Taylor Series Expansion

Let us consider the following SDE where the volatility σ and the drift µ are constants:

dS = Sµdt + SσdW (3)

Suppose that f (S) is a smooth function of S. If we vary S by a small amount dS, then clearly f also varies by a small amount. From the Taylor series expansion, we can write:

This result can be further generalized by considering a function of the random variable S and of time, f (S, t). We can expand f (S+dS, t+dt) in a Taylor series of (S, t) to get:

(4)

where the dots denote a remainder (O(dS3)) which is smaller than any of the terms we have retained. Now recall that dS is given by the SDE {3}. Here dS is simply a number, although random, and so squaring it, we find that:

(5)

We now examine the order of magnitude of each of the terms in {5}

Since:

dW² --> dt, as dt --> 0

the third term is the largest for small dt and dominates the other two terms. Therefore:

dS² = S²σ²dt

If we substitute this result into {4} and retain only those terms which are relevant, we get:

(6)

We can obtain the same result in an easy way by applying Ito’s lemma {1} directly to our SDE {3}.

Prof. Klaus Schmitz