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2.1 MARKET PRICE FORMATION
Millions of different financial assets (stocks, bonds, currencies,
options, and others) are traded around the world. Some financial
markets are organized in exchanges or bourses (e.g., New York
Stock Exchange (NYSE)). In other, so-called over-the-counter
(OTC) markets, participants operate directly via telecommunication
systems. Market data are collected and distributed by markets them-
selves and by financial data services such as Bloomberg and Reuters.
Modern electronic networks facilitate access to huge volumes of
market data in real time.
Market prices are formed with the trader orders (quotes) submitted
on the bid (buy) and ask (sell) sides of the market. Usually, there is a



5
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spread between the best (highest) bid and the best (lowest) ask prices,
which provides profits for the market makers. The prices seen on the
tickers of TV networks and on the Internet are usually the transaction
prices that correspond to the best prices. The very presence of trans-
actions implies that some traders submit market orders; they buy at
current best ask prices and sell at current best bid prices. The trans-
action prices represent the mere tip of an iceberg beneath which prices
of the limit orders reside. Indeed, traders may submit the sell orders at
prices higher than the best bid and the buy orders at prices lower than
the best ask. The limit orders reflect the trader expectations of future
price movement. There are also stop orders designated to limit pos-
sible losses. For an asset holder, the stop order implies selling assets if
the price falls to a predetermined value.
Holding assets, particularly holding derivatives (see Section 9.1), is
called long position. The opposite of long buying is short selling, which
means selling assets that the trader does not own after borrowing
them from the broker. Short selling makes sense if the price is
expected to fall. When the price does drop, the short seller buys the
same number of assets that were borrowed and returns them to the
broker. Short sellers may also use stop orders to limit their losses in
case the price grows rather than falls. Namely, they may submit the
stop order for triggering a buy when the price reaches a predeter-
mined value.
Limit orders and stop orders form the market microstructure: the
volume-price distributions on the bid and ask sides of the market. The
concept market liquidity is used to describe price sensitivity to market
orders. For instance, low liquidity means that the number of securities
available at the best price is smaller than a typical market order. In this
case, a new market order is executed within a range of available prices
rather than at a single best price. As a result, the best price changes its
value. Securities with very low liquidity may have no transactions and
few (if any) quotes for some time (in particular, the small-cap stocks off
regular trading hours). Market microstructure information usually is
not publicly available. However, the market microstructure may be
partly revealed in the price reaction to big block trades.
Any event that affects the market microstructure (such as submis-
sion, execution, or withdrawal of an order) is called a tick. Ticks are
recorded along with the time they are submitted (so-called tick-by-tick
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data). Generally, tick-by-tick data are not regularly spaced in time,
which leads to additional challenges for high-frequency data analysis
[1, 2]. Current research of financial data is overwhelmingly conducted
on the homogeneous grids that are defined with filtering and aver-
aging tick-by-tick data.
Another problem that complicates analysis of long financial time
series is seasonal patterns. Business hours, holidays, and even daylight
saving time shifts affect market activity. Introducing the dummy
variables into time series models is a general method to account for
seasonal effects (see Section 5.2). In another approach, ˜˜operational
time™™ is employed to describe the non-homogeneity of business activ-
ity [2]. Non-trading hours, including weekends and holidays, may be
cut off from operational time grids.

2.2 RETURNS AND DIVIDENDS
2.2.1 SIMPLE COMPOUNDED RETURNS
AND

While price P is the major financial variable, its logarithm,
p ¼ log (P) is often used in quantitative analysis. The primary reason
for using log prices is that simulation of a random price innovation
can move price into the negative region, which does not make sense.
In the mean time, negative logarithm of price is perfectly acceptable.
Another important financial variable is the single-period return (or
simple return) R(t) that defines the return between two subsequent
moments t and tÀ1. If no dividends are paid,
R(t) ¼ P(t)=P(t À 1) À 1 (2:2:1)
Return is used as a measure of investment efficiency.1 Its advantage is
that some statistical properties, such as stationarity, may be more
applicable to returns rather than to prices [3]. The simple return of a
portfolio, Rp (t), equals the weighed sum of returns of the portfolio
assets
X X
N N
Rp (t) ¼ wip ¼ 1,
wip Rip (t), (2:2:2)
i¼1 i¼1

where Rip and wip are return and weight of the i-th portfolio asset,
respectively; i ¼ 1, . . . , N.
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The multi-period returns, or the compounded returns, define the
returns between the moments t and t À k þ 1. The compounded
return equals
R(t, k) ¼ [R(t) þ 1] [R(t À 1) þ 1] . . . [R(t À k þ 1) þ 1] þ 1
¼ P(t)=P(t À k) þ 1 (2:2:3)
The return averaged over k periods equals
" #1=k
Y
kÀ1
ˇ
R(t, k) ¼ (R(t À i) þ 1) À1 (2:2:4)
i¼0

