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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)

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