve mispricing

in

0.864

0.932 0.946 0.783

0.837 0.891

0.821 0.960

0.872 0.739 0.773 0.882

0.503 0.903 0.758 0.879

0.808

9 15 9 9 2 7 5 21 11 _ 3 11 _ 4 _ 110

62 42 79 90 62 65 86 87 78 75 105 73 102 84 45 1172

number of reversals

37

number of obs.

mean time before reversal (trading hrs)

number of obs.

number of reversals

Far contract

and reversals in mispricing

Near contract

Jun 86

Quarter ending

Table 3 Persistence

62.2

198.4 27.2 _ 6.3 _

41.6 42.6 49.1 80.4 281.3 58.7 133.0 13.0 8.3 _ _

+ve mispricing

80.4

170.1 _

18.6 46.5

_

20.0 14.7 40.5 7.8 3.3 55.0 15.9 27.6 62.8

-ve mispricing

mean time before reversal (trading hours)

0.958

0.942 0.754 0.586 0.851 0.750 0.936 0.906 0.867 0.802 0.586 0.713 0.780 0.935 0.866 0.926

first order serial correlation mispricing

in

PK. Yadau, P.F. Pope/Journal

ofBanking & Finance 18 (1994) 921-953

935

presents the matrix of possibilities for the relative direction of near and far contract mispricing. The number of cases in both the cells along the diagonal (which correspond to near and far contract mispricing being in the same direction) are significantly greater than the number of cases expected if there was no relationship between near and far contract mispricing (p-value < 0.0001). To further investigate the association between near and far contract mispricing, the mispricing variable X is classified into five bands as follows:

I - 21 f-1) (01 {+1) (+21

{X6

- 1.0%) {-1.0%
Out of 899 simultaneous observations, mispricings in the near and far contracts are in the same band in 368 (40.9%) cases, one band away from each other in 379 (42.2%) cases, two bands away from each other in 147 (16.4%) cases, and three bands away from each other in the remaining 5 (0.5%) cases. There are no cases in which mispricing is in one outermost mispricing band for the near contract and in the opposite direction outermost band for the far contract. Table 2 also presents the matrix of possibilities for the relative association between the band of near contract mispricing and the band of far contract mispricing. The number of cases in each of the cells along the diagonal, which correspond to near and far mispricing being in the same band, and most of the cells which correspond to a difference of one band, are significantly greater than the number of cases expected in the absence of any association between near and far contract mispricing. In fact, the number of cases in the cells which are far from the diagonal, and, in particular, in the cells which correspond to differences of three or four bands, is significantly less than the number of cases expected in the absence of any association. Clearly, the mispricing of the far contract tends to be in the same direction, and “close” to the mispricing in the near contract. Mispricing also tends to persist over long periods. The degree of this persistence can be estimated non-parametrically in terms of the average length of a run (or the average time before a mispricing reversal) and parametrically in terms of the serial correlation in the mispricing time series. Table 3 reports results for both the near and the far contracts. For the aggregate sample, the average time before mispricing reversal for the near contract is about 15 trading hours when mispricing is positive and about 19 trading hours when mispricing is negative. For each contract the “runs” test showed that the number of runs is significantly less than the number of runs expected if successive observations were independent. The hypothesis of no persistence is conclusively rejected (with p-value < 0.001) in every case. The first order autocorrelation is significantly greater than zero in

Jun 86 Sep 86 Dee 86 Mar 87 Jun 87 Sep 87 Dee 87 Mar 88 Jun 88 Sep 88 Dee 88 Mar 89 Jun 89 Sep 89 Dee 89 Mar 90 All

Contract expiring

113 160 148 235 264 264 272 221 193 241 240 313 295 281 310 312

total number of observations

48 79 60 101 134 90 141 67 90 138 99 167 78 109 188 100 1689

number of arbitrage oooortunities .. ex ante simulations 1.4 0.4 2.6 8.5 14.7 12.0 59.3 2.6 8.9 26.4 9.6 31.7 11.4 16.1 60.5 12.5 278.7

ex post simulations 10.3 22.6 15.6 23.1 36.4 26.6 91.1 12.9 21.1 51.8 27.1 45.9 19.4 31.1 84.2 25.3 544.4

arbitrage profits/contract (fOO0J

Category A arbitrageurs

Trading rule 1 (hold to expiration)

Table 4A Arbitrage profit simulations:

39.6 27.8 40.0 59.4 62.7 66.7 65.2 53.7 64.4 58.0 55.6 79.6 76.9 67.0 78.7 75.0 63.9

percentage of profitable trades in ex ante simulations

47 78 7 0 81 88 140 49 90 0 57 0 73 47 182 0 939

number of early unwindings

5.3 19.2 1.5 0.0 17.7 16.2 21.6 7.8 11.3 0.0 9.9 0.0 9.8 8.4 55.0 0.0 183.9

ex post simulations

- 0.2 9.6 -0.1 0.0 12.8 - 0.2 - 0.4 3.8 -8.1 0.0 8.6 0.0 8.4 0.7 54.5 0.0 89.2

ex ante simulations

additional arbitrage profits/ contract (EOOO)

Trading rule 2 (early unwinding)

258.0

261.5 150.2 302.0

439.2

105.5 116.5 79.7 70.5 335.2

260.8 389.3 74.8

mean time to early unwinding (trading hours)

Jun 86 Sep 86 Dee 86 Mar 87 Jun 87 Sep 87 Dee 87 Mar 88 Jun 88 Sep 88 Dee 88 Mar 89 Jun 89 Sep 89 Dee 89 Mar90 All

