## A One-Line Viewpoint on Trading

Given a time-series of one security’s price-train P[t], a low-frequency trader’s job (forgetting trading costs) is to find a step function S[t] to convolve against price changes P′[t]

$\large \dpi{200} \bg_white \int^{\tau}_0 dP_t \ast S_t \ dt \quad = \quad \text{profit}$

with the proviso that the other side to the trade exists.

S[t] represents the bet size long or short the security in question. The trader’s profit at any point in time τ is then given by the above definite integral.


• I haven’t seen anyone talk this way about the problem, perhaps because I don’t read enough or because it’s not a useful idea. But … it was a cool thought, representing a >0 amount of cogitation.
• This came to mind while reading a discussion of “Monkey Style Trading” on NuclearPhynance. My guess is that monkey style is a Brownian ratchet and as such should do no useful work.
• If I were doing a paper investigating the public-welfare consequences of trading, this is how I’d think about the problem.

Each hedge fund / central bank / significant player is reduced to a conditional response strategy, chosen from the set of all step functions uniformly less than a liquidity constraint. This endogenously coughs up the trading volume which really should be fed back into the conditional strategies.
• Does this viewpoint lead to new risk metrics?
• Should be mechanical to expand to multiple securities. Would anything interesting come from that?

I wouldn’t usually think that multiplication of functions has anything to do with trading. Maybe some theorems can do a bit of heavy lifting here; maybe not.

It at least feels like an antidote to two wrongful axiomatic habits. For economists who look for real value, logic, and Information Transmission, it says The market does whatever it wants, and the best response is a response to whatever that is. For financial engineering graduates who spent too long chanting the mantraμ dt + σ dBt" this is just another way of emphasising: you can’t control anything except your bet size.

UPDATE: Thanks to an anonymous commenter for a correction.

5 notes

1. isomorphismes posted this