If the simple returns are small, the right-hand side of (2.2.4) can be
reduced to the first term of its Taylor expansion:
1XkÀ1
ˇ
R(t, k) % R(t, i) (2:2:5)
k i¼1

The continuously compounded return (or log return) is defined as:
r(t) ¼ log [R(t) þ 1] ¼ p(t) À p(t À 1) (2:2:6)
Calculation of the compounded log returns is reduced to simple
summation:
r(t, k) ¼ r(t) þ r(t À 1) þ . . . þ r(t À k þ 1) (2:2:7)
However, the weighing rule (2.2.2) is not applicable to the log returns
since log of sum is not equal to sum of logs.

2.2.2 DIVIDEND EFFECTS
If dividends D(t þ 1) are paid within the period [t, t þ 1], the simple
return (see 2.2.1) is modified to
R(t þ 1) ¼ [P(t þ 1) þ D(t þ 1) ]=P(t) À 1 (2:2:8)
The compounded returns and the log returns are calculated in the
same way as in the case with no dividends.
Dividends play a critical role in the discounted-cash-flow (or pre-
sent-value) pricing model. Before describing this model, let us intro-
duce the notion of present value. Consider the amount of cash K
invested in a risk-free asset with the interest rate r. If interest is paid
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every time interval (say every month), the future value of this cash
after n periods is equal to
FV ¼ K(1 þ r)n (2:2:9)
Suppose we are interested in finding out what amount of money will
yield given future value after n intervals. This amount (present value)
equals
PV ¼ FV=(1 þ r)n (2:2:10)
Calculating the present value via the future value is called discounting.
The notions of the present value and the future value determine the
payoff of so-called zero-coupon bonds. These bonds sold at their
present value promise a single payment of their future value at ma-
turity date.
The discounted-cash-flow model determines the stock price via its
future cash flow. For the simple model with the constant return
E[R(t) ] ¼ R, one can rewrite (2.2.8) as
P(t) ¼ E[{P(t þ 1) þ D(t þ 1)}=(1 þ R)] (2:2:11)
If this recursion is repeated K times, one obtains
" #
X
K
D(t þ i)=(1 þ R)i þ E[P(t þ K)=(1 þ R)K ]
P(t) ¼ E (2:2:12)
i¼1

In the limit K ! 1, the second term in the right-hand side of (2.2.12)
can be neglected if

lim E[P(t þ K)=(1 þ R)K ] ¼ 0 (2:2:13)
K!1

Then the discounted-cash-flow model yields
" #
X1
D(t þ i)=(1 þ R)i
PD (t) ¼ E (2:2:14)
i¼1

