NONTRADABLE MARKET INDEX AND ITS
DERIVATIVES
by
Peng Xu
A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
Graduate Department of Economics
University of Toronto
c©Copyright by Peng Xu (2009)
Nontradable Market Index and Its Derivatives
Peng Xu
Ph.D. 2009
Graduate Department of Economics
University of Toronto
Abstract
The S&P 500 Index is a leading indicator of U.S. equities and is meant to reflect the
risk and return on the U.S. stock market. Many derivatives based on the S&P 500 are
available to investors. The S&P 500 Futures of the Chicago Mercantile Exchange and
the S&P 500 Index Options of the Chicago Board Options Exchange are both actively
traded.
This thesis argues that the S&P 500 Index is only a summary statistic designed to
reflect the evolution of the stock market. It is not the value of a self-financed tradable
portfolio, and its modifications do not coincide with changes of the value of any mimicking
portfolio, due to the particular way the S&P 500 Index is computed and maintained.
Therefore, the Spot-Futures Parity and the Put-Call Parity do not hold for the S&P 500
Index and its derivatives. Furthermore, its derivatives cannot be priced by using the
standard option pricing models, which assume that the underlying asset is tradable.
ii
Chapter One analyzes why the S&P 500 Index does not represent the value of a self-
financed tradable portfolio and why it cannot be replaced by the value of a tracker such
as the SPDR. In particular, we show that the nonlinear and extreme risk dynamics of
the SPDR and of the S&P 500 Index are very different.
Chapter Two provides empirical evidence that the non-tradability of the S&P 500
Index can explain the Put-Call Parity deviations. Even after controlling for the liquidity
risk of the options, we find that the Put-Call Parity implied dividends depend significantly
on the option strike.
In Chapter Three, we develop an affine multi-factor model to price coherently various
derivatives such as forwards and futures written on the S&P 500 Index, and European put
and call options written on the S&P 500 Index and on the S&P 500 futures. We consider
the cases when the underlying asset is self-financed and tradable and when it is not,
and show the difference between them. When the underlying asset is self-financed and
tradable, an additional arbitrage condition has to be introduced and implies additional
parameter restrictions.
iii
Acknowledgements
This dissertation could not have been accomplished without Professor Christian Gourier-
oux, who not only served as my supervisor but also encouraged and challenged me
throughout my academic program. I sincerely appreciate his guidance and patience.
I would also like to thank Professor Alan White, who gave me many useful suggestions
throughout the dissertation process.
I am grateful to Professor Varouj Aivazian, Professor Frank Mathewson, Professor
Angelo Melino and Professor James Pesando, who encouraged me to enter the Ph.D.
program and helped me in various aspects.
A special thank you goes to Professor Joann Jasiak for her enormous encouragements
and her kindness in reading through the entire thesis and having many helpful discussions
with me.
Finally, I want to thank my parents, Chunhong Wang and Xingli Xu, and my husband,
Yu Chen, for their infinite patience and moral support.
iv
Contents
1 Is the S&P 500 Index tradable? 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 The S&P 500 Index, the Trackers and the Index Derivatives . . . . . . . 7
1.2.1 Description of the S&P 500 Index . . . . . . . . . . . . . . . . . . 7
1.2.2 Mimicking the S&P 500 . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.2.1 S&P 500 Index Funds . . . . . . . . . . . . . . . . . . . 16
1.2.2.2 Exchange Traded Funds . . . . . . . . . . . . . . . . . . 17
1.2.2.3 Static Comparison of the Relative Price Changes of the
SPDR and of the S&P 500 Index . . . . . . . . . . . . . 22
1.2.2.4 Dynamic Comparison of the SPDR and the S&P 500 Index 26
1.3 The Effects of the Non-Tradability of the Index . . . . . . . . . . . . . . 29
1.3.1 Derivative Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.3.2 Spot-Futures Parity and Put-Call Parity . . . . . . . . . . . . . . 30
1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2 The S&P 500 Index Options and the Put-Call Parity 37
v
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 S&P 500 Index Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.1 The Characteristics of Traded Options . . . . . . . . . . . . . . . 40
2.2.2 The Activity on the Option Market . . . . . . . . . . . . . . . . . 42
2.3 The Put-Call Parity Implied Dividends of the S&P 500 Index . . . . . . 49
2.3.1 Calculation of the Implied Dividends . . . . . . . . . . . . . . . . 49
2.3.2 Data and Empirical Results . . . . . . . . . . . . . . . . . . . . . 51
2.3.3 Discussions of the Empirical Results . . . . . . . . . . . . . . . . 55
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Non-tradable S&P 500 Index and the Pricing of Its Traded Derivatives 58
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 The Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.1.1 Historical Dynamics of the Index . . . . . . . . . . . . . 61
3.2.1.2 Specification of the Stochastic Discount Factor . . . . . 64
3.2.2 Pricing Formulas for European Derivatives Written on the Index . 66
3.2.2.1 Power Derivatives Written on the Index . . . . . . . . . 67
3.2.2.2 The Risk-free Term Structure . . . . . . . . . . . . . . . 67
3.2.2.3 Forward Prices for the S&P 500 Index . . . . . . . . . . 69
3.2.2.4 Futures Prices . . . . . . . . . . . . . . . . . . . . . . . 70
3.2.2.5 European Call and Put Options Written on the Index . 71
vi
3.2.3 Pricing Formulas for European Derivatives Written on Futures . . 72
3.2.3.1 Derivatives Written on Futures . . . . . . . . . . . . . . 72
3.2.3.2 European Call Options Written on Futures . . . . . . . . 74
3.3 Parameter Restrictions for a Tradable Index . . . . . . . . . . . . . . . . 74
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
A Official Description of the Index 78
B Proofs of Propositions 85
B.1 Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
B.2 Proof of Proposition 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
B.3 Proof of Proposition 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
B.4 Proof of Proposition 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
B.5 Proof of Proposition 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
B.6 Proof of Proposition 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
B.7 Proof of Proposition 3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
B.8 Proof of Proposition 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
C Tables and Figures 98
vii
List of Tables
C.1 Summary Statistics of the SPDR Traded from Jan 2, 2001 to Dec 30, 2005. 99
C.2 Ljung-Box Statistics for Daily Relative Price Changes of the S&P 500
Index and the SPDR ( XR and DRR) from Jan 2, 2001 to Dec 30, 2005. . 100
C.3 Ljung-Box Statistics for Daily Holding Period Returns of the S&P 500
Index and the SPDR ( XR and DRRd) from Jan 2, 2001 to Dec 30, 2005. 100
C.4 Ljung-Box Statistics for Squared Daily Relative Price Changes of the
S&P 500 Index and the SPDR ( XR2 and DRR
2) from Jan 2, 2001 to
Dec 30, 2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
C.5 Ljung-Box Statistics for Squared Daily Holding Period Returns of the
S&P 500 Index and the SPDR ( XR2 and DRR
2) from Jan 2, 2001 to
Dec 30, 2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
C.6 Times-to-Maturity of the S&P 500 Index Options . . . . . . . . . . . . . 102
C.7 Summary Statistics of the Options Traded from Jan 2, 2003 to Dec 31,
2003 (252 days). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
viii
C.8 Summary Statistics of the Actively Traded Options (with traded volume
≥ 2,000 contracts) from Jan 2, 2003 to Dec 31, 2003 (252 days). . . . . . 105
C.9 Summary Statistics of the Actively Traded Call and Put Options with the
Same Strike Prices and Times-to-Maturity from Jan 2, 2003 to Dec 31,
2003 (252 days). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
C.10 Summary Statistics of Implied Dividends, Q(t,K,T-t), with Treasury Bill
Rates as Proxies for the Risk-free Rates . . . . . . . . . . . . . . . . . . . 106
C.11 Summary Statistics of Implied Dividends, Q(t,K,T-t), with Zero Rates as
Proxies for the Risk-free Rates . . . . . . . . . . . . . . . . . . . . . . . . 107
ix
List of Figures
C.1 Bid-Ask Spread and Trading Volume (shares) of the SPDR from Jan 2,
2001 to Dec 30, 2005 (1,256 observations) . . . . . . . . . . . . . . . . . . 108
C.2 Histograms of Bid-Ask Spread and Trading Volume (shares) of the SPDR
from Jan 2, 2001 to Dec 30, 2005 . . . . . . . . . . . . . . . . . . . . . . 109
C.3 The S&P 500 Index and the SPDR×10 from Jan 2, 2001 to Dec 30, 2005
(1,256 observations). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
C.4 Level Difference between the S&P 500 Index and the SPDR × 10 from
Jan 2, 2001 to Dec 30, 2005. . . . . . . . . . . . . . . . . . . . . . . . . . 111
C.5 Level Difference between the S&P 500 Index and the SPDR × 10 from
Feb 1, 1993 to Dec 30, 2005. . . . . . . . . . . . . . . . . . . . . . . . . . 111
C.6 Daily Relative Price Changes (log xt−log xt−1) of the S&P 500 Index (XRt)
and the SPDR (DRRt) from Jan 2, 2001 to Dec 30, 2005 (1,255 observations).112
C.7 Daily Holding Period Returns (log(xt + dt) − log xt−1) of the S&P 500
Index (XRdt ) and the SPDR (DRR
dt ) from Jan 2, 2001 to Dec 30, 2005
(1,255 observations). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
x
C.8 Daily Relative Price Change Difference between the S&P 500 Index and
the SPDR from Jan 2, 2001 to Dec 30, 2005. . . . . . . . . . . . . . . . . 114
C.9 Daily Holding Period Return Difference between the S&P 500 Index and
the SPDR from Jan 2, 2001 to Dec 30, 2005. . . . . . . . . . . . . . . . . 114
C.10 Histogram of Daily Relative Price Changes of the S&P 500 Index (XR)and
the SPDR (DRR) and Daily Holding Period Returns of the SPDR (DRRd)from
Jan 2, 2001 to Dec 30, 2005. . . . . . . . . . . . . . . . . . . . . . . . . . 115
C.11 Histogram of Relative Price Change Difference between the S&P 500 Index
and the SPDR and Histogram of Daily Holding Period Return Difference
between the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30,
2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
C.12 Squared Relative Price Changes of the S&P 500 Index (XR2t ) and the
SPDR (DRR2t ) from Jan 2, 2001 to Dec 30, 2005. . . . . . . . . . . . . . . 117
C.13 Squared Daily Holding Period Returns of the S&P 500 Index and the
SPDR from Jan 2, 2001 to Dec 30, 2005. . . . . . . . . . . . . . . . . . . 117
C.14 Autocorrelation Function for Squared Daily Relative Price Changes of the
S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30, 2005. . . . . . 118
C.15 Autocorrelation Function for Squared Daily Holding Period Returns of the
S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30, 2005. . . . . . 118
xi
C.16 Cross-Correlation Function for Squared Relative Price Changes of the
S&P 500 Index and the SPDR and Autocorrelation Function for Squared
Relative Price Changes of the S&P 500 Index from Jan 2, 2001 to Dec 30,
2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
C.17 Cross-Correlation Function for Squared Daily Holding Period Returns of
the S&P 500 Index and the SPDR and Autocorrelation Function for Squared
Daily Holding Period Returns of the S&P 500 Index from Jan 2, 2001 to
Dec 30, 2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
C.18 Autocorrelation Function of Max(exp(Rt)− k, 0) for Daily Relative Price
Changes of the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30,
2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
C.19 Autocorrelation Function of Max(exp(Rdt )− k, 0) for Holding Period Re-
turns of the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30,
2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
C.20 Autocorrelation Function of Max(k− exp(Rt), 0) for Daily Relative Price
Changes of the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30,
2005. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
C.21 Autocorrelation Function of Max(k − exp(Rdt ), 0) for Holding Period Re-
turns of the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30,
2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
C.22 Number of Times-to-Maturity on Each Trading Day in 2003 . . . . . . . 125
C.23 Times-to-Maturity on Each Trading Day in 2003 . . . . . . . . . . . . . . 125
xii
C.24 Times-to-Maturity of Options with High Volume on Each Trading Day in
2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
C.25 Strike Prices of Options with High Volume on Each Trading Day in 2003 127
C.26 Moneyness of Options with High Volume on Each Trading Day in 2003 . 127
C.27 Prices of Traded Options with Traded Volume ≥ 2,000 Contracts in 2003 127
C.28 Number and Proportion of Highly Traded Options on Each Trading Day
in 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
C.29 Number of Highly Traded Put Options and Its Proportion in Highly Traded
Options on Each Trading Day in 2003 . . . . . . . . . . . . . . . . . . . . 129
C.30 Number of Actively Traded Options with Time-to-Maturity T1 and Its
Proportion in the Actively Traded Options on Each Trading Day in 2003 130
C.31 Number of Actively Traded Options with Time-to-Maturity T2 and Its
Proportion in the Actively Traded Options on Each Trading Day in 2003 131
C.32 Number of Actively Traded Options with Time-to-Maturity T3 and Its
Proportion in the Actively Traded Options on Each Trading Day in 2003 132
C.33 Number of Actively Traded Options with Time-to-Maturity T4 and Its
Proportion in the Actively Traded Options on Each Trading Day in 2003 133
C.34 Moneyness, Total Volume and Total Value of Actively Traded Options
with Time-to-Maturity T1 on Each Trading Day in 2003 . . . . . . . . . 134
C.35 Moneyness, Total Volume and Total Value of Actively Traded Options
with Time-to-Maturity T2 on Each Trading Day in 2003 . . . . . . . . . 135
xiii
C.36 Moneyness, Total Volume and Total Value of Actively Traded Options
with Time-to-Maturity T3 on Each Trading Day in 2003 . . . . . . . . . 136
C.37 Moneyness of Actively Traded Call and Put Options with the Same Strike
Prices and Times-to-Maturity on Each Trading Day in 2003 . . . . . . . 137
C.38 Time-to-Maturity of Actively Traded Call and Put Options with the Same
Strike Prices and Times-to-Maturity on Each Trading Day in 2003 . . . . 137
C.39 Implied Dividends, Q(t,K,T-t), with Treasury Bill Rates as Proxies for the
Risk-free Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
C.40 Implied Dividends, Q(t,K,T-t), with the same t and T-t, but different K,
and with Treasury Bill Rates as Proxies for the Risk-free Rates . . . . . 138
C.41 Difference between the maximum and minimum Q(t,K,T-t), with the same
t and T-t, but different K, and with Treasury Bill Rates as Proxies for the
Risk-free Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
C.42 Implied Dividends, Q(t,K,T-t), with the same t and K, but different T-t,
and with Treasury Bill Rates as Proxies for the Risk-free Rates . . . . . 139
C.43 Difference between the maximum and minimum Q(t,K,T-t), with the same
t and K, but different T-t, and with Treasury Bill Rates as Proxies for the
Risk-free Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
C.44 Implied Dividend, Q(t,K,T-t), with Zero Rates as Proxies for the Risk-free
Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
C.45 Implied Dividends, Q(t,K,T-t), with the same t and T-t, but different K,
and with Zero Rates as Proxies for the Risk-free Rates . . . . . . . . . . 141
xiv
C.46 Difference between the maximum and minimum Q(t,K,T-t), with the same
t and T-t, but different K, and with Zero Rates as Proxies for the Risk-free
Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
C.47 Implied Dividends, Q(t,K,T-t), with the same t and K, but different T-t,
and with Zero Rates as Proxies for the Riskfree Rates . . . . . . . . . . 142
C.48 Difference between the maximum and minimum Q(t,K,T-t), with the same
t and K, but different T-t, and with Zero Rates as Proxies for the Risk-free
Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
xv
Chapter 1
Is the S&P 500 Index tradable?
1
1.1 Introduction
The S&P 500 Index is one of the most commonly used benchmarks for the overall U.S.
stock market. It is a leading indicator of U.S. equities and is meant to reflect the risk and
return of the whole market. A number of derivatives based on the S&P 500 are available
to investors. In particular, the S&P 500 futures traded in the Chicago Mercantile Ex-
change (CME) and the S&P 500 options traded in the Chicago Board Options Exchange
are actively traded because they “extend the range of investment and risk management
strategies available to investors by offering them the possibility of unbundling the market
and nonmarket components of risk and return in their portfolios” [Figlewski (1984a)].
The success of these derivatives has not only attracted the investors, but also drawn
a lot of attention from researchers who study their pricing. For example, the S&P 500
Index and its derivatives have been widely used to test the Spot-Futures Parity, the
Put-Call Parity and different kinds of pricing models.
The Spot-Futures Parity is based on the argument that the spot asset and the for-
ward contract written on it can form a risk-free portfolio in a frictionless market. This
portfolio must earn risk-free interest because otherwise the arbitrageurs will make profit
from this opportunity and close the gap quickly. The futures contract is generally dif-
ferent from the forward contract because of the mark-to-market rule. However, when
the interest rates are non-stochastic, the futures and forward prices are the same [see
Cox, Ingersoll and Ross (1981) and French (1983)], and the parity relationship holds also
for the spot asset and the futures written on it. Figlewski wrote a series of papers on
2
the hedging performance and basis risk in stock index futures and on the index-futures
arbitrage. One key result of these papers is that the Spot-Futures Parity implied by the
No-Arbitrage condition cannot be verified using the S&P 500 Index and the index futures
data. Mackinlay and Ramaswamy (1988) confirmed this result using intraday transaction
data for the S&P 500 stock index futures prices and intraday quotes for the underlying
index. Five explanations have been proposed for the deviation between the futures prices
and index values. First, the index is not self-financed in the sense that the component
stocks of the index pay dividends, while the S&P 500 Index reflects the evolution of the
market value of the 500 listed firms without reflecting the dividends paid by those stocks.
The dividends paid by the index component stocks are not known ex-ante, so arbitrageurs
have to bear additional carrying cost and risk of the dividends[See Figlewski (1984a,b)].
This implies that the arbitrage portfolio is not risk-free and hence does not necessarily
earn risk-free interest. Second, in the real world due to transaction costs such as commis-
sions and bid-ask spreads, the arbitrage trading required to reach the Spot-Futures Parity
cannot be easily completed. Spot and futures prices can deviate from their theoretical
prices without inducing an arbitrage opportunity [See Figlewski (1984a,b), Mackinlay
and Ramaswamy (1988) and reference therein]. Third, the mark-to-market rule of the
futures differentiates it from the forward contract [See Cox, Ingersoll and Ross (1981)
and French (1983)]. Daily cash flows from the futures position may include unanticipated
interest earnings or costs. The covariance between the interest rates and the futures is
almost always nonzero. As a consequence, in general the price of a forward contract with
a certain delivery date is not the same as the price of a futures with the same delivery
3
date. Fourth, there are different taxes for spot and futures earnings, and a tax-timing
option exists for a spot position, but not for a futures position1. This may reduce the
futures price below its predicted level. And fifth, a risk-free arbitrage between a large
portfolio of 500 stocks and a futures contract is impossible, since it is impossible to buy or
sell all of the stocks simultaneously at the desired size to take advantage of the short-run
deviation of the futures price from its implied level [See Figlewski (1984a)]. The first four
explanations have been carefully studied either theoretically and/or empirically. There
was no conclusion about whether they are sufficient to explain the mispricing of futures
contracts relative to the spot index. By contrast, little is known about the fifth explana-
tion. The issue here is that the S&P 500 is a pure index of stocks, that is, an artificial
number, and is non-tradable. Specifically, it is not the price of a portfolio traded in the
financial market, because the investors cannot buy a portfolio at the price of the index
one day and be guaranteed that they can sell it anytime in the future at the future price
of the index. This is because no portfolio can replicate the index perfectly due to the
1Cornell and French (1983) suggested that the discrepancy between the actual and theoretical futuresprices is caused by taxes. Capital gains and losses are only taxed when they are realized. If stock holdershave capital losses, they can transfer part of their losses to the government by selling the stocks anddeduct losses at the ordinary short term rate. If there are capital gains, the investors can delay the sellingand take advantage of the long-term capital gain rate. This so-called “timing option” is not available forfutures traders. All capital gains and losses must be realized either at the end of the fiscal year, or at theexpiry of the futures. Including the tax option in the Cornell and French model reduces the predictedfutures prices. Cornell (1985) tested this conjecture empirically and his results showed that the patternof discrepancy between the actual and predicted futures prices is not consistent with the prediction ofthe timing option model. Figlewski (1984b) and Cornell (1985) discussed the reasons why it was difficultto judge the importance of the timing option in explaining the discrepancy. The value of the timingoption is dependent on unknown parameters such as the average holding period and taxable basis forthe market stock portfolio and the average investor’s marginal tax rate. A huge number of stocks areheld by tax-exempt institutions or by taxable investors whose holding periods are already greater thanone year. If the timing option is the main reason for the futures discount, these investors should takeadvantage of it by selling the stocks and buying the futures and risk-free securities. Furthermore, thetransaction costs or tax-related constraints may also reduce the value of the timing option.
4
composition and management of the index as well as because of the frictions and the lack
of perfect liquidity of the market. In Section 1.2.1, we elaborate on the reasons why the
S&P 500 is non-tradable. We also show that some trackers such as the SPDR can mimic
the linear dynamics of the index. However, the nonlinear and extreme risk dynamics
of the SPDR are very different from those of the index. If the index is non-tradable,
investors cannot form a risk-free portfolio including the index and the futures contract,
so the Spot-Futures Parity does not hold.
The non-tradability of the index can also help explain why the Put-Call Parity does
not hold for the S&P 500 Index and the options written on it. The Put-Call Parity
implied by the No-Arbitrage condition was first formalized by Stoll (1969) and Merton
(1973a). It is based on the argument that the spot asset and its put and call options can
form a risk-free portfolio in a frictionless market. The existence of arbitrageurs ensures
that there is no arbitrage opportunity remaining in the market and the portfolio will earn
a risk-free rate. Using daily and intradaily prices of the S&P 500 and its options from
1986-1989, Kamara and Miller (1995) observed parity deviations even after controlling
for the effects of dividends and transaction costs. Ackert and Tian (2001) found a similar
price deviation using data from 1992 to 1994. They both tested whether this was due
to the “liquidity risk”, that is, the difficulty of buying or selling a portfolio of 500 stocks
simultaneously before adverse price movements occurred. They both observed supporting
evidence that the “liquidity risk” can help explain the parity deviation. However, they
both treated the S&P 500 Index as a tradable asset, which is not very liquid. However,
the index is not just illiquid, but instead completely non-tradable in the market. As
5
for the Spot-Futures Parity, if the index is not tradable in the market, investors cannot
make risk-free arbitrage transactions using the index and the options. Therefore, the
No-Arbitrage condition does not imply the Put-Call Parity for the index and its options.
Let us also consider option pricing. The S&P 500 Index options are frequently used
for testing the option pricing models for the following reasons: First, the S&P options
are among the most liquid options. Second, they are European options. And third, the
underlying asset is an index whose values are less likely to jump due to the effect of
diversification. Therefore, the index satisfies several assumptions underlying the Black-
Scholes model. However, the options are written on an index which is not traded on the
market, and thus the index itself cannot be used in the arbitrage strategy. For example,
the well-known Black-Scholes model [See Black and Scholes (1973) and Merton (1973b)]
assumes that the underlying asset price follows a geometric Brownian Motion process.
The underlying asset and the option can be used to construct an instantaneous risk-
free portfolio, which should earn an instantaneous risk-free return by the No-Arbitrage
argument. The No-Arbitrage argument along with the terminal condition of the option
determines the option price. However, if the underlying asset is not traded, the Black-
Scholes formula could be violated without creating arbitrage opportunities. This is also
true for the option pricing formulas derived from the general equilibrium models. If the
investors account for the non-tradability of the asset, consumption, investment decisions
and equilibrium prices differ from their model prediction.
The rest of this paper is organized as follows: In Section 1.2, we introduce the S&P 500
Index and its derivatives, and explain why the S&P 500 Index is non-tradable and cannot
6
be perfectly mimicked. We analyze the effect of non-tradability on the Spot-Futures
Parity, the Put-Call Parity and the option pricing in Section 1.3. We conclude in Section
1.4. Technical results and details are gathered in the Appendices.
1.2 The S&P 500 Index, the Trackers and the Index
Derivatives
1.2.1 Description of the S&P 500 Index
The S&P 500 Index is a recognized barometer of the U.S. stock market, which has been
widely used in asset pricing studies and is a benchmark for most professional investors.
The S&P 500 Index is based on the stock prices of 500 different companies – including
about 80% industrials, 3% utilities, 1% transportation companies, and 15% financial
institutions. The market value of the 500 firms of the index represents approximately
80% of the value of all stocks traded on the New York Stock Exchange.
