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.

Mathematician (PhD in Probability Theory).
Art lover (spent one excellent year studying painting and ceramics at Batley Art College).
Ex investment banker (2yrs of fixed-income exotics trading, 5 yrs of quantitative research, 2 yrs of inflation structuring).
Now busy as a quantitative software developer.
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2 thoughts on “Ito’s product and quotient rules as described by a trader”

Reblogged this on Human Mathematics.

Thanks.