Contract expiring

15 72 57 101 132 88 141 67 90 138 57 167 71 108 184 100

number of rollovers

2.1 26.5 6.7 27.6 47.1 21.6 56.8 13.7 11.6 49.4 22.3 44.5 11.5 30.8 56.9 33.1 462.3

ex post simulations

UJOOJ

additional arbitrage profits/contract

Trading rule 3 (rollover)

-0.9 -3.8 - 6.2 16.0 18.7 0.9 11.6 6.6 1.1 0.6 5.6 4.7 11.0 14.5 54.9 3.3 138.5

ex ante simulations 64.1 84.2 227.6 81.1 84.8 338.5 301.6 180.8 53.5 43.2 67.3 99.0 198.2 41.0 39.5 22.8 136.2

mean time to rollover (trading hrs)

47 78 57 101 134 90 141 67 90 138 57 167 73 109 184 100 1633

number of early unwindings/ rollovers

32 9 7 0 25 51 105 27 6 0 0 0 2 14 0 0 278

number of early unwindings

15 69 50 101 109 39 36 40 84 138 57 167 71 95 184 100 1355

number of rollovers

Trading rule 4 (early unwinding/rollover)

5.6 27.9 7.4 27.6 42.1 19.5 45.1 11.6 11.4 49.4 22.3 44.5 11.7 27.7 56.9 33.1 443.9

ex post simulations

(fOO0)

additional arbitrage profits/contract

-0.8 -4.0 -5.5 16.0 6.1 0.9 21.9 2.8 1.4 0.6 5.6 4.7 11.3 17.1 54.9 3.3 136.1

ex ante simulations

178.0 77.9 125.3 81.1 55.6 81.2 54.7 47.4 52.7 43.2 67.3 99.0 194.5 31.7 39.5 22.8 81.4

mean time to early unwinding/ rollover

Jun 86 Sep 86 Dee 86 Mar 87 Jun 87 Sep 87 Dee 87 Mar 88 Jun 88 Sep 88 Dee 88 Mar 89 Jun 89 Sep 89 Dee 89 Mar 90 All

Contract expiring

113 160 148 235 264 264 272 221 193 241 240 313 295 281 310 312

7 32 20 12 20 12 82 2 8 26 28 24 9 17 96 8 403

1.1 7.5 3.7 2.0 3.8 2.8 61.8 0.3 1.3 20.3 5.8 6.1 1.6 4.6 30.2 1.5 154.5

ex post simulations

ex ante simulations - 0.5 -4.6 - 2.8 -1.6 -1.3 0.6 38.3 -0.6 0.1 8.2 - 2.0 2.0 0.4 3.1 13.1 -0.4 52.0

57.1 28.1 35.0 41.7 50.0 75.0 67.1 0.0 75.0 50.0 53.6 70.8 77.8 88.2 76.0 50.0 61.8

7 32 0 0 16 12 82 2 8 0 26 0 9 17 96 0 307

0.8 8.0 0.0 0.0 3.6 2.2 11.8 0.4 1.0 0.0 4.5 0.0 1.2 3.0 29.0 0.0 65.6

simulations

ex post

0.0 4.2 0.0 0.0 2.1 0.7 -1.0 0.5 - 0.7 0.0 4.0 0.0 1.1 0.2 28.6 0.0 39.7

ex ante simulations

additional arbitrage profits/ contract (fOO0)

number of early unwindings

percentage of profitable trades in ex ante simulations

Trading rule 2 (early unwinding)

number of arbitrage opportunities

arbitrage profits/contract (LOOO)

B arbitrageurs

total number of observations

Category

Trading rule 1 (hold to expiration)

Table 4B Arbitrage profit simulations:

288.3

326.5 158.0 282.0

494.8

81.8 207.9 99.3 91.2 410.6

378.0 333.3

mean time to early unwinding (trading hours)

Jun 86 Sep 86 Dee 86 Mar 87 Jun 87 Sep 87 Dee 87 Mar 88 Jun 88 Sep 88 Dee 88 Mar 89 Jun 89 Sep 89 Dee 89 Mar 90 All

Contract expiring

5 32 20 12 20 12 82 2 8 26 26 24 9 17 96 8 399

0.8 16.7 2.5 3.9 6.8 3.9 36.2 0.5 1.0 8.3 10.6 5.5 1.5 3.8 28.4 2.1 132.4

ex post simulations

WOO)

-0.6 - 1.6 - 2.0 2.2 4.7 1.9 15.7 0.4 1.0 - 1.7 4.1 -0.3 1.5 6.9 31.2 0.8 64.1

ex ante simulations 22.9 29.6 100.5 50.3 122.0 230.6 347.2 68.9 52.0 73.4 65.8 118.8 257.7 17.2 55.5 4.5 162.0

7 32 20 12 20 12 82 2 8 26 26 24 9 17 96 8 401

2 0 0 0 8 3 62 0 0 0 0 0 0 0 0 0 75

number of early unwindings

5 32 20 12 12 9 20 2 8 26 26 24 9 17 96 8 326

number of rollovers

number of early unwindings/ rollovers

mean time to rollover (trading hrs)

number of rollovers

additional arbitrage profits/contract

Trading rule 4 (early unwinding/rollover)

Trading rule 3 (rollover)