The S&P 500 Composite Index is calculated as follows2:
indext+1
indext=
∑i pi(t+1),t+1qi(t+1),t+1∑i pi(t+1),tqi(t+1),t+1
(1.1)
=∑i
pi(t+1),tqi(t+1),t+1∑i pi(t+1),tqi(t+1),t+1
pi(t+1),t+1
pi(t+1),t
, (1.2)
where pi(t+1),t+1 is the last trading price of security i(t + 1) at time t+1, qi(t+1),t+1 is its
number of shares available to public at time t+1, and pi(t+1),t is the last trading price of
security i(t+ 1) at time t. The S&P 500 Index is updated every 30 seconds on a trading
2Appendix A discusses S&P’s presentation of the index, where a so-called ”divisor” is directly intro-duced in the calculation. It is shown that their method and the method presented here are identical.
7
day, so t+1 is generally 30 seconds later than t, except that, if t is the closing time of one
day, then t+1 is the opening time of the next day. Since September 19, 2005, the S&P 500
Index has been moving to float adjustment, that is, the index only includes shares that
are easily available to investors. The shares that are not in the float include those closely
held by other publicly traded companies, control groups, or government agencies, if they
total more than 10%. The Standard and Poor’s Agency determines what percentage
of its shares is available to the public for each company in the index. This percentage
is called the investable weight factor or IWF. The quantity qi(t),t is equal to the Index
Shares, that is, the share count that S&P uses in its index calculations, multiplied by
the IWF. The composition of the index can change over time due to the deletion and
inclusion of stock(s). Hence the set of index component stocks at t, {i(t)}500i=1, may not
be the same as that at t+1, {i(t + 1)}500i=1
3. The index ratio is computed as the sum of
the products of prices and quantities at t + 1 of all stocks, i = 1, · · · , 500, from the set
{i(t+ 1)}500i=1, divided by the sum of the products of prices at t and quantities at t+ 1 of
all stocks, i = 1, · · · , 500, from the set {i(t+ 1)}500i=1.
Equation (1.1) shows that the S&P 500 Index is essentially a Paasche chain index for
prices, where the quantities of the index component stocks at t+ 1 are used, that is,
indextindex0
=t−1∏τ=0
∑i pi(τ+1),τ+1qi(τ+1),τ+1∑i pi(τ+1),τqi(τ+1),τ+1
,
3The changes to the index composition only happen after the market closes. If time t and time t+ 1are within the same day, {i(t)}500i=1 is the same as {i(t + 1)}500i=1. If t is the closing time of one day andt+ 1 is the opening time of the next day, then {i(t)}500i=1 may be different from {i(t+ 1)}500i=1.
8
with 4index0 = 10. The change in the index is defined as a percentage change in the
total market value from one point in time to the next.
Equation (1.2) implies that the rate of return of the index equals the rate of return
that would be earned by an investor holding a portfolio that consists of all 500 stocks
in the index and is weighted in proportion to each stock’s float-adjusted market value
(which is defined here as the product of each component stock’s price and the number of
shares available to public for that company), except that the index does not reflect cash
dividends paid out by those stocks5. Indeed, the S&P 500 is not self-financed and is a
float-adjusted-capitalization-weighted (FACap-weighted) index reflecting the evolution of
the (float-adjusted) market value of the 500 listed firms. Because of the manner in which
the index is weighted, a price change in any stock will affect the index in proportion to
the stock’s relative (float-adjusted) market value.
The S&P 500 is maintained by the S&P Index Committee, which is a team of Standard
& Poor’s economists and index analysts who meet on a regular basis. The identities of
index component stocks and their share numbers used in the index computation may be
adjusted on each trading day after the market closes. In other words, {qi(t),t}500i=1 may be
different from {qi(t+1),t+1}500i=1, if t is the closing time of one day and t+1 is the opening
time of the next day. The adjustment can lead to the following effects: First, after
the market closes at t, {i(t + 1)} is different from {i(t)} and hence {qi(t+1),t+1} differs
from {qi(t),t}. This could be caused by the identity change of index components due to
4The S&P 500 index has a Base Value of 10 and a Base Period of 1941-1943.5In the S&P 500 Index, about 380 stocks pay dividends.
9
the deletion and inclusion of stock(s). For example, 176 index identity changes have
been observed from Jan 5, 2000 to March 31, 2006. Second, after the market closes at
t, {i(t + 1)} can be the same as {i(t)}, but {qi(t+1),t+1} can be different from {qi(t),t}.
That effect occurs when the stock composition of the index remains the same, but the
numbers of shares available to the public change. This happens when the total number
of outstanding shares available to the public of one or more component index securities
changes due to secondary offering, repurchases, conversions, or other corporate actions.
It is possible that both effects occur simultaneously. Changes to the S&P 500 Index are
made whenever they are needed. There is no annual or semi-annual updating frequency.
Instead, changes in response to corporate actions and market developments can be made
at any time. These changes are typically announced after the closure of a trading day,
which is two to five days before the changes are scheduled to be implemented. The Index
Committee also lays down the policies about share changes6.
The above discussion shows that the S&P 500 index is an artificial number constructed
to reflect the evolution of the market. In the remaining part of this subsection, we explain
why no self-financed portfolio can be constructed to replicate the index perfectly due to
the particular way the S&P 500 Index is calculated and maintained.
Let us consider two consecutive trading days. We use t to denote a point in time on
the first day, and t* and t’ the closing time on the first and second day, respectively.
Let us first suppose that there is no transaction cost in trading the component stocks
of the S&P 500, and that none of the stocks pays dividends. Consider an investor who
6See Appendix A for details.
10
buys a portfolio withqi(t),t∑
i pi(t),t−1qi(t),tindext−1
of each stock i(t) at time t. If the investor can buy
each stock at exactly pi(t),t, then the cost of the portfolio is equal to∑
i pi(t),tqi(t),t∑i pi(t),t−1qi(t),t
indext−1
or
indext by Equation (1.1). Suppose there is no adjustment to the index after the market
closes at t*, that is, {i(t′)} is the same as {i(t)} and {i(t∗)}, and {qi(t′),t′} is the same as
{qi(t),t} and {qi(t∗),t∗}. If the investor can sell the portfolio at exactly {pi(t′),t′}500i=1, then at
t′ s/he will get
∑i pi(t),t′qi(t),t∑i pi(t),t−1qi(t),tindext−1
=indext−1
∑i pi(t∗),t∗qi(t∗),t∗∑i pi(t),t−1qi(t),t
∑i pi(t∗+1),t∗qi(t∗+1),t∗+1∑
i pi(t∗),t∗qi(t∗),t∗
×∑
i pi(t′),t′qi(t′),t′∑i pi(t∗+1),t∗qi(t∗+1),t∗+1
=indext−1indext∗
indext−1
∑i pi(t∗+1),t∗qi(t∗+1),t∗+1∑
i pi(t∗),t∗qi(t∗),t∗
indext′
indext∗
=indext′ .
The holding period return of the portfolio isindext′indext
. Thus, without any adjustment to
the index composition, the S&P 500 can be replicated using the buy-and-hold portfolio
strategy if all the 500 stocks are perfectly liquid and the investor can buy and sell all
of them at the last trading prices at the desired time. However, not all of the stocks
in the S&P 500 are very liquid, and it is almost impossible for all of the 500 stocks’
transactions to occur at exactly the last trading prices7. Therefore, replicating the index
with the buy-and-hold strategy is almost impossible.
Suppose now that the index composition is adjusted after the stock market closes on
the first day, that is, {qi(t′),t′} is different from {qi(t),t} and {qi(t∗),t∗}, and/or {i(t + 1)}7Indeed, the smallest stocks included in the index are not always very liquid, which explains why the
time interval between two trading can be quite large.
11
is different from {i(t)} and {i(t∗)}. Let us consider again the investor who buys at time
t an index replicating portfolio ofqi(t),t∑
i pi(t),t−1qi(t),tindext−1
of each stock i(t) with portfolio value
indext by Equation (1.1). The value of the portfolio at t′ is
∑i pi(t),t′qi(t),t∑i pi(t),t−1qi(t),tindext−1
=indext−1
∑i pi(t∗),t∗qi(t∗),t∗∑i pi(t),t−1qi(t),t
∑i pi(t∗+1),t∗qi(t∗+1),t∗+1∑
i pi(t∗),t∗qi(t∗),t∗
×∑
i pi(t),t′qi(t),t∑i pi(t∗+1),t∗qi(t∗+1),t∗+1
=indext∗
∑i pi(t∗+1),t∗qi(t∗+1),t∗+1∑
i pi(t∗),t∗qi(t∗),t∗
∑i pi(t),t′qi(t),t∑
i pi(t∗+1),t∗qi(t∗+1),t∗+1
6=indext∗indext′
indext∗
=indext′ .
After the adjustment, the portfolio that replicates the index at time t no longer replicates
the index at time t′. In order for the portfolio to replicate the index both before and
after the adjustment, the investor has to adjust his/her portfolio allocation and hold
qi(t′),t′∑i pi(t′),t∗qi(t′),t′
indext∗
of each stock i(t′) at t*. The sales and purchases of the stocks have to be
made at the closing prices of the first day, {pi(t),t∗} and {pi(t′),t∗}, so that this portfolio can
always have the same value as the index. To see this, we have to show that the value of
the portfolio withqi(t),t∑
i pi(t),t−1qi(t),tindext−1
of each stock i(t) at time t is equal to indext by Equation
(1.1). The value of the same portfolio at t* is∑
i pi(t),t∗qi(t),t∑i pi(t),t−1qi(t),t
indext−1
, which is equal to indext∗ .
The value of the portfolio withqi(t′),t′∑
i pi(t′),t∗qi(t′),t′indext∗
of each stock i(t′) at t* is∑
i pi(t′),t∗qi(t′),t′∑i pi(t′),t∗qi(t′),t′
indext∗
,
which is still equal to indext∗ . The value of the portfolio withqi(t′),t′∑
i pi(t′),t∗qi(t′),t′indext∗
of each stock
i(t′) at t’ is∑
i pi(t′),t′qi(t′),t′∑i pi(t′),t∗qi(t′),t′
indext∗
, which is equal to indext′ . If there is an adjustment to the
index composition and the investor can make the corresponding adjustment to his/her
12
replicating portfolio using the above mentioned self-financing strategy, the index can still
be replicated. If there is one adjustment to the index, the adjustment to the portfolio has
to involve 500 stocks or more. The new quantity,qi(t′),t′∑
i pi(t′),t∗qi(t′),t′indext∗
, of each stock that the
investor has to hold depends on {pi(t′),t∗}, {qi(t′),t′}, {pi(t∗),t∗} and {qi(t∗),t∗}. Quantities
{qi(t′),t′} and {qi(t∗),t∗} are usually known before the market is closed. Prices {pi(t′),t∗}
and {pi(t∗),t∗} are the closing prices of {i(t′)} and {i(t∗)} at t*, and are not known until
the market is closed. Therefore, if the investor decides to adjust the portfolio before t∗8,
s/he has to estimate the closing prices pi(t′),t∗ and pi(t∗),t∗ for about 500 stocks, determine
the quantity of each stock to be held based on his/her estimations, and trade the stocks
at prices as close as possible to the closing prices. To minimize the hedging error, the
estimation and the trading should be made as near as possible to the end of the trading
day. Because the market is not perfectly liquid, it may not be possible to trade the stocks
instantaneously and it may take some time to complete the transactions for all stocks.
The lack of perfect foresight and perfect liquidity may lead both the quantities and prices
away from those desired, and this divergence could involve all the relevant stocks. This
can cause large tracking errors. If the investor instead decides to adjust the portfolio at
the beginning of the second day, the opening prices of the second day are generally not
the same as the closing prices of the first day due to unexpected overnight information
flow and market adjustment9, so the investors still cannot obtain the desired quantity
8Empirical evidence shows that in order to minimize tracking errors most institutional investors suchas index funds choose to adjust the portfolio on the effective day, that is, the day after the closure atwhich the change to the index becomes effective [See e.g. Beneish and Whaley (1996) and Cusick (2002)].
9As shown in Beneish and Whaley (1996), Lynch and Mendenhall (1997) and Cusick (2002), after theannouncement of the index addition and/or deletion, there is a significant positive (resp. negative) meanabnormal return until the effective day for the stock to be added (resp. deleted). The overnight and
13
of each stock i(t′) at the desired price. In summary, without perfect foresight and a
perfectly liquid market, it is impossible for the investor to obtain the desired quantity of
each stock i(t′) at the closing price of t*. Since the investor’s portfolio allocation on the
second day is not equal to the index allocation, we still conclude that the S&P 500 index
cannot be perfectly replicated.
Moreover, because the adjustment to the index occurs whenever it is needed, investors
cannot perfectly anticipate how many changes there will be and what changes will take
place during their holding period of the portfolio. This adds more uncertainty to the
future value of the replicating portfolio. Investors cannot buy a portfolio at the price
equal to the index value today and expect to sell it at the price equal to the future index
value later10.
Now let us also take into account the transaction costs and dividends paid by the
component stocks. When an investor holds a portfolio that replicates the index, s/he not
only has to pay the transaction costs when purchasing and selling the portfolio of about
500 stocks, but s/he also needs to pay for the transaction costs at each time the index and
hence the portfolio are adjusted. This makes the replicating strategy described above no
daily mean abnormal returns after the effective day are negative (resp. positive). The trading volume ofthe stock starts rising after the announcement day and peaks on the effective day. What could happen isthat, after the announcement of the change, the risk arbitrageurs know that there will be a huge demand(resp. supply) for the stock to be added (resp. deleted) on the effective day. Then, they start to buy(resp. sell) the stock, driving the price up (resp. down). The index fund managers do not want to buy(resp. sell) the stock earlier because they may not be rewarded for creating “tracking errors”. On theeffective day, the fund managers buy (resp. sell) the stock. The next day, the risk arbitrageurs sell (resp.buy) the extra stocks due to their overestimation of the demand (resp. supply) of the fund manager.
10This is consistent with the empirical results in Mackinlay and Ramaswamy (1988), where the mis-pricing of the futures relative to the spot is positively related to time-to-maturity. For a very shorttime-to-maturity, there is little chance that an adjustment to the index will occur. So investors canuse the buy-and-hold strategy to replicate the index and take advantage of the arbitrage opportunity, ifthere is one after considering the transaction cost, dividend, etc.
14
longer self-financing. Indeed, if most of the index mimicking fund managers adjust their
portfolio about the same time, the transaction cost could be very high. Moreover, the
number of changes is uncertain, which increases the risk on the portfolio and makes it
even more difficult for an investor to replicate the index.
There are about 380 stocks in the S&P 500 Index that pay dividends. The payment
dates and amounts of dividends are generally not perfectly predictable. This by itself
does not cause the non-tradability of the index since a lot of securities traded in the
market pay dividends which are not known ex-ante11. If the market is frictionless, with
perfect liquidity and there is no adjustment to the index, the portfolio with quantity
qi(t),t∑i pi(t),t−1qi(t),t
indext−1
of each stock i(t) purchased at price qi(t),t at time t replicates the index
at time t with value indext. Whenever there is an ex-dividend of the index component
stocks, the relevant stock prices and the index level may adjust accordingly. If the
replicating portfolio pays exactly the same amount of dividends as the index component
stocks, the value of the portfolio will always mimic the index exactly. That is, the index
can be seen as the price of the replicating portfolio which pays dividends and is not
self-financing itself. But as discussed above, even without dividends, the index cannot
be replicated perfectly in a market with friction, lack of perfect liquidity, and periodical
adjustments to the index. The presence of dividends introduces additional risks on the
replicating portfolio. When an investor buys a replicating portfolio at t, s/he not only is
uncertain about the dates and sizes of the future dividends of each component stock, but
11Although the uncertainty of the dividends does not necessarily cause the non-tradability of the index,it may cause Spot-Futures Parity and Put-Call Parity deviations.
15
also about the identities and quantities of the stocks that will be held and pay him/her
dividends during the holding period. Because investors do not know ex-ante what change
will occur to the index, how many changes there will be and when they will occur, they
do not know what stocks and how many shares of these stocks they will have to hold
later on to replicate the index. This aggravates the uncertainty about the dividends of
the replicating portfolio.
Since the S&P 500 Index is non-tradable, many financial products have been intro-
duced on the market to mimic the index. We discuss below the so-called S&P 500 Index
Funds and Exchange Traded Funds12.
1.2.2 Mimicking the S&P 500
1.2.2.1 S&P 500 Index Funds
S&P 500 Index Funds are mutual funds seeking to replicate and track the performance
of the S&P 500 Index. This is accomplished by holding either all of the securities in the
index in the appropriate proportions, or by holding a selected sample of securities that
closely track the desired index. Like any other fund, the index funds entail operating
expenses and transaction costs. Typically, an index fund distributes to shareholders
its net income (interest and dividends, less expenses) as well as any net capital gains
realized from the sales of its holdings. For example, the largest index fund, Vanguard
Index Fund, distributes its income dividends in March, June, September and December,
and its capital gains in December. In addition, the fund may occasionally be required
12Investors also use index futures and risk-free securities to mimic the spot index, although there areviolations of the Spot-Futures Parity.
16
to make supplemental distributions at some other time during the year. Like any open-
end fund, index funds can only be purchased or sold at their end-of-day net asset value.
This implies that the index funds are illiquid within the day and cannot mimic the
S&P 500 Index continuously, while the S&P 500 Index value changes continuously within
the trading day. Therefore, the existence of index funds does not solve the problem of
non-tradability of the S&P 500 Index. Furthermore, in this paper we are interested in
how the non-tradability of the index affect its derivative pricing. Both the futures and
options written on the index are actively traded in the markets and have the opening
prices on the maturity dates as the settle prices. Therefore, we will now focus our analysis
on the mimicking funds which can be traded any time within the trading day.
1.2.2.2 Exchange Traded Funds
The Standard and Poor’s Depositary Receipt (SPDR), often referred to as the “spider”,
is an exchange traded fund which, as stated in the prospectus, holds all of the S&P 500
Index stocks and is designed to reflect the price and yield performance of the S&P 500
Index. As of September 30, 2005, SPDR had $47 billion in total assets. The SPDR Trust
issues and redeems SPDRs only in multiples of 50,000 SPDRs (referred to as “Creation
Units”) in exchange for S&P 500 Index stocks and cash. For example, to create (resp.
redeem) 50,000 SPDRs, the investor will deposit with (resp. be delivered by) the Trustee
a specified portfolio of Index Securities and a cash payment generally equal to dividends
(net of expenses) accumulated up to the time of deposit. The first creation units were
deposited on January 22, 1993. The Trust is scheduled to terminate no later than January
17
22, 2118, but may terminate earlier under certain circumstances. The Creation Units are
redeemable only in kind and not redeemable for cash. The creation and redemption
take place after the closure of the trading day. Regardless of the number of Creation
Units created or redeemed, the investor has to pay $3,000 for each transaction. Each
SPDR represents an undivided ownership interest in the SPDR Trust and has a price
approximately equal to one-tenth of the Index level. It is traded on the American Stock
Exchange (AMEX) like any other equity security at any time within the trading day.
Thus the purchases and sales of the SPDRs are subject to transaction costs such as
bid-ask spread and ordinary brokerage commissions and charges. The minimum trading
unit is one SPDR. The SPDR is an actively traded security. As shown in Table C.1, the
average daily trading volume is over 38 million shares and the average trading value per
day is over $4 billion. About 11.2% of the total shares are traded every day. The Trust
charges a very low expense fee (18.45 basis point in the past13) to the SPDR holders.
The SPDR Trust pays dividends. The ex-dividend date is the third Friday in March,
June, September and December. Beneficial owners are entitled to receive the dividends
accumulated through the quarterly dividend period which ends on the business day before
the ex-dividend day. The dividend is the amount of any cash dividends declared on the
SPDR portfolio during the corresponding period14, net of fees and expenses associated
with the operations of the Trust, and taxes, if applicable. It is paid on the last business
day of April, June, October and January. Due to the fees and expenses, the dividend
13See Elton, Gruber, Comer and Li (2002).14The dividends that the trust receives from the SPDR portfolio are held in an nonearning account
before being distributed to the investors.
18
yield for SPDRs is usually less than that of the stocks in the S&P 500 Index. The trustees
may also distribute the capital gains in January and require an additional distribution
shortly after the end of the year.
To match the composition and weights of stocks held by the Trust with component
stocks of the S&P 500 Index, the trustee adjusts the composition and weights of stocks
held by the Trust periodically to respond to changes in the identity and/or weighting of
the Index. The trustee aggregates certain of these adjustments and makes changes at least
monthly or even more frequently in the case of significant changes to the S&P 500 Index.
Any change in the identity or weighting of an Index Security will cause an adjustment to
the portfolio held by the Trust effective on any day that the New York Stock Exchange
(NYSE) is open for business following the day on which after the close of the market the
change to the S&P 500 Index takes effect.
There are several reasons why the SPDR may not mimic the index perfectly. First,
the SPDR pays dividends later than the index component stocks and holds the cash
dividends in a non-earning account, that is, the cash dividends are not reinvested in the
SPDR. The SPDR also charges management expenses, which lower the amount of the
dividends to be paid. So the timing and the amount of the cash flows from the SPDR are
different from those of the index securities, which may cause the value of the SPDR to
diverge from the index level. Second, the SPDR needs to make occasional adjustments
to its portfolio to respond to the adjustment to the index. As discussed in the last
subsection, this kind of adjustments may be expensive and lead to differences between
the SPDR portfolio and the index portfolio. Third, the index is not traded on the market
19
while the SPDR is, which can induce a liquidity premium for the SPDR. In summary,
due to the different timing and amount of dividends paid by the Index securities and the
SPDR, the way the SPDR deals with the dividends, the expenses of the SPDR Trust,
the maintenance of the index and SPDR portfolio, and the different forces that drive the
price of the SPDR and the prices of the securities components of the index, the dynamics
of the S&P 500 Index and the SPDR can diverge.
In the remainder of this subsection, we examine how closely the SPDR mimics the
S&P 500 Index using daily data from January 2, 2001 to December 30, 200515.
We will check the mimicking performance of the SPDR in two ways. First, we examine
how the relative price change of the SPDR mimics the relative change of the index.
Second, we study how the holding period return of the SPDR mimics the return on the
index. In the first case, the relative price change of the SPDR is defined as
Rt+1 = log(SPDRt+1)− log(SPDRt). (1.3)
In the second case, the dividend from the SPDR is reinvested on the ex-dividend day and
the holding period return over one day is defined as
Rdt+1 = log(SPDRt+1 + dt+1)− log(SPDRt), (1.4)
15Beaulieu and Morgan (2000) studied the high-frequency relationships between the S&P 500 Indexand the SPDR by using minute-by-minute data for November 1997 through February 1998. The authorsused both the covariance estimator of De Jong and Nijman (1997) and the GMM estimation of systems ofsimultaneous equations using imputed data (both filtered and unfiltered). Many one minute intervals didnot contain a SPDR transaction and the two methods dealt with the missing observations in differentways, so the conclusion from the two methods were slightly different. Specifically, they found thatthe SPDR led the index by one minute from the De Jong and Nijman estimator and the filtered dataindicated that the index led the SPDR by one minute with the GMM. In either case, the SPDR did nottrack the index perfectly.
20
where SPDRt is the price of the SPDR at t and dt+1 is the dividend with t + 1 as the
ex-dividend day. Since the index itself pays no dividend, its holding period return is the
same as the relative change defined by Equation (1.1). If the holding period return can
replicate the return of the index, the index can be replicated by the SPDR using the
self-financing strategy.
The data source is the Center for Research in Security Prices (CRSP).
For the S&P 500 Index, we use the level of the Standard & Poor’s 500 Composite
Index at the end of the trading day. These data were collected from publicly available
sources such as the Dow Jones News Service, The Wall Street Journal or the Standard
& Poor’s Statistical Service. While the index does not include dividends, it indicates the
change in price of the component securities.
For the SPDR price, we use the average of the closing bid and ask prices16, which is
the average of the bid and ask prices from the last representative quote before the market
closes. The relative price change is calculated using this data. Figure C.1 presents the
bid-ask spread and the trading volume of the SPDR from January 1, 2001 to December
30, 2005. More than fifteen percent of observations had a negative bid-ask spread. We
do not know whether this is due to data recording errors or whether there are arbitrage
opportunities to be taken advantage of in the market. We do not remove these observa-
tions because their averages do not seem to be outliers. The bid-ask spread of the earlier
sample period is very volatile. This could be caused by the so-called “dotcom bubble”,
16There are 17 days which do not have quotation prices and we use closing prices to replace the averageof bid and ask prices. We set the bid-ask spreads of these days equal to 0.
