ñòð. 2 |

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

6 Financial Markets

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

7

Financial Markets

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, â€˜â€˜operational

timeâ€™â€™ 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 Â¼ 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 tÃ€1. If no dividends are paid,

R(t) Â¼ P(t)=P(t Ã€ 1) Ã€ 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) Â¼ wip Â¼ 1,

wip Rip (t), (2:2:2)

iÂ¼1 iÂ¼1

where Rip and wip are return and weight of the i-th portfolio asset,

respectively; i Â¼ 1, . . . , N.

8 Financial Markets

The multi-period returns, or the compounded returns, define the

returns between the moments t and t Ã€ k Ã¾ 1. The compounded

return equals

R(t, k) Â¼ [R(t) Ã¾ 1] [R(t Ã€ 1) Ã¾ 1] . . . [R(t Ã€ k Ã¾ 1) Ã¾ 1] Ã¾ 1

Â¼ P(t)=P(t Ã€ k) Ã¾ 1 (2:2:3)

The return averaged over k periods equals

" #1=k

Y

kÃ€1

Ë‡

R(t, k) Â¼ (R(t Ã€ i) Ã¾ 1) Ã€1 (2:2:4)

iÂ¼0

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:

1XkÃ€1

Ë‡

R(t, k) % R(t, i) (2:2:5)

k iÂ¼1

The continuously compounded return (or log return) is defined as:

r(t) Â¼ log [R(t) Ã¾ 1] Â¼ p(t) Ã€ p(t Ã€ 1) (2:2:6)

Calculation of the compounded log returns is reduced to simple

summation:

r(t, k) Â¼ r(t) Ã¾ r(t Ã€ 1) Ã¾ . . . Ã¾ r(t Ã€ k Ã¾ 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 Ã¾ 1) are paid within the period [t, t Ã¾ 1], the simple

return (see 2.2.1) is modified to

R(t Ã¾ 1) Â¼ [P(t Ã¾ 1) Ã¾ D(t Ã¾ 1) ]=P(t) Ã€ 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

9

Financial Markets

every time interval (say every month), the future value of this cash

after n periods is equal to

FV Â¼ K(1 Ã¾ 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 Â¼ FV=(1 Ã¾ 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) ] Â¼ R, one can rewrite (2.2.8) as

P(t) Â¼ E[{P(t Ã¾ 1) Ã¾ D(t Ã¾ 1)}=(1 Ã¾ R)] (2:2:11)

If this recursion is repeated K times, one obtains

" #

X

K

D(t Ã¾ i)=(1 Ã¾ R)i Ã¾ E[P(t Ã¾ K)=(1 Ã¾ R)K ]

P(t) Â¼ E (2:2:12)

iÂ¼1

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 Ã¾ K)=(1 Ã¾ R)K ] Â¼ 0 (2:2:13)

K!1

Then the discounted-cash-flow model yields

" #

X1

D(t Ã¾ i)=(1 Ã¾ R)i

PD (t) Â¼ E (2:2:14)

iÂ¼1

Further simplification of the discounted-cash-flow model is based on

the assumption that the dividends grow linearly with rate G

E[D(t Ã¾ i) ] Â¼ (1 Ã¾ G)i D(t) (2:2:15)

Then (2.2.14) reduces to

10 Financial Markets

1Ã¾G

PD (t) Â¼ D(t) (2:2:16)

RÃ€G

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) Â¼ PD (t) Ã¾ B(t), B(t) Â¼ E[B(t Ã¾ 1)=(1 Ã¾ 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 growthâ€”the â€˜â€˜bubble.â€™â€™ Yet, this

bubble is â€˜â€˜rationalâ€™â€™ since it is based on rational expectations of future

returns. In the popular Blanchard-Watson model

(

1 Ã¾ R B(t) Ã¾ e(t Ã¾ 1) with probability p, 0 < p < 1

p

B(t Ã¾ 1) Â¼ (2:2:18)

e(t Ã¾ 1) with probability 1 Ã€ p

where e(t) is an independent and identically distributed process (IID)2

with E[e(t) ] Â¼ 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 Â¼ K exp (rt), PV Â¼ FV exp (Ã€rt) (2:2:19)

In the following chapters, both the discrete and the continuous pre-

sentations will be used.

11

Financial Markets

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 â€˜â€˜bestâ€™â€™ price. In our example, it is a finite

number of Yens available at the exchange rate USD/JPY Â¼ 120. As

soon as they all are taken, the exchange rate USD/JPY falls to the

equilibrium value 1 USD/JPY Â¼ 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.

12 Financial Markets

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 companyâ€™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 â€˜â€˜weakâ€™â€™ form of EMH, current prices reflect all information on

past prices. Then the technical analysis seems to be helpless. In the

â€˜â€˜strongâ€™â€™ 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 â€˜â€˜semi-

strongâ€™â€™ 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 Bachelierâ€™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 [9â€“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

13

Financial Markets

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 1962â€“1986

[12]. Interestingly, similar analysis for the period of 1986â€“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 â€˜â€˜fair

priceâ€™â€™ and â€˜â€˜rational investorsâ€™â€™ do not stand criticism in the light of

the financial market booms and crashes. The â€˜â€˜irrational exuberanceâ€™â€™

in 1999â€“2000 can hardly be attributed to rational behavior [10]. In

fact, empirical research in the new field â€˜â€˜behavioral financeâ€™â€™ 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 Keynesâ€™ explanation that â€˜â€˜animal spiritsâ€™â€™ 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 â€˜â€˜What do we mean by saying markets

are efficientâ€™â€™ of his book â€˜â€˜A Random Walk down Wall Streetâ€™â€™ [9]:

â€˜â€˜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.â€™â€™

Thus, polemics on EMH changes the discussion from whether

prices follow the random walk to the practical ability to consistently

â€˜â€˜beat the market.â€™â€™

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 â€˜â€˜Dogs of the Dowâ€™â€™ (buying 10 stocks of the Dow

Jones Industrial Average with highest dividend yield), â€˜â€˜January

Effectâ€™â€™ (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

â€˜â€˜channel,â€™â€™ â€˜â€˜head and shoulders,â€™â€™ 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., [17â€“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

15

Financial Markets

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 [9â€“11].6 On the

ñòð. 2 |