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Ito's Lemma and its Derivation

Ito's Lemma is named for its discoverer, the brilliant Japanese mathematician Kiyoshi Ito. The human race lost this extraordinary individual on November 10, 2008. He died at age 93. His work created a field of mathematics that is a calculus of stochastic variables.

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 given by


dS = adt + bdz,

where a and b may 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 random influences over the interval dt. The Central Limit Theorem then 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.

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:


dC = pdt + qdz.

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

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


dS = adt + bdz.

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


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.


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


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 stock price dS is given by

dS = adt + bdz

where w is a standard normal random variable and v is the scale of the variability of the random element; i.e., its standard deviation. Substitution of adt + bvw(dt)½ for dS in the above equation (5) yields:

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 squared term and the product term the result is:

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.

Taking into account the infinitesimal nature of dt so that dt to any power higher than unity vanishes, (7) reduces to:

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

Noting that the expected value of w² is unity, the expected value of dC is:

[∂f/∂t + (∂f/∂S)a + ½(∂²f/∂S²)b²]dt.

This is the deterministic component of dC. The stochastic component is the term that depends upon dz, which in (8) is represented as vw(dt)½. Therefore the stochastic component is:


From the above derivation it would seem that there is an additional stochastic term that arises from the random deviations of w² from its expected value of 1; i.e., the additional term


However the variance of this additional term is proportional to (dt)² whereas the variance of the stochastic term given in (10) is proportional to (dt). Thus the stochastic term given in (11) vanishes in comparison with the stochastic term given in (10).

Ito's Lemma is essential in the derivation of Black and Scholes Equation.

An immediate question is whether is an extension of Ito's Lemma for stable distributions of z other than the normal distribution. This question is investigated in a page on stable distributions.

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