21
when the stock prices, the index and the SPDR were very volatile. As we can see from
the second graph of Figure C.1, which covers June 1, 2001 to December 30, 2005, most
bid-ask spreads are within the interval (-0.1, 0.1). The mean of the spread is 0.13 and
the variance is 0.27. There are 379 out of 1,256 observations with a bid-ask spread equal
to 0.01. The result is confirmed in Figure C.2 where the historical distributions of the
bid-ask spread and trading volume of SPDR as well as their joint distribution are pre-
sented. The average daily trading volume of the SPDR is about 38 million shares. The
correlation coefficient between the spread and daily volume is -0.31, which is consistent
with the stylized fact that the higher the trading volume, the lower the bid-ask spread.
For the daily holding return of the SPDR, we use the log of ”holding period return”
data from CRSP which is calculated as shown in Equation (1.4) except that SPDRt is
the last sale price of the SPDR at time t.
1.2.2.3 Static Comparison of the Relative Price Changes of the SPDR andof the S&P 500 Index
Figure C.3 shows the SPDR price and the S&P 500 Index level. The SPDR price is
multiplied by 10 and denoted SPDR× 10. As expected, the historical means of the two
time series are very close, as are the historical variances, skewnesses and kurtosis, since
the SPDR is intended to mimic the value of the S&P 500 Index. Figure C.4 shows the
time series of the difference between the SPDR× 10 and the S&P 500. We observe that
the difference diminishes regularly on the ex-dividend days of the SPDR, which creates
a periodic non-stationary feature. In Figure C.4, we see that the difference between the
22
SPDR× 10 and the S&P 500 can be as large as 19.76 index points17. On February 6th,
2001, the SPDR×10 was 19.76 index points below the S&P 500 Index. On that day, the
Standard and Poor’s added (and correspondingly deleted) one stock component. This
change could explain a spike in the series of difference. Another cause of the spike in the
difference between the SPDR × 10 and the S&P 500 on that day can be the so-called
“dotcom bubble”. Indeed during this bubble, the stock prices were very volatile and so
were the index and the SPDR. If the index and the SPDR do not move simultaneously,
this can create a large difference. To highlight this phenomenon, we also plot the dif-
ference series using a longer period from February 1, 1993 through December 30,2005
in Figure C.5. During the “dotcom bubble” period, i.e., roughly between 1997 to 2001,
the difference series was much more volatile than during the other period. The index
dropped from 1354.31 on February 5, 2001 to 1352.26 on February 6, 2001 and continued
to decrease to 1340.89 on February 7, 2001. The SPDR × 10 dropped from 1358.50 on
February 5, 2001 to 1332.50 on February 6, 2001 and increased to 1347.05 on February 7,
2001. It could be that on February 6, 2001, investors who held the SPDR overreacted to
the negative information and underpriced the SPDR. The value of the SPDR rebounded
on February 7, 2001. In conclusion, Figures C.4 and C.5 show that the SPDR does not
mimic perfectly the index.
So far we have examined the daily closing prices. Figures C.6 and C.8 provide the daily
relative price changes of the SPDR and the S&P 500 Index, as defined in Equations (1.3)
17There are 7 observations in which the difference between the two time series is larger than 10 indexpoints. There are 5 cases in which the SPDR× 10 is larger and 2 cases in which the S&P 500 Index islarger.
23
and (1.1), and their difference series. The sample mean of the SPDR return is 0.05 basis
points lower than the sample mean of the index return, which accounts for 24% of the
average daily return of the S&P 500 Index. The historical correlation coefficient between
the two return series is 0.98, suggesting that the SPDR mimics the index rather well as
far as a linear static analysis is concerned. When we consider the higher order moments,
we see that the sample skewnesses of the two time series are different and take values
0.17 for the index and 0.13 for the SPDR, respectively. Again, the differences shown
in Figure C.8 decrease regularly on the ex-dividend days of the SPDR. The difference
series of daily returns can be as large as 223 basis points per day, as occurred on April
4, 2001. From April 3 to April 5, 2001, the index jumped from 1106.46 to 1151.44 and
the SPDR× 10 jumped from 1100 to 1150.45. However, the two series did not increase
simultaneously in that period. On April 4, the index did not change much and closed
at 1103.25 while the SPDR moved to 1121.55. Hence, the different evolutions of the
two processes caused the huge daily return difference. Figures C.7 and C.9 provide the
daily holding period returns, as defined in Equations (1.4) and (1.1), of the SPDR and
the S&P 500 Index, respectively, and their daily difference. The differences between the
holding period returns are bigger than the relative price changes.
Figure C.10 shows the histograms of daily relative price changes of the index and
the SPDR, and the histogram of daily holding period returns of the SPDR. From the
first two histograms, we see that the relative price changes of the index and the SPDR
have different unconditional distributions. The 5% and 95% quantiles corresponding to
the left and right tails of the distribution differ by 11 basis points, which account for
24
6.04% and 6.32% of the corresponding daily index returns, respectively. The medians
themselves differ by about 10% of the index return. The first and third histograms show
the differences between the holding period returns of the index and of the SPDR. The fact
that the quantiles of the SPDR and the S&P 500 distributions do not match can have
important effects on European call prices for which the price thresholds (strikes) play
an important role. Figure C.11 shows the sample distribution of relative price change
differences and daily holding period return differences between the S&P 500 Index and
the SPDR. The relative price change differences are between -0.018 and 0.022, compared
to the relative price changes which are, as shown in Figure C.10, between -0.051 and
0.056. This confirms the error on relative price changes. There are 9.8% of observations
with absolute daily geometric return difference greater than 25 basis point, 28.21% of
observations with absolute daily geometric return differences greater than 10 basis point,
and 59.12% of observations with absolute daily geometric return differences greater than
4 basis points. This is a huge difference given that the medians of the daily geometric
returns of the index and the SPDR are less than 4 basis points. The second histogram
shows that the daily holding period return differences are even more frequently away
from zero and have a relatively larger range.
Figures C.12 and C.13 show the squared daily geometric returns. It is not surprising
that the SPDR and the S&P 500 plots are different, given that they have different daily
relative price changes and holding period returns. We will compare the squares of daily
relative price changes and holding period returns in greater detail in the next subsection
on dynamic analysis.
25
1.2.2.4 Dynamic Comparison of the SPDR and the S&P 500 Index
The analysis above is a first step in a comparison of the properties of the SPDR and the
S&P 500 Index, since it is based only on historical summary statistics and thus neglects
the serial dependence. It is necessary to compare the serial dependence of the series,
especially if we have in mind a continuously updated portfolio to hedge a derivative.
Since the SPDR is intended to mimic the S&P 500, we can expect that both series
will have similar dynamic features, as long as linear dynamics are considered. However,
it is less likely that their nonlinear dynamics such as the volatility or the extreme return
dynamics are compatible.
Tables C.2 and C.3 report the Ljung-Box statistics of daily relative price changes and
holding period returns of the S&P 500 Index and the SPDR. Table C.2 suggests that the
white noise hypothesis at 5% significance level for the first 15 lags cannot be rejected
in both series. However, when more lags are included in the test statistics, the white
noise hypothesis at 5% significance level is rejected in both series. At higher lag, the
SPDR returns show stronger evidence against the white noise hypothesis. For example,
if 25 lags are included in the test, the white noise hypothesis is rejected by the geometric
return of the SPDR at 1% confidence level, but is not rejected by the geometric return
of the Index at the same significance level. Table C.3 shows the results for the two series
of holding period returns and suggests that the white noise hypothesis at 5% significance
level for the first 15 lags cannot be rejected. If more lags are included in the test, the
white noise hypothesis at 5% significance level is rejected for both series. However, up to
26
lag 10, the SPDR shows stronger evidence against the white noise hypothesis. For longer
lags, the S&P 500 shows stronger evidence against the white noise hypothesis18.
Let us now examine the risk dynamics. Figures C.14 and C.15 show the autocorrela-
tion functions of the squares of relative price changes and holding period returns of the
S&P 500 Index and the SPDR. As reported in the ARCH literature, there is more linear
serial dependence in the squared returns than in the returns. This is confirmed by the
Ljung-Box statistics for squared returns reported in Tables C.4 and C.5. The difference in
the autocorrelation functions between these series suggests that they have very different
risk dynamics. Figures C.16 and C.17, where different cross-correlation functions of the
index and the SPDR are compared with corresponding autocorrelation functions of the
index, confirm this result. The graphs show that these functions are clearly different. We
conclude that the risk dynamics of the SPDR and the S&P 500 are different.
As seen from the histograms of daily relative price changes and holding period returns,
the tail quantiles of all series are different and this can have important effects on European
call and put prices for which the price thresholds (i.e. strikes) play an important role. To
see the dynamics of the right tails, we plot the autocorrelation functions ofMax(exp(Rt)−
k, 0) and Max(exp(Rdt )− k, 0) of the S&P 500 Index and the SPDR in Figures C.18 and
C.19, respectively. We also plot the autocorrelation functions of Max(k − exp(Rt), 0)
and Max(k− exp(Rdt ), 0) for the S&P 500 Index and SPDR in Figures C.20 and C.21 to
18We also plot the autocorrelation (ACF) and partial autocorrelation function (PACF) for daily relativeprice changes and holding period returns of the S&P 500 Index and the SPDR up to 135 lags as wellas their cross correlation functions (XCF). For the sake of brevity, we do not report them here. It canbe easily seen from the graphs that, although the signs of the ACF and the PACF of the series arealmost always the same, the sizes are very different, which implies that they have different conditionalevolutions.
27
describe the dynamics of the left tails. For the right tails of index returns, since the 97.5%
quantile is about 0.025, the 95% quantile is about 0.02 and the 90% quantile is about
0.015, we pick k=1.015, 1.02 and 1.025 for the call, as shown in Figures C.18 and C.19.
For the same reason, we pick k=0.985, 0.98 and 0.975 for the put, as shown in Figures C.20
and C.21. The autocorrelation functions of Max(exp(Rt)−k, 0) and Max(k−exp(Rt), 0)
are very different for the S&P 500 Index and the SPDR. The SPDR has higher first-order
autocorrelation than the Index in all the plots, which implies that the SPDR has higher
volatility clustering than the index. The difference in the autocorrelation functions for
higher orders reflects that the two series have very different volatility clustering properties.
A similar analysis suggests that the holding period returns of the S&P 500 Index and the
SPDR also have different volatility properties.
The analysis shows that the S&P 500 index and the SPDR price have different histor-
ical distributions and different dynamics. Therefore, the SPDR price does not replicate
the S&P 500 perfectly. A similar analysis performed for the holding period returns of
the SPDR and the index suggests that the daily holding return of the SPDR does not
replicate the return of the Index effectively, either. This implies that the index cannot
be replicated by the SPDR using self-financing strategy.
The facts that S&P 500 is not traded in the market and cannot be replicated have a
lot of implications which are discussed in the next section.
28
1.3 The Effects of the Non-Tradability of the Index
1.3.1 Derivative Pricing
The non-tradability of the S&P 500 Index has significant implications on risk hedging
and pricing restrictions. For example, the well-known Black-Scholes model [See Black
and Scholes (1973) and Merton (1973b)] assumes that the underlying asset is tradable
and follows a geometric Brownian Motion process with constant volatility. Therefore,
the market is completed by the underlying asset itself. By the No-Arbitrage condition,
the market price of risk is determined uniquely by the price of the underlying asset. All
derivatives written on the underlying asset can be evaluated uniquely with this market
price of risk, combined with the terminal condition of the respective derivatives. But, if
the underlying asset is non-tradable, the underlying asset cannot be used as part of the
arbitrage strategy and hence the value of the underlying asset does not need to satisfy
the No-Arbitrage condition. The knowledge of the value of the underlying asset alone
does not reveal the price of risk. Therefore, the prices of options written on a non-
traded underlying asset, whose value follows a geometric Brownian Motion process, are
not constrained to satisfy the Black-Scholes formula. To evaluate the illiquid assets in
this framework, we need either to introduce an additional tradable asset to complete the
market, or to assume some form of risk-neutral distribution, i.e., of market price of risk.
Since the S&P 500 Index is not a tradable asset in the market, the index and its options
cannot be used in standard option pricing models that are based on the assumption that
the underlying asset is a security traded in the market.
29
The pricing formulas derived from general equilibrium models are also no longer valid.
If the investor cannot trade the underlying asset in the market, s/he has fewer choices for
hedging future risks, so s/he has to make different consumption and investment decisions,
which will affect the equilibrium prices.
In Chapter 3, we derive a coherent multi-factor model for pricing various derivatives
such as forwards, futures and European options written on the S&P 500. We consider
two cases when the underlying asset is tradable and when it is not. The model explains
why the prices of derivatives written on a tradable asset and a non-tradable asset can be
different.
In the next subsection, we study how the pricing parities are violated when the un-
derlying asset is not tradable.
1.3.2 Spot-Futures Parity and Put-Call Parity
In this subsection, we use the following notations:
- It is the value of a non-tradable underling asset at t;
- St is the value of a tradable underling asset at t;
- F it,T is the forward price at t of a contract written on the non-tradable asset expiring
at T , with t ≤ T ;
- F st,T is the forward price at t of a contract written on the tradable asset expiring at
T ;
30
- Git denotes the price at time t of a European call option written on the non-tradable
underlying asset with strike K and maturity T;
- Gst denotes the price at time t of a European call option written on the tradable
underlying asset with strike K and maturity T;
- H it denotes the price at time t of a European put option written on the non-tradable
underlying asset with strike K and maturity T;
- Hst denotes the price at time t of a European put option written on the tradable
underlying asset with strike K and maturity T;
- rt,T is the risk-free interest rate from t to T .
Let us first suppose that there is no dividend. Consider Portfolio A: At time t, let
us take a long position in the forward contract written on the tradable underlying asset
and invest an amount of cash equal to F st,T exp[−rt,T (T − t)] into the risk-free asset. The
value of this portfolio at T is ST . By the No-Arbitrage condition, the current value of
Portfolio A should be St. This implies the Spot-Futures Parity19 relationship for the
tradable underlying asset and the forward contract written on it,
F st,T = St exp[rt,T (T − t)]. (1.5)
Next consider Portfolio B, which is the same as Portfolio A except that the forward
contract is written on the non-tradable underlying asset and the invested cash amount is
19We follow the convention and call Equation (1.5) Spot-Futures Parity [see e.g. Ait-Sahalia and Lo(1993)], although it generally holds for the spot and forward contracts. Due to the mark-to-market rule,the futures contract is different from the forward contract in general. However, when the interest ratesare non-stochastic, the futures price and the forward price are equal[see Cox, Ingersoll and Ross (1981)and French (1983)], and Equation (1.5) holds also for the spot and futures contracts.
31
F it,T exp[−rt,T (T −t)]. The value of Portfolio B at T is IT . It is very tempting to conclude
that the value of Portfolio B at t is It. But this is not true because the underlying asset
is non-tradable. Let us consider the S&P 500 Index as an example. As discussed above,
when an investor buys a portfolio at t for indext, there is no guarantee that s/he can
sell it for indexT at T. In other words, the value at t of some asset which will be worth
indexT at T is not necessarily indext. Thus in general, the Spot-Futures Parity does not
hold for the non-tradable underlying asset and the forward contract written on it:
F it,T 6= It exp[rt,T (T − t)]. (1.6)
By the same token, let us consider Portfolio C: At time t, let us buy one European
call option with strike K written on the tradable underlying asset, sell one European
put option with strike K on the same asset, and invest an amount of cash equal to
K exp[−rt,T (T − t)] in the risk-free asset. The value of this portfolio at T is ST . By
the No-Arbitrage condition, the current value of Portfolio A should be St. This is the
Put-Call Parity:
Gst +K exp[−rt,T (T − t)] = Hs
t + St. (1.7)
Let us now consider Portfolio D, which is the same as Portfolio C except that the
options are written on a non-tradable asset. Although the value of this asset at T is IT ,
for the same reason as above, Equation (1.7) is not necessarily satisfied for options on
non-tradable assets:
Git +K exp[−rt,T (T − t)] 6= H i
t + It. (1.8)
32
Although, the Spot-Futures Parity and the Put-Call Parity do not generally hold for
the S&P 500 Index and its derivatives, the following equation still holds when the forward
contract and options have the same maturity date20.
Git +K exp[−rt,T (T − t)] = H i
t + F it,T exp[−rt,T (T − t)]. (1.9)
When dividends are taken into account, there are at least three reasons why the
Spot-Futures Parity and the Put-Call Parity may not hold for the S&P 500 Index and
its derivatives.
First, the component stocks of the index pay dividends while the S&P 500 Index
reflects the evolution of the market value of the 500 listed firms without reflecting cash
dividends paid out by those stocks. Therefore, the index is not self-financed and Equa-
tions (1.5) and (1.7) do not hold for the index and its derivatives.
Second, when the index is non-tradable, the Spot-Futures Parity and the Put-Call
Parity for dividend-paying tradable asset do not need to hold. For example, let us
consider a tradable underlying asset that pays a continuous dividend yield at a rate qt,T
per annum. If the dividend is reinvested continuously, then one share at t will become
exp[qt,T (T −t)] shares at T . Thus, if a portfolio pays ST at T and if qt,T is known ex-ante,
the value of the portfolio at t must be St exp[−qt,T (T − t)] to be consistent with the No-
Arbitrage condition. So the tradable dividend-paying underlying asset and its forward
20This can be seen by considering the following portfolio: At t, buy one European call option writtenon the index, sell one European put option, sell one forward contract written on the same asset andinvest an amount of cash equal to (K −F it,T ) exp[−rt,T (T − t)] into the risk-free asset. The value of thisportfolio at T is 0. By No-Arbitrage, the current value of Portfolio A should be 0.
33
price should satisfy the following relationship:
F st,T = St exp[(rt,T − qt,T )(T − t)], (1.10)
and similarly, the Put-Call Parity for options written on the underlying asset paying
dividend is:
Gst +K exp[−rt,T (T − t)] = Hs
t + St exp[−qt,T (T − t)]. (1.11)
However, for the same reason as above, the arbitrageurs cannot earn a risk-free arbitrage
return by trading a non-tradable asset. That is, even if Equations (1.10) and (1.11) do
not hold, there does not necessarily exist a risk-free arbitrage opportunity. So Equations
(1.10) and (1.11) do not have to hold for non-tradable assets under No-Arbitrage.
Third, the timing and size of dividends paid by the index component stocks are not
known ex-ante. Even if the index could be mimicked perfectly by its replicating portfolio,
Equations (1.10) and (1.11) do not need to hold for the index and its derivatives since the
arbitrage opportunity is not risk-free. Moreover, the uncertainty involved in the change
of index increases the uncertainty on the dividend. Therefore the Spot-Futures parity
and the Put-Call parity may not hold.
The Spot-Futures Parity and Put-Call Parity are widely used in derivative pricing.
The deviations from both parities by the S&P 500 Index and its derivatives implies that
the two parities cannot be used for various purposes. For example, to calculate the
dividends on indices, the Ivy DB for OptionMetrics Dataset assumes that the security
pays dividends continuously and the existence of a Put-Call relationship. The implied
index dividend is calculated from a linear regression model based on the Put-Call Parity.
34
If the Put-Call Parity does not hold for the S&P 500 Index and its options, the dividends
computed from this method may be incorrect and yield misleading results.
We have argued above that because the S&P 500 Index is not tradable and cannot
be mimicked perfectly by its replicating portfolios, the Spot-Futures Parity and Put-Call
Parity should not hold for the index and its derivatives. However, if investors traded as if
the index were tradeable, the Parities might hold in practice. In Chapter 2, we take the
Put-Call Parity as an example and test whether the Put-Call Parity holds for S&P 500
Index and its options in reality.
1.4 Conclusion
The S&P 500 Index is not a self-financed or a tradable portfolio. This is because the
index itself is not traded in the market and no portfolio can be constructed to perfectly
replicate the index due to the composition and maintenance of the index as well as
the frictions and the lack of perfect liquidity on the market. Although some mimicking
portfolios such as the SPDR can effectively mimic the linear dynamics of the index ,
the nonlinear and extreme risk dynamics of the SPDR are very different from those of
the index. If the index is non-tradable, investors cannot form a risk-free portfolio using
the index and futures contract or using the index and options. Hence, the No-Arbitrage
condition does not imply the Spot-Futures Parity or the Put-Call Parity for the index
and its derivatives. The non-tradability of the index also implies that the index and its
options cannot be used for testing the option pricing models based on the assumption
35
that the underlying asset is a security traded in the market. These consequences of the
non-tradability of S&P 500 Index are explained and tested in detail in Chapter 2 and
Chapter 3.
36
Chapter 2
The S&P 500 Index Options and the
Put-Call Parity
37
2.1 Introduction
The market for S&P 500 Index Options has grown very quickly since they appeared on
July 1, 1983. The options on S&P 500 are standardized financial contracts which have
deterministic issuing dates, maturity dates and strikes. Although this standardization
enhances the liquidity, it also creates seasonal trading patterns and may cause market
incompleteness on certain days. For example, the options that mature in a quarterly
cycle in March, June, July and December are more actively traded than other options
with the same time-to-maturity that mature in different months.
The S&P 500 Index Options are widely used by researchers for testing the option
pricing theories. However, their underlying asset, i.e., the S&P 500 Index, is an artificial
summary statistic constructed to reflect the evolution of the market [see Chapter 1]. It
is not a self-financed tradable portfolio and its modifications do not coincide with the
changes of a mimicking portfolio (tracker) such as the SPDR, due to the particular way
the S&P 500 Index is calculated and maintained. Therefore, pricing theories based on the
assumptions of tradable assets may not hold for the S&P 500 Index and its derivatives.
For example, the non-tradability of the index can help explain why the Put-Call Parity
does not hold for the S&P 500 Index and its options.
The Put-Call Parity implied by the No-Arbitrage condition was first formalized by
Stoll (1969) and Merton (1973a). It is based on the argument that in a frictionless market
the spot asset and the put and call options written on it can form a risk-free portfolio.
The presence of arbitrageurs ensures that there is no remaining arbitrage opportunity
38
and this portfolio will earn the risk-free rate. Using the daily and intradaily prices of the
S&P 500 and its options from 1986-1989, Kamara and Miller (1995) observed deviations
from the Put-Call Parity even after controlling for dividends and transaction costs. Ackert
and Tian (2001) found similar evidence using data from 1992 to 1994. Both articles
tested whether this was due to the ”liquidity risk”, that is, to the difficulty in buying
or selling a portfolio of 500 stocks and index options simultaneously before any adverse
price movement. The evidence found in these articles suggested that the ”liquidity risk”
can help explain the parity deviation. In both articles, the S&P 500 Index was a tradable
asset, which is not very liquid. However, the index is not only illiquid, it is not tradable
as we have seen in Chapter 1. As a consequence, investors cannot make risk-free arbitrage
transactions using the index and the options. Therefore, the No-Arbitrage condition does
not imply Put-Call Parity for the index and its options. Furthermore, the index is not
self-financed since it reflects the price changes of its component stocks without reflecting
the cash dividends. The uncertainty of the dividends can also help explain why the Put-
Call Parity does not hold for the S&P 500 Index. In this paper, we perform an empirical
analysis, based on the most liquid options to eliminate the liquidity risk. We use the
Put-Call Parity equation to derive and compute the Put-Call Parity implied dividends.
We find that the implied dividends depend on time, maturity and strike price. We also
analyze why deviations from the Put-Call Parity do not imply the existence of risk-free
arbitrage opportunities.
In Section 2.2, we consider the S&P 500 Index Options. We first describe the char-
acteristics of traded options and their issuing procedure. Then we provide stylized facts
39
concerning the S&P 500 options. We compute and analyze the implied dividends in
Section 2.3. Section 2.4 concludes.
2.2 S&P 500 Index Options
2.2.1 The Characteristics of Traded Options
The S&P 500 Index Options traded on the Chicago Board Options Exchange (CBOE) are
European options. They are among the most liquid exchange-traded options and are ex-
tensively used for testing the option pricing models. The exchange-traded S&P 500 Index
Options differ from over-the-counter options and have a standardized format described
below. This standardization is introduced to enhance the liquidity.
