Ito's Lemma gives that for a function $F$ of a stochastic variable $X$, $dF = \frac{dF}{dX}dX + \frac{1}{2}\frac{d^2F}{dX^2}dt$

Given a stochastic differential equation $dS = a(S) dt + b(S) dX$ and a function $V(S)$, the differential equation satisfied by $V(S)$ is stated to be $dV = \frac{dV}{dS}dS + \frac{1}{2}b^2 \frac{d^2V}{dS^2}dt $. The book says this can be derived properly or the cheating substitution of $dX^2 = dt$ could be used.

But I just don't understand how the expression for $dV$ is obtained - either properly or by using the Taylor series. While using the Taylor series, how should I expand $V$? Should I use the Taylor series for a composite function, as shown, for instance, here? And how might I do it "properly"?

The book is Quantitative Finance of Paul Wilmott, I have made a screenshot of the relevant section.

I make a completely naive attempt by trying to apply Ito's lemma to $V$ treating $S$ as the random variable (since it is, in effect the function of one).

$dV = \frac{dV}{dS}dS + \frac{1}{2}\frac{d^2V}{dS^2}dS^2 $

and then substituting the given expression of $dS$ in the second term of the RHS. $dV = \frac{dV}{dS}dS + \frac{1}{2}\frac{d^2V}{dS^2}(a(S) dt + b(S) dX)^2$

This can directly give the correct expression for $dV$ if the $dt$ term inside the braces can be made to go to zero. But why would that be? So obviously there's something subtle going on that I am completely missing. Any pointers would be appreciated. Also regarding how to derive it properly.

enter image description here


In analysis, you generally consider (on the infinitesimal scale) only first-order variation ($dt$, $dx$) since continuously differentiable functions have bounded variation on any finite interval, hence zero quadratic variation: " $\left(dt\right)^2 = 0$".

However, this is not the case of Brownian motion, which has infinite variation on any finite interval; yet, it has finite quadratic variation, $d\langle W\rangle_t = dt$. This is why you have to consider its second-order variation, or quadratic variation (see Quadratic Variation on Wikipedia). You also have $d\langle t, W_t\rangle \approx \left(dt\right)\left(dW_t\right)$, which has zero mean and a negligible variance of order $\left(dt\right)^3$.

At the end of the day, the only nonnegligible term in the $\left(dS_t\right)^2$ term is the $b\left(S\right)^2 dt$ term.

Paul Wilmott's books are absolutely great, but they are definitely not the right ones to take for mathematical rigor ;) if you want a good finance textbook with a fair stochastic analysis introduction, I recommend Martingale Methods in Financial Modelling from Musiela and Rutkowski.

| improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.