UNCERTAINTY, RANDOM WALKS AND STOCHASTIC PROCESSES

4.3 UNCERTAINTY, RANDOM WALKS AND STOCHASTIC PROCESSES

A stochastic process is an indexed pair {events, time} expressed in terms of a function – a random variable indexed to time. This defines a sample path, i.e. a set of values that the process can assume over time. For example, it might be a stock price denoting events, indexed to a time scale. The study of stochastic processes has its origin in the study of the kinetic behaviour of molecules in gas by physicists in the nineteenth century. It was only in the twentieth century, following work by Einstein, Kolmogorov, Levy, Wiener and others, that stochastic processes were studied in some depth. In finance, however, Bachelier, in his dissertation in 1900, had already provided a study of stock exchange speculation using a fundamental stochastic process we call the ‘random walk’, establishing a connection between price fluctuations in the stock exchange and Brownian motion – a continuous-time expression of the random walk assumptions.

4.3.1 The random walk

The random walk model of price change is based on two essential behavioural hypotheses.

(1) In any given time interval, prices may increase with a known probability

0 < p < 1, or decrease with probability 1 − p. (2) Price changes from period to period are statistically independent.

UNCERTAINTY , RANDOM WALKS AND STOCHASTIC PROCESSES

w.p. p (t) =

w.p. 1− p

Thus, if x(t) is the price at the discrete time t, and if it is only a function of the prices is given by:

values denote a stochastic process x(t) which is also written as {x(t), t ≥ 0}. The price at time t, x(t), assumes in this case a binomial distribution since events

are independent and of fixed probability, as we shall see next. Say that we start

at a given price x 0 at time t 0 = 0. At time t 1 =t 0

Namely, x (t 1 ) = x(t 1 1 , or x (t 1 ) = x(t 0 1 . We can also write this equation in terms of the number of times i 1 the price

x (t 1 )=x 0 +i 1 x − (1 − i 1 1 ∼ B(1, p) where i 1 assumes two values i 1 = 0, 1 given by the binomial probability distri-

i 1 = 0, 1 and parameter (1, p), 0 < p < 1. An instant of time later t 2 =t 1 + t=t 0

x (t 2 ) = x(t 2 2 or x(t 2 ) = x(t 1 2 or x(t 2 ) = x(t 0 1 2 which we can write as follows (see also Figure 4.1): x (t 2 )=x 0 +i 2 x − (2 − i 2 2 ∼ B(2, p) and generally, for n successive intervals of time (t n

by:

n ∼ B(n, p) where:

x (t n )=x 0 +i n x − (n − i n

B (n, p) =

(1 − p) ;i n = 0, 1, 2, . . . , n

n−i n

86 PROBABILITY AND FINANCE

x+ 2∆x; p 2

x + ∆x; p

x; 2p(1− p)

x − ∆x; p

x −2∆x;(1− p) 2

Figure 4.1

A two-period tree.

The price process can thus be written by:

where x(t n ) − x(t 0 j ( j = 1, . . . , n). Since price changes are of equal size, we can state that the number of times prices have increased is given by the binomial distribution B(n, p). The expected price and its variance can now be calculated easily. The expected price at time t n is:

n ); E(i n ) = np; E(n − i n ) = n(1 − p) Set d = [i n − (n − i n

E (x(t n )) = x 0 n

and var(d), with q = 1 − p are then,

This is easily proved. Note that E(i ) = np and var(i) = npq, with i replacing i n for simplicity. Thus,

E Also

The results above are expressed in terms of small distance (which we shall hence- increments be very small, we can obtain continuous time and continuous state

limits for the equation of motion. Explicitly, in a time interval [0, t], let the number

UNCERTAINTY , RANDOM WALKS AND STOCHASTIC PROCESSES

E (d) =

var(d) =

2 t must exist, however. In other words, we are specifying a priori that the stochastic

process, has at the limit, finite mean and finite variance growth rates. If we let these limits be:

It is also possible to express the probability of a price increase in terms of these parameters which we choose for convenience to be:

Inserting this probability in the mean and variance equations, and moving to the limit, we obtain the mean and variance functions m(t) and σ 2 (t) which are linear in time:

(t) = 2Ct; σ 2 (t) = 2Dt

where C is called the ‘drift’ of the process expressing its tendency over time while

D is its diffusion, expressing the process variability. The proof of these is simple to check. First note that:

equation above to lead to:

2 var(d) = t 2

At the time limit, we obtain the variance var(d) = σ 2 (t) = 2Dt stated above.