New S&P 500 Index Options are issued on the Monday following the third Friday of
each month. These new options have the same maturity date but different strikes. Table
C.6 shows the times-to-maturity of the options introduced each month. For example, the
options issued in March and September have a 12 month time-to-maturity, the options
issued in June and December have a 24 month time-to-maturity, and the options issued
in the other months have a time-to-maturity of 3 months. Based on this standardized
deterministic issuing procedure, there are generally eight maturity dates on any trading
day. The expiration months are the three near-term months followed by three additional
months from the March quarterly cycle, which are March, June, September and Decem-
ber, plus two additional months from June and December1. When options with a new
1In the product specification of the S&P 500 Index Options from CBOE, it is stated that the ex-piration months are the three near-term months followed by three additional months from the March
40
maturity are issued, the in-, at- and out-of-the-money strike prices are initially listed.
The underlying asset is the index level multiplied by 100. The index level is computed
using the last transaction prices of its component stocks. The options are quoted in
index points, where one point equals $100. The options have a minimum tick of 0.05
points for options trading below 3.00 and 0.10 points for the others. Strike price intervals
are 5 points and 25 points for long term contracts. The expiration date is the Saturday
following the third Friday of the expiration month. As the options are European, they
can only be exercised on the last business day before expiration. Trading in S&P 500
Index Options will generally terminate on the trading day preceding the day on which
the exercise-settlement value is computed, which is usually a Thursday. The exercise-
settlement value is computed using the opening (first) reported sales price in the primary
market of each component stock on the last business day before the expiration date, i.e.,
usually a Friday. The last reported sales price in the primary market is used in computing
the exercise-settlement value if a stock in the index does not open on the day on which
the exercise and settlement value is determined. Exercise will result in delivery of cash,
which amounts to the difference between the exercise-settlement value and the strike
price of the option multiplied by 100, on the business day following expiration. There are
no effective position and exercise limits. Purchases of puts or calls with 9 months or less
until expiration must be paid in full. Writers of uncovered puts or calls must maintain a
certain margin to cover their position if their investment value declines over time.
quarterly cycle (March, June, September and December). The data show that there are generally twoadditional longer term options traded in the market.
41
2.2.2 The Activity on the Option Market
The option data used in this paper are from the Wharton Research Data Services (WRDS)
and prepared by Ivy DB OptionMetrics. The data consists of the highest closing bid
prices, the lowest closing ask prices and the traded volumes of all the call and put options
with various strike prices and expiration dates for each trading day from January 2nd,
2003 to December 31st, 2003. There are 130,336 observations.
Figure C.22 shows the number of different times-to-maturity on each trading day
in 2003. As noted above, there are generally eight maturity dates for options on any
trading day. However, since expiring options are usually not traded at maturity, but
settled on these days, while new options are not issued until the next Monday, there are
seven maturity dates only on the last trading days before expiration dates2. This could
potentially cause market incompleteness, since there are fewer traded options on the last
trading days before expiration dates.
Because the exchanged traded options have fixed maturity dates, we observe a de-
clining pattern of times-to-maturity on each trading day, as shown in Figure C.23. This
figure and Table C.6 show that options traded in the first six months have the same
time-to-maturity pattern as those traded in the following six months. For example, the
pattern on the left side of Figure C.23 is the same as the pattern on the right side. We
2The exceptions are the following: on Jan 17, 2003, the data shows 8 maturity dates, one of whichis Jan 18, 2003. But the volume for options expiring on this date is 0. The same thing happens onSeptember 19, 2003. On Jan 2nd, the data does not show any options maturing on December 18, 2004,so there are only 7 maturity dates on that day. For June 19 and June 20, the data shows the optionmaturing on June 18, 2005, which usually will not be traded until the next Monday, although the volumeis 0 on both days. The data does not show options maturing on December 17, 2005 for December 22,2003.
42
conclude that there are two cycles per year. Figure C.23 also shows that new options
which are not issued from the March quarterly cycle will always expire in three months,
while the options issued in March and September will expire in a year and the options
issued in June and December will expire in 2 years. Therefore, all options with a time-to-
maturity longer than three months have expiration dates in the March quarterly cycle,
and all the options with a time-to-maturity longer than 1 year have expiration dates in
June and December.
Table C.7 presents the summary statistics of call and put options traded in 2003.
There are, on average, 517 options traded every day with different strike prices and
times-to-maturity. The daily number ranges from 422 to 572 with a standard deviation
of 28. For each call option with a given strike and time-to-maturity, we almost always
find a put option with the same strike and time-to-maturity. In the 252 trading days
in 2003, there are only two days when the number of call options is different from the
number of put options and that difference is just 1 option.
The market for S&P 500 Index Options is fast-growing and these options are among
the most actively traded derivatives. The average total daily traded volume in 1993 was
65,476 contracts [see Ait-Sahalia and Lo (1998)], and it increased to 137,143 contracts in
2003. The daily traded volume of S&P 500 Index Options ranges from 37,428 contracts
to 322,711 contracts, with a standard deviation of 43,213 contracts. The average daily
traded volume is higher for put options than for call options, while the opposite is true
for the average daily traded value. The average total daily traded value for all options
is $327,816,156 with a minimum of $51,359,310 and a maximum of $1,600,668,170. The
43
standard deviation is $197,363,809.
The second panel of Table C.7 describes the summary statistics of volumes and prices
in the sample of 130,336 observations. The options with the same maturity and strike
price, but traded on different days, are regarded as different options. The price is com-
puted as the average of the best closing bid and ask prices. More than 60% of the options
have zero traded volume in 2003 and only about 4% of the options have a traded volume
greater than 2,000 contracts. Prices move from $0.025 to $993.5 per 1/100 contract, with
a mean of $111.85 and a standard deviation of $146.53.
Figure C.24 displays the times-to-maturity of both call and put options with a traded
volume greater than 2,000 contracts on each day in 2003. By comparing this figure with
Figure C.23, we see again that a large number of options are not actively traded each
day, especially those with longer times-to-maturity. The options expiring in the March
quarterly cycle are more frequently traded after their times-to-maturity reduce to about
100 days. Figures C.25 and C.26 show strike prices and moneyness3 of options with high
traded volumes on each trading day. Most of the actively traded options are at- and
out-of-the-money. These are call options with strike prices equal to or higher than the
index level, and put options with strike prices equal to or lower than the index level.
This is consistent with the results reported by Kamara and Miller (1995) and Ackert and
Tian (2001). The observed asymmetry reflects the strong demand of portfolio managers
for protective calls and puts. Figure C.27 shows the prices of options with high traded
volumes on each day in 2003. The most actively traded options are those with low prices.
3The moneyness is defined as the strike price divided by the price of the underlying index.
44
This is not surprising since most options with high volumes are at- and out-of-money.
Figures C.28 to C.33 investigate the number and proportion of highly traded options
on each trading day, and their summary statistics are presented in Table C.8.
Figure C.28 shows the number of highly traded options on each trading day and their
daily relative frequencies of trades (in %). As shown in Table C.8, there are on average
20 options actively traded on each day, ranging from 3 options to 43 options with a
standard deviation of 6.57 options. On average, these actively traded options account for
only 3.89% of all traded options, with a minimum of 0.59%, a maximum of 7.62% and a
standard deviation of 1.21%. It can also be observed that the number of actively traded
options on the last trading day before the expiration date is always lower than that on the
previous day. There is no such regular decrease in the relative frequencies of trades (in
%) since the total number of options on these days are also lower. Although the actively
traded options account for only a small part of all traded options, their traded volumes
account on average for 61.65% of all the options, ranging from 34.24% to 85.65%, with a
standard deviation of 9.88%.
Figure C.29 shows the number of actively traded put options on each trading day and
its daily part of trades. Among those highly traded options, on average 58.92% are put
options , with a minimum of 28.57% and a maximum of 100%, as shown in the second
panel of Table C.8. The standard deviation is 11.11%.
Let us now examine the options in terms of their times-to-maturity on each day. All
the options traded in 2003 are classified into four categories for each trading day: T1
with the shortest time-to-maturity, T2 with the second shortest time-to-maturity, T3
45
with the third shortest time-to-maturity, and T4 with longer times-to-maturity. From
Figure C.22, there are generally eight traded maturity dates on each trading day. The
T1 options are mainly those options with the closest maturity date, except that on the
last trading day before the expiration date in each month, the options to expire the next
month are regarded as T1 since the options expiring this month are not traded on that
day. A similar rule applies to the T2 and T3 options, which usually include the options
with the second and third closest maturity date, except on the last trading day before
the expiration date in each month. The T4 options contain all the options with other
maturity dates. We focus our attention on the actively traded options.
As shown in the second panel of Table C.8, among actively traded options, about 46%
on average are T1, while T2 represents 25%, T3 13% and T4 16%, respectively. Thus
the most actively traded options are those with the shortest time-to-maturity.
We observe periodic patterns from Figures C.30 to C.33. For example, in Figure C.32,
the number of T3 options is mostly zero on the last trading day before the expiration date
in the months other than the March cycle, while the number of T3 options on the last
trading day before the expiration date in the March cycle is positive and higher than that
on the previous day. The T3 options on the last trading day before the expiration date in
the March cycle are the options with 3 months of time-to-maturity and to expire in the
March cycle. We observed in Figure C.24 that options expiring in the March quarterly
cycle seem to be more frequently traded in the last 100 days of their times-to-maturity.
On the other hand, on the last trading day before the expiration date in the months
other than the March cycle, the T3 options are the less actively traded options with
46
four or five months to maturity, since the 3-month options are yet to be introduced on
the following Monday. Therefore, it is not surprising to observe in Figure C.32 a higher
number of actively traded T3 options on the last trading day before the expiration date
in the March quarterly cycle than in the other months.
The first graph in Figure C.32 also shows that the T3 options expiring in the March
quarterly cycle (the first, fourth, seventh, tenth and thirteenth column of the graph)
have a higher number of actively traded options than the T3 options expiring in the
other months. One possible explanation is that the T3 options expiring in the March
quarterly cycle are issued much earlier before they become T3 options, and thus may have
a larger number of actively traded strike prices, while the T3 options expiring in the other
months are newly issued and may have a smaller number of actively traded strike prices.
However, as shown in the first graph of Figure C.36, although the T3 options expiring in
the March quarter cycle have a greater number of actively traded options with different
moneyness, the range of their moneyness is not very different from that of the other
T3 options. Moreover, the second and third graphs in Figure C.36 show that the total
volume and total value of the actively traded T3 options expiring in the March cycle
are higher as well. So there may be other reasons that explain the pattern observed in
the first graph of Figure C.32. For example, the S&P 500 Futures traded in the Chicago
Mercantile Exchange expire only in the March quarterly cycle. The more active trading
of the T3 options expiring in the March cycle could be caused by the interaction between
the S&P 500 option and the S&P 500 futures markets. A similar pattern can also be
observed in the second graphs of Figures C.30 and C.31 and in Figures C.34 and C.35,
47
that is, the options that mature in the March quarterly cycle are more actively traded
than other options with the same time-to-maturity that mature in different months.
We also observe a periodic pattern in the first graph of Figure C.33, where the number
of actively traded T4 options is generally higher during the days close to but before the
expiration day in the March cycle, and decreases sharply on the last trading day before
the expiration date in the March cycle. This observation is consistent with the pattern
observed in Figure C.32. During the days close to but before the expiration day in the
March cycle, the options maturing in the March cycle have less than 100 days of time-
to-maturity and are actively traded, while on the last trading day before the expiration
date in the March cycle, these options become T3.
In summary, the S&P 500 index options are standardized contracts, with determin-
istic issuing dates, and a limited number of maturity dates and strikes available at the
issuing. Although this standardization enhances liquidity, it also creates certain trading
seasonality. The S&P 500 index options are among the most actively traded options in
the world. However, most of these actively traded options are at- or out-of-the-money
options with times-to-maturity of less than three months, and hence with relatively low
prices. In the next section, we will use these option data to test the Put-Call Parity for
the S&P 500 Index.
48
2.3 The Put-Call Parity Implied Dividends of the
S&P 500 Index
2.3.1 Calculation of the Implied Dividends
The Put-Call Parity implied by No-Arbitrage was first formalized by Stoll (1969) and
Merton (1973a). When the underlying asset is tradable and the dividends paid by the
underlying asset are known ex-ante, the Put-Call Parity is expressed in the following
inequality form:
Ca − P b +K exp[−rt,T (T − t)]− St exp[−qt,T (T − t)] ≥ 0, (2.1)
P a − Cb −K exp[−rt,T (T − t)] + St exp[−qt,T (T − t)] ≥ 0, (2.2)
where Cb and Ca are the bid and ask prices of the call option at time t, P b and P a are
the bid and ask prices of the put option, K is the strike price, T is the maturity date,
rt,T is the risk-free interest rate from t to T , and qt,T is a continuous dividend yield paid
by the underlying asset. The two inequalities need to hold simultaneously. When the
first inequality is violated, arbitrageurs can generate earnings by buying the call, selling
the put and the underlying asset, and investing the net proceeds into a risk-free interest
account. When the second inequality is violated, arbitrage profits can be created by
selling the call, borrowing K exp[−rt,T (T − t)] at the risk-free interest rate, buying the
put, and the underlying asset.
The S&P 500 Index is an artificial number constructed to reflect the performance of
the entire market. As argued in Chapter 1, it is not a traded asset in the market. Due
49
to the frictions of the market, the illiquidity of the component stocks and the changes
of composition, the index cannot be replicated by any traded portfolio. Therefore, it is
impossible to form a riskless arbitrage position with the options and the index. That is,
the Put-Call Parity does not have to hold for the S&P 500 Index and its options. The
deviation from the Put-Call Parity by the S&P 500 Index and its options does not imply
that there exist arbitrage opportunities.
Furthermore, the dividends paid by the index component stocks are not known ex-
ante. Even if the index could represent the price of a tradable portfolio, Inequalities (2.1)
and (2.2) do not have to hold for the index and its options because the arbitrage is risky.
Testing whether Inequalities (2.1) and (2.2) hold empirically for the S&P 500 Index
and its options is difficult, if not impossible, in a model-free environment because the
dividend yield, qt,T , is not observable, and there is no reason to assume that the dividends
observed ex-post are the same as the expected future dividend [see Harvey and Whaley
(1991) for example]. However, the financial literature often assumes that the following
equation holds for the S&P 500 Index and its options [See e.g. Ait-Sahalia and Lo (1998)]:
Git +K exp[−rt,T (T − t)] = H i
t + It exp[−qt,T (T − t)], (2.3)
where It is the value of the S&P 500 index at time t, and Git and H i
t denote the average
of the bid and ask prices at time t of a European call option and a European put option
written on the index with strike K and maturity T, respectively. In this paper, we will
test whether Equation (2.3) holds for S&P 500 Index and its options.
50
Let us define the ”Implied Dividend”, Q, as
Q(t,K, T ) =−1
T − tlog
Git +K exp[−rt,T (T − t)]−H i
t
It, (2.4)
where T − t is measured in years.
The position, which consists of buying a European call option, selling a European
put option on the index with strike price K and maturity T, and investing an amount
of cash equal to K exp[−rt,T (T − t)] into the risk-free asset at t, will result in receiving
IT at T. Thus the market evaluates the current value of the future index IT as Git +
K exp[−rt,T (T − t)] −H it . Because the index is non-tradable and not self-financed, this
value is not necessarily the same as the current index level, It. The implied dividend,
Q, represents the annualized ratio between the current value of the future index IT and
the current index, It. If the Put-Call Parity (2.3) holds, the implied dividends should
not depend on the strike price of the options. In the next subsection, we analyze the
properties of the implied dividend.
2.3.2 Data and Empirical Results
In order to calculate the implied dividends, we have to consider put and call options
with the same strike price and expiration date that are both actively traded on the same
day. We select those options with a daily traded volume of more than 2,000 contracts.
There are 798 pairs of such put and call options. As shown in Table C.9, there are on
average only 3 pairs of actively traded put and call options with the same strike price and
maturity on each day. The daily number of actively traded pairs of options ranges from
0 to 10 pairs, with a standard deviation of 1.78 pairs. These options account on average
51
for only 1.22% of all the traded options, with a standard deviation of 0.67%. However,
in terms of traded volume, these option pairs account on average for about 22.25% of
all traded options, with a range from 0 to 72.99% and a standard deviation of 11.55%.
Figures C.37 and C.38 show the moneyness and times-to-maturity of these options on
each trading day in 2003. It is not surprising that the moneyness is around 1 and most
of the times-to-maturities are less than 100 days.
The data for the S&P 500 Index come from the Center for Research in Security Prices
(CRSP). We use the level of the Standard & Poor’s 500 Composite Index at the end of
the trading day. These data are collected from publicly available sources such as the
Dow Jones News Service, The Wall Street Journal or the Standard & Poor’s Statistical
Service. The index does not include dividends.
We use two proxies for the risk-free rate. The U.S. Treasury Bill rate is from the US
Treasury. For options with a time-to-maturity less (resp. more) than 28 days, we use
the closing over-the-counter market quotation on recently issued 4-week (resp. 13-week)
Treasury Bills. In 2003, the averages of the 4-week and 13-week Treasury Bill rates are
1.02% and 1.03%, with a standard deviation of 0.13% and 0.11%, respectively. Since the
investors generally use the LIBOR rate as a benchmark, we also consider the zero rates
provided by OptionMetrics as alternative proxies for the risk-free rates. They are based
on the BBA LIBOR rates and the Eurodollar strip implied future rates.
Let us first use the Treasury Bill rates as proxies for the risk-free rates. The implied
dividend, Q(t,K,T), is computed following Equation (2.4) and shown in Figure C.39. The
implied dividend is displayed on a daily basis, with the horizontal axis representing the
52
day when the options are traded. There are days without significant trades of put and
call options, while on other days there are up to 10 traded pairs. The summary statistics
are presented in the first column of Table C.10. The implied dividends range from −0.51
to 0.43 with a median of 1.40× 10−2 and a standard deviation of 6.26× 10−2. For a pair
of call and put options with time-to-maturity of 50 days, the ratio between the current
value of the future index and the current index value ranges from 94.27% to 107.24%
with a median of 99.81%. The t-statistic of 3.62 suggests that the null hypothesis of zero
dividend is rejected. That is, the current value of the future index is not the same as the
current index value.
In order to see whether the implied dividend depends on the strike price and time-
to-maturity, we consider i) implied dividends with the same trading day t and the same
time-to-maturity T-t, but at least two different strike prices K, and ii) implied dividends
with the same date t and the same strike K, but at least two different T-t.
In Figure C.40, the implied dividends computed from options traded on the same day
and with the same time-to-maturity but different strike prices share the same point on
the horizontal axis and are regarded as one observation. There are 218 such observations.
For each observation, there are at least two implied dividends computed with different
strike prices. As shown in Figure C.40, options with the same date t and time-to-
maturity T-t, but different strikes K, yield different implied dividends. Figure C.41
displays the difference between the maximum and the minimum of the implied dividends.
The summary statistics are presented in the second column of Table C.10. The difference
has a mean of 1.03 × 10−2 and a standard deviation of 3.79 × 10−3 with a range from
53
6.44 × 10−5 to 0.19. The t-statistic of 6.92 implies that the null hypothesis that the
options with the same date t and the same time-to-maturity T-t have the same implied
dividend is rejected.
In Figure C.42, the implied dividends are computed from options traded on the same
day and with the same strike price, but different expiration dates. There are 142 such
sets of observations. For each set, there are at least two implied dividends computed
with different expiration dates. As shown in Figure C.42, the options with the same
date t and the same strike K, but different times-to-maturity T-t, yield different implied
dividends. Figure C.43 displays the difference between the maximum and the minimum
of the implied dividends for each set. The summary statistics are presented in the third
column of Table C.10. The difference has a mean of 6.61×10−2 and a standard deviation
of 9.57× 10−2 with a range from 8.91× 10−6 to 0.51. The t-statistic of 13.06 means that
the null hypothesis that the options traded on the same day, with the same strike price,
will have the same implied dividend is rejected.
When the zero rates are used as proxies of the risk-free rates, the empirical results
are reported in Table C.11 and Figures C.44 to C.48. We restrict our attention to the
actively traded put and call options with time-to-maturity greater than 30 days. Indeed,
small errors in the prices of the options and the index translate into large variations in
implied dividends for small times-to-maturity. The results are similar to those where the
Treasury Bill rates are proxies of the risk-free rates. For example, the t-statistics in Table
C.11 show that the hypotheses that the implied dividend does not depend on either the
time-to-maturity, or the strike price, are rejected.
54
2.3.3 Discussions of the Empirical Results
The above empirical results show that the current value of the future index is not equal
to the current index value, and that the implied dividends depend not only on the time
and maturity, but also on the strike. Several explanations for these stylized facts can be
offered.
Firstly, since the index is non-tradable, and the changes of its composition as well
the dividends to be paid by its component stocks are uncertain ex-ante, the arbitrages
in this option market can be risky. The violation of Inequalities (2.1) and (2.2) does
not imply risk-free arbitrage opportunities. When the deviations become too large, a
traded portfolio of a small subset of stocks in the index is selected and traded against
the options. Because this arbitrage is not risk-free, the options prices can move freely in
a wide range without introducing arbitrage profits. This allows implied dividends to be
different for options with the same time t and the same maturity T.
Secondly, transaction costs may also play a role. Transaction costs for arbitrage trad-
ing include both the commissions and the bid-ask spreads. The commissions for arbitrage
traders may be small, while the bid-ask spreads can be large. The transaction costs for
trading options and Treasury Bills in the risky arbitrage mentioned above are easily de-
fined. They include the commissions and bid-ask spreads of trading the call options,
the put options and the Treasury Bills. However, it is hard to measure the transaction
costs of trading the stock portfolios, since the costs depend on the portfolios used by
the arbitrageurs. Without explicitly modeling the evolution process of the index and
55
its component stocks, specifying the dividend paying process, and making assumptions
about the arbitrageurs’ risk preferences, we do not know what portfolios are used in the
risky arbitrage and how to measure the relevant transaction costs. Moreover, in this
model-free framework, it is hard to distinguish the effects of the non-tradability of the
index, the uncertainty of the dividends and the transaction costs.
Thirdly, timing is of importance. In our paper, all prices and index levels are recorded
at the end of the trading day. However, the CBOE closes at 3:15pm Central Time, while
the underlying stock market closes at 3:00pm Central Time. Thus, the implied dividend
reflects the ratio between the current value of the future index at 3:15pm and the current
index level at 3:00pm. Any new information entering the market between 3:00pm and
3:15pm will be reflected in the option prices, but not in the index level. Whether this
timing effect is significant is not well understood. Harvey and Whaley (1991) find that the
non-synchronous price problem may induce negative first-order serial correlation in the
implied volatility changes from day to day, while Evnine and Rudd (1985) and Kamara
and Miller (2001) both find that intraday and daily closing data yield similar results for
testing the Put-Call Parity.
A similar remark can be made about the maturities. We use the closing over-the-
counter market quotation on recently issued 4-week (resp. 13-week) Treasury Bills for
options with time-to-maturity less (resp. more) than 28 days. Although these are among
the most actively traded Treasury Bills, there is a mismatch between the maturities of
the Treasury Bills and of the index options.
56
2.4 Conclusion
In this chapter, we analyzed some characteristics of the S&P 500 Index Options. The
S&P 500 options are standardized exchange traded options. This standardization en-
hances liquidity for short- and medium-term, at- and out-of-the-money options, but is
not sufficient to recover the Put-Call Parity. The S&P 500 Index is not a self-financed
tradable portfolio and cannot be replaced by a mimicking portfolio. This property of
the index may cause deviations from the Put-Call Parity, without producing arbitrage
opportunities. This paper empirically tests this argument by considering the series of the
Put-Call Parity implied dividends in 2003. The implied dividends depend significantly
on the strike, which indicates that the standard Put-Call Parity equalities taking into
account the dividends do not hold.
