Introduction

A random walk is a mathematical formalization of a path that consists of a succession of

random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the

search path of a foraging animal, the price of a fluctuating stock and the financial status of a

gambler can all be modeled as random walks, although they may not be truly random in

reality. The term random walk was first introduced by Karl Pearson in 1905. Random walks

have been used in many fields: ecology, economics, psychology, computer science, physics,

chemistry, and biology. Random walks explain the observed behaviors of processes in these

fields, and thus serve as a fundamental model for the recorded stochastic activity.

With "random walk", Malkiel asserts that price movements in securities are unpredictable.

Because of this random walk, investors cannot consistently outperform the market as a

whole. Applying fundamental analysis or technical analysis to time the market is a waste of

time that will simply lead to underperformance. Investors would be better off buying and

holding an index fund.

Random walk theory jibes with the semi-strong efficient hypothesis in its assertion that it is

impossible to outperform the market on a consistent basis. This theory argues that stock

prices are efficient because they reflect all known information (earnings, expectations,

dividends). Prices quickly adjust to new information and it is virtually impossible to act on

this information. Furthermore, price moves only with the advent of new information and this

information is random and unpredictable. The theory that stock price changes have the same

distribution and are independent of each other, so the past movement or trend of a stock price

or market cannot be used to predict its future movement.

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In short, this is the idea that stocks take a random and unpredictable path. A follower of the

random walk theory believes it's impossible to outperform the market without assuming

additional risk. Critics of the theory, however, contend that stocks do maintain price trends

over time - in other words, that it is possible to outperform the market by carefully selecting

entry and exit points for equity investments.

This theory raised a lot of eyebrows in 1973 when author Burton Malkiel wrote "A Random

Walk Down Wall Street", which remains on the top-seller list for finance books.

According to the theory, the successive price changes or changes in return are independent

and these successive price changes are randomly distributed. Random walk model argues that

all publicly available information is fully reflected on the stock prices and further the stock

prices instantaneously adjust themselves to the available new information. The theory mainly

deals with the successive changes rather than the price or return levels.

The investors should note that the random theory says nothing about the relative price

changes that is the changes that are occurring across the securities. The random walk

hypothesis deals with the absolute price changes and not with the relative prices.

Random walk theory gained popularity in 1973 when Burton Malkiel wrote "A Random

Walk Down Wall Street", a book that is now regarded as an investment classic. Random

walk is a stock market theory that states that the past movement or direction of the price of a

stock or overall market cannot be used to predict its future movement. Originally examined

by Maurice Kendall in 1953, the theory states that stock price fluctuations are independent of

each other and have the same probability distribution, but that over a period of time, prices

maintain an upward trend.

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In short, random walk says that stocks take a random and unpredictable path. The chance of a

stock's future price going up is the same as it going down. A follower of random walk

believes it is impossible to outperform the market without assuming additional risk. In his

book, Malkiel preaches that both technical analysis and fundamental analysis are largely a

waste of time and are still unproven in outperforming the markets.

Malkiel constantly states that a long-term buy-and-hold strategy is the best and that

individuals should not attempt to time the markets. Attempts based on technical,

fundamental, or any other analysis are futile. He backs this up with statistics showing that

most mutual funds fail to beat benchmark averages like the S&P 500.

While many still follow the preaching of Malkiel, others believe that the investing landscape

is very different than it was when Malkiel wrote his book nearly 30 years ago. Today,

everyone has easy and fast access to relevant news and stock quotes. Investing is no longer a

game for the privileged. Random walk has never been a popular concept with those on Wall

Street, probably because it condemns the concepts on which it is based such as analysis and

stock picking.

It's hard to say how much truth there is to this theory; there is evidence that supports both

sides of the debate. Our suggestion is to pick up a copy of Malkiel's book and draw your own

conclusions.

Types of Random Walks

Various different types of random walks are of interest. Often, random walks are assumed to

be Markov chains or Markov processes, but other, more complicated walks are also of

interest. Some random walks are on graphs, others on the line, in the plane, or in higher

dimensions, while some random walks are on groups. Random walks also vary with regard to

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the time parameter. Often, the walk is in discrete time, and indexed by the natural numbers,

as in . However, some walks take their steps at random times, and in that

case the position is defined for the continuum of times . Specific cases or limits of

random walks include the Lévy flight. Random walks are related to the diffusion models and

are a fundamental topic in discussions of Markov processes. Several properties of random

walks, including dispersal distributions, first-passage times and encounter rates, have been

extensively studied.

Lattice random walk

A popular random walk model is that of a random walk on a regular lattice, where at each

step the location jumps to another site according to some probability distribution. In a simple

random walk, the location can only jump to neighboring sites of the lattice. In simple

symmetric random walk on a locally finite lattice, the probabilities of the location jumping to

each one of its immediate neighbours are the same.

Gaussian random walk

A random walk having a step size that varies according to a normal distribution is used as a

model for real-world time series data such as financial markets. The Black–Scholes formula

for modeling option prices, for example, uses a Gaussian random walk as an underlying

assumption.

Here, the step size is the inverse cumulative normal distribution where

0 ≤ z ≤ 1 is a uniformly distributed random number, and μ and σ are the mean and standard

deviations of the normal distribution, respectively.

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If μ is nonzero, the random walk will vary about a linear trend. If v s is the starting value of

the random walk, the expected value after n steps will be vs + nμ.

For the special case where μ is equal to zero, after n steps, the translation distance's

probability distribution is given by N(0, nσ2), where N() is the notation for the normal

distribution, n is the number of steps, and σ is from the inverse cumulative normal

distribution as given above. The Gaussian random walk can be thought of as the sum of a

series of independent and identically distributed random variables, X i from the inverse

cumulative normal distribution with mean equal zero and σ of the original inverse cumulative

normal distribution:

Z = ,

Applications of Random Walk

The following are some applications of random walk:

In economics, the "random walk hypothesis" is used to model shares prices and other factors.