1.0 16.7 2.5 3.9 4.9 3.7 32.8 0.5 1.0 8.3 10.6 5.5 1.5 3.8 28.4 2.1 127.0

ex post simulations

WJO)

additional arbitrage profits/contract

- 0.6 - 1.6 -2.0 2.2 0.4 2.3 22.8 0.4 1.0 -1.7 4.1 -0.3 1.5 6.9 31.2 0.8 67.3

ex ante simulations

107.4 29.6 100.5 50.3 56.6 79.8 65.2 68.9 52.0 73.4 65.8 118.8 257.7 17.2 55.5 4.5 81.0

mean time to early unwinding/ rollover

940

P.K. Yadau, P.F. Pope/Journal

of Banking & Finance 18 (1994) 921-953

every case. In fact it is often very close to unity-varying from 0.503 (0.586) to 0.960 (0.958) for the near (far) contract. The extent of persistence in mispricing-whether measured in terms of serial correlation or the average length of a run-may be regarded as an inverse measure of the elasticity of arbitrage services. ” Hence, it is clear that the elasticity of arbitrage services has not increased over time. The high degree of persistence in mispricing suggests that the possibility of delayed execution may not be a serious risk for index arbitrageurs. l9 On the other hand, since the average run is less than about three trading days for the near contract, it appears that the early unwinding option should be significantly valuable. 3.3. Index arbitrage profitability

simulations

3.3.1. Ex post profit simulations Tables 4A and 4B present, inter alia, the results of simulating the profitability of index arbitrage based on simple ex-post trading rules which assume that it is possible to use the prices at any time to execute a trade at the same price at the same time. Let TC% be the transaction cost relevant for the arbitrageur, and Y% be the additional profit required by the arbitrageur to be motivated to consider an arbitrage trade. Four trading rules are considered. Trading Rule 1 is the simple hold-to-expiration trading rule with the arbitrage position initiated when the absolute magnitude of mispricing exceeds (TC + Y>%. In Trading Rule 2, instead of waiting until contract expiration, the position is unwound as soon as early unwinding generates a marginal profit of Y% after providing for the incremental transaction costs involved in such early unwinding. In Trading Rule 3, the position is rolled forward to next available maturity as soon as rollover generates a marginal profit of Y% after providing for the incremental transaction costs involved in such rollover. ” *’ Trading Rule 4 is a combination of Trading Rules 1, 2 and 3. The arbitrage position is initiated as in Trading Rule 1, but is unwound early as per Trading Rule 2 or rolled forward, as per Trading Rule 3, whichever option becomes activated at an earlier date. Y is assumed to be 0.25%. Two values of TC are used in each case: 0.25% and 0.75%. These correspond to the estimated transaction costs of Category A and Category B arbitrageurs, respectively.

‘* See e.g. Garbade and Silber (1983) and Pope and Yadav (1992). I9 The high degree of persistence in the overall mispricing series can mask significant localised variations conditional on the absolute magnitude of mispricing. Pope and Yadav (1992) argue that the degree of persistence in mispricing should be very high when the magnitude of mispricing is so small that no arbitrageurs are tempted into the market, but should become substantially lower when the magnitude of mispricing is high enough to attract arbitrage capital. The evidence in Yadav and Pope (1992b) is consistent with such a hypothesis. ” Compound rollovers are ignored. The rolled over position is assumed to be carried to expiration. 21 See Yadav and Pope (1990, pp. 576 and 593) for the magnitude of the incremental transaction costs involved in early unwinding or rollover in the UK.

P.K. Yadau, P.F. Pope/Journal

of Banking & Finance 18 (1994) 921-953

941

Tables 4A and 4B report the profits (in & 000) earned from the different trading rules by Category A and Category B arbitrageurs assuming that each arbitrage position involves only one contract. The first point to note is that the additional profits arising from the rollover option or the early unwinding option are a significant proportion of the total arbitrage profits, and often exceed the arbitrage profits arising from the simple hold to expiration strategy. These high additional profits imply a heavy transaction cost “discount” and should generate “risky” arbitrage activity even when mispricing is within transaction cost bounds. Secondly, arbitrage positions are almost never held to expiration. With Trading Rule 4, more than 97% positions are closed before expiration and the median time to early unwinding/rollover is less than two weeks for both transaction cost levels. 22 Thus index arbitrage should not create significant expiration day price volume and volatility effects in underlying stocks. Thirdly, the median time to early unwinding is substantially greater than the median time to rollover. 23 Thus arbitrage positions are more likely to be rolled over into the next available maturity rather than being unwound early. This is because, as we have seen earlier, the far contract tends to be mispriced in the same direction as the near contract. 3.3.2. Ex ante profit simulations Tables 4A and 4B also report index arbitrage profits for transaction cost bounds of 0.25% and 0.75%, based again on Trading Rules 1, 2, 3 and 4 but implemented on an ex-ante basis. Thus, the calculation of ex-ante profits in Tables 4A and 4B assumes that if there is an arbitrage opportunity perceived on the basis of cash and futures prices in hourly period t, the required arbitrage strategy is executed only on the basis of the prices in hourly period (t + 1). Similarly it is assumed that if an early unwinding or rollover trade is indicated on the basis of cash and futures prices in hourly period t’, the trade is actually executed only on the basis of prices in hourly period (t’ + 1). With the easy availability of basket trades on competitive terms, the assumed execution delay of one hour is a reasonably conservative assumption for assessing the potential risk in index arbitrage due to the possibility of delayed execution of trades. In this context, it is important to emphasise that the cash index in London is based on firm quotations on which respective market makers are obliged to trade up to fairly large sizes. Thus the cash and futures prices one hour later clearly represent actually tradable prices. 22

The time to early unwinding and the time to rollover indicated in Tables 4A and 4B are upward biased conservative estimates of the true time to early unwinding/rollover, because, in our calculations we have assumed that mispricing is zero at the open and on the hours corresponding to our “missing” values, i.e. whenever a futures price was not available in the 60 seconds prior to the cash index quote. zs The upward bias cited in the immediately preceding footnote is also likely to be greater for the time to rollover than for the time to early unwinding since there were many more “missing” values for the far contract than for the near contract.