Further simplification of the discounted-cash-flow model is based on
the assumption that the dividends grow linearly with rate G
E[D(t þ i) ] ¼ (1 þ G)i D(t) (2:2:15)
Then (2.2.14) reduces to
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1þG
PD (t) ¼ D(t) (2:2:16)
RÀG
Obviously, equation (2.2.16) makes sense only for R > G. The value
of R that may attract investors is called the required rate of return.
This value can be treated as the sum of the risk-free rate and the asset
risk premium. While the assumption of linear dividend growth is
unrealistic, equation (2.2.16) shows the high sensitivity of price to
change in the discount rate R when R is close to G (see Exercise 2). A
detailed analysis of the discounted-cash-flow model is given in [3].
If the condition (2.2.13) does not hold, the solution to (2.2.12) can
be presented in the form
P(t) ¼ PD (t) þ B(t), B(t) ¼ E[B(t þ 1)=(1 þ R) ] (2:2:17)
The term PD (t) has the sense of the fundamental value while the
function B(t) is often called the rational bubble. This term implies
that B(t) may lead to unbounded growth”the ˜˜bubble.™™ Yet, this
bubble is ˜˜rational™™ since it is based on rational expectations of future
returns. In the popular Blanchard-Watson model
(
1 þ R B(t) þ e(t þ 1) with probability p, 0 < p < 1
p
B(t þ 1) ¼ (2:2:18)
e(t þ 1) with probability 1 À p
where e(t) is an independent and identically distributed process (IID)2
with E[e(t) ] ¼ 0. The specific of this model is that it describes period-
ically collapsing bubbles (see [4] for the recent research).
So far, the discrete presentation of financial data was discussed.
Clearly, market events have a discrete nature and price variations
cannot be smaller than certain values. Yet, the continuum presenta-
tion of financial processes is often employed [5]. This means that the
time interval between two consecutive market events compared to the
time range of interest is so small that it can be considered an infini-
tesimal difference. Often, the price discreteness can also be neglected
since the markets allow for quoting prices with very small differen-
tials. The future value and the present value within the continuous
presentation equal, respectively
FV ¼ K exp (rt), PV ¼ FV exp (Àrt) (2:2:19)
In the following chapters, both the discrete and the continuous pre-
sentations will be used.
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2.3 MARKET EFFICIENCY
2.3.1 ARBITRAGE
Asset prices generally obey the Law of One Price, which says that
prices of equivalent assets in competitive markets must be the same
[6]. This implies that if a security replicates a package of other
securities, the price of this security and the price of the package it
replicates must be equal. It is expected also that the asset price must
be the same worldwide, provided that it is expressed in the same
currency and that the transportation and transaction costs can be
neglected. Violation of the Law of One Price leads to arbitrage, which
means buying an asset and immediate selling it (usually in another
market) with profit and without risk. One widely publicized example
of arbitrage is the notable differences in prices of prescription drugs in
the USA, Europe, and Canada. Another typical example is the so-
called triangle foreign exchange arbitrage. Consider a situation in
which a trader can exchange one American dollar (USD) for one
Euro (EUR) or for 120 Yen (JPY). In addition, a trader can exchange
one EUR for 119 JPY. Hence, in terms of the exchange rates, 1 USD/
JPY > 1 EUR/JPY * 1 USD/EUR.3 Obviously, the trader who
operates, say 100000 USD, can make a profit by buying 12000000
JPY, then selling them for 12000000/119 % 100840 EUR, and then
buying back 100840 USD. If the transaction costs are neglected, this
operation will bring profit of about 840 USD.
The arbitrage with prescription drugs persists due to unresolved
legal problems. However, generally the arbitrage opportunities do not
exist for long. The triangle arbitrage may appear from time to time.
Foreign exchange traders make a living, in part, by finding such
opportunities. They rush to exchange USD for JPY. It is important
to remember that, as it was noted in Section 2.1, there is only a finite
number of assets at the ˜˜best™™ price. In our example, it is a finite
number of Yens available at the exchange rate USD/JPY ¼ 120. As
soon as they all are taken, the exchange rate USD/JPY falls to the
equilibrium value 1 USD/JPY ¼ 1 EUR/JPY * 1 USD/EUR, and the
arbitrage vanishes. In general, when arbitrageurs take profits, they act
in a way that eliminates arbitrage opportunities.
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2.3.2 EFFICIENT MARKET HYPOTHESIS (EMH)
Efficient market is closely related to (the absence of) arbitrage. It
might be defined as simply an ideal market without arbitrage, but there
is much more to it than that. Let us first ask what actually causes price
to change. The share price of a company may change due to its new
earnings report, due to new prognosis of the company performance, or
due to a new outlook for the industry trend. Macroeconomic and
political events, or simply gossip about a company™s management,
can also affect the stock price. All these events imply that new infor-
mation becomes available to markets. The Efficient Market Theory
states that financial markets are efficient because they instantly reflect
all new relevant information in asset prices. Efficient Market Hypoth-
esis (EMH) proposes the way to evaluate market efficiency. For
example, an investor in an efficient market should not expect earnings
above the market return while using technical analysis or fundamental
analysis.4
Three forms of EMH are discerned in modern economic literature.
In the ˜˜weak™™ form of EMH, current prices reflect all information on
past prices. Then the technical analysis seems to be helpless. In the
˜˜strong™™ form, prices instantly reflect not only public but also private
(insider) information. This implies that the fundamental analysis
(which is what the investment analysts do) is not useful either. The
compromise between the strong and weak forms yields the ˜˜semi-
strong™™ form of EMH according to which prices reflect all publicly
available information and the investment analysts play important role
in defining fair prices.
Two notions are important for EMH. The first notion is the
random walk, which will be formally defined in Section 5.1. In short,
market prices follow the random walk if their variations are random
and independent. Another notion is rational investors who immedi-
ately incorporate new information into fair prices. The evolution of
the EMH paradigm, starting with Bachelier™s pioneering work on
random price behavior back in 1900 to the formal definition of
EMH by Fama in 1965 to the rigorous statistical analysis by Lo
and MacKinlay in the late 1980s, is well publicized [9“13]. If prices
follow the random walk, this is the sufficient condition for EMH.
However, as we shall discuss further, the pragmatic notion of market
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efficiency does not necessarily require prices to follow the random
walk.
Criticism of EMH has been conducted along two avenues. First, the
thorough theoretical analysis has resulted in rejection of the random
walk hypothesis for the weekly U.S. market returns during 1962“1986
[12]. Interestingly, similar analysis for the period of 1986“1996 shows
that the returns conform more closely to the random walk. As the
authors of this research, Lo and MacKinlay, suggest, one possible
reason for this trend is that several investment firms had implemented
statistical arbitrage trading strategies5 based on the market inefficien-
cies that were revealed in early research. Execution of these strategies
could possibly eliminate some of the arbitrage opportunities.
Another reason for questioning EMH is that the notions of ˜˜fair
price™™ and ˜˜rational investors™™ do not stand criticism in the light of
the financial market booms and crashes. The ˜˜irrational exuberance™™
in 1999“2000 can hardly be attributed to rational behavior [10]. In
fact, empirical research in the new field ˜˜behavioral finance™™ demon-
strates that investor behavior often differs from rationality [14, 15].
Overconfidence, indecisiveness, overreaction, and a willingness to
gamble are among the psychological traits that do not fit rational
behavior. A widely popularized example of irrational human behav-
ior was described by Kahneman and Tversky [16]. While conducting
experiments with volunteers, they asked participants to make choices
in two different situations. First, participants with $1000 were given a
choice between: (a) gambling with a 50% chance of gaining $1000 and
a 50% chance of gaining nothing, or (b) a sure gain of $500. In the
second situation, participants with $2000 were given a choice be-
tween: (a) a 50% chance of losing $1000 and a 50% of losing nothing,
and (b) a sure loss of $500. Thus, the option (b) in both situations
guaranteed a gain of $1500. Yet, the majority of participants chose
option (b) in the first situation and option (a) in the second one.
Hence, participants preferred sure yet smaller gains but were willing
to gamble in order to avoid sure loss.
Perhaps Keynes™ explanation that ˜˜animal spirits™™ govern investor
behavior is an exaggeration. Yet investors cannot be reduced to
completely rational machines either. Moreover, actions of different
investors, while seemingly rational, may significantly vary. In part,
this may be caused by different perceptions of market events and
14 Financial Markets



