Quantitative Finance for Physicists

2.3 MARKET EFFICIENCY

2.3.1 A

RBITRAGE

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.

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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.

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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

12 Financial Markets