88 PROBABILITY AND FINANCE

Since this limit results from limiting arguments to the underlying binomial process describing the random walk, we can conclude that the parameters (m(t), σ 2 (t)) are normally distributed, or:

f (x, t) = [x − m(t)] √ exp −

2 σ 2 (t) This equation turns out to be also a particular solution of a partial differential

2π σ (t)

equation expressing the continuous time–state evolution of the process probabil- ities and called the Fokker–Planck equation. Using the elementary observation that a linear transformation of normal random variables are also normal, we can √ write the price equation in terms of its drift 2C and diffusion 2D (also called volatility), by:

t . Such processes, in continuous time are called stochastic differential equations

is essentially a zero mean normally distributed random variable with independent increments and a linear variance in time t. It is named after Robert Brown (1773– 1858), a botanist who discovered the random motion of colloid-sized particles found in experiments performed in June–August 1825 with pollen. If we were to take a stock price, it would be interesting to estimate both the drift and the diffusion of the process. Would it fit? Would the residual error be indeed a normal probability distribution with mean zero, and a linear time variance with no correlation? Such

a study would compare stock data taken every minute (tickertape), daily, weekly and monthly. Probably, results will differ according to the time scale taken for the estimation and thereby violate the assumptions of the model. Such studies are important in financial statistics when they seek to justify the assumption of ‘error normality’ in financial time series.

The Wiener process is of fundamental importance in mathematical finance because it is used to model the uncertainty associated with many economic pro- cesses. However, it is well known in finance that such a process underestimates the probability of the price not changing, and overestimates the mid-range value price fluctuations. Further, extreme price jumps are grossly underestimated by the Wiener (normal) process. The search for distributions that can truly reflect stock market behaviour has thus became an important preoccupation. Mandelbrot and Fama for example have suggested that we use Pareto–Levy distributions as well as leptokurtic distributions to describe the statistics of price fluctuations. Explic-

itly, say that a distribution has mean m and variance σ 2 and define the following coefficients ζ 1 =m 3 /σ 3 and ζ 2 =m 4 /σ 4 − 3 where m 3 and m 4 are the third and the fourth moment respectively. The first index is an index of asymmetry pointing to leptokurtic distributions while the second is ‘an excess coefficient’ point to

platokurtic distributions. For the Normal distribution we have ζ 1 = 0 and ζ 2 = 0,

89 thus any departure from these reference values will also indicate a departure from

UNCERTAINTY , RANDOM WALKS AND STOCHASTIC PROCESSES

normality. Pareto–Levy stable distributions exhibit, however, an infinite variance, practically referred to as ‘fat tail distributions’ that also violate the underlying assumptions of ‘Normal–Wiener’ processes. When weekly or monthly data is used (rather than daily and intraday data), a smoothing of the data allows the use of the Normal distribution. This observation thus implies that the time scale we choose to characterize uncertainty is an important factor to deal with. When the time scale increases, the use of Normal distributions is justified because in such cases, we gradually move from leptokurtic to Normal distributions. What statistical distribution can one assume over different periods of consideration? The random walk is by far the most used and the easiest to work with and agrees well for larger periods of time. Other distributions are mathematically more challenging, especially since different results are seen for various assets. Part of the problem can be explained by the deviations from the efficient markets hypothesis and external influences on the market, as we shall see in subsequent chapters.

Formally, it is a Markov stochastic process x = {x(t); t ≥ 0} whose non-

x are stationary, independently and normally distributed with mean zero and vari-

often written as:

√ dx = 2C dt = 2D dw(t)

Such equations are known as stochastic differential equations. Generalization to far more complex movements can also be constructed by changing the modelling hypotheses regarding the drift and the diffusion processes. When the diffusion– volatility is also subject to uncertainty, this leads to processes we call stochastic volatility models, leading to incomplete markets (as will be seen in Chapter 5). In many cases, volatility can be a function of the process itself. For example, say that σ = σ (x), then evidently,

which need not lead, necessarily, to a Normal probability distribution for x. For example, in some cases, it is convenient to presume that rates of returns are

(the expected rate of return) and known diffusion (the rates of returns volatility). Thus, the following hypothesis is stated:

xx

90 PROBABILITY AND FINANCE

This is equivalent to stating that the log of return y = ln (x) has a Normal proba- bility distribution:

with mean αt and variance σ 2 t and therefore, x has a lognormal probability distribution. In many economic and financial applications stochastic processes are driven by a Wiener process leading to models of the form:

X t +1 =X t +f t (X ) + σ t (X )ε t ,ε t ∼ N (0, 1), t = 0, 1, 2, . . . where ε t is a zero mean, unit variance and normally distributed random variable.

