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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

MBA level, much of the material pertinent to this chapter is given

in [3].

EXERCISES

1. Familiarize yourself with the financial market data available on

the Internet (e.g., http://www.finance.yahoo.com). Download the

weekly closing prices of the exchange-traded fund SPDR that

replicates the S&P 500 index (ticker SPY) for 1996â€“2003. Cal-

culate simple weekly returns for this data sample (we shall use

these data for other exercises).

2. Calculate the present value of SPY for 2004 if the asset risk

premium is equal to (a) 3% and (b) 4%. The SPY dividends in

2003 were $1.63. Assume the dividend growth rate of 5% (see

Exercise 5.3 for a more accurate estimate). Assume the risk-free

rate of 3%. What risk premium was priced in SPY in the end of

2004 according to the discounted-cash-flow theory?

3. Simulate the rational bubble using the Blanchard-Watson

model (2.2.18). Define e(t) Â¼ PU (t) Ã€ 0:5 where PU is standard

uniform distribution (explain why the relation e(t) Â¼ PU (t)

cannot be used). Use p Â¼ 0:75 and R Â¼ 0:1 as the initial values

for studying the model sensitivity to the input parameters.

4. Is there an arbitrage opportunity for the following set of the

exchange rates: GBP/USD Â¼ 1.7705, EUR/USD Â¼ 1.1914,

EUR/GBP Â¼ 0.6694?

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Chapter 3

Probability Distributions

This chapter begins with the basic notions of mathematical statistics

that form the framework for analysis of financial data (see, e.g.,

[1â€“3]). In Section 3.2, a number of distributions widely used in statis-

tical data analysis are listed. The stable distributions that have become

popular in Econophysics research are discussed in Section 3.3.

3.1 BASIC DEFINITIONS

Consider the random variable (or variate) X. The probability dens-

ity function P(x) defines the probability to find X between a and b

Ã°

b

b) Â¼ P(x)dx

Pr(a X (3:1:1)

a

The probability density must be a non-negative function and must

satisfy the normalization condition

XÃ°

max

P(x)dx Â¼ 1 (3:1:2)

Xmin

where the interval [Xmin , Xmax ] is the range of all possible values of X.

In fact, the infinite limits [Ã€1, 1] can always be used since P(x) may

17

18 Probability Distributions

be set to zero outside the interval [Xmin , Xmax ]. As a rule, the infinite

integration limits are further omitted.

Another way of describing random variable is to use the cumulative

distribution function

Ã°

b

b) Â¼

Pr(X P(x)dx (3:1:3)

Ã€1

Obviously, probability satisfies the condition

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