% of profitable positions

Transaction

wJOOJ

total profit/contract

costs = 0.75%

Trading rule threshold = 0.5% 97.9 4.7 Jun 86 100.0 25.5 Sep 86 95.0 9.3 Dee 86 100.0 23.0 Mar 87 100.0 35.8 Jun 87 100.0 15.8 Sep 87 100.0 103.2 Dee 87 100.0 7.2 Mar 88 100.0 10.5 Jun 88 100.0 69.5 Sep 88 57.6 27.1 Dee 88 100.0 21.4 Mar 89 94.9 10.7 Jun 89 100.0 27.0 Sep 89 98.9 88.7 Dee 89 100.0 29.3 Mar 90 96.9 508.7 All

Quarter ending

Table 4C Risky arbitrage strategies

97.9 97.5 85.0 100.0 100.0 94.4 100.0 95.5 92.2 100.0 57.6 100.0 46.2 100.0 97.3 100.0 93.1

profitable positions

% of

Transaction

2.4 28.7 3.7 18.5 24.5 7.9 90.3 3.5 2.3 53.7 15.9 22.4 -5.4 19.5 63.1 16.2 367.3

(fOO0)

total profit/contract

costs = 1.0%

58.3 91.1 55.0 100.0 100.0 71.1 100.0 55.2 51.1 100.0 57.6 100.0 16.7 99.1 96.3 99.0 84.0

profitable positions

% of

Transaction

-3.8 29.4 -2.9 13.0 21.9 3.5 83.0 -7.4 - 10.5 42.4 4.8 22.1 - 19.3 13.8 40.8 11.0 241.7

(fOO0)

total profit/contract

costs = 1.25%

29.2 87.3 41.7 100.0 100.0 62.2 100.0 7.5 13.3 100.0 51.5 100.0 1.3 73.4 96.3 57.0 72.9

profitable positions

% of

Transaction

- 10.2 28.0 - 8.4 10.6 22.4 -8.3 77.5 - 22.8 - 25.8 33.8 -8.5 12.8 -31.5 - 10.2 27.7 - 17.8 69.3

(fOOtI)

total profit/contract

costs = 1.5%

% of profitable positions

Transaction

total profit/contract WOO)

costs = 1.25%

100.0 100.0 90.0 100.0 100.0 83.3 100.0 100.0 87.5 100.0 92.9 100.0 11.1 100.0 100.0 100.0 96.3

% of profitable positions

Transaction

0.2 15.9 0.3 2.1 1.9 1.1 68.2 0.1 -0.1 19.5 6.9 2.0 -1.7 1.6 20.0 0.4 138.4

(fOO0)

total profit/contract

costs = 1.5%

28.6 100.0 35.0 100.0 100.0 75.0 100.0 0.0 0.0 100.0 92.9 100.0 0.0 100.0 100.0 50.0 88.6

% of profitable positions

Transaction

-1.3 14.5 -3.3 1.1 1.2 0.1 61.8 - 0.6 -2.3 17.0 3.7 2.6 -3.1 1.7 15.1 - 1.1 107.1

WOO)

total profit/contract

costs = 1.75%

0.0 100.0 10.0 100.0 100.0 50.0 100.0 0.0 0.0 100.0 75.0 100.0 0.0 88.2 100.0 12.5 83.6

% of profitable positions

Transaction

- 2.5 14.7 -6.1 1.0 1.8 - 2.5 58.8 -0.8 - 3.2 22.0 -1.5 1.7 -4.2 0.3 15.4 -3.7 91.3

WOO)

total profit/contract

costs = 2.0%

The arbitrage position is initiated whenever mispricing exceeds X% in magnitude (X = 0.5%, 1.0%) even when actual transaction costs exceed X%. The position is unwound early, or rolled forward, as soon as the additional profit from early unwinding, or the additional profit from rollover, makes the overall position profitable after inclusion of the incremental transaction costs involved in early unwinding/rollover.

Trading rule threshold = 1.0% Jun 86 100.0 0.6 100.0 Sep 86 12.1 Dee 86 100.0 1.7 Mar 87 100.0 2.7 100.0 Jun 87 2.9 Sep 87 100.0 2.9 Dee 87 100.0 76.2 100.0 Mar 88 0.3 100.0 Jun 88 0.3 Sep 88 100.0 22.5 92.9 Dee 88 10.1 Mar 89 100.0 2.7 Jun 89 100.0 0.7 100.0 Sep 89 3.7 Dee 89 100.0 32.1 Mar 90 100.0 1.3 All 99.5 172.8

