Ito’s product and quotient rules as described by a trader

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:

\text{d}(XY) = Y \, \text{d}X + X \, \text{d}Y + \text{d}X \, \text{d}Y

If we instead write this in terms of the relative changes (eg dX/X or dY/Y) then we get:

\frac{\text{d}(XY)}{XY} = \frac{\text{d}X}{X} + \frac{\text{d}Y}{Y} + \frac{\text{d}X}{X} \, \frac{\text{d}Y}{Y}

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:

\frac{\text{d}(X/Y)}{X/Y} = \frac{\text{d}X}{X} - \frac{\text{d}Y}{Y} + \left(\frac{\text{d}Y}{Y}\right)^2 - \frac{\text{d}X}{X} \, \frac{\text{d}Y}{Y}

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.


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