trends (heterogeneous beliefs). In addition, investors may have differ-
ent resources for acquiring and processing new information. As a
result, the notion of so-called bounded rationality has become popular
in modern economic literature (see also Section 12.2).
Still the advocates of EMH do not give up. Malkiel offers the
following argument in the section ˜˜What do we mean by saying markets
are efficient™™ of his book ˜˜A Random Walk down Wall Street™™ [9]:
˜˜No one person or institution has yet to provide a long-term,
consistent record of finding risk-adjusted individual stock
trading opportunities, particularly if they pay taxes and
incur transactions costs.™™
Thus, polemics on EMH changes the discussion from whether
prices follow the random walk to the practical ability to consistently
˜˜beat the market.™™
Whatever experts say, the search of ideas yielding excess returns
never ends. In terms of the quantification level, three main directions
in the investment strategies may be discerned. First, there are qualita-
tive receipts such as ˜˜Dogs of the Dow™™ (buying 10 stocks of the Dow
Jones Industrial Average with highest dividend yield), ˜˜January
Effect™™ (stock returns are particularly high during the first two Janu-
ary weeks), and others. These ideas are arguably not a reliable profit
source [9].
Then there are relatively simple patterns of technical analysis, such as
˜˜channel,™™ ˜˜head and shoulders,™™ and so on (see, e.g., [7]). There has
been ongoing academic discussion on whether technical analysis is able
to yield persistent excess returns (see, e.g., [17“19] and references
therein). Finally, there are trading strategies based on sophisticated
statistical arbitrage. While several trading firms that employ these strat-
egies have proven to be profitable in some periods, little is known about
persistent efficiency of their proprietary strategies. Recent trends indi-
cate that some statistical arbitrage opportunities may be fading [20].
Nevertheless, one may expect that modern, extremely volatile markets
will always provide new occasions for aggressive arbitrageurs.

2.4 PATHWAYS FOR FURTHER READING
In this chapter, a few abstract statistical notions such as IID and
random walk were mentioned. In the next five chapters, we take a short
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tour of the mathematical concepts that are needed for acquaintance
with quantitative finance. Those readers who feel confident in their
mathematical background may jump ahead to Chapter 8.
Regarding further reading for this chapter, general introduction to
finance can be found in [6]. The history of development and valid-
ation of EMH is described in several popular books [9“11].6 On the

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