Quarter ending

944

P.K. Yadau, P.F. Pope/Journal

of Banking & Finance 18 (1994) 921-953

The results of ex-ante trading rules are qualitatively similar to those of ex-post trading rules, but the magnitude of arbitrage profits are reduced, as can be expected. The reduction in profits from Trading Rule 1 is, on average, about one half for Category A arbitrageurs and two-thirds for Category B arbitrageurs. About two-thirds of trades are still profitable in both cases. The reduction in total profits from Trading Rule 4 is, on average, of similar order of magnitude. Even on an ex-ante basis the profits per contract are surprisingly large. Clearly, the profitability of index arbitrage has been fairly robust to delays in execution of even one hour. 3.3.3. Profit simulations for risky arbitrage The high profit generated by the early unwinding option and the rollover option suggest potential for risky “arbitrage” trades which initiate the arbitrage trade with the magnitude of mispricing less than the actual transaction cost of the potential arbitrageur in the hope that the marginal profit generated by early unwinding/rollover will not only cover the loss involved in initiating a trade within the transaction cost window, but also generate a net profit. There can be trades. The profit simulations reported in Table several forms of risky “arbitrage” 4C are based on implementing the following risky arbitrage trading rule: the arbitrage position is initiated whenever mispricing exceeds X% in magnitude (X = 0.5%, 1.0%) even when actual transaction costs exceed X% (by an amount equal to 0.25%, 0.50%, 0.75%, 1.00%); and the position is unwound early, or rolled forward, as soon as the additional profit from early unwinding, and the additional profit from rollover, make the overall position profitable after inclusion of the incremental transaction costs involved in early unwinding/rollover. The results in Table 4C are fairly striking. With a trading rule threshold of 0.5%, even after excluding the contract spanning Black Monday, index arbitrage has been profitable for arbitrageurs with transaction costs of 1.25% with more than 80% of positions being profitable. With a trading rule threshold of l.O%, index arbitrage has been profitable even for arbitrageurs with transaction costs of 2.0% again with more than 80% positions being profitable. In each case, the options to unwind early or rollover have provided a transaction cost discount of about 1% in magnitude and have reduced the effective width of the arbitrage window by more than 60%. 24 Since such trades are clearly not risk-free, it appears that those index arbitrageurs who do not want to take on any risk in addition to the inherently unavoidable risks arising due to stochastic interest rates and dividends, should never be able to enter the arbitrage market if market participants actively engaged in “risky” arbitrage have sufficient arbitrage capital available.

24 Once again, the simulated profits from risky arbitrage are likely to be understated because of our including only those futures prices which were posted less than 60 seconds before the cash index quote.

P.K. Yadav, P.F. Pope/Journal

3.4. Execution

of Banking & Finance 18 (1994) 921-953

945

risks

The results of the ex-ante profit simulations show clearly that the profitability of index arbitrage has been robust to delays in execution. This section addresses another kind of execution risk, namely whether the profitability of index arbitrage is overstated on account of index futures being significantly overpriced only in sharply rising markets and significantly underpriced only in sharply falling markets, making it difficult to execute the cash leg of the initial arbitrage trade due to market makers being short of stock when the arbitrageur needs to buy stocks and having surplus stock when the arbitrageur needs to sell stocks. This possibility is suggested inter alia by Gould (1988) and the Stock Exchange Quality Markets Quarterly (1989). To investigate this, it is necessary to examine whether arbitrage opportunities arising due to significant index futures overpricing (underpricing) at time t are accompanied by significantly high (low) cash market returns from time (t - 1) to t. The mispricing variable X is again classified into the five bands ( - 2}, { - l}, {0), { + l}, and { +2} defined earlier. For the five sub-samples sorted on the basis of these five mispricing bands, Table 5 reports the mean (median) cash returns and the percentage of observations in the band which also correspond to the top and bottom deciles and the top and bottom quartiles of cash returns in the overall sample. It also reports the results of testing relevant hypotheses in this regard.

of

Table 5 Arbitrage Mispricing

(-2:

opportunities

and cash returns

bands

x d -1.0%)

(- 1: - 1.0% < x d -0.5%) (0: -0.5%

< x Q 0.5%)

(+l:O.S%


(2: 1.0% < XI

Percentage correspond

of observations which also to deciles/quartiles of cash returns

lowest decile a

lowest quartile a

10.5 (0.717) 10.1 (0.938) 10.0 (0.953) 10.0 (0.990) 9.1 (0.628)

28.2 (0.138) 26.0 (0.482) 24.6 (0.657) 25.0 (0.983) 20.4 (0.081)

highest decile a

highest quartile a

&4)

YE37)

(;::23, 9.9 (0.850) 11.3 (0.256) 13.4 (0.034)

::::35) 24.9 (0.936) 28.0 (0.060) 32.5 (0.004)

Mean cash return b

Median cash return ’

(o/o)

(%)

- 0.031 (0.076) - 0.013 (0.230) 0.006 (0.760) 0.027 (0.130) 0.022 (0.740)

- 0.013 (0.043) 0.000 (0.112) 0.011 (0.711) 0.006 (0.324) 0.039 (0.010)

a In parentheses is the p-value for the hypothesis that, in the subsample corresponding to that mispricing band, the percentage of observations which also correspond to the particular decile (quartile) of cash returns is equal to 10% (25%). b In parentheses is the p-value for the hypothesis that the mean cash return in the subsample of that mispricing band is equal to the mean cash return in the whole sample (0.003%). ’ In parentheses is the p-value for the hypothesis that the median cash return in the subsample of that mispricing band is equal to the median cash return in the whole sample (0.006%0).

946

P.K. Yadau, P.F. Pope/Journal

ofBanking & Finance 18 (1994) 921-953

The mean (median) cash return in the subsample corresponding to the mispricing band {- 1: - 1.0%
P.K. Yadau, P.F. Pope /Journal

of Banking & Finance 18 (1994) 921-953

947

Over our sample period, the mean (median) absolute return on the index between 11 a.m. and 12 noon on expiration days was 0.12% (0.06%), and the maximum value of this absolute return was 0.77%. If index values follow a random walk, the variance of prices between 11.10 a.m. and 11.20 a.m. should be one-sixth of the variance between 11 a..m and 12 noon. Therefore, as a first approximation, the mean (median) absolute index return between 11.10 a.m. and 11.20 a.m. should have been about 0.05% (0.03%). z This is clearly insignificant even if it is assumed that the closing execution index price received by the arbitrageur goes against him to the maximum extent in relation to the settlement price at expiration. 26 3.6. Misspecification

