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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

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