57
Chapter 3
Non-tradable S&P 500 Index and
the Pricing of Its Traded Derivatives
58
3.1 Introduction
As argued in Chapter 1, the S&P 500 Index is an artificial number constructed to reflect
the evolution of the market. It is not a self-financed or a tradable portfolio and it cannot
be replaced by a mimicking portfolio such as the SPDR due to the particular way the
S&P 500 Index is calculated and maintained. The non-tradability of the S&P 500 Index
has significant implications on risk hedging and pricing constraints. For example, the well-
known Black-Scholes model [See Black and Scholes (1973) and Merton (1973)] assumes
that the underlying asset is tradable and follows a geometric Brownian Motion process
with constant volatility. Therefore, the market is completed by the underlying asset itself
and the risk involved can be fully hedged by the underlying asset. By the No-Arbitrage
condition, the market price of risk is determined uniquely by the price of the underlying
asset. All derivatives written on the underlying asset can be evaluated uniquely with this
market price of risk, combined with the terminal condition of the respective derivatives.
If the underlying asset is non-tradable, the underlying asset cannot be used as part of
the arbitrage strategy and the value of the underlying asset does not need to satisfy the
No-Arbitrage condition. The risk associated with the underlying asset is not hedged by
itself and the expected return of the underlying asset under the risk-neutral probability
is not necessarily equal to the risk-free rate. Knowing only the value of the underlying
asset, we do not know the price of the risk. Therefore the prices of options written on
a non-traded underlying asset whose price follows a geometric Brownian Motion process
do not have to be evaluated by the Black-Scholes formula. Similar ideas apply to many
59
other models. For instance, the stochastic volatility models in Heston (1993) and Ball
and Roma (1994) assume that the expected return of the underlying asset is equal to the
risk-free rate under the risk-neutral probability, i.e., that the underlying asset is tradable
and the risk associated with the underlying asset is hedged by itself. In general, because
of the non-tradability of the S&P 500 Index, the prices of its options do not have to
satisfy the restrictions imposed by the pricing models that are based on the assumption
that the underlying asset is a security traded in the market.
In this paper we introduce a coherent multi-factor model for pricing various derivatives
such as forwards, futures and European options written on the non-tradable S&P 500
Index. The model illustrates the relationship between the index and its futures, and the
relationship between the index and its put and call options, when the underlying asset
is non-tradable. We also consider what the prices of the derivatives should be, if the
index were self-financed and tradable. The model explains why the prices of derivatives
written on a tradable asset and a non-tradable asset can be different. Moreover, the model
provides a framework to test whether the S&P 500 derivatives are priced by the investors
as if the index were self-financed and tradable. This model can be easily extended to
price derivatives written on other non-tradable indices such as a retail price index, a
meteorological index, or an index summarizing the results of a set of insurance companies.
The rest of the paper is organized as follows: In Section 3.2, we present a coherent
pricing model for pricing derivatives written on the S&P 500. The Spot-Futures Parity
and Put-Call Parity are also derived for the case of a non-tradable underlying index. In
Section 3.3, we derive the parameter restrictions which characterize the derivative pricing
60
if the index were tradable. This allows us to discuss how derivative prices can differ for
tradable and non-tradable underlying assets. We conclude in Section 3.4. Technical
results and details are gathered in the Appendices.
3.2 The Pricing Model
Under the absence of arbitrage opportunity (AAO), the market prices have to be com-
patible with a valuation system based on stochastic discounting [Harrison and Kreps
(1979)]. The pricing formulas can be written either in discrete time or in continuous
time, according to the assumptions of discrete or continuous trading (and information
sets). The modern pricing methodology requires a joint coherent specification of these
historical and risk neutral distributions. For this purpose, we follow the practice initially
introduced by Constantinides (1992), which specifies a parametric historical distribution
and a parametric stochastic discount factor.
3.2.1 Assumptions
3.2.1.1 Historical Dynamics of the Index
The value of the index at date t is denoted by It. We assume that the log-index satisfies a
diffusion equation with affine drift and volatility functions of K underlying factors {xk,t},
k = 1, · · · , K:
Assumption 3.1.
d log It = (µ0 +K∑k=1
µkxk,t)dt+ (γ0 +K∑k=1
γkxk,t)1/2dwt, (3.1)
where {µk} and {γk} , k = 0, · · · , K are constants, and {wt} is a Brownian motion.
61
The underlying factors summarize the dynamic features of the index. As seen in
Equation (3.2), they are assumed to be independent Cox, Ingersoll and Ross (CIR)
processes, independent of the standard Brownian motion {wt}. Since the CIR processes
are nonnegative, the volatility of the log-index is positive whenever parameters {γk}, k =
0, · · · , K, are positive. This positive parameter restriction is imposed in the rest of the
paper.
Assumption 3.2. The CIR processes {xk,t}, k = 1, · · · , K satisfy the stochastic differ-
ential equations:
dxk,t = ξk(ζk − xk,t)dt+ νk√xk,tdwk,t, k = 1, · · · , K, (3.2)
where ξk, ζk and νk are positive constants and {wk,t}, k = 1, · · · , K are standard inde-
pendent Brownian motions, independent of {wt}.
The condition ξkζk > 0 ensures the nonnegativity of the CIR process (for a positive
initial value x0 > 0), while the conditions ξk > 0 and ζk > 0 imply the stationarity of the
CIR process. The condition νk > 0 can always be assumed for identifiability reason.
This general specification of the index dynamics includes the Black-Scholes model
[Black and Scholes (1973)], when µk = γk = 0, k = 1, · · · , K, the stochastic volatility
model considered by Heston (1993) and Ball and Roma (1994), when K = 1 and x1
is interpreted as a stochastic volatility, or the model with stochastic dividend yield [see
Schwartz (1997) for example], when K = 1 and x1 appears in the drift only.
The transition distribution of the integrated CIR process is required for deriva-
tive pricing. This distribution is characterized by the conditional Laplace transform
Et[exp(−z∫ t+h
txk,τdτ)], where Et denotes the conditional expectation given the past
62
values of the process and z is a nonnegative constant (or more generally a complex num-
ber), which belongs to the domain of the existence of the conditional Laplace transform.
This domain does not depend on past factor realizations, that is, on the information
set. The conditional Laplace transform admits a closed form expression [see e.g. Cox,
Ingersoll and Ross (1985b)]. The conditional Laplace transform of the integrated CIR
process is an exponential affine function of the current factor value. It is given by
Et[exp(−z∫ t+h
t
xk,τdτ)] = exp[−Hk1 (h, z)xk,t −Hk
2 (h, z)], (3.3)
where
Hk1 (h, z) =
2z(exp[εk(z)h]− 1)
(εk(z) + ξk)(exp[εk(z)h]− 1) + 2εk(z),
Hk2 (h, z) =
−2ξkζkν2k
{log[2εk(z)] +h
2[εk(z) + ξk] (3.4)
− log[(εk(z) + ξk)(exp(εk(z)h)− 1) + 2εk(z)]},
εk(z) =√ξ2k + 2zν2
k .
This formula also holds for a complex number z = u+ iv, whenever u > −1 and v ∈ R.
63
The joint dynamics of factors and log-index can be represented by means of the
stochastic differential system:
d
x1,t
...
xK,t
log It
=
ξ1(ζ1 − x1,t)
...
ξK(ζK − xK,t)
µ0 +∑K
k=1 µkxk,t
dt
+
ν1√x1,t 0 · · · 0
0. . .
...
... νK√xK,t 0
0 · · · 0 (γ0 +∑K
k=1 γkxk,t)1/2
dw1,t
...
dwK,t
dwt
,
where both the drift vector and the volatility-covolatility matrix are affine functions of
the current values of the joint process (x1,t, · · · , xK,t, log It)′. Thus, the stacked process
(x1,t, · · · , xK,t, log It)′ is an affine process [see Duffie and Kan (1996)], and the conditional
Laplace transform of the integrated process Et[exp∫ t+h
t(z1x1,τ + · · ·+zKxK,τ +z log Iτ )dτ ]
will also admit an exponential affine closed form expression.
3.2.1.2 Specification of the Stochastic Discount Factor
The model is completed by a specification of a stochastic discount factor (SDF), which
is used later on to price all derivatives written on the index.
64
Assumption 3.3. The stochastic discount factor (SDF) for period (t, t+dt) is
Mt,t+dt = exp(dmt) = exp[(α0 +K∑k=1
αkxk,t)dt+ βd log It]
= exp{[α0 + βµ0 +K∑k=1
(αk + βµk)xk,t]dt+ β(γ0 +K∑k=1
γkxk,t)1/2dwt}. (3.5)
This SDF explains how to correct for risk when pricing derivatives. The “risk premia”
depend on the factors and index values, whereas the sensitivities of this correction with
respect to these risk variables are represented by the α and β parameters. The market
price of risk associated with wt is1 −β(γ0 +∑K
k=1 γkxk,t)1/2. This specification of the
SDF implicitly assumes that the market prices of the risk factors {wk,t}, k = 1, · · · , K
are 0. Equivalently, Equation (3.2) also describes the risk-neutral distribution of {xk,t},
k = 1, · · · , K. Under the risk-neutral probability, the joint dynamics of the underlying
factors and log-index can be represented by means of the stochastic differential system:
d
x1,t
...
xK,t
log It
=
ξ1(ζ1 − x1,t)
...
ξK(ζK − xK,t)
µ0+βγ0+∑
(µk+βγk)xk,t)
dt
+
ν1√x1,t 0 · · · 0
0. . .
...
... νK√xK,t 0
0 · · · 0√γ0+
∑γkxk,t
dw1,t
...
dwK,t
dw∗t
,
1This can be seen easily from the short rate computed in Equation (3.14).
65
where {wk,t}, k = 1, · · · , K, and {w∗t } are standard independent Brownian motions under
the risk-neutral probability. Thus only the last row is corrected for risk. This differential
stochastic system is still an affine process.
3.2.2 Pricing Formulas for European Derivatives Written on
the Index
As mentioned above, the arbitrage pricing proposes a valuation approach, which is
compatible with observed market prices and proposes coherent quotes for non-highly-
traded derivatives. More precisely, the value (price) at t of a European derivative paying
g(x1,t+h, · · · , xK,t+h, It+h) at time t+ h is
c(t, t+ h, g) = Et[exp(
∫ t+h
t
dmτ )g(x1,t+h, · · · , xK,t+h, It+h)]. (3.6)
The valuation formula is not assumed to be unique. Indeed, the SDF has been
parameterized by α0, α1, · · · , αK , β, but the parameter values have not been fixed ex-
ante. Thus, we propose implicitly different possible valuations and by comparing with
observed derivative prices, we estimate ex-post which one(s) is(are) compatible with
observed market prices.
The aim of this section is to derive explicit valuation formulas for European index
derivatives. All the formulas are derived from the valuation of European Index derivatives
with power payoff. Such derivatives are not traded or more generally quoted. But these
basic computations are used to derive:
• the risk-free term structure of interest rates
66
• the forward and futures prices of the index
• the prices of European options written on the index.
3.2.2.1 Power Derivatives Written on the Index
The following proposition is proved in Appendix B.1.
Proposition 3.1. The value at t of the European derivative paying exp[ulog(It+h)] =
(It+h)u at maturity t+h is
C(t, t+ h, u) = Et[(It+h)u exp(
∫ t+h
t
dmτ )]
= Et[exp(
∫ t+h
t
dmτ + u log It+h)]
= exp(u log It) exp[−hz0(u)−K∑k=1
Hk1 (h, zk(u))xk,t −
K∑k=1
Hk2 (h, zk(u))],
(3.7)
where
zk(u) = −αk − (β + u)µk −γk2
(β + u)2, ∀ k = 0, · · · , K, (3.8)
= zk(0) + ulk +γk2u(1− u), (3.9)
lk = −µk −1 + 2β
2γk, ∀ k = 0, · · · , K, (3.10)
and Hk1 (·, ·) and Hk
2 (·, ·) are given in equation(3.4).
Proposition 3.1 holds, if and only, if zk(u) > −1, ∀ k = 1, · · · , K. When we apply this
formula to different traded derivatives, i.e., different values of u, the inequalities above
imply restrictions on parameters α, β and γ.
3.2.2.2 The Risk-free Term Structure
The zero-coupon bonds correspond to a unitary payoff, and their prices B(t, t + h) cor-
respond to the special case of C(t, t+ h, u) where u = 0. The continuously compounded
67
risk-free interest rates are defined by r(t, t + h) = − 1h
logB(t, t + h). We make the
following proposition:
Proposition 3.2. The prices of the zero-coupon bonds are:
B(t, t+ h) = C(t, t+ h, 0)
= exp[−hz0(0)−K∑k=1
Hk1 (h, zk(0))xk,t −
K∑k=1
Hk2 (h, zk(0))], (3.11)
where zk(·) is defined in Equation (3.8), and Hk1 (·, ·) and Hk
2 (·, ·) are given in equa-
tion(3.4). We deduce the expressions of the interest rates:
r(t, t+ h) = −1
hlogB(t, t+ h)
= −1
h[−hz0(0)−
K∑k=1
Hk1 (h, zk(0))xk,t −
K∑k=1
Hk2 (h, zk(0))]
= z0(0) +1
h
K∑k=1
Hk1 (h, zk(0))xk,t +
1
h
K∑k=1
Hk2 (h, zk(0)). (3.12)
The risk-free interest rates are affine functions of the CIR risk factors. This specifi-
cation is the standard affine term structure model introduced in Duffie and Kan (1996)
[see also Dai, Singleton (2000)]. It includes the one-factor CIR model [Cox, Ingersoll and
Ross (1985b)] as well as the multi-factor term structure model of Chen and Scott (1993).
As explained in subsection 3.2.2.1, the following restrictions are imposed on the pa-
rameters:
zk(0) = −αk − βµk −γk2β2 > −1, ∀ k = 1, · · · , K. (3.13)
The short rate is defined by r(t) = limh→0− 1h
logB(t, t+h). The following proposition
is proved in Appendix B.2.
68
Proposition 3.3. The short rate is given by
r(t) = limh→0−1
hlogB(t, t+ h) =
d[− logB(t, t+ h)]
dh|h=0
= z0(0) +K∑k=1
zk(0)xk,t. (3.14)
3.2.2.3 Forward Prices for the S&P 500 Index
A forward contract is an agreement to deliver or receive a specified amount of the under-
lying asset (or equivalent cash value) at a specified price and date. A forward contract
always has zero value when it is initiated. There is no money exchange initially or during
the life of the contract, except at the maturity date when the price paid is equal to the
specified forward price. The following proposition is proved in Appendix B.3.
Proposition 3.4. The forward prices are given by
f(t, t+ h) =C(t, t+ h, 1)
C(t, t+ h, 0)
= It exp{−hl0 −K∑k=1
Hk1 (h, zk(1))xk,t +
K∑k=1
Hk1 (h, zk(0))xk,t
−K∑k=1
Hk2 (h, zk(1)) +
K∑k=1
Hk2 (h, zk(0))}, (3.15)
where zk(·) is defined in Equation (3.8), l0 is defined in Equation (3.10), and Hk1 (·, ·) and
Hk2 (·, ·) are given in equation(3.4).
In addition to the restrictions in (3.13), the following restrictions are imposed on the
parameters:
zk(1) = −αk − (β + 1)µk −γk2
(β + 1)2 = zk(0) + lk > −1, ∀ k = 1, · · · , K. (3.16)
69
3.2.2.4 Futures Prices
Let us now consider the price at t of a futures contract written on It+h. The major dif-
ference between a futures contract and a forward contract is the mark-to-market practice
for the futures. A futures contract has also zero value when it is issued and there is no
money exchange initially. However, at the end of each trading day during the life of the
contract, the party against whose favor the price changes must pay the amount of change
to the winning party. That is, a futures contract always has zero value at the end of each
trading day during the life of the contract. If the interest rate is stochastic, the forward
price and futures price are generally not the same [See Cox, Ingersoll and Ross (1981)
and French (1983)]. The following proposition is proved in Appendix B.4.
Proposition 3.5. The prices at t of futures written on It+h are given by
Ft,t+h = Et[exp(
∫ t+h
t
dmτ ) exp(
∫ t+h
t
rτdτ)It+h]
= It exp[−hl0 −K∑k=1
Hk1 (h, lk)xk,t −
K∑k=1
Hk2 (h, lk)], (3.17)
where lk is defined in Equation (3.10), and Hk1 (·, ·) and Hk
2 (·, ·) are given in equation(3.4).
As explained earlier, in addition to the restrictions in (3.13), the following restrictions
are imposed on the parameters:
lk = −µk −1 + 2β
2γk > −1 ∀ k = 1 · · ·K. (3.18)
Propositions 3.4 and 3.5 show that
f(t, t+ h) = Et[exp(
∫ t+h
t
dmτ ) exp(r(t, t+ h)h)It+h]
70
and
Ft,t+h = Et[exp(
∫ t+h
t
dmτ ) exp(
∫ t+h
t
rτdτ)It+h].
Since the short rate is stochastic, the forward and futures prices are not equal in general.
A sufficient condition for the forward and futures prices to be identical is zk(0) = 0,
∀ k = 1 · · ·K, i.e., the interest rates are non-stochastic. This is Proposition 3 in Cox,
Ingersoll and Ross (1981).
3.2.2.5 European Call and Put Options Written on the Index
The prices of the European options are deduced by applying a transform analysis to
function C(t, t + h, u) computed for pure imaginary argument u [see Duffie, Pan and
Singleton (2000) and Appendix B.5].
Proposition 3.6.
i) The European call prices are given by
G(t, t+ h,X) = Et{exp(
∫ t+h
t
dmτ )[exp(log It+h)−X]+} (3.19)
=C(t, t+ h, 1)
2− 1
π
∫ ∞
0
Im[C(t, t+ h, 1− iv) exp(iv logX)]
vdv
−X{C(t, t+ h, 0)
2− 1
π
∫ ∞
0
Im[C(t, t+ h,−iv) exp(iv logX)]
vdv}
(3.20)
where X is the strike price, h is the time-to-maturity, i denotes the pure imaginary number
and Im(·) is the imaginary part of a complex number.
71
ii) The European put prices are given by
H(t, t+ h,X) = Et{exp(
∫ t+h
t
dmτ )[X − exp(log It+h)]+} (3.21)
= −C(t, t+ h, 1)
2+
1
π
∫ ∞
0
Im[C(t, t+ h, 1 + iv) exp(−iv logX)]
vdv
+X{C(t, t+ h, 0)
2− 1
π
∫ ∞
0
Im[C(t, t+ h, iv) exp(−iv logX)]
vdv}.
(3.22)
Again, the restrictions (3.13) and (3.16) are imposed2.
In particular, the relationship between the prices of European call and put options is
G(t, t+ h,X)− C(t, t+ h, 1) = H(t, t+ h,X)−XC(t, t+ h, 0) (3.23)
− 1
π
∫ ∞
0
Im[C(t, t+ h, 1− iv) exp(iv logX)]
vdv
+1
π
∫ ∞
0
Im[C(t, t+ h, 1 + iv) exp(−iv logX)]
vdv
+X
π
∫ ∞
0
Im[C(t, t+ h,−iv) exp(iv logX)]
vdv
− X
π
∫ ∞
0
Im[C(t, t+ h, iv) exp(−iv logX)]
vdv
Equation (3.23) provides the deviation to the Put-Call Parity due to the non-tradability
of the underlying index and shows that this deviation is stochastic.
3.2.3 Pricing Formulas for European Derivatives Written onFutures
3.2.3.1 Derivatives Written on Futures
As for derivatives written on the index, we first consider European derivatives written
on futures with exponential payoffs. More precisely, we introduce three different dates:
2Re(zk(1− iv)) > −1, Re(zk(−iv)) > −1, Re(zk(1 + iv)) > −1 and Re(zk(iv)) > −1,∀ k = 1, · · · ,K,where Re(·) denotes the real part of a complex number, should also hold in order for the pricing formulasto exist. Re(zk(1 − iv)) = Re(zk(1 + iv)) = −αk − (β + 1)µk − γk
2 (β + 1)2 + γk
2 v2 = zk(1) + γk
2 v2 and
Re(zk(−iv)) = Re(zk(iv)) = −αk − βµk − γk
2 β2 + γk
2 v2 = zk(0) + γk
2 v2. So the restrictions (3.13)and
(3.16)are sufficient.
72
• t is the current date
• t+h is the maturity date of the derivatives on futures
• t+h+m is the maturity date of the futures on which the derivatives are written.
CF (t, t + h, t + h + m,u) denotes the price at t of the European derivative paying
(Ft+h,t+h+m)u at t+ h. The following proposition is proved in Appendix B.6.
Proposition 3.7. The prices at t of the European derivatives paying (Ft+h,t+h+m)u at
t+ h are given by
CF (t, t+ h, t+ h+m,u)
=Et[exp
∫ t+h
t
dmτ (Ft+h,t+h+m)u]
= exp(u log It) exp{m(uµ0 −1 + 2β
2uγ0) + h[(β + u)µ0 +
(β + u)2
2γ0 + α0]
−K∑k=1
uHk1 (m, lk)hξkζk −
K∑k=1
uHk2 (m, lk)−
K∑k=1
Hk2 (h, pk(m,u))
−K∑k=1
[uHk1 (m, lk) +Hk
1 (h, pk(m,u))]xk,t} (3.24)
where
pk(m,u) = −αk−(β+u)µk−uHk1 (m, lk)ξk−
γk2
(β+u)2−u2
2[Hk
1 (m, lk)]2ν2k ∀ k = 1 · · ·K,
(3.25)
lk is given in equation(3.10), and Hk1 (·, ·) and Hk
2 (·, ·) are given in equation(3.4).
Again, we impose (3.13) and (3.18) as well as the following restrictions on the param-
eters:
pk(m,u) = −αk − (β + u)µk − uHk1 (m, lk)ξk −
γk2
(β + u)2 − u2
2[Hk
1 (m, lk)]2ν2k > −1,
∀ k = 1, · · · , K. (3.26)
73
3.2.3.2 European Call Options Written on Futures
The following proposition is proved in Appendix B.7.
Proposition 3.8. The prices of European calls written on futures are given by
GF (t, t+ h, t+ h+m,X)
=Et{exp(
∫ t+h
t
dmτ )[exp(logFt+h,t+h+m)−K]+}
=CF (t, t+ h, t+ h+m, 1)
2
− 1
π
∫ ∞
0
Im[CF (t, t+ h, t+ h+m, 1− iv) exp(iv logX)]
vdv
−X{CF (t, t+ h, t+ h+m, 0)
2
− 1
π
∫ ∞
0
Im[CF (t, t+ h, t+ h+m,−iv) exp(iv logX)]
vdv} (3.27)
The parameters are subject to the restrictions in (3.13), (3.18) and3
pk(m, 1) = −αk − (β + 1)µk −Hk1 (m, lk)ξk −
γk2
(β + 1)2 − 1
2[Hk
1 (m, lk)]2ν2k > −1,
∀ k = 1, · · · , K. (3.28)
3.3 Parameter Restrictions for a Tradable Index
In Section 3.2, the pricing formulas are valid for tradable or non-tradable indices. In this
section, we derive the restrictions implied by the tradability of the underlying index.
When the benchmark index is a self-financed and tradable asset, the pricing formula
is also valid for the index itself. In this case, we have
It = Et[exp(
∫ t+h
t
dmτ )It+h] = C(t, t+ h, 1),
3Restrictions (3.13) imply pk(m, 0) = zk(0) > −1. The inequalities pk(m, 0) > −1 and pk(m, 1) > −1imply Re(pk(m,−iv)) > −1 and Re(pk(m, 1− iv)) > −1.
74
and
C(t, t+ h, 1) = It exp[−hz0(1)−K∑k=1
Hk1 (h, zk(1))xk,t −
K∑k=1
Hk2 (h, zk(1))], (3.29)
where zk(·) is defined in Equation (3.8), and Hk1 (·, ·) and Hk
2 (·, ·) are given in equation
(3.4). zk(1) > −1 is imposed, ∀ k = 1, · · · , K.
By considering the expression of C(t, t + h, 1) and identifying the different terms in
the decomposition, we see that the dynamic parameters are constrained byHk
1 (h, zk(1)) = 0, ∀ k = 1, · · · , K, ∀h,
−hz0(1)−∑K
k=1Hk2 (h, zk(1)) = 0, ∀h,
(3.30)
or equivalently by the conditions shown in Proposition 3.9 (See the proof in Appendix
B.8).
Proposition 3.9. When the benchmark index is a self-financed and tradable asset, the
dynamic parameters are constrained by
zk(1) = αk + (β + 1)µk +γk2
(β + 1)2 = zk(0) + lk = 0, ∀ k = 0, · · · , K. (3.31)
These restrictions fix the parameters {αk}, k = 0, · · · , K of the SDF as functions of
the parameters of the index dynamics.
