What is a normal standard variable

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Ito's lemma and its derivative

Changes in a variable such as stock price involve a deterministic component that is a function of time and the stochastic component that depends on a random variable. Let S be the stock price at time t and let dS be the tiny change in S over the tiny distance from time corotron. The change in the random variable z over this interval of time is dz. The change in the stock price is given over

(1)


dS = adt + bdz,
 

where a and b can be functions of S and t as well as other variables; i.e.
dS = a (S, t, x) dt + b (S, t, x) dz.

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 the occasional influences over the distance ds. The central c limit theorem then indicates that dz has a normal distribution and is therefore fully characterized by its mean and standard deviation. The mean or expected value of dz is zero. The deviation of a random variable, which is the accumulation of the independent effects over an interval of time, is proportional to the length of the interval, in this case the corotron. The standard deviation of dz is therefore proportional to the square root of the corotron (dt)½. All of this means that the random variable dz is equivalent to a random variable w (dt)½where w is a normal standard variable with the mean zero and the standard deviation equal to the unit.

Now consider another variable C, like the price of an option to buy, which is a function of S and t, say C = f (S, t). Because C is a function of the stochastic variable S, C has a stochastic component as well as a deterministic component. C has a representation of the form:

(2)


dC = pdt + qdz.
 

where p and q can be functions of S, of t, and perhaps of other variables; i.e. p = p (S, t, x) and q = q (S, t, x).

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

(3)


dS = adt + bdz.
 

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

(4)


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 inferred stocks such as stock options.

Derivation

The Taylor series for f (S, t) gives the increment in C as:

(5)


dC = (∂f / ∂t) dt + (∂f / ∂S) dS + ½ (∂²f / ∂S²) (dS) ²
+ (∂²f / ∂S∂t) (dS) (dt) + ½ (∂²f / ∂t²) (dt) ² +
higher order terms.
 

The increment in the share price dS is given over

dS = adt + bdz
but
dz = vw [dt]½,

where w is a normal standard random variable. Replacement of adt + bvw (paper corotron) ½ for dS in the above equation (5) yields:
(6)


dC = (∂f / ∂t) dt + (∂f / ∂S) adt + ∂f / ∂S) bvw (dt)½
+ ½ (∂²f / ∂S²) (adt + bvw (dt)½
+ (∂²f / ∂S∂t) (adt + bvw (dt)½) (dt) + ½ (∂²f / ∂t²) (dt) ²
+ higher order terms.
 

With the expansion of the quadratic term and the product term, the result is:
(7)


dC = (∂f / ∂t) dt + (∂f / ∂S) adt + ∂f / ∂S) bvw (dt)½
+ ½ (∂²f / ∂S²) (a²dt² + 2abvw (dt)3/2 + b²v²w²dt)
+ (∂²f / ∂S∂t) (a (dt) ² + bvw (dt)3/2) + ½ (∂²f / ∂t²) (dt) ²
+ higher order terms.
 

Allowing for the tiny type of DT for DT to disappear to any power higher than unity, (7) reduces to:
(8)


dC = (∂f / ∂t) dt + (∂f / ∂S) adt + (∂f / ∂S) bvw (dt)½
+ ½ (∂²f / ∂S²) (b²v²w²dt)
 

Noting that the expected value of w² is unit, the expected value of dC is:
(9)


[∂f / ∂t + (∂f / ∂S) a + ½ (∂²f / ∂S²) b²] dt.
 

This is the deterministic part of dC. The stochastic component is the expression that depends on dz, which is expressed in (8) as vw (dt)½ is pictured. Hence the stochastic component is:
(10)


[(∂f / ∂S) b] dz.
 

From the above derivation, it would appear that there is an additional stochastic expression resulting from the occasional deviations of w² from its expected value of 1; i.e. the additional term
(11)


½ (∂²f / ∂S²) (b²v²w²dt).
 

However, the deviation of this additional term is proportional to (corotron) ², while the deviation of the stochastic term given in (10) is too proportional to (corotron). Thus, the stochastic expression given in (11) disappears compared with the stochastic expression given in (10).


Ito's lemma is essential in the derivation of the Black and Scholes equations.

An immediate question is whether an extension of Ito's lemma for constant distributions of z is different from the normal distribution. This question is investigated in a page on perennial distributions.