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 Stochastic Processes

Changes in a variable such as stock price involve a deterministic component which is a function of time and a stochastic component which depends upon a random variable. Let S be the stock price at time t and let dS be the infinitesimal change in S over the infinitesimal interval of time dt. The change in the random variable z over this interval of time is dz. The change in stock price is then given by

(1)

#### dS = adt + bdz,

where a and b may be functions of S and t as well as other variables.
The expected value of dz is zero so the expected value of dS is equal to the deterministic component, adt.

The random variable dz represents an accumulation of random influences over the interval dt. If the random influences are of finite variance then the Central Limit Theorem implies that dz has a normal distribution and hence is completely characterized by its mean and standard deviation.

The mean or expected value of dz is zero. The variance of a random variable which is the accumulation of independent effects over an interval of time is proportional to the length of the interval, in this case dt. The standard deviation of dz is thus proportional to the square root of dt, (dt)½. All of this means that the random variable dz is equivalent to a random variable w(dt)½, where w is a standard normal variable with mean zero and standard deviation equal to unity.

Some particular stochastic processes which are important are:

• Additive random walk
(2)

#### dS = μdt + σdz,

where μ and σ are constants. This is a stochastic process in which the variable exhibits a trend with slope μ and a constant volatility of σ. The problem with this process is that the variable S could take on negative values.
• Multiplicative random walk
(3)

#### dS = μSdt + σSdz.

In this stochastic process the variable S has an exponential trend, growing at a rate of μ.
• Mean-reverting process
(4)

#### dr = λ(ρ - r)dt + σdz.

In this process the variable r, say the interest rate, fluctuates but always tends to ultimately more toward the value ρ.
• Poisson or jump process,
(5)

#### dq = 1 with a probability λdt       = 0 with a probability 1 - λdt

Now consider another variable C, such as the price of a call option, which is a function of S and t, say C = f(S, t). Because C is a function of the stochastic variable S, C will have a stochastic component as well as a deterministic component. C will have a representation of the form:

(6)

#### dC = pdt + qdz.

The crucial problem is how the functions p and q are related to the functions a and b in the equation

(7)

#### dS = adt + bdz.

Ito's Lemma gives the answer. The deterministic and stochastic components of dC are given by:

(8)

#### p = ∂f/∂t + (∂f/∂S)a + ½(∂²f/∂S²) b² q = (∂f/∂S)b.

Ito's Lemma is crucial in deriving differential equations for the value of derivative securities such as stock options.

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