When the benchmark index is tradable, the risk-neutral dynamics of log It can also
be written as
d log It = [r(t)− γ0 +∑γkxk,t
2]dt+
√γ0+
∑γkxk,tdw
∗t , (3.32)
or
dItIt
= r(t)dt+√γ0+
∑γkxk,tdw
∗t . (3.33)
75
In other word, conditional on the underlying factors, the risk wt can be hedged by the
index and the short rate if the index is tradable.
If the benchmark index is tradable, the formulas of derivative prices can be simplified.
In particular, the forward price derived in Proposition 3.4 simplifies to the standard
formula:
f(t, t+ h) =It
B(t, t+ h), (3.34)
and the Spot-Futures Parity will hold for the index and its forward price.
3.4 Conclusion
In this paper we consider a coherent multi-factor affine model to price various derivatives
such as forwards, futures and European options written on the non-tradable S&P 500
Index, and derivatives written on the S&P 500 futures.
We consider both cases when the underlying index is self-financed and tradable and
when it is not, and show the difference between them. When the underlying asset is
self-financed and tradable, an additional arbitrage condition has to be introduced and
implies additional parameter restrictions. These restrictions can be tested in practice to
check whether the derivatives are priced as if the underlying index were self-financed and
tradable.
The S&P 500 Index is not the only non-tradable index on which various derivatives
have been written. Our model can be easily extended to price derivatives written on
76
other non-tradable indices such as a retail price index, a meteorological index, or an
index summarizing the results of a set of insurance companies.
77
Appendix A
Official Description of the Index
The S&P presents the computation of the S&P 500 Index in the following way1. The
S&P 500 Composite Index is calculated with divisors using the so-called “base-aggregative
approach” as follows:
Indext =
∑i pi(t),tqi(t),tDt
, (A.1)
where Dt is the base value or divisor at time t. That is, each component stock’s price is
multiplied by the number of outstanding common shares available to the public for that
company, and the resulting float-adjusted market values are summed for all 500 stocks
and divided by a predetermined base value. Equation (A.1) implies that
Indext+1
Indext=
∑i pi(t+1),t+1qi(t+1),t+1∑
i pi(t),tqi(t),t
Dt
Dt+1
(A.2)
The stocks in the index change from time to time because of mergers, acquisitions or
bankruptcies. For example, in 2005, there were 20 component changes to the S&P 500
Index. The changes are effective after the closure of the trading day. When the index
1Most of the material from this appendix is from www.standardandpoors.com.
78
composition changes, the index value should not change, as otherwise market moves would
be confused by index maintenance. The divisor is used to keep the index value unchanged.
The float adjusted market capitalization of the index is calculated before and after the
change. Before the change, we know the market capitalization2,∑
i pi(t),tqi(t),t, the index
level and the divisor of time t. After the change, we know the market capitalization with
the new stocks {i(t + 1)}, which is∑
i pi(t+1),tqi(t+1),t+1, and the index level (which will
not change); Thus, we can solve for the new divisor, that is,
Indext =
∑i pi(t),tqi(t),tDt
=
∑i pi(t+1),tqi(t+1),t+1
Dt+1
⇒Dt+1
Dt
=
∑i pi(t+1),tqi(t+1),t+1∑
i pi(t),tqi(t),t. (A.3)
The S&P performs the calculations after the market closes. The index opens the next
day with the new stock components and the new divisor. The index level changes only
if the prices of the stocks change.
We can derive from above that
Indext+1
Indext=
∑i pi(t+1),t+1qi(t+1),t+1∑
i pi(t),tqi(t),t
Dt
Dt+1
=
∑i pi(t+1),t+1qi(t+1),t+1∑i pi(t+1),tqi(t+1),t+1
.
The second equation is the same as Equation(1.1).
Equation (A.3) holds generally no matter whether t is a market closing time or not.
If t is not a closing time,
Dt+1
Dt
= 1 =
∑i pi(t+1),tqi(t+1),t+1∑i pi(t+1),tqi(t+1),t+1
=
∑i pi(t+1),tqi(t+1),t+1∑
i pi(t),tqi(t),t,
2Since changes only happen after the market closes, t denotes the closing time and t+1 the openingtime on the next day.
79
since {i(t+ 1)} is the same as {i(t)} and hence qi(t+1),t+1 is the same as qi(t),t for all i.
The divisor is also adjusted periodically (typically, several times per quarter), when
the total of outstanding shares have changed in one or more component index securities
due to secondary offering, repurchases, conversions or other corporate actions, that is,
{qi(t),t} is different from {qi(t+1),t+1}, although {i(t + 1)} is the same as {i(t)}. If the
number of shares of a stock changes, the divisor is adjusted in the same way as when
we have stock changes so the index level remains unchanged. The index can be adjusted
in this way when companies issue new shares or buy back their shares. The same basic
approach is used for stock price adjustments, such as when a company spins off a unit.
Adjustments to the divisor assure that changes in the index’s level reflect the changes in
the market and not the corporate actions. There are many other corporate actions that
require divisor adjustments, for instance, a change of the available float shares of index
securities. One common action that does not require a divisor adjustment is a stock split
when the number of shares increases and their price decreases in proportion.
The S&P 500 is maintained by the S&P Index Committee, that is, a team of Standard
& Poor’s economists and index analysts who meet on a regular basis.
i) Eligibility criteria for inclusion in the S&P 500 are:
• U.S. companies.
• Adequate liquidity and reasonable per-share price – the ratio of annual dollar
value traded to market capitalization should be 0.3 or greater. Very low stock
prices can affect a stock liquidity.
80
• Market capitalization of $4 Billion or more for the S&P 500. The market
capitalization of a potential addition to an index is looked at in the context of
its short- and medium-term historical trends, as well as those of its industry.
The range is reviewed from time to time to ensure consistency with market
conditions.
• Financial viability, usually measured as four consecutive quarters of positive
as-reported earnings. As-reported earnings are GAAP Net Income excluding
discontinued operations and extraordinary items.
• Public float of at least 50% of the stock.
• Maintaining sector balance for each index, as measured by a comparison of
the GICS sectors in each index and in the market, in the relevant market
capitalization ranges.
• Initial public offerings (IPOs) should be seasoned for 6 to 12 months before
being considered for addition to indices.
• Operating company and not a closed-end fund, holding company, partnership,
investment vehicle or royalty trust. Real Estate Investment Trusts are eligible
for inclusion in Standard & Poor’s U.S. indices.
ii) Eligibility criteria for deletions from the S&P 500 are:
• Companies involved in mergers, being acquired or significantly restructured
such that they no longer meet inclusion criteria.
81
• Companies which substantially violate one or more of the addition criteria.
The Standard & Poor’s believes turnover in index membership should be avoided
when possible. The addition criteria are for addition to an index, not for continued
membership. As a result, a company in an index that appears to violate the criteria for
addition to that index will not be deleted unless ongoing conditions warrant an index
change. When a company is removed from an index, the Standard & Poor’s will explain
the basis for the removal.
There are totally 176 index composition changes from Jan 5, 2000 to March 31, 2006.
In 162 of these changes, the deleted stock is immediately replaced with another stock
and the simultaneous deletion and addition are counted as a single change. In the other
14 changes, what typically happens is a stock is deleted from the index after the market
closes on the first day and another stock is added to the index after the market closes
on the next one or two days. The deletion and the addition are counted as two changes.
During the period when the old stock is deleted but the new stock is not added yet,
the index is usually calculated with the closing price of the deleted stock on the day the
deletion happens. For example, on January 3, 2006, the Standard & Poors announced
that Estee Lauder Companies, Inc. would be added to the S&P 500 after the close of
trading on January 4. Estee Lauder would take the place of S&P 500 constituent Mercury
Interactive Corp., which would be removed from the index after the close of trading on
January 3. On January 4, the S&P 500 is calculated with the closing price of Mercury
Interactive on January 3. Then, when Estee Lauder is added on January 4, the divisor
82
is adjusted to reflect the change of the index.
Changes to the S&P 500 Index are made on an as needed basis. There is no annual or
semi-annual reconstitution. Rather, changes in response to corporate actions and market
developments can be made at any time. Constituent changes are typically announced two
to five days before they are scheduled to be implemented. Announcements are available
to the public via the web site, www.indices.standardandpoors.com, before or at the same
time they are available to clients or the affected companies.
In cases where there is no achievable market price for a stock being deleted, it can be
removed at a zero or minimal price at the index committee’s discretion, in recognition of
the constraints faced by investors in trading bankrupt or suspended stocks.
The index committee also lays down policies about share changes. Changes in a com-
pany’s outstanding shares of less than 5% due to its acquisition of another company in
the same headline index (e.g., both are in the S&P 500) are made as soon as reasonably
possible. All other changes of less than 5% are accumulated and made quarterly on the
third Friday of March, June, September, and December; they are usually announced two
days prior. Changes in a company’s outstanding shares of 5% or more due to mergers, ac-
quisitions, public offerings, private placements, tender offers, Dutch auctions or exchange
offers are made as soon as reasonably possible. Other changes of 5% or more (due to,
for example, company stock repurchases, redemptions, exercise of options, warrants, con-
version of preferred stock, notes, debt, equity participations or other recapitalizations)
are made weekly, and are announced on Tuesday for implementation after the close of
trading on Wednesday. In the case of certain rights issuances, in which the number of
83
rights issued and/or terms of their exercise are deemed substantial, a price adjustment
and share increase may be implemented immediately. Corporate actions such as stock
splits, stock dividends, spinoffs and rights offerings are applied after the close of trading
on the day prior to the ex-date.
Changes in IWF’s of more than ten percentage points caused by corporate actions
(such as M&A activity, restructurings or spinoffs) are made as soon as reasonably possible.
Other changes in IWF’s are made annually, in September, when IWF’s are reviewed.
84
Appendix B
Proofs of Propositions
B.1 Proof of Proposition 3.1
The price of the call option is:
C(t, t+ h, u)
=Et[exp(
∫ t+h
t
dmτ + u log It+h)]
=Et{exp[
∫ t+h
t
dmτ + u(log It +
∫ t+h
t
d log Iτ )]}
= exp(u log It)Et[exp(
∫ t+h
t
(dmτ + ud log Iτ )]
= exp(u log It)Et{exp
∫ t+h
t
([α0 + βµ0 +K∑k=1
(αk + βµk)xk,τ ]dτ
+ β(γ0 +K∑k=1
γkxk,τ )1/2dwτ + u[(µ0 +
K∑k=1
µkxk,τ )dτ + (γ0 +K∑k=1
γkxk,τ )1/2dwτ ])}
= exp(u log It)Et{exp
∫ t+h
t
([α0 + (β + u)µ0 +K∑k=1
(αk + (β + u)µk)xk,τ ]dτ
+ (β + u)(γ0 +K∑k=1
γkxk,τ )1/2dwτ )}
85
= exp(u log It)Et{exp
∫ t+h
t
[α0 + (β + u)µ0 +K∑k=1
(αk + (β + u)µk)xk,τ ]dτ
× Et(exp[(β + u)
∫ t+h
t
(γ0 +K∑k=1
γkxk,τ )1/2dwτ ] | Xk,τ )},
where Xk,τ denotes the set {xk,τ}k=1···Kτ=t···t+h.
Since exp[(β + u)
∫ t+h
t
(γ0 +K∑k=1
γkxk,τ )1/2dwτ ] | Xk,τ
∼ LN(0, (β + u)2
∫ t+h
t
(γ0 +K∑k=1
γkxk,τ )dτ),
we deduce that:
C(t, t+ h, u)
= exp(u log It)Et{exp
∫ t+h
t
[α0 + (β + u)µ0 +K∑k=1
(αk + (β + u)µk)xk,τ ]dτ
× exp[(β + u)2
2
∫ t+h
t
(γ0 +K∑k=1
γkxk,τ )dτ ]}
= exp(u log It)Et{exp
∫ t+h
t
([α0 + (β + u)µ0 +K∑k=1
(αk + (β + u)µk)xk,τ ]dτ
+(β + u)2
2(γ0 +
K∑k=1
γkxk,τ )dτ)}
= exp(u log It) exp
∫ t+h
t
[α0 + (β + u)µ0 +(β + u)2
2γ0]dτ
× Et{exp
∫ t+h
t
K∑k=1
[(αk + (β + u)µk +(β + u)2
2γk)xk,τ ]dτ}
= exp(u log It) exp{h[α0 + (β + u)µ0 +(β + u)2
2γ0]}
× Et{expK∑k=1
∫ t+h
t
[αk + (β + u)µk +(β + u)2
2γk]xk,τdτ}
= exp(u log It) exp{h[α0 + (β + u)µ0 +(β + u)2
2γ0]}
86
×K∏k=1
Et{exp−∫ t+h
t
−[αk + (β + u)µk +(β + u)2
2γk]xk,τdτ},
since factors {xk,t}, k = 1, · · · , Kare independent,
= exp(u log It) exp{h[α0 + (β + u)µ0 +(β + u)2
2γ0]}
×K∏k=1
exp[−Hk1 (h, zk(u))xk,t −Hk
2 (h, zk(u))]
= exp(u log It) exp{h[α0 + (β + u)µ0 +γ0
2(β + u)2]
−K∑k=1
Hk1 (h, zk(u))xk,t −
K∑k=1
Hk2 (h, zk(u))},
where
zk(u) = −αk − (β + u)µk −γk2
(β + u)2,
and Hk1 (·, ·) and Hk
2 (·, ·) are given in equation(3.4)
B.2 Proof of Proposition 3.3
The instantaneous interest rate is defined by:
r(t) = limh→0−1
hlogB(t, t+ h) =
d[− logB(t, t+ h)]
dh|h=0 .
We have:
− logB(t, t+ h) = −h(α0 + βµ0 +γ0
2β2) +
K∑k=1
Hk1 (h, zk(0))xk,t +
K∑k=1
Hk2 (h, zk(0)).
We deduce that:
d[− logB(t, t+ h)]
dh= −α0 − βµ0 −
γ0
2β2 +
K∑k=1
dHk1 (h, zk(0))
dhxk,t +
K∑k=1
dHk2 (h, zk(0))
dh,
where
87
Hk1 (h, zk(0)) =
2zk(0)(exp[εk(zk(0))h]− 1)
(εk(zk(0)) + ξk)(exp[εk(zk(0))h]− 1) + 2εk(zk(0)),
Hk2 (h, zk(0)) =
−2ξkζkν2k
{log[2εk(zk(0))] +h
2[εk(zk(0)) + ξk]
− log[(εk(zk(0)) + ξk)(exp[εk(zk(0))h]− 1) + 2εk(zk(0))]}.
Let us denote (εk(zk(0)) + ξk)(exp[εk(zk(0))h]− 1) + 2εk(zk(0)) ≡ A. We get:
dHk1 (h, zk(0))
dh|h=0
=2zk(0) exp[εk(zk(0))h]εk(zk(0))A
A2
− 2zk(0)(exp[εk(zk(0))h]− 1)(εk(zk(0)) + ξk) exp[εk(zk(0))h]εk(zk(0))
A2|h=0
=2zk(0)εk(zk(0))2εk(zk(0))− 0
[2εk(zk(0))]2= zk(0),
and
dHk2 (h, zk(0))
dh|h=0
=−2ξkζkν2k
{1
2[εk(zk(0)) + ξk]−
1
A(εk(zk(0)) + ξk) exp[εk(zk(0))h]εk(zk(0))} |h=0
=−2ξkζkν2k
{1
2[εk(zk(0)) + ξk]−
(εk(zk(0)) + ξk)εk(zk(0))
2εk(zk(0))} = 0.
We deduce:
r(t) = −α0 − βµ0 −γ0
2β2 +
K∑k=1
zk(0)xk,t.
B.3 Proof of Proposition 3.4
Since
E[exp(
∫ t+h
t
dmτ )(f(t, t+ h)− It+h)] = 0,
88
we get:
B(t, t+ h)f(t, t+ h) = E[exp(
∫ t+h
t
dmτ )It+h],
and
f(t, t+ h) =C(t, t+ h, 1)
C(t, t+ h, 0).
B.4 Proof of Proposition 3.5
Since Et[
∫ t+h
t
(exp
∫ t+τ
t
dms)dFτ ] = 0, we get:
Ft,t+h
=Et[exp(
∫ t+h
t
dmτ ) exp(
∫ t+h
t
rτdτ)It+h]
=Et{exp[
∫ t+h
t
(dmτ + rτdτ) + log It +
∫ t+h
t
d log Iτ ]}
=ItEt[exp(
∫ t+h
t
(dmτ + rτdτ + d log Iτ )]
=ItEt{exp
∫ t+h
t
[(α0 + βµ0 +K∑k=1
(αk + βµk)xk,τ )dτ
+ β(γ0 +K∑k=1
γkxk,τ )1/2dwτ + (µ0 +
K∑k=1
µkxk,τ )dτ
+ (γ0 +K∑k=1
γkxk,τ )1/2dwτ + (−α0 − βµ0 −
γ0
2β2 +
K∑k=1
zk(0)xk,τ )dτ ]}
=ItEt{exp
∫ t+h
t
[(α0 + (β + 1)µ0 − α0 − βµ0 −γ0
2β2)dτ
+K∑k=1
(αk + βµk + µk + zk(0))xk,τdτ + (β + 1)(γ0 +K∑k=1
γkxk,τ )1/2dwτ ]}
=It exp[h(µ0 −γ0
2β2)]Et{exp
∫ t+h
t
K∑k=1
(αk + (β + 1)µk + zk(0))xk,τdτ
× Et[exp
∫ t+h
t
(β + 1)(γ0 +K∑k=1
γkxk,τ )1/2dwτ | Xk,τ ]}.
89
Since exp
∫ t+h
t
(β + 1)(γ0 +K∑k=1
γkxk,τ )1/2dwτ | Xk,τ
∼ LN(0, (β + 1)2
∫ t+h
t
(γ0 +K∑k=1
γkxk,τ )dτ),
we get:
Ft,t+h
=It exp[h(µ0 −γ0
2β2)]
× Et{exp
∫ t+h
t
[K∑k=1
(αk + (β + 1)µk + zk(0))xk,τdτ +(β + 1)2
2(γ0 +
K∑k=1
γkxk,τ )dτ ]}
=It exp[h(µ0 −γ0
2β2 +
(β + 1)2
2γ0)]
× Et{exp
∫ t+h
t
[K∑k=1
(αk + (β + 1)µk + zk(0) +(β + 1)2
2γk)xk,τ ]dτ}
=It exp[h(µ0 −γ0
2β2 +
(β + 1)2
2γ0)]
×K∏k=1
Et{exp−∫ t+h
t
−[αk + (β + 1)µk + zk(0) +(β + 1)2
2γk]xk,τdτ}
=It exp[h(µ0 −γ0
2β2 +
(β + 1)2
2γ0)]
K∏k=1
exp[−Hk1 (h, lk)xk,t −Hk
2 (h, lk)]
=It exp[h(µ0 +1 + 2β
2γ0)−
K∑k=1
Hk1 (h, lk)xk,t −
K∑k=1
Hk2 (h, lk)],
where
lk = −αk − (β + 1)µk − zk(0)− γk2
(β + 1)2
= −µk −1 + 2β
2γk.
90
B.5 Proof of Proposition 3.6
Let us first consider the call option with price G(t, t+ h,X). Its price is given by:
G(t, t+ h,X)
= Et{exp(
∫ t+h
t
dmτ )[exp(log It+h)−X]+}
= Et{exp(
∫ t+h
t
dmτ )[exp(log It+h)−X]1− log It+h≤− logX}
= A1,−1(− logX;x1,t, · · · , xK,t, log It, h)−XA0,−1(− logX;x1,t, · · · , xK,t, log It, h),
whereAa,b(y;x1,t, · · · , xK,t, log It, h) = Et[exp(
∫ t+h
t
dmτ ) exp(a log It+h)1b log It+h≤y].
The Fourier-Stieltjes transform of Aa,b(y;x1,t, · · · , xK,t, log It, h) is
∫<
exp(ivy)dAa,b(y;x1,t, · · · , xK,t, log It, h)
= Et{exp(
∫ t+h
t
dmτ ) exp[(a+ ivb) log It+h]} = C(t, t+ h, a+ ivb).
We deduce that:
Aa,b(y;x1,t, · · · , xK,t, log It, h)
=C(t, t+ h, a)
2− 1
π
∫ ∞
0
Im[C(t, t+ h, a+ ivb) exp(−ivy)]
vdv
[see Duffie, Pan and Singleton (2000), p1352].
By substitution we get the call price
G(t, t+ h,X) =C(t, t+ h, 1)
2− 1
π
∫ ∞
0
Im[C(t, t+ h, 1− iv) exp(iv logX)]
vdv
−X{C(t, t+ h, 0)
2− 1
π
∫ ∞
0
Im[C(t, t+ h,−iv) exp(iv logX)]
vdv}.
91
Similarly, for the put option with price H(t, t+ h,X), we have
H(t, t+ h,X)
= Et{exp(
∫ t+h
t
dmτ )[X − exp(log It+h)]+}
= Et{exp(
∫ t+h
t
dmτ )[X − exp(log It+h)]1 log It+h≤logX}
= −A1,1(logX;x1,t, · · · , xK,t, log It, h) +XA0,1(logX;x1,t, · · · , xK,t, log It, h)
= −C(t, t+ h, 1)
2+
1
π
∫ ∞
0
Im[C(t, t+ h, 1 + iv) exp(−iv logX)]
vdv
+X{C(t, t+ h, 0)
2− 1
π
∫ ∞
0
Im[C(t, t+ h, iv) exp(−iv logX)]
vdv}.
B.6 Proof of Proposition 3.7
We have:
Ft+h,t+h+m = It+h exp[m(µ0 −1 + 2β
2γ0)−
K∑k=1
Hk1 (m, lk)xk,t+h −
K∑k=1
Hk2 (m, lk)],
logFt+h,t+h+m = log It+h +m(µ0 −1 + 2β
2γ0)−
K∑k=1
Hk1 (m, lk)xk,t+h −
K∑k=1
Hk2 (m, lk).