of dividends

The fair value of the futures contract depends on three ex-ante dividend relating to the future dividend profile on the index: the magnitude of dividends, the ex-dividend date, and the actual dividend payment date. Our basic mispricing estimates are based on the assumption that the dividend forecast used by the arbitrageur is equal to the actual ex-post dividend inflow on the index. 27 The perfect foresight assumption is clearly unrealistic. Typically, market participants estimate future dividends by applying a fixed percentage growth factor to past dividends and use identical (or corresponding 28) ex-dividend and payment dates. We first examine a variable d,& y) defined as the difference between mispricing estimated from actual ex-post dividend inflows and mispricing estimated using last year’s dividend plus a y% growth factor where y = O%, 10% and 20%. These growth factors represent conservative assumptions on either side. The mean/ median value of ddivl is less than 0.1% for every contract and cannot explain the

forecasts

z The mean absolute return is approximately equal to the standard deviation of returns since squared mean returns are typically very small relative to the mean of squared returns. 26 Strictly speaking, the worst case scenario will be when the expiration index value is fixed on the basis of (say) the maximum prices of each individual stock in the interval, and the closing trade of the arbitrageur in the individual stock executes at the minimum prices of that stock in that interval, and vice versa. Considering the easy availability of basket trades, this was considered too remote a possibility to deserve investigation. ” Ex-post dividend data on the S&P500 index was not publicly available until recently. Published studies based on US data have potentially serious dividend misspecification problems. For example, Ma&inlay and Ramaswamy (1988) and Bhatt and Caciki (1990) mix cash/futures prices on the S&P500 index with dividend yields from the small stock dominated NYSE/AMEX portfolio. Klemkosky and Lee (1991) assume constant semi-monthly dividend yields and estimate these by using the annual dividend yield together with the dividend flow seasonalities documented for 1982 by Gastineau and Madansky (1983). 28 In the UK stocks go ex-dividend only on Mondays and hence the exact ex-dividend date changes from year to year.

948

P.K. Yadau, P.F. Pope/Journal

ofBanking & Finance 18 (1994) 921-953

systematic biases in observed mispricing. The magnitude of ddivl is below 0.1% in about two-thirds of cases. The interquartile range of the ddiv,(y) variable-a possible measure of the risk faced by the arbitrageur on account of dividend uncertainty-is also always about O.l%, suggesting that such risks are not important. Our basic mispricing estimates are also based on the assumption that the average time interval between the ex-dividend date and the actual dividend payment date is equal to the average ex-post value of this time interval for all index constituents over the sample period. Hence, we also examine a variable ddiv2 defined as the difference in estimated mispricing resulting from a misspecification of this time interval by up to 8 weeks. The mean value of ddiv2 variable, and its interquartile range, are of the order of 0.01% for all contracts and hence clearly insignificant, showing that this factor is important neither for systematic price biases nor as a determinant of arbitrage risks. Furthermore, the error in forecasting the ex-dividend date will affect only the values of mispricing during the interval between the forecast and actual ex-dividend dates. Thus the contribution to mean mispricing ouer a contract of an estimation error in this regard will be much smaller than that estimated by ddiv2. 3.7. Marking to market cash flows The impact of marking to market cash flows on arbitrage risk depends on the path of ex-post futures price changes up to maturity. Ex-post, this impact can be quantified on the basis of Cox et al. (1981, Proposition 6, p. 326). We accordingly use day-end settlement prices to estimate the day t ex-post misspecification of mispricing on this account by the following variable:

411nl(t)=

~T+T,T-r+l,T~ i

exP[?+,,,+,,&--

r 7--f

exP[r,,&--

r-

111

T>I

-1

1

d,,,(t) is a valid measure only if the futures position is rebalanced daily as indicated in Proposition 6 of Cox et al. However, arbitrageurs typically price the futures contract as if it is a forward contract except for an estimated price adjustment reflecting the additional cost of financing the futures position. A realistic estimate of the ex-post impact arising from marking to market cash flows can be made by assuming that the arbitrage position is established and held to expiration as per Trading Rule 1 with futures contracts bought or sold accordingly only at the time the arbitrage position was first established and all (possibly negative) marking to market cash flows invested in zero coupon bonds maturing at futures maturity. Accordingly, to capture the ex-post misspecification of the mispricing variable under these circumstances, the following variable is also estimated for each day t in the sample period, using day-end settlement prices:

P.K. Yadav, P.F. Pope/Journal

ofBanking & Finance 18 (1994) 921-953

949

The mean (median) impact of marking to market cash flows, whether measured by dmml or dmm2, is below 0.02%. The mean (median) impact over different contracts is also not significantly correlated with the observed average systematic bias. The contribution of marking to market cash flows to the risk of the arbitrage position, as measured by the interquartile range of d_, or dmm2, is also well below O.l%, and hence insignificant. However, while the maximum magnitude of d _l is below 0.04%, the magnitude of dmm2 exceeds 0.1% in 12% cases, and the is as high as about 0.4% in the quarter spanning maximum magnitude of d_, Black Monday. The large magnitude of the maximum and minimum values of d mm2 shows that if the futures position is not rebalanced with reference to daily price changes, the risk on this account can occasionally be substantial. Depending on the degree of risk aversion, the extreme values could potentially be important. 3.8. Absolute mispricing and time to maturity The impact on mispricing of the uncertainty about dividends, marking-to-market cash flows and future volatility should be greater for longer times to maturity leading to a wider arbitrage window and hence a higher absolute magnitude of mispricing as time to maturity increases. However, it is difficult to specify a precise functional form for the dependence of absolute mispricing on time to maturity. Hence, to investigate this dependence we sort the data into 5 groups based on time to maturity. Groups 1 to 4 correspond to monthly intervals. Group 5 is a two-month interval in view of the relatively few observations for such long times to maturity. The mean (median) absolute magnitude of mispricing increases monotonically from 0.3% (0.3%) for Group 1, to about 0.6% (0.5%) for Groups 2 and 3, to 0.7% (0.6%) for Group 4 and 0.9% (0.8%) for Group 5. Using a dummy variable based regression procedure, the hypothesis of equality of mean absolute mispricing across different groups is conclusively rejected with an F-statistic of 151.6. Similarly, using the Mood (1954) non-parametric test, the hypothesis of equality of median absolute mispricing across different groups is conclusively rejected with a chi-square statistic of 458.4. Finally, using the Jonckheere (1954) and Terpstra (1952) non-parametric test for ordered alternatives, the hypothesis of equality of medians is conclusively rejected (z-statistic 78.4) against the specific alternative that median absolute magnitude of mispricing increases monotonically with time to maturity. Consistent with the results of Ma&inlay and Ramaswamy (1988), the absolute magnitude of mispricing clearly appears to be greater for longer times to maturity. 29 This is consistent with arbitrage being perceived as more risky for longer times to maturity.