When the time interval is infinitely small, in continuous time, we have a stochastic differential equation:

dx(t) = f (x, t) dt + σ (x, t) dw(t), x(0) = x 0 , 0≤t≤T The variable x(t) is defined, however, only if the above equation is meaningful

in a statistical sense. In general, existence of a solution for the stochastic differ- ential equation cannot be taken for granted and conditions have to be imposed to guarantee that such a solution exists. Such conditions are provided by the Lip- schitz conditions assuming that: f , σ and the initial condition x(0) are real and continuous and satisfy the following hypotheses:

r f and σ satisfy uniform Lipschitz conditions in x. That is, there is a K > 0

such that for x 2 and x 1 ,

| f (x 2 , t ) − f (x 1 , t )| ≤ K |x 2 −x 1 | |σ (x 2 , t ) − σ (x 1 , t )| ≤ K |x 2 −x 1 |

r f and σ are continuous in t on [0, T ], x(0) is any random variable with

E (x(0)) 2 < ∞, independent of the increment stochastic process. Then: (1) The stochastic differential equation has, in the mean square limit sense,

a solution on

t t ∈ [0, T ] , x(t) − x(0) =

f (x, τ ) dτ + σ (x, τ ) dw(τ )

(2) x(t) is mean square continuous on [0, T ]

(3) E (x(0)) 2 < M, for all t ∈ [0, T ] and arbitrary M, T

E ((x(t)) 2 ) dt < ∞

91 (4) x(t) − x(0) is independent of the stochastic process {dw(τ ); τ > t} for

UNCERTAINTY , RANDOM WALKS AND STOCHASTIC PROCESSES

t ∈ [0, T ]. The stochastic process x(t), t ∈ [0, T ], is then a Markov process and, in a mean

square sense, is uniquely determined by the initial condition x(0). The Lipschitz and the growth conditions, meaning ( f (x, t)) 2 2 2 + (σ (x, t)) 2 ≤K (1 + |x| ), pro- vide both a uniqueness and existence non-anticipating solution x(t) of the stochas- tic differential equation in the appropriate range [0, T ]. In other words, if these conditions are not guaranteed, as is the case when the variance of processes increases infinitely, a solution to the stochastic differential equation cannot be assured.

Clearly, there is more than one way to conceive and formalize stochastic models of prices. In this approach, however, the evolution of prices was entirely indepen- dent of their past history. And further, a position at an instant of time depends only on the position at the previous instant of time. Such assumptions, compared to the real economic, financial and social processes we usually face, are extremely simplistic. They are, however, required for analytical tractability and we must therefore be aware of their limitations. The stringency of the assumptions re- quired to construct stochastic processes, thus, point out that these can be useful to study systems which exhibit only small variations in time. Models with large and unpredictable variations must be based therefore on an intuitive understanding of the problem at hand or some other modelling techniques.

4.3.2 Properties of stochastic processes

The characteristics of time series are mostly expressed in terms of, ‘stationarity, ergodicity, correlation and independent increments’. These terms are often en- countered in the study of financial time series and we ought therefore to understand them.

Stationarity

A time series is stationary when the evolution of its mean (drift) and variance (volatility–diffusion) are not a function of time. If f (x, t) is the probability dis- tribution of x at time t, then: f (x, t) = f (x, t + τ ) = f (x) for all t and τ . This property is called strict stationarity. In this case, for a two random variables process, we have:

f (x 1 , x 2 , t 1 , t 2 ) = f (x 1 , x 2 , t 1 , t 1 + τ ) = f (x 1 , x 2 , t 2 −t 1 ) = f (x 1 , x 2 ,τ ) That is, for the joint distribution of a strict stationary process, the distribution is

a function of the time difference τ of the two (prices) random variables. As a re- sult, the correlation function B(t 1 , t 2 ) = E(x(t 1 )x(t 2 )), describing the correlation between (x 1 , x 2 ) at instants of time (t 1 , t 2 ), is a function of the time difference t 2 −t 1 = τ only. The autocovariance function (the correlation function about the mean) is then given by K (t 1 , t 2 ), with K (t 1 , t 2 ) = B(t 1 , t 2 )−Ex 1 (t 1 )E x 2 (t 2 ). By the same token, the correlation coefficient R 1 (τ ) of the random variable x 1 is a

92 PROBABILITY AND FINANCE

function of the time difference τ only, or

cov[x 1 (t), x 1

(t + τ )]

var[x 1 (t)] var[x 1 (t + τ )] For stationary processes we have necessarily var[x(t)] = var [x(t + τ )] and there-

fore the correlation coefficient is a function of the time difference only, or

[B(τ ) − m]

2 K (τ )

var[x(t)]

K (0)

Independent increments

tributed. This property leads to well-known processes such as the Poisson Jump and the Wiener process we saw earlier and can, sometimes, be necessary for the mathematical tractability of stochastic processes. The first two moments of non- overlapping independent and stationary increments point to a linear function of time (hence the term of linear finance, associated with using Brownian motion in financial model building). This is shown by the simple equalities:

E [X (t)] = t E[X(1)] + (1 − t)E[X(0)]; var[X (t)] = t var[X(1)] + (1 − t) var[X(0)]

The proof is straightforward and found by noting that if we set f (t) = E[X(t)] −

E [X (0)], then, non-overlapping stationary increments imply that:

f (t + s) = E[X(t + s)] − E[X(0)] = E[X(t + s) − X(t)] + E[X(t) − X(0)]

= E[X(s) − X(0)] + E[X(t) − X(0)] = f (t) + f (s)

And the only solution is f (t) = t f (1), which is used to prove the result for the expectation. The same technique applies to the variance.