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]**

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 mantra “

**μ 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.

Tags: bet sizing, Brownian ratchet, convolution, economics, finance, financial economics, financial engineering, functional analysis, functionals, integrals, integration, Kelly rule, liquidity, macroeconomics, mathematics, monkey style trading, profit, risk metrics, trading, welfare economics

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