29 The mean/median mispricing in each of these groups was also analysed, and consistent with the results of Cornell (1985) and Yadav and Pope (1990) there did not appear to be a monotonic relationship between mispricing and time to maturity.

9.50

P.K. Yadau, P.F. Pope/Journal

of Banking & Finance 18 (1994) 921-953

To investigate the dependence between mispricing and expected future index volatility, the data are sorted into 5 equally sized groups on the basis of the volatility implied in option prices. Group 1 corresponds to the lowest volatility and Group 5 to the highest voIatility. So The mean ~median) mispricing increases monotonically from - 0.6% ( - 0.5%) for Group 1, to - 0.2% ( - 0.2%) for Group 2, -0.1% (-0.1%) for Group 3, 0.1% (0.1%) for Group 4 and 0.4% (0.4%) for Group 5. Similarly, the proportion of positive mispricing values increases steadily from 0.16 for Group 1, 0.36 for Group 2, 0.42 for Group 3, 0.53 for Group 4 and 0.71 for Group 5. The mispricing clearly appears to be higher for higher levels of implied index volatility. 3* Using a dummy variable based regression procedure, the hypothesis of equality of mean mispricing across different groups is conclusively rejected with an F-statistic of 260.9. Similarly, using the Mood (1954) test, the hypothesis of equality of median mispricing across different groups is conclusively rejected with a chi-square statistic of 619.9. Finally, using the Jonckheere (1954) and Terpstra (1952) test for ordered alternatives, the hypothesis of equality of medians is conclusively rejected (z-statistic 33.3) against the specific alternative that median mispricing increases monotonically with volatility implied in index option prices. The positive association between mispricing and implied volatility is opposite to what can be expected if implied call volatility affects futures mispricing only because of the possible value of the tax timing option. However, it is important to note that the strong positive dependence of futures mispricing on call implied volatility can potentially be driven by “‘mispricing” of the index call (relative to the cash index) tending to be in the same direction as the “mispricing” of the futures contract relative to cash (just as the far futures contract has been found to be mispriced in the same direction as the near futures contract). Because of the difficulties involved in trading the index basket of stocks, market makers in options markets are likely to use index futures, rather than the cash index, to hedge their positions, and this can lead to index calls being mispriced in the same direction as index futures.

3o To avoid potential distortion due to outliers, a period of twenty trading days starting on Black October 16, 1989, are omitted from the Monday, October 19, 1987, and the day of the “mini-crash”, data. 3’ On the other hand, the mean/median absolute magnitude of m&pricing in each of these groups was also examined, and there did not appear to be a monotonic relationship between the absolute magnitude of mispricing and the volatility implied in call option prices.

P.K. Yadau, P.F. Pope/Journal

of Banking & Finance 18 (1994) 921-953

9.51

4. Conclusions This paper has analysed empirical evidence on stock index futures pricing based on about four years of synchronous hourly data from the UK. The index is based on obligatory market-maker quotes and the arbitrage opportunities identified are hence economically significant. The paper controls for cash market settlement procedures. Several interesting results are documented. First, the absolute magnitude of mispricing has often exceeded the estimated transaction costs of the more favourably positioned categories of arbitrageurs. Simulations show that even ex-ante trading rules have provided attractive profits after transaction costs. Both the early unwinding option and the rollover option have been very valuable suggesting potential for “risky arbitrage” and little possibility of expiration day effects. The magnitude of mispricing cannot be explained by dividend uncertainties, or marking-to-market cash flows, or possible delays in trade execution. Second, mean mispricing over a contract has often been significantly different from zero, but the direction of significant mean mispricing has varied substantially from period to period in a manner which cannot be explained by variation in transaction costs. Third, the direction of mispricing in the near contract at any particular time has tended to be the same as the direction of mispricing in the far contract at that time. Fourth, though there has been a mild tendency for futures to be underpriced in sharply falling markets and overpriced in sharply rising markets, this has been essentially of no economic significance for index arbitrageurs. Fifth, the absolute magnitude of mispricing has been greater for longer times to maturity consistent with arbitrage being perceived as more risky when time to maturity is greater. Finally, there appears to have been a strong positive relationship between futures mispricing and the ex-ante market volatility implied by index call option prices. The direction of this relationship is found to be opposite to that predicted by the existence of a tax timing option, but is consistent with index calls being “mispriced” (relative to the cash index) in the same direction as index futures contracts. Overall, the results reported in this paper suggest that mispricing is more likely to represent profit opportunities rather than risk premia.