Empirical studies found some deviations from this theoretical model, especially in short term

and long term correlations.

In population genetics, random walk describes the statistical properties of genetic drift

In physics, random walks are used as simplified models of physical Brownian motion and

diffusion such as the random movement of molecules in liquids and gases. See for example

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diffusion-limited aggregation. Also in physics, random walks and some of the self interacting

walks play a role in quantum field theory.

In mathematical ecology, random walks are used to describe individual animal movements,

to empirically support processes of biodiffusion, and occasionally to model population

dynamics.

In polymer physics, random walk describes an ideal chain. It is the simplest model to study

polymers.

In other fields of mathematics, random walk is used to calculate solutions to Laplace's

equation, to estimate the harmonic measure, and for various constructions in analysis and

combinatorics.

In computer science, random walks are used to estimate the size of the Web. In the World

Wide Web conference-2006, bar-yossef et al. published their findings and algorithms for the

same.

In image segmentation, random walks are used to determine the labels (i.e., "object" or

"background") to associate with each pixel. This algorithm is typically referred to as the

random walker segmentation algorithm.

In all these cases, random walk is often substituted for Brownian motion.

In brain research, random walks and reinforced random walks are used to model cascades of

neuron firing in the brain.

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In vision science, fixational eye movements are well described by a random walk.

In psychology, random walks explain accurately the relation between the time needed to

make a decision and the probability that a certain decision will be made.

Random walks can be used to sample from a state space which is unknown or very large, for

example to pick a random page off the internet or, for research of working conditions, a

random worker in a given country.

When this last approach is used in computer science it is known as Markov Chain Monte

Carlo or MCMC for short. Often, sampling from some complicated state space also allows

one to get a probabilistic estimate of the space's size. The estimate of the permanent of a

large matrix of zeros and ones was the first major problem tackled using this approach.

Random walks have also been used to sample massive online graphs such as online social

networks.

In wireless networking, a random walk is used to model node movement.

Motile bacteria engage in a biased random walk.

Random walks are used to model gambling.

In physics, random walks underlie the method of Fermi estimation.

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Variants of random walks

A number of types of stochastic processes have been considered that are similar to the pure

random walks but where the simple structure is allowed to be more generalized. The pure

structure can be characterized by the steps being defined by independent and identically

distributed random variables.

Random walk on graphs

A random walk of length k on a possibly infinite graph G with a root 0 is a stochastic process

with random variables such that and is a vertex chosen

uniformly at random from the neighbors of . Then the number is the

probability that a random walk of length k starting at v ends at w. In particular, if G is a graph

with root 0, is the probability that a -step random walk returns to 0.

Assume now that our city is no longer a perfect square grid. When our drunkard reaches a

certain junction he picks between the various available roads with equal probability. Thus, if

the junction has seven exits the drunkard will go to each one with probability one seventh.

This is a random walk on a graph

In a transient system, one only needs to overcome a finite resistance to get to infinity from

any point. In a recurrent system, the resistance from any point to infinity is infinite.

This characterization of recurrence and transience is very useful, and specifically it allows us

to analyze the case of a city drawn in the plane with the distances bounded.

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A random walk on a graph is a very special case of a Markov chain. Unlike a general Markov

chain, random walk on a graph enjoys a property called time symmetry or reversibility.

Roughly speaking, this property, also called the principle of detailed balance, means that the

probabilities to traverse a given path in one direction or in the other have a very simple

connection between them (if the graph is regular, they are just equal). This property has

important consequences.

Starting in the 1980s, much research has gone into connecting properties of the graph to

random walks. In addition to the electrical network connection described above, there are

important connections to isoperimetric inequalities, see more here, functional inequalities

such as Sobolev and Poincaré inequalities and properties of solutions of Laplace's equation.

A significant portion of this research was focused on Cayley graphs of finitely generated

groups. For example, the proof of Dave Bayer and Persi Diaconis that 7 riffle shuffles are

enough to mix a pack of cards (see more details under shuffle) is in effect a result about

random walk on the group Sn, and the proof uses the group structure in an essential way. In

many cases these discrete results carry over to, or are derived from manifolds and Lie groups.

Self-interacting random walks

There are a number of interesting models of random paths in which each step depends on the

past in a complicated manner. All are more complex for solving analytically than the usual

random walk; still, the behavior of any model of a random walker is obtainable using

computers. Examples include:

The self-avoiding walk (Madras and Slade 1996).

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The self-avoiding walk of length n on Z^d is the random n-step path which starts at the

origin, makes transitions only between adjacent sites in Z^d, never revisits a site, and is

chosen uniformly among all such paths. In two dimensions, due to self-trapping, a typical

self-avoiding walk is very short,while in higher dimension it grows beyond all bounds. This

model has often been used in polymer physics (since the 1960s).

The loop-erased random walk (Gregory Lawler).

The reinforced random walk (Robin Pemantle 2007).

The exploration process.

The multiagent random walk.

Long-range correlated walks

Long-range correlated time series are found in many biological, climatological and economic

systems.

Heartbeat records

Non-coding DNA sequences

Volatility time series of stocks

Temperature records around the globe

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Bibliography

Textbooks

Punithavathy Pandian, Security Analysis and Portfolio Management, Vikas Publishing House

Ltd.

Donald E. Fischer & Ronald J. Jordan, Security Analysis and Portfolio Management, Pearson

Internet

www.investopedia.com/terms/r/randomwalktheory.asp

www.investopedia.com/university/concepts/concepts5.asp

en.wikipedia.org/wiki/Random_walk_hypothesis

en.wikipedia.org/wiki/Random_walk

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