Therefore,
CF (t, t+ h, t+ h+m,u)
= Et(exp
∫ t+h
t
dmτ (Ft+h,t+h+m)u)
= Et[exp
∫ t+h
t
dmτ exp(u logFt+h,t+h+m)]
= Et[exp(
∫ t+h
t
dmτ + u logFt+h,t+h+m)]
92
= Et{exp(
∫ t+h
t
[(α0 + βµ0 +K∑k=1
(αk + βµk)xk,τ )dτ + β(γ0 +K∑k=1
γkxk,τ )1/2dwτ ]
+ u[log It +
∫ t+h
t
d log It +m(µ0 −1 + 2β
2γ0)
−K∑k=1
Hk1 (m, lk)(xk,t +
∫ t+h
t
dxk,τ )−K∑k=1
Hk2 (m, lk)])}
= exp[u log It + um(µ0 −1 + 2β
2γ0)− u
K∑k=1
Hk1 (m, lk)xk,t − u
K∑k=1
Hk2 (m, lk)]
× Et{exp
∫ t+h
t
[(α0 + βµ0 +K∑k=1
(αk + βµk)xk,τ )dτ + β(γ0 +K∑k=1
γkxk,τ )1/2dwτ
+ u(µ0 +K∑k=1
µkxk,τ )dτ + u(γ0 +K∑k=1
γkxk,τ )1/2dwτ
− uK∑k=1
Hk1 (m, lk)(ξk(ζk − xk,τ )dτ + νk
√xk,τdwk,τ )]}
= exp[u log It + um(µ0 −1 + 2β
2γ0)− u
K∑k=1
Hk1 (m, lk)xk,t − u
K∑k=1
Hk2 (m, lk)]
× Et{exp
∫ t+h
t
[(α0 + βµ0 + uµ0 − uK∑k=1
Hk1 (m, lk)ξkζk)dτ
+K∑k=1
(αk + βµk + uµk + uHk1 (m, lk)ξk)xk,τ )dτ
+ (β + u)(γ0 +K∑k=1
γkxk,τ )1/2dwτ − u
K∑k=1
Hk1 (m, lk)νk
√xk,τdwk,τ ]}
= exp{u[log It +m(µ0 −1 + 2β
2γ0)−
K∑k=1
Hk1 (m, lk)xk,t −
K∑k=1
Hk2 (m, lk)]
+ h[α0 + (β + u)µ0 − uK∑k=1
Hk1 (m, lk)ξkζk]}
× Et{exp(
∫ t+h
t
K∑k=1
[αk + (β + u)µk + uHk1 (m, lk)ξk]xk,τdτ)
× Et(exp
∫ t+h
t
[(β + u)(γ0 +K∑k=1
γkxk,τ )1/2dwτ − u
K∑k=1
Hk1 (m, lk)νk
√xk,τdwk,τ ] | Xk,τ )}
93
= exp{u log It + (um+ hβ + hu)µ0 −1 + 2β
2umγ0 + hα0
−K∑k=1
uHk1 (m, lk)(xk,t + hξkζk)−
K∑k=1
uHk2 (m, lk)}
× Et{exp(
∫ t+h
t
K∑k=1
[αk + (β + u)µk + uHk1 (m, lk)ξk]xk,τdτ)
× (exp
∫ t+h
t
((β + u)2
2(γ0 +
K∑k=1
γkxk,τ ) +K∑k=1
u2
2[Hk
1 (m, lk)]2ν2kxk,τ )dτ)}
= exp{u log It + (um+ hβ + hu)µ0 −1 + 2β
2umγ0 + hα0
−K∑k=1
uHk1 (m, lk)(xk,t + hξkζk)−
K∑k=1
uHk2 (m, lk)}
× Et{exp
∫ t+h
t
((β + u)2
2γ0
+K∑k=1
[αk + (β + u)µk + uHk1 (m, lk)ξk +
(β + u)2
2γk +
u2
2[Hk
1 (m, lk)]2ν2k ]xk,τ )dτ}
= exp{u log It + (um+ hβ + hu)µ0 +(β + u)2h− (1 + 2β)um
2γ0 + hα0
−K∑k=1
uHk1 (m, lk)(xk,t + hξkζk)−
K∑k=1
uHk2 (m, lk)}
×K∏k=1
Et[exp−∫ t+h
t
−(αk + (β + u)µk + uHk1 (m, lk)ξk
+(β + u)2
2γk +
u2
2[Hk
1 (m, lk)]2ν2k)xk,τdτ ]
= exp{u log It + (um+ hβ + hu)µ0 +(β + u)2h− (1 + 2β)um
2γ0 + hα0
−K∑k=1
uHk1 (m, lk)(xk,t + hξkζk)−
K∑k=1
uHk2 (m, lk)}
×K∏k=1
exp[−Hk1 (h, pk(m,u))xk,t −Hk
2 (h, pk(m,u))]
= exp{u log It + (um+ hβ + hu)µ0 +(β + u)2h− (1 + 2β)um
2γ0 + hα0
94
−K∑k=1
uHk1 (m, lk)hξkζk −
K∑k=1
uHk2 (m, lk)−
K∑k=1
Hk2 (h, pk(m,u))
−K∑k=1
[uHk1 (m, lk) +Hk
1 (h, pk(m,u))]xk,t}
= exp(u log It) exp{m(uµ0 −1 + 2β
2uγ0) + h[(β + u)µ0 +
(β + u)2
2γ0 + α0]
−K∑k=1
uHk1 (m, lk)hξkζk −
K∑k=1
uHk2 (m, lk)−
K∑k=1
Hk2 (h, pk(m,u))
−K∑k=1
[uHk1 (m, lk) +Hk
1 (h, pk(m,u))]xk,t},
where
pk(m,u) = −αk − (β + u)µk − uHk1 (m, lk)ξk −
γk2
(β + u)2 − u2
2[Hk
1 (m, lk)]2ν2k ,
lk is given in equation(3.10), and Hk1 (·, ·) and Hk
2 (·, ·) are given in equation(3.4).
B.7 Proof of Proposition 3.8
The price of the call optioin written on the futures is:
GF (t, t+ h, t+ h+m,X)
= Et{exp(
∫ t+h
t
dmτ )[exp(logFt+h,t+h+m)−X]+}
= Et{exp(
∫ t+h
t
dmτ )[exp(logFt+h,t+h+m)−X]1− logFt+h,t+h+m≤− logX}
= A1,−1(− logX;x1,t, · · · , xK,t, logFt,t+h+m, h)
−XA0,−1(− logX;x1,t, · · · , xK,t, logFt,t+h+m, h),
where
Aa,b(y;x1,t, · · · , xK,t, logFt,t+h+m, h)
95
=Et[exp(
∫ t+h
t
dmτ ) exp(a logFt+h,t+h+m)1b logFt+h,t+h+m≤y].
The Fourier-Stieltjes transform of Aa,b(y;x1,t, · · · , xK,t, logFt,t+h+m, h) is:∫<
exp(ivy)dAa,b(y;x1,t, · · · , xK,t, logFt,t+h+m, h)
= Et{exp(
∫ t+h
t
dmτ ) exp[(a+ ivb) logFt+h,t+h+m]} = CF (t, t+ h, t+ h+m, a+ ivb).
Therefore, we have:
Aa,b(y;x1,t, · · · , xK,t, logFt,t+h+m, h)
=CF (t, t+ h, t+ h+m, a)
2− 1
π
∫ ∞
0
Im[CF (t, t+ h, t+ h+m, a+ ivb) exp(−ivy)]
vdv,
and
GF (t, t+ h, t+ h+m,X)
=CF (t, t+ h, t+ h+m, 1)
2− 1
π
∫ ∞
0
Im[CF (t, t+ h, t+ h+m, 1− iv) exp(iv logX)]
vdv
−X{CF (t, t+ h, t+ h+m, 0)
2
− 1
π
∫ ∞
0
Im[CF (t, t+ h, t+ h+m,−iv) exp(iv logX)]
vdv}.
B.8 Proof of Proposition 3.9
The first restriction in Equation (3.30) holds, if and only, if
zk(1) = −αk − (β + 1)µk −γk2
(β + 1)2 = 0, ∀k = 1, · · · , K.
This implies εk(zk(1)) =|ξk |, ∀k = 1, · · · , K, and
Hk2 (h, zk(1)) =
−2ξkζkν2k
{log|2ξk |+h
2(|ξk |+ξk)− log[(|ξk |+ξk)(exp(|ξk |h)− 1) + 2 |ξk |]}
96
= 0, no matter if ξk > 0 or ξk < 0, ∀k = 1, · · · , K.
This, with the second restriction in Equation (3.30), implies that
α0 + (β + 1)µ0 +γ0
2(β1)
2 = 0.
Therefore, Equation (3.30) is equivalent to
αk + (β + 1)µk +γk2
(β1)2 = 0, ∀k = 0, · · · , K.
97
Appendix C
Tables and Figures
98
Table C.1: Summary Statistics of the SPDR Traded from Jan 2, 2001 to Dec 30, 2005.
Daily Traded Volume (1) No. of Publicly Held Shares (2) (1) as a % of (2)a
Mean 38,740,562 Shares 336,267,000 Shares 11.20%
Median 37,608,250 Shares 374,888,000 Shares 10.55%
Stddev 21,223,754 Shares 111,369,000 Shares 4.73%
Min 3,303,100 Shares 149,422,000 Shares 2.21%
Max 141,120,800 Shares 471,080,000 Shares 39.16%
Daily Traded Valueb (3) Value of Publicly Held Sharesc (4) (3) as a % of (4)d
Mean $4,216,321,787 $36,606,862,398 11.20%
Median $3,852,668,649 $39,438,671,390 10.55%
Stddev $416,685,383 $12,345,657,946 4.73%
Min $409,914,710 $14,535,772,160 2.21%
Max $16,821,599,360 $58,654,170,800 39.16%
aFor each day, the daily traded volume as a percentage of the number of publicly held shares iscalculated first, then the statistics are calculated.
bThe daily traded value is computed as the daily traded volume multiplied by the closing price.cThe value of publicly held shares is computed as the number of publicly held shares multiplied by
the closing price.dFor each day, the daily traded value as a percentage of the value of publicly held shares is calculated
first, then the statistics are calculated. So it is not surprising that the numbers in these columns areexactly the same as the ones above.
99
Table C.2: Ljung-Box Statistics for Daily Relative Price Changes of the S&P 500 Index
and the SPDR ( XR and DRR) from Jan 2, 2001 to Dec 30, 2005.
XR DRR
Lags Qstat p-V alue Qstat p-V alue
5 2.78 .734 2.80 .730
10 11.03 .355 14.32 .159
15 20.20 .165 22.72 .090
20 34.46 .023 36.78 .012
25 40.83 .024 45.64 .007
30 49.41 .014 57.16 .002
35 61.91 .003 68.60 .001
40 64.74 .008 71.95 .001
45 75.51 .003 83.13 .000
50 78.68 .006 85.95 .001
Table C.3: Ljung-Box Statistics for Daily Holding Period Returns of the S&P 500 Index
and the SPDR ( XR and DRRd) from Jan 2, 2001 to Dec 30, 2005.
XR DRRd
Lags Qstat p-V alue Qstat p-V alue
5 2.78 .734 2.83 .726
10 11.03 .355 11.18 .343
15 20.20 .165 19.15 .207
20 34.46 .023 32.22 .041
25 40.83 .024 38.10 .045
30 49.41 .014 48.03 .020
35 61.91 .003 61.20 .004
40 64.74 .008 65.01 .007
45 75.51 .003 74.68 .004
50 78.68 .006 76.87 .009
100
Table C.4: Ljung-Box Statistics for Squared Daily Relative Price Changes of the S&P 500
Index and the SPDR ( XR2 and DRR
2) from Jan 2, 2001 to Dec 30, 2005.
XR2
DRR2
Lags Qstat p-V alue Qstat p-V alue
5 379.98 0.000 393.33 0.000
10 642.35 0.000 683.72 0.000
15 841.33 0.000 909.53 0.000
20 988.01 0.000 1062.17 0.000
25 1058.96 0.000 1143.93 0.000
30 1159.51 0.000 1243.25 0.000
35 1235.91 0.000 1321.98 0.000
40 1308.42 0.000 1407.93 0.000
45 1394.45 0.000 1505.98 0.000
50 1540.66 0.000 1661.52 0.000
Table C.5: Ljung-Box Statistics for Squared Daily Holding Period Returns of the S&P 500
Index and the SPDR ( XR2 and DRR
2) from Jan 2, 2001 to Dec 30, 2005.
XR2
DRR2
Lags Qstat p-V alue Qstat p-V alue
5 379.98 0.000 367.30 0.000
10 642.35 0.000 679.47 0.000
15 841.33 0.000 869.90 0.000
20 988.01 0.000 1009.88 0.000
25 1058.96 0.000 1075.38 0.000
30 1159.51 0.000 1162.60 0.000
35 1235.91 0.000 1230.64 0.000
40 1308.42 0.000 1307.56 0.000
45 1394.45 0.000 1394.66 0.000
50 1540.66 0.000 1556.68 0.000
101
Tab
leC
.6:
Tim
es-t
o-M
aturi
tyof
the
S&
P50
0In
dex
Opti
ons
Mon
th01
0203
0405
0607
0809
1011
1201
0203
0405
0607
0809
1011
1201
0203
0405
06D
ec-J
an1m
2m3m
6m9m
12m
18m
24m
Jan-
Feb
1m2m
3m5m
8m11
m17
m23
mFe
b-M
ar1m
2m3m
4m7m
10m
16m
22m
Mar
-Apr
1m2m
3m6m
9m12
m15
m21
mA
pr-M
ay1m
2m3m
5m8m
11m
14m
20m
May
-Jun
1m2m
3m4m
7m10
m13
m19
mJu
n-Ju
l1m
2m3m
6m9m
12m
18m
24m
Jul-
Aug
1m2m
3m5m
8m11
m17
m23
mA
ug-S
ep1m
2m3m
4m7m
10m
16m
22m
Sep-
Oct
1m2m
3m6m
9m12
m15
m21
mO
ct-N
ov1m
2m3m
5m8m
11m
14m
20m
Nov
-Dec
1m2m
3m4m
7m10
m13
m19
m
Not
e: 1.T
he
left
colu
mn
show
sth
etr
adin
gm
onth
.E
ach
item
repre
sents
the
per
iod
from
the
firs
ttr
adin
gday
afte
rth
eop
tion
expir
atio
ndat
eof
one
mon
thto
the
opti
onex
pir
atio
nday
ofth
enex
tm
onth
.F
orex
ample
,“D
ec-J
an”
mea
ns
from
the
Mon
day
follow
ing
the
thir
dF
riday
ofD
ecem
ber
toth
eSat
urd
ayfo
llow
ing
the
thir
dF
riday
ofth
eJan
uar
y.
2.T
he
top
row
show
sth
eex
pir
atio
nm
onth
.F
orex
ample
,th
efirs
tro
wca
nb
ere
adas
:O
pti
ons
trad
edduri
ng
the
per
iod
from
the
Mon
day
follow
ing
the
thir
dF
riday
ofD
ecem
ber
(the
day
onw
hic
ha
set
ofop
tion
sw
ith
new
expir
atio
ndat
ear
ein
troduce
d)
toth
eth
ird
Fri
day
ofJan
uar
yw
ill
expir
ein
Jan
uar
y(0
1),
Feb
ruar
y(0
2),
Mar
ch(0
3),
June
(06)
,Sep
tem
ber
(09)
and
Dec
emb
er(1
2)of
the
sam
eye
aran
dJune
(06)
and
Dec
emb
er(1
2)in
the
follow
ing
year
.T
he
tim
e-to
-mat
uri
tyfo
rea
chof
them
onth
eM
onday
follow
ing
the
thir
dF
riday
ofD
ecem
ber
is1
mon
th,
2m
onth
s,3
mon
ths,
6m
onth
s,9
mon
ths,
12m
onth
s,18
mon
ths
and
24m
onth
s,re
spec
tive
ly.
3.It
can
easi
lyb
ese
enth
atop
tion
str
aded
inth
efirs
tsi
xm
onth
shav
eth
esa
me
tim
es-t
o-m
aturi
tyas
thos
etr
aded
inth
ese
cond
six
mon
ths,
soth
ere
are
two
cycl
esp
erye
ar.
102
Tab
leC
.7:
Sum
mar
ySta
tist
ics
ofth
eO
pti
ons
Tra
ded
from
Jan
2,20
03to
Dec
31,
2003
(252
day
s).
Dai
ly#
of
Opt
ions
Dai
lyT
otal
Tra
ding
Vol
ume
(Con
trac
ts)
Dai
lyT
otal
Tra
ding
Val
uea
($)
Cal
lP
utB
oth
Cal
lP
utB
oth
Mea
n51
7.21
57,4
64.9
279
,678
.26
137,
143.
1917
8,76
4,12
4.37
149,
052,
032.
0032
7,81
6,15
6.37
Med
ian
518.
0053
,988
.00
76,3
18.5
013
1,86
9.00
133,
965,
026.
2513
0,61
7,44
3.75
286,
344,
682.
50
Stdd
ev28
.05
23,2
66.5
025
,232
.18
43,2
12.9
116
6,54
1,44
1.06
76,6
46,5
31.5
619
7,36
3,80
9.44
Min
422.
007,
819.
0029
,609
.00
37,4
28.0
017
,861
,787
.50
21,5
85,5
67.5
051
,359
,310
.00
Max
572.
0016
3,80
7.00
164,
600.
0032
2,71
1.00
1,43
0,18
7,71
7.50
523,
773,
205.
001,
600,
668,
170.
00
(Con
tinued
onnex
tpag
e)
aD
aily
trad
edva
lue
isco
mpu
ted
asda
ilytr
aded
volu
me
tim
esth
eav
erag
eof
best
clos
ing
bid
and
ask
pric
esm
ulti
plie
dby
100.
103
(Table C.7 continued)
Traded Volume of Each Optiona Price of Each Optionb
Mean 265 Contracts $111.85
Mode 0 Contracts $0.25
Stddev 1,053 Contracts $146.53
Min 0 Contracts $0.025
10% quantile 0 Contracts $0.25
20% quantile 0 Contracts $2.20
30% quantile 0 Contracts $9.40
40% quantile 0 Contracts $23.70
Median 0 Contracts $47.10
60% quantile 0 Contracts $82.60
70% quantile 6 Contracts $132.40
80% quantile 67 Contracts $210.90
90% quantile 565 Contracts $329.20
95% quantile 1,545 Contracts $427.80
96% quantile 1,953 Contracts $456.51
97% quantile 2,500 Contracts $491.87
98% quantile 3,334 Contracts $539.10
99% quantile 5,020 Contracts $618.01
Max 64,611 Contracts $993.50
aThere are 130,336 observations. Options with the same maturity date and strike price, but tradedon different days are regarded as different options.
bThe price is computed as the average of bid and ask prices. The value of each contract is equal tothe price multiplied by 100.
104
Tab
leC
.8:
Sum
mar
ySta
tist
ics
ofth
eA
ctiv
ely
Tra
ded
Opti
ons
(wit
htr
aded
volu
me≥
2,00
0co
ntr
acts
)fr
omJan
2,20
03
toD
ec31
,20
03(2
52day
s).
Dai
lyN
umbe
rof
Act
ivel
yT
rade
d
Opt
ions
Dai
lyT
rade
dV
olum
eof
Act
ivel
y
Tra
ded
Opt
ions
Dai
lyT
rade
dV
alue
ofA
ctiv
ely
Tra
ded
Opt
ions
Dai
lyN
umbe
rA
sa
%of
All
the
Opt
ions
Num
ber
of
Con
trac
ts
As
a%
ofA
llth
e
Opt
ions
Dai
lyT
radi
ngV
alue
($)
As
a%
ofA
llth
e
Opt
ions
Mea
n20
.15
3.89
%87
,842
.46
61.6
5%19
6,60
4,06
2.57
53.5
3%
Med
ian
20.0
03.
87%
83,0
57.0
061
.62%
150,
121,
532.
5053
.33%
Stdd
ev6.
571.
21%
39,7
26.0
09.
88%
183,
475,
723.
4216
.82%
Min
3.00
0.59
%16
,301
.00
34.2
4%8,
874,
810.
0014
.80%
Max
43.0
07.
62%
246,
621.
0085
.65%
1,40
0,20
9,01
7.50
93.2
4%
Dai
lyA
ctiv
ely
Tra
ded
Put
Opt
ions
Dai
ly#
ofA
ctiv
ely
Tra
ded
Opt
ions
Dai
ly%
ofA
ctiv
ely
Tra
ded
Opt
ions
#of
Con
trac
ts%
T1
T2
T3
T4
T1
T2
T3
T4
Mea
n11
.73
58.9
2%9.
145.
162.
623.
2446
.35%
25.0
2%12
.67%
15.9
7%
Med
ian
11.0
058
.82%
9.00
4.00
1.00
2.00
45.0
8%22
.40%
7.28
%13
.33%
Stdd
ev4.
0611
.11%
3.69
3.70
2.97
2.79
15.2
8%15
.74%
13.6
5%12
.41%
Min
2.00
28.5
7%1.
000.
000.
000.
0015
.00%
0.00
%0.
00%
0.00
%
Max
28.0
010
0.00
%20
.00
19.0
013
.00
15.0
094
.12%
66.6
7%56
.25%
66.6
7%
105
Table C.9: Summary Statistics of the Actively Traded Call and Put Options with the
Same Strike Prices and Times-to-Maturity from Jan 2, 2003 to Dec 31, 2003 (252 days).
Daily Number of Op-
tions
Daily Traded Vol-
ume
Daily Traded Value
Number
of Pairs
As a %
of All the
Options
Number of
Contracts
As a %
of All the
Options
Traded Value
($)
As a %
of All the
Options
Mean 3.17 1.22% 33,320.52 22.25% 101,492,573.96 28.22%
Median 3.00 1.16% 26,833.50 20.93% 67,370,921.25 24.62%
Stddev 1.78 0.67% 26,261.42 11.55% 109,100,474.65 16.77%
Min 0.00 0.00% 0.00 0.00% 0.00 0.00%
Max 10.00 3.95% 205,883.00 72.99% 858,955,617.50 89.98%
Table C.10: Summary Statistics of Implied Dividends, Q(t,K,T-t), with Treasury Bill
Rates as Proxies for the Risk-free Rates .
Implied
Q(t,K,T-t)
maxQ(t,K, T − t) −
minQ(t,K, T − t) with
same t and T-t but
different K
maxQ(t,K, T − t) −
minQ(t,K, T − t) with
same t and K but different
T-t
# of Observations 798 218 142
Mean 8.02× 10−3 1.13× 10−2 6.61× 10−2
Median 1.40× 10−2 3.79× 10−3 2.71× 10−2
Stddev 6.26× 10−2 2.41× 10−2 9.57× 10−2
Min −0.51 6.44× 10−5 8.91× 10−6
Max 0.43 0.19 0.51
t-statistic 3.62 6.93 8.23
106
Table C.11: Summary Statistics of Implied Dividends, Q(t,K,T-t), with Zero Rates as
Proxies for the Risk-free Rates .
Implied
Q(t,K,T-t)
maxQ(t,K, T − t) −
minQ(t,K, T − t) with
same t and T-t but
different K
maxQ(t,K, T − t) −
minQ(t,K, T − t) with
same t and K but different
T-t
# of Observations 423 109 33
Mean 1.68× 10−2 2.43× 10−3 5.17× 10−3
Median 1.68× 10−2 1.62× 10−3 4.13× 10−3
Stddev 1.12× 10−2 2.42× 10−3 4.11× 10−3
Min −2.93× 10−2 4.03× 10−6 3.76× 10−4
Max 6.62× 10−2 1.10× 10−2 1.57× 10−2
t-statistic 30.80 10.49 7.23
107
Figure C.1: Bid-Ask Spread and Trading Volume (shares) of the SPDR from Jan 2, 2001
to Dec 30, 2005 (1,256 observations)
0 200 400 600 800 1000 1200 1400−1
0
1
2
3
4
5
6
7
Date
Bid
−A
sk S
prea
d of
SP
DR
Jan 2, 2001 to Dec 30, 2005
0 200 400 600 800 1000 1200 1400−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
Date
Bid
−A
sk S
prea
d of
SP
DR
Jun 1, 2001 to Dec 30, 2005
0 200 400 600 800 1000 1200 14000
5
10
15x 10
7
Date
Vol
ume
Jan 2, 2001 to Dec 30, 2005
Ask-Bid of SPDR Trading Volume
Mean 0.13 38,740,562 shares
cov 0.27 −3.4513× 106
−3.4513× 106 4.5045× 1014
108
Figure C.2: Histograms of Bid-Ask Spread and Trading Volume (shares) of the SPDR
from Jan 2, 2001 to Dec 30, 2005
−1 0 1 2 3 4 5 6 70
100
200
300
400
500
600
700
800
900
1000
Spread
Fre
quen
cy
SPDR
0 5 10 15
x 107
0
5
10
15
20
25
30
35
40
Volume
Fre
quen
cy
SPDR
−1
0
1
2
3
4
5
6
7
0
5
10
15
x 107
0
20
40
60
80
SpreadVolume
10
20
30
40
50
60
70
Frequency
Quantiles of the Bid-Ask Spread and Trading Volume:
Prob. 0 1% 2.5% 5% 10% 15% 20%Spread −0.52 −0.08 −0.05 −0.03 −0.02 −0.01 0Volume 3.30× 106 6.06× 106 7.18× 106 8.42× 106 1.14× 107 1.48× 107 1.78× 107
Prob. 25% 30% 35% 40% 45% 50% 55%Spread 0 0 0.01 0.01 0.01 0.01 0.01Volume 2.14× 107 2.61× 107 2.98× 107 3.26× 107 3.51× 107 3.76× 107 4.00× 107
Prob. 60% 65% 70% 75% 80% 85% 90%Spread 0.01 0.02 0.02 0.03 0.04 0.05 0.08Volume 4.32× 107 4.60× 107 4.92× 107 5.18× 107 5.51× 107 5.94× 107 6.53× 107
Prob. 95% 97.5% 99% 100%Spread 1 1.76 2.98 6.5Volume 7.49× 107 8.66× 107 1.00× 108 1.41× 108
109
Figure C.3: The S&P 500 Index and the SPDR × 10 from Jan 2, 2001 to Dec 30, 2005
(1,256 observations).
0 200 400 600 800 1000 1200 1400700
800
900
1000
1100
1200
1300
1400
Date
Leve
l
SPX
0 200 400 600 800 1000 1200 1400700
800
900
1000
1100
1200
1300
1400
Date
Leve
l
SPDR×10
S&P 500 SPDR× 10
Mean 1097.94 1101.20
Var/Cov 15849.59 15779.02
15779.02 15714.26
Skewness -0.53 -0.54
Kurtosis 2.52 2.54
110
Figure C.4: Level Difference between the S&P 500 Index and the SPDR× 10 from Jan
2, 2001 to Dec 30, 2005.
