Ito’s product and quotient rules are a corollary of the Ito lemma, and are one of the most important parts of the stochastic-calculus toolkit.
When I first started working as a quant I managed to find an alternative form for the rules which sits well in a Black-Scholes type of world and corresponds more closely to a trader’s way of describing a trade.
My versions were often useful to me in terms of developing an intuitive understanding of the risk positions in the derivatives trading book (eg for quanto-ed or payment-delay trades).
Firstly, the standard way to write the product rule is something like:
If we instead write this in terms of the relative changes (eg dX/X or dY/Y) then we get:
At this point I read this expression in terms of risks:
we are long both X and Y, and are long correlation.
Explanation: we are effectively saying that the expectation of the product will increase as X and Y increase (or rather their starting points for our diffusion increase), and will also increase if we had a higher correlation between the two variables that diffuse.
The same trick works for the quotient rule:
I would read this as saying:
we are long X, and short Y
we are long vol of Y,
we are short correlation.
So here are two alternative ‘trader-like’ wordy versions of the two Ito rules:
Ito product rule: we buy correlation when we have a product
Ito quotient rule: we sell correlation when we have a ratio, and we are long vol of the denominator.