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random variable. The components of this system evolve spontan-

eously into the power law distribution f(w, t) $ wÃ€(1Ã¾a) . In the

mean time, evolution of W(t) exhibits intermittent fluctuations that

can be parameterized using the truncated Levy distribution with the

same index a [14].

Seeking universal properties of the financial market crashes is

another interesting problem explored by Sornette and others (see

[16] for details). The main idea here is that financial crashes are

caused by collective trader behavior (dumping stocks in panic),

which resembles the critical phenomena in hierarchical systems.

Within this analogy, the asymptotic behavior of the asset price S(t)

has the log-periodic form

S(t) Â¼ A Ã¾ B(tc Ã€ t)a {1 Ã¾ C cos [w ln (tc Ã€ t) Ã€ w]} (8:2:2)

where tc is the crash time; A, B, C, w, a, and w are the fitting

parameters. There has been some success in describing several market

crashes with the log-periodic asymptotes [16]. Criticism of this ap-

proach is given in [17] and references therein.

8.3 NEW DEVELOPMENTS

So, do the findings listed in the preceding section solve the problem

of scaling in finance? This remains to be seen. First, B. LeBaron has

shown how the price distributions that seem to have the power-law

form can be generated by a mix of the normal distributions with

91

Scaling in Financial Time Series

different time scales [18]. In this work, the daily returns are assumed

to have the form

R(t) Â¼ exp [gx(t) Ã¾ m]e(t) (8:3:1)

where e(t) is an independent random normal variable with zero mean

and unit variance. The function x(t) is the sum of three processes with

different characteristic times

x(t) Â¼ a1 y1 (t) Ã¾ a2 y2 (t) Ã¾ a3 y3 (t) (8:3:2)

The first process y1 (t) is an AR(1) process

y1 (t Ã¾ 1) Â¼ r1 y1 (t) Ã¾ Z1 (t Ã¾ 1) (8:3:3)

where r1 Â¼ 0:999 and Z1 (t) is an independent Gaussian adjusted so

that var[y1 (t)] Â¼ 1. While AR(1) yields exponential decay, the chosen

value of r1 gives a long-range half-life of about 2.7 years. Similarly,

y2 (t Ã¾ 1) Â¼ r2 y2 (t) Ã¾ Z2 (t Ã¾ 1) (8:3:4)

where Z2 (t) is an independent Gaussian adjusted so that

var[y2 (t)] Â¼ 1. The chosen value r2 Â¼ 0:95 gives a half-life of about

2.5 weeks. The process y3 (t) is an independent Gaussian with unit

variance and zero mean, which retains volatility shock for one day.

The normalization rule is applied to the coefficients ai

a1 2 Ã¾ a2 2 Ã¾ a3 2 Â¼ 1: (8:3:5)

The parameters a1 , a2 , g, and m are chosen to adjust the empirical data.

This model was used for analysis of the Dow returns for 100 years

(from 1900 to 2000). The surprising outcome of this analysis is retrieval

of the power law with the index in the range of 2.98 to 3.33 for the data

aggregation ranges of 1 to 20 days. Then there are generic comments by

T. Lux on spurious scaling laws that may be extracted from finite

financial data samples [19]. Some reservation has also been expressed

about the graphical inference method widely used in the empirical

research. In this method, the linear regression equations are recovered

from the log - log plots. While such an approach may provide correct

asymptotes, at times it does not stand up to more rigorous statistical

hypothesis testing. A case in point is the distribution in the form

f(x) Â¼ xÃ€a L(x) (8:3:6)

where L(x) is a slowly-varying function that determines behavior of

the distribution in the short-range region. Obviously, the â€˜â€˜universalâ€™â€™

92 Scaling in Financial Time Series

scaling exponent a Â¼ Ã€log [f(x)]= log (x) is as accurate as L(x) is close

to a constant. This problem is relevant also to the multifractal scaling

analysis that has become another â€˜â€˜hotâ€™â€™ direction in the field.

The multifractal patterns have been found in several financial time

series (see, e.g., [20, 21] and references therein). The multifractal

framework has been further advanced by Mandelbrot and others.

They proposed compound stochastic process in which a multifractal

cascade is used for time transformations [22]. Namely, it was assumed

that the price returns R(t) are described as

R(t) Â¼ BH [u(t)] (8:3:7)

where BH [] is the fractional Brownian motion with index H and u(t) is

a distribution function of multifractal measure (see Section 6.2). Both

stochastic components of the compound process are assumed inde-

pendent. The function u(t) has a sense of â€˜â€˜trading timeâ€™â€™ that reflects

intensity of the trading process. Current research in this direction

shows some promising results [23â€“26]. In particular, it was shown

that both the binomial cascade and the lognormal cascade embedded

into the Wiener process (i.e., into BH [] with H Â¼ 0:5) may yield a more

accurate description of several financial time series than the GARCH

model [23]. Nevertheless, this chapter remains â€˜â€˜unfinishedâ€™â€™ as new

findings in empirical research continue to pose new challenges for

theoreticians.

8.4 REFERENCES FOR FURTHER READING

Early research of scaling in finance is described in [2, 6, 7, 9, 17].

For recent findings in this field, readers may consult [10â€“13, 23â€“26].

8.5 EXERCISES

**1. Verify how a sum of Gaussians can reproduce a distribution

with the power-law tails in the spirit of [18].

**2. Discuss the recent polemics on the power-law tails of stock

prices [27â€“29].

**3. Discuss the scaling properties of financial time series reported

in [30].

Chapter 9

Option Pricing

This chapter begins with an introduction of the notion of financial

derivative in Section 9.1. The general properties of the stock options

are described in Section 9.2. Furthermore, the option pricing theory is

presented using two approaches: the method of the binomial trees

(Section 9.3) and the classical Black-Scholes theory (Section 9.4).

A paradox related to the arbitrage free portfolio paradigm on which

the Black-Scholes theory is based is described in the Appendix section.

9.1 FINANCIAL DERIVATIVES

In finance, derivatives1 are the instruments whose price depends

on the value of another (underlying) asset [1]. In particular, the

stock option is a derivative whose price depends on the underlying

stock price. Derivatives have also been used for many other assets,

including but not limited to commodities (e.g., cattle, lumber,

copper), Treasury bonds, and currencies.

An example of a simple derivative is a forward contract that obliges

its owner to buy or sell a certain amount of the underlying asset at a

specified price (so-called forward price or delivery price) on a specified

date (delivery date or maturity). The party involved in a contract as a

buyer is said to have a long position, while a seller is said to have a short

position. A forward contract is settled at maturity when the seller

93

94 Option Pricing

delivers the asset to the buyer and the buyer pays the cash amount at

the delivery price. At maturity, the current (spot) asset price, ST , may

differ from the delivery price, K. Then the payoff from the long

position is ST Ã€ K and the payoff from the short position is K Ã€ ST .

Future contracts are the forward contracts that are traded on

organized exchanges, such as the Chicago Board of Trade (CBOT)

and the Chicago Mercantile Exchange (CME). The exchanges deter-

mine the standardized amounts of traded assets, delivery dates, and

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