References Bhatt, S. and N. Cakici, 1990, Premiums on stock index futures-some evidence, Journal of Futures Markets 10, 367-375. Brennan, M.J. and ES. Schwartz, 1990, Arbitrage in stock index futures, Journal of Business 63, s7-s31. Canina, L. and S. Figlewski, 1991, The information content of implied volatility, New York University Stem School of Business Working Paper. Chung, Y.P., 1991, A transactions data test of stock index futures market efficiency and index arbitrage profitability, Journal of Finance 46, 1791-1809.

952

P.K. Yadau, P.F. Pope/Journal

ofBanking & Finance 18 (1994) 921-953

Cohen, K.J., SF. Maier, R.A. Schwartz and D.K. Whitcomb, 1986, The microstructure of securities markets (Prentice Hall, Englewood Cliffs, N.J.) Cornell, B., 1985, Taxes and the pricing of stock index futures, Journal of Futures Markets 5, 89-101. Cornell, B. and K.R. French, 1983, Taxes and the pricing of stock index futures, Journal of Finance 38, 675-694. Cox, J.C., J.E. Ingersoll Jr. and S.A. Ross, 1981, The relation between forward prices and futures prices, Journal of Financial Economics 9, 321-346. Day, T.E. and CM. Lewis, 1992, Stock market volatility and the information content of stock index options, Journal of Econometrics 52, 267-287. Figlewski, S., 1984, Hedging performance and basis risk in stock index futures, Journal of Finance 39, 657-669. Garbade, K.D. and W.L. Silber, 1983, Price movements and price discovery in futures and cash markets, The Review of Economics and Statistics 65, 289-297. Gastineau, G. and A. Madansky, 1983, S&P 500 stock index futures evaluation tables, Financial Analysts Journal, 30, 68-76. Gould, F.J., 1988, Stock index futures: the arbitrage cycle and portfolio insurance, Financial Analysts Journal, July. Harvey, C.R. and R.E. Whaley, 1992, Dividends and S&P 100 index option valuation, Journal of Futures Markets, 12, 123-137. Hemler, M.L. and F.A. Longstaff, 1991, General equilibrium stock index futures prices: theory and empirical evidence, Journal of Financial and Quantitative Analysis 26, 287-308. Jonckheere, A.R., 1954, A distribution free k-sample test against ordered alternatives, Biometrika 41, 133-145. Klemkosky, R.C. and J.H. Lee, 1991, The intraday expost and ex ante profitability of index arbitrage, Journal of Futures Markets 11, 291-311. Ma&inlay, C. and K. Ramaswamy, 1988, Program trading and the behavior of stock index futures prices, Review of Financial Studies 1, 137-158. Merrick, J.J., Jr., 1989, Early unwindings and rollovers of stock index futures arbitrage programs: analysis and implications for predicting expiration day effects, Journal of Futures Markets 9, 101-110. Modest, M. and M. Sundaresan, 1983, The relationship between spot and futures in stock index futures: some preliminary evidence, Journal of Futures Markets 3, 15-41. Mood, A.M., 1954, On the asymptotic efficiency of certain nonparametric two-sample tests, Annals of Mathematical Statistics 25, 514-522. Pope, P.F. and P.K. Yadav, 1992, Transaction cost thresholds, arbitrage activity, and index futures pricing, NYU Salomon Center Working Paper S92-38. Pope, P.F. and P.K. Yadav, 1994, The impact of short sales constraints on stock index futures prices: Evidence from FTSElOO futures, Journal of Derivatives 1, 15-26. Puttonen, V. and T. Martikainen, 1991, Short sales restrictions: implications for stock index arbitrage, Economics Letters 37, 159-163. Scholes, M. and J. Williams, 1977, Estimating betas from non-synchronous data, Journal of Financial Economics 5, 309-328. Sofianos, G., 1990, Index arbitrage profitability, NYSE Working Paper #90-04 (draft dated November 27, 1990). Stall, H.R., 1989, Inferring the components of the bid-ask spread: theory and empirical tests, Journal of Finance 44, 115-134. Stall, H.R. and R.E. Whaley, 1987, Program trading and expiration day effects, Financial Analysts’ Journal, March, 16-27. Strickland, C.R. and X. Xu, 1992, Behaviour of the FTSElOO basis, Review of Futures Markets (forthcoming).

P.K. Yadav, P.F. Pope/Journal

ofBanking & Finance 18 (1994) 921-953

953

Terpstra, T.J., 1952, The asymptotic normality and consistency of Kendall’s test against trend, when ties are present in one ranking, Indag Math 14, 327-333. Yadav, P.K. and P.F. Pope, 1990, Stock index futures pricing: international evidence, Journal of Futures Markets 10, 573-603. Yadav, P.K. and P.F. Pope, 1991, Testing index futures market efficiency using price differences: a critical analysis, Journal of Futures Markets, 11, 239-252. Yadav, P.K. and P.F. Pope, 1992a, Intraweek and intraday seasonalities in stock market risk premia: cash vs futures, Journal of Banking and Finance 16, 233-270. Yadav, P.K. and P.F. Pope, 1992b, Mean reversion in stock index futures mispricing: evidence from the US and the UK, Conference Paper, American Finance Association Annual Meeting, January 1993. Yadav, P.K. and P.F. Pope, 1992c, Pricing of stock index futures spreads: theory and evidence, Conference Paper, Western Finance Association Annual Meeting, June 1993. Yadav, P.K., P.F. Pope and K. Paudyal, 1994, Threshold autoregressive modelling in finance: The price differences of equivalent assets, Mathematical Finance 4, 205-221.