0 200 400 600 800 1000 1200 1400−20
−15
−10
−5
0
5
10
15
20
Date
Leve
l
SPDR×10−SPX
SPDR× 10− S&P 500
Mean 3.26
Var 5.81
skewness -1.26
kurtosis 14.34
Figure C.5: Level Difference between the S&P 500 Index and the SPDR× 10 from Feb
1, 1993 to Dec 30, 2005.
0 500 1000 1500 2000 2500 3000 3500−30
−25
−20
−15
−10
−5
0
5
10
15
20
Number of Observations
Leve
l
SPDR×10−SPX
SPDR× 10− S&P 500
Mean 2.28
Var 6.35
skewness -6.26
kurtosis 1.32
111
Figure C.6: Daily Relative Price Changes (log xt− log xt−1) of the S&P 500 Index (XRt)
and the SPDR (DRRt) from Jan 2, 2001 to Dec 30, 2005 (1,255 observations).
0 200 400 600 800 1000 1200 1400−0.06
−0.04
−0.02
0
0.02
0.04
0.06
Date
Geo
met
ric R
etur
n
XR
0 200 400 600 800 1000 1200 1400−0.06
−0.04
−0.02
0
0.02
0.04
0.06
Date
Geo
met
ric R
etur
n
DRR
S&P 500 SPDR
Mean −2.20× 10−5 −2.74× 10−5
Var/Cov 1.31× 10−4 1.30× 10−4
1.30× 10−4 1.34× 10−4
Skewness 1.73× 10−1 1.29× 10−1
Kurtosis 5.41 5.34
112
Figure C.7: Daily Holding Period Returns (log(xt + dt)− log xt−1) of the S&P 500 Index
(XRdt ) and the SPDR (DRR
dt ) from Jan 2, 2001 to Dec 30, 2005 (1,255 observations).
0 200 400 600 800 1000 1200 1400−0.06
−0.04
−0.02
0
0.02
0.04
0.06
Date
Geo
met
ric R
etur
n
XR
0 200 400 600 800 1000 1200 1400−0.06
−0.04
−0.02
0
0.02
0.04
0.06
Date
Geo
met
ric R
etur
n
DRR
S&P 500 SPDR
Mean −2.20× 10−5 3.75× 10−5
Var/Cov 1.31× 10−4 1.29× 10−4
1.30× 10−4 1.34× 10−4
Skewness 1.73× 10−1 6.66× 10−2
Kurtosis 5.41 5.34
113
Figure C.8: Daily Relative Price Change Difference between the S&P 500 Index and the
SPDR from Jan 2, 2001 to Dec 30, 2005.
0 200 400 600 800 1000 1200 1400−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
0.025
Date
Geo
met
ric R
etur
nDR
R−XR
DRRt −X Rt
Mean −5.34× 10−6
Var 4.52× 10−6
skewness 5.02× 10−1
kurtosis 3.42× 10
Figure C.9: Daily Holding Period Return Difference between the S&P 500 Index and the
SPDR from Jan 2, 2001 to Dec 30, 2005.
0 200 400 600 800 1000 1200 1400−0.01
−0.005
0
0.005
0.01
0.015
Date
Geo
met
ric R
etur
n
DRR−
XR
DRRdt −X Rdt
Mean 5.96× 10−5
Var 5.86× 10−6
skewness 2.07× 10−2
kurtosis 4.53
114
Figure C.10: Histogram of Daily Relative Price Changes of the S&P 500 Index (XR)and
the SPDR (DRR) and Daily Holding Period Returns of the SPDR (DRRd)from Jan 2,
2001 to Dec 30, 2005.
−0.06 −0.04 −0.02 0 0.02 0.04 0.060
10
20
30
40
50
60
70
80
Geometric Return
Fre
quen
cy
XR
−0.06 −0.04 −0.02 0 0.02 0.04 0.060
10
20
30
40
50
60
70
80
Geometric Return
Fre
quen
cy
DRR
−0.06 −0.04 −0.02 0 0.02 0.04 0.060
10
20
30
40
50
60
70
80
90
Geometric Return
Fre
quen
cy
DRRd
Quantiles of the Geometric Returns:
Prob. 0 1% 2.5% 5% 10%XR −5.05× 10−2 −3.00× 10−2 −2.40× 10−2 −1.82× 10−2 −1.40× 10−2
DRR −4.82× 10−2 −3.02× 10−2 −2.39× 10−2 −1.93× 10−2 −1.41× 10−2
DRRd −5.37× 10−2 −3.19× 10−2 −2.46× 10−2 −1.98× 10−2 −1.32× 10−2
Prob. 15% 20% 25% 30% 35%XR −1.03× 10−2 −8.18× 10−3 −6.43× 10−3 −4.80× 10−3 −3.16× 10−3
DRR −1.04× 10−2 −8.08× 10−3 −6.49× 10−3 −4.66× 10−3 −3.33× 10−3
DRRd −1.03× 10−2 −8.41× 10−3 −6.42× 10−3 −4.49× 10−3 −2.85× 10−3
Prob. 40% 45% 50% 55% 60%XR −1.88× 10−3 −7.71× 10−4 3.60× 10−4 1.30× 10−3 2.29× 10−3
DRR −2.02× 10−3 −7.61× 10−4 3.26× 10−4 1.41× 10−3 2.37× 10−3
DRRd −1.80× 10−3 −3.32× 10−4 7.06× 10−4 1.59× 10−3 2.72× 10−3
Prob. 65% 70% 75% 80% 85%XR 3.42× 10−3 4.61× 10−3 5.95× 10−3 7.64× 10−3 9.89× 10−3
DRR 3.50× 10−3 4.60× 10−3 6.07× 10−3 7.61× 10−3 9.74× 10−3
DRRd 3.65× 10−3 4.80× 10−3 6.17× 10−3 7.70× 10−3 1.01× 10−2
Prob. 90% 95% 97.5% 99% 100%XR 1.28× 10−2 1.74× 10−2 2.27× 10−2 3.48× 10−2 5.57× 10−2
DRR 1.31× 10−2 1.85× 10−2 2.27× 10−2 3.45× 10−2 6.00× 10−2
DRRd 1.29× 10−2 1.75× 10−2 2.21× 10−2 3.39× 10−2 5.79× 10−2
115
Figure C.11: Histogram of Relative Price Change Difference between the S&P 500 Index
and the SPDR and Histogram of Daily Holding Period Return Difference between the
S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30, 2005.
−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.0250
50
100
150
200
250
300
Geometric Return Difference
Fre
quen
cy
DRR−
XR
−0.01 −0.005 0 0.005 0.01 0.0150
10
20
30
40
50
60
Geometric Return Difference
Fre
quen
cy
DRRd−
XR
Quantiles of Geometric Return Difference:
Prob. 0 1% 2.5% 5% 10%DRR−X R −1.78× 10−2 −7.47× 10−3 −4.19× 10−3 −2.66× 10−3 −1.34× 10−3
DRRd −X R −9.32× 10−3 −6.44× 10−3 −5.21× 10−3 −3.93× 10−3 −2.88× 10−3
Prob. 15% 20% 25% 30% 35%DRR−X R −9.65× 10−4 −6.45× 10−4 −5.00× 10−4 −3.54× 10−4 −2.34× 10−4
DRRd −X R −2.09× 10−3 −1.63× 10−3 −1.26× 10−3 −9.21× 10−4 −6.12× 10−4
Prob. 40% 45% 50% 55% 60%DRR−X R −1.33× 10−4 −4.52× 10−5 4.17× 10−5 1.39× 10−4 2.48× 10−4
DRRd −X R −4.01× 10−4 −1.71× 10−4 4.48× 10−5 2.77× 10−4 5.82× 10−4
Prob. 65% 70% 75% 80% 85%DRR−X R 3.39× 10−4 4.18× 10−4 5.41× 10−4 6.84× 10−4 8.95× 10−4
DRRd −X R 7.93× 10−4 1.09× 10−3 1.40× 10−3 1.78× 10−3 2.31× 10−3
Prob. 90% 95% 97.5% 99% 100%DRR−X R 1.35× 10−3 2.31× 10−3 3.44× 10−3 5.91× 10−3 2.23× 10−2
DRRd −X R 2.96× 10−3 3.76× 10−3 4.87× 10−3 6.82× 10−3 1.01× 10−2
116
Figure C.12: Squared Relative Price Changes of the S&P 500 Index (XR2t ) and the SPDR
(DRR2t ) from Jan 2, 2001 to Dec 30, 2005.
0 200 400 600 800 1000 1200 14000
0.5
1
1.5
2
2.5
3
3.5x 10
−3
Date
Squ
are
of G
eom
etric
Ret
urn
XR2
0 200 400 600 800 1000 1200 14000
0.5
1
1.5
2
2.5
3
3.5
4x 10
−3
Date
Squ
are
of G
eom
etric
Ret
urn
DRR2
XR2t DRR
2t
Mean 1.31× 10−4 1.34× 10−4
Var/Cov 7.56× 10−8 7.47× 10−8
7.47× 10−8 7.81× 10−8
Skewness 5.11 5.17
Kurtosis 3.83× 10 4.12× 10
Figure C.13: Squared Daily Holding Period Returns of the S&P 500 Index and the SPDR
from Jan 2, 2001 to Dec 30, 2005.
0 200 400 600 800 1000 1200 14000
0.5
1
1.5
2
2.5
3
3.5x 10
−3
Date
Squ
are
of G
eom
etric
Ret
urn
XR2
0 200 400 600 800 1000 1200 14000
0.5
1
1.5
2
2.5
3
3.5x 10
−3
Date
Squ
are
of G
eom
etric
Ret
urn
DRR2
XR2t DRR
2t
Mean 1.31× 10−4 1.34× 10−4
Var/Cov 7.56× 10−8 7.42× 10−8
7.47× 10−8 7.78× 10−8
Skewness 5.11 5.10
Kurtosis 3.83× 10 3.94× 10
117
Figure C.14: Autocorrelation Function for Squared Daily Relative Price Changes of the
S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30, 2005.
0 20 40 60 80 100 120 140−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Lag
AC
FXR2
DRR2
Figure C.15: Autocorrelation Function for Squared Daily Holding Period Returns of the
S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30, 2005.
0 20 40 60 80 100 120 140−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Lag
AC
F
XR2
DRR2
118
Figure C.16: Cross-Correlation Function for Squared Relative Price Changes of the
S&P 500 Index and the SPDR and Autocorrelation Function for Squared Relative Price
Changes of the S&P 500 Index from Jan 2, 2001 to Dec 30, 2005.
−140 −120 −100 −80 −60 −40 −20 0−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Lag
XC
F/A
CF
ACF of XR2
XCF of XR2
t &
DRR2
t+lag
0 20 40 60 80 100 120 140−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Lag
XC
F/A
CF
ACF of XR2
XCF of XR2
t &
DRR2
t+lag
119
Figure C.17: Cross-Correlation Function for Squared Daily Holding Period Returns of the
S&P 500 Index and the SPDR and Autocorrelation Function for Squared Daily Holding
Period Returns of the S&P 500 Index from Jan 2, 2001 to Dec 30, 2005.
−140 −120 −100 −80 −60 −40 −20 0−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Lag
XC
F/A
CF
ACF of XR2
XCF of XR2
t &
DRR2
t+lag
0 20 40 60 80 100 120 140−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Lag
XC
F/A
CF
ACF of XR2
XCF of XR2
t &
DRR2
t+lag
120
Figure C.18: Autocorrelation Function of Max(exp(Rt) − k, 0) for Daily Relative Price
Changes of the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30, 2005.
0 20 40 60 80 100 120 140
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Lag
AC
F o
f [ex
p(R
t)−k]
+
k=1.015
XR
t
DRR
t
0 20 40 60 80 100 120 140−0.02
0
0.05
0.1
0.15
0.2
Lag
AC
F o
f [ex
p(R
t)−k]
+
k=1.02
XR
t
DRR
t
0 20 40 60 80 100 120 140
0
0.05
0.1
0.15
0.2
0.25
Lag
AC
F o
f [ex
p(R
t)−k]
+
k=1.025
XR
t
DRR
t
121
Figure C.19: Autocorrelation Function ofMax(exp(Rdt )−k, 0) for Holding Period Returns
of the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30, 2005.
0 20 40 60 80 100 120 140
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Lag
AC
F o
f [ex
p(R
t)−k]
+
k=1.015
XR
t
DRR
td
0 20 40 60 80 100 120 140
0
0.05
0.1
0.15
0.2
Lag
AC
F o
f [ex
p(R
t)−k]
+
k=1.02
XR
t
DRR
td
0 20 40 60 80 100 120 140
0
0.05
0.1
0.15
0.2
Lag
AC
F o
f [ex
p(R
t)−k]
+
k=1.025
XR
t
DRR
td
122
Figure C.20: Autocorrelation Function of Max(k − exp(Rt), 0) for Daily Relative Price
Changes of the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30, 2005.
0 20 40 60 80 100 120 140−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Lag
AC
F o
f [k−
exp(
Rt)]
+
k=0.985
XR
t
DRR
t
0 20 40 60 80 100 120 140−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Lag
AC
F o
f [k−
exp(
Rt)]
+
k=0.98
XR
t
DRR
t
0 20 40 60 80 100 120 140
0
0.05
0.1
0.15
0.2
0.25
Lag
AC
F o
f [k−
exp(
Rt)]
+
k=0.975
XR
t
DRR
t
123
Figure C.21: Autocorrelation Function ofMax(k−exp(Rdt ), 0) for Holding Period Returns
of the S&P 500 Index and the SPDR from Jan 2, 2001 to Dec 30, 2005
0 20 40 60 80 100 120 140
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Lag
AC
F o
f [k−
exp(
Rt)]
+
k=0.985
XR
t
DRR
td
0 20 40 60 80 100 120 140
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Lag
AC
F o
f [k−
exp(
Rt)]
+
k=0.98
XR
t
DRR
td
0 20 40 60 80 100 120 140
0
0.05
0.1
0.15
0.2
0.25
Lag
AC
F o
f [k−
exp(
Rt)]
+
k=0.975
XR
t
DRR
td
124
Figure C.22: Number of Times-to-Maturity on Each Trading Day in 2003
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/197
7.2
7.4
7.6
7.8
8
8.2
8.4
8.6
8.8
9
date
Num
ber
of M
atur
ities
Number of Maturities on Each Trading Day
Note: The label on the x-axis shows the Friday before expiration date in each month except for April17, which is the Thursday, since there was no trading on April 18.
Figure C.23: Times-to-Maturity on Each Trading Day in 2003
01/17 02/21 03/21 04/17 05/16 06/2007/02 07/18 08/15 09/19 10/17 11/21 12/190
100
200
300
400
500
600
700
800
date
Tim
e to
Mat
urity
Time to Maturities on Each Trading Day
125
Figure C.24: Times-to-Maturity of Options with High Volume on Each Trading Day in
2003
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
100
200
300
400
500
600
700
800
date
Tim
e to
Mat
urity
Call and Put Options with Daily Volume >= 2000
callput
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
100
200
300
400
500
600
700
800
date
Tim
e to
Mat
urity
Call Options with Daily Volume >= 2000
call
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
100
200
300
400
500
600
700
date
Tim
e to
Mat
urity
Put Options with Daily Volume >= 2000
put
126
Figure C.25: Strike Prices of Options with High Volume on Each Trading Day in 2003
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/19400
600
800
1000
1200
1400
1600
1800
date
SP
X &
Str
ike
Call Options with Daily Volume >= 2000
SPXcall
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/19400
600
800
1000
1200
1400
1600
1800
date
SP
X &
Str
ike
Put Options with Daily Volume >= 2000
SPXput
Figure C.26: Moneyness of Options with High Volume on Each Trading Day in 2003
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
date
Mon
eyne
ss
Call Options with Daily Volume >= 2000
call
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
date
Mon
eyne
ss
Put Options with Daily Volume >= 2000
put
Figure C.27: Prices of Traded Options with Traded Volume ≥ 2,000 Contracts in 2003
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
100
200
300
400
500
600
700
800
900
date
Opt
ion
Pric
es
Prices of Call and Put Options with Daily Volume >= 2000
127
Figure C.28: Number and Proportion of Highly Traded Options on Each Trading Day in
2003
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
5
10
15
20
25
30
35
40
45
date
Num
ber
of O
ptio
nsNumber of Highly Traded Options
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
date
Pro
port
ion
Proportion of Highly Traded Options
128
Figure C.29: Number of Highly Traded Put Options and Its Proportion in Highly Traded
Options on Each Trading Day in 2003
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
5
10
15
20
25
30
date
Num
ber
of O
ptio
nsNumber of Highly Traded Put Options
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
date
Pro
port
ion
Highly Traded Put as a Proportion of Highly Traded Options
129
Figure C.30: Number of Actively Traded Options with Time-to-Maturity T1 and Its
Proportion in the Actively Traded Options on Each Trading Day in 2003
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
2
4
6
8
10
12
14
16
18
20
date
Num
ber
of O
ptio
nsNumber of Options with Daily Volume >= 2000 and Time to Maturity T1
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
date
Pro
port
ion
Proportion of Options with Daily Volume >= 2000 and Time to Maturity T1
130
Figure C.31: Number of Actively Traded Options with Time-to-Maturity T2 and Its
Proportion in the Actively Traded Options on Each Trading Day in 2003
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
2
4
6
8
10
12
14
16
18
20
date
Num
ber
of O
ptio
nsNumber of Options with Daily Volume >= 2000 and Time to Maturity T2
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
date
Pro
port
ion
Proportion of Options with Daily Volume >= 2000 and Time to Maturity T2
131
Figure C.32: Number of Actively Traded Options with Time-to-Maturity T3 and Its
Proportion in the Actively Traded Options on Each Trading Day in 2003
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
2
4
6
8
10
12
14
16
18
20
date
Num
ber
of O
ptio
nsNumber of Options with Daily Volume >= 2000 and Time to Maturity T3
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
date
Pro
port
ion
Proportion of Options with Daily Volume >= 2000 and Time to Maturity T3
132
Figure C.33: Number of Actively Traded Options with Time-to-Maturity T4 and Its
Proportion in the Actively Traded Options on Each Trading Day in 2003
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
2
4
6
8
10
12
14
16
18
20
date
Num
ber
of O
ptio
nsNumber of Options with Daily Volume >= 2000 and Time to Maturity T4
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
date
Pro
port
ion
Proportion of Options with Daily Volume >= 2000 and Time to Maturity T4
133
Figure C.34: Moneyness, Total Volume and Total Value of Actively Traded Options with
Time-to-Maturity T1 on Each Trading Day in 2003
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
date
Mon
eyne
ss
Moneyness of Options with Daily Volume >= 2000 and Time to Maturity T1
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
2
4
6
8
10
12
14x 10
4
date
Vol
ume
of O
ptio
ns
Daily Total Volume of Options with Daily Volume >= 2000 and Time to Maturity T1
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
1
2
3
4
5
6
7x 10
8
date
Val
ue o
f Opt
ions
Daily Total Value of Options with Daily Volume >= 2000 and Time to Maturity T1
134
Figure C.35: Moneyness, Total Volume and Total Value of Actively Traded Options with
Time-to-Maturity T2 on Each Trading Day in 2003
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
date
Mon
eyne
ss
Moneyness of Options with Daily Volume >= 2000 and Time to Maturity T2
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
2
4
6
8
10
12x 10
4
date
Vol
ume
of O
ptio
ns
Daily Total Volume of Options with Daily Volume >= 2000 and Time to Maturity T2
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
1
2
3
4
5
6
7x 10
8
date
Val
ue o
f Opt
ions
Daily Total Value of Options with Daily Volume >= 2000 and Time to Maturity T2
135
Figure C.36: Moneyness, Total Volume and Total Value of Actively Traded Options with
Time-to-Maturity T3 on Each Trading Day in 2003
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
date
Mon
eyne
ss
Moneyness of Options with Daily Volume >= 2000 and Time to Maturity T3
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
5
10
15x 10
4
date
Vol
ume
of O
ptio
ns
Daily Total Volume of Options with Daily Volume >= 2000 and Time to Maturity T3
01/17 02/21 03/21 04/17 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
1
2
3
4
5
6
7
8x 10
8
date
Val
ue o
f Opt
ions
Daily Total Value of Options with Daily Volume >= 2000 and Time to Maturity T3
136
Figure C.37: Moneyness of Actively Traded Call and Put Options with the Same Strike
Prices and Times-to-Maturity on Each Trading Day in 2003
01/17 02/21 03/21 04/18 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
date
Mon
eyne
ss
Call and Put Options with the Same Strike Prices and Time to Maturities That Have Daily Volume >= 2000
Figure C.38: Time-to-Maturity of Actively Traded Call and Put Options with the Same
Strike Prices and Times-to-Maturity on Each Trading Day in 2003
01/17 02/21 03/21 04/18 05/16 06/20 07/18 08/15 09/19 10/17 11/21 12/190
100
200
300
400
500
600
700
date
Tim
e to
Mat
urity
Call and Put Options with the Same Strike Prices and Time to Maturities That Have Daily Volume >= 2000
137
Figure C.39: Implied Dividends, Q(t,K,T-t), with Treasury Bill Rates as Proxies for the
Risk-free Rates
252−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
t
Impl
ied
Div
iden
d, Q
Implied Dividend, Q(t,K,T−t)
Figure C.40: Implied Dividends, Q(t,K,T-t), with the same t and T-t, but different K,
and with Treasury Bill Rates as Proxies for the Risk-free Rates
218−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
(t, T−t)
Impl
ied
Div
iden
d, Q
Implied Dividend, Q(t,K,T−t), with same t and T−t but different K
138
Figure C.41: Difference between the maximum and minimum Q(t,K,T-t), with the same
t and T-t, but different K, and with Treasury Bill Rates as Proxies for the Risk-free Rates
2180
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
(t, T−t)
Diff
eren
ce o
f Im
plie
d D
ivid
end
Difference between the maximum and minimum Q(t,K,T−t), with same t and T−t but different K
Figure C.42: Implied Dividends, Q(t,K,T-t), with the same t and K, but different T-t,
and with Treasury Bill Rates as Proxies for the Risk-free Rates
142−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
(t, K)
Impl
ied
Div
iden
d, Q
Implied Dividend, Q(t,K,T−t), with same t and K but different T−t
139
Figure C.43: Difference between the maximum and minimum Q(t,K,T-t), with the same
t and K, but different T-t, and with Treasury Bill Rates as Proxies for the Risk-free Rates
1420
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(t, K)
Diff
eren
ce o
f Im
plie
d D
ivid
end
Difference between the maximum and minimum Q(t,K,T−t), with same t and K but different T−t
Figure C.44: Implied Dividend, Q(t,K,T-t), with Zero Rates as Proxies for the Risk-free
Rates
252−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
t
Impl
ied
Div
iden
d, Q
Implied Dividend, Q(t,K,T−t)
140
Figure C.45: Implied Dividends, Q(t,K,T-t), with the same t and T-t, but different K,
and with Zero Rates as Proxies for the Risk-free Rates
109−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
(t, T−t)
Impl
ied
Div
iden
d, Q
Implied Dividend, Q(t,K,T−t), with same t and T−t but different K
Figure C.46: Difference between the maximum and minimum Q(t,K,T-t), with the same
t and T-t, but different K, and with Zero Rates as Proxies for the Risk-free Rates
1090
0.002
0.004
0.006
0.008
0.01
0.012
(t, T−t)
Diff
eren
ce o
f Im
plie
d D
ivid
end
Difference between the maximum and minimum Q(t,K,T−t), with same t and T−t but different K
141
Figure C.47: Implied Dividends, Q(t,K,T-t), with the same t and K, but different T-t,
and with Zero Rates as Proxies for the Riskfree Rates
33−0.02
−0.01
0
0.01
0.02
0.03
0.04
(t, K)
Impl
ied
Div
iden
d, Q
Implied Dividend, Q(t,K,T−t), with same t and K but different T−t
Figure C.48: Difference between the maximum and minimum Q(t,K,T-t), with the same
t and K, but different T-t, and with Zero Rates as Proxies for the Risk-free Rates
330
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
(t, K)
Diff
eren
ce o
f Im
plie
d D
ivid
end
Difference between the maximum and minimum Q(t,K,T−t), with same t and K but different T−t
142
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