Technology📐Mathematics
TheoryKamil Szczerba

Would it not be won­der­ful if, by look­ing at the mar­ket, we could in­fer the prob­a­bil­ity of the next or­der ? That is the goal of this ar­ti­cle. This is our roadmap:

1. The Kelly cri­te­rion
   What ex­actly we must max­imise
2. Com­put­ing op­ti­mal bets
   How to com­pute the op­ti­mal bet, know­ing the win­ning prob­a­bil­ity
3. Re­verse-com­put­ing prob­a­bil­i­ties
   How to re­verse that com­pu­ta­tion
4. The or­der book
   What pub­lic data to re­verse

The first step is to find what cri­te­rion to op­ti­mise for. We can start with a sim­ple ex­am­ple. You have $1. You can bet any frac­tion $f$ of your money for a lim­ited num­ber of rounds.

• 60% ($p$) of time, your bet will dou­ble.
• 40% ($1 - p$) of the time, you will lose your bet.

What is the op­ti­mal bet? Let us for­malise the prob­lem:

$wealth_{n}=\left(1+f\right)^{wins_{n}}\times\left(1-f\right)^{losses_{n}}$

Where $n$ is the step, and $wins_{n}$ and $losses_{n}$ are the num­ber of wins and losses at that step. In­deed, every win mul­ti­plies your wealth by $1+f$, and every loss, by $1-f$. Per the law of large num­bers, $wins_{n}\approx np$ and $losses_{n}\approx n\left(1 - p\right)$. So:

$wealth_{n}\approx\left(1+f\right)^{np}\times\left(1-f\right)^{n\left(1 - p\right)}$

Here is what $wealth_{n}$ looks like de­pend­ing on the frac­tion $f$:

We see that the op­ti­mal bet does not de­pend on the num­ber of rounds $n$. Since $\ln$ is mo­not­o­n­i­cally in­creas­ing, find­ing the max­i­mum of $wealth$ is equiv­a­lent to find­ing the max­i­mum of $\ln\left(wealth\right)$: $=p\times\ln\left(1+f\right)+\left(1−p\right)\ln\left(1−f\right)$.

That is ex­actly the ex­pec­ta­tion of the log­a­rithm of our wealth! That is the Kelly cri­te­rion: we should max­imise the ex­pected log­a­rithm of our wealth.

Oth­er­wise, if we max­imised the ex­pected wealth, we would bet every­thing, and with every round, the prob­a­bil­ity of bank­ruptcy would tend to 1.

Now that we have a value to max­imise, imag­ine we had the prob­a­bil­ity of wins and losses for any in­vest­ment in the mar­ket (would it not be won­der­ful?). Could we find the op­ti­mal bets? For ex­am­ple:

• You have one dol­lar. You can bet on two coins:
• You can bet any frac­tion $f_{0}$ on a coin with prob­a­bil­ity $p_{0} = 0.6$.
• You can bet any frac­tion $f_{1}$ on a coin with prob­a­bil­ity $p_{1} = 0.5$.

Let us com­pute the Kelly frac­tions:

Run­ning this code will down­load ~30MB of com­pressed de­pen­den­cies.

Per this code, we should bet 20% on the first coin and 0% on the sec­ond (the right value for the 60% coin, if you look at the graph above). You can fid­dle with the val­ues of $p_{i}$ to see what hap­pens.

Even bet­ter now: can we find the prob­a­bil­i­ties, know­ing the op­ti­mal frac­tions? Let us re­mem­ber that $f_{i}$ should max­imise the ex­pec­ta­tion, so its de­riv­a­tive should be zero:

It works! In other words, when we know the frac­tions of the op­ti­mal bets, we can in­fer the ini­tial prob­a­bil­i­ties of win­ning.

Once a price in­te­grates an in­for­ma­tion, it is too late to gain profit from it. In other words, the mar­ket re­flects past in­for­ma­tion. But how can we pre­dict the fu­ture?

Let us ex­am­ine how a mar­ket de­ter­mines a price. Imag­ine that cur­rently, 1 BTC = 1 USD, and you be­lieve that the BTC will fall, then rise. You can set an or­der to buy BTC. It re­quires a unit price (in USD) and a quan­tity (how much BTC). The goal is to pre­dict the prob­a­bil­ity of the next spot or­der.

• If you ask for 1 BTC for 0.9 USD, but some­one is ready to sell for 0.8 USD, you have wasted money.
• How­ever, if you ask for 0.7 USD, and no­body is ready to sell for that price, then you have locked your money for noth­ing.

So, you want to pre­dict that is not too high, nor too low. And since you may con­sider many pos­si­ble prices, you will spread your money across them. That is what traders do: pre­dict the prob­a­bil­ity for all pos­si­ble prices, then set or­ders. The record of such or­ders is called the or­der book, and it looks like this:

This is the cur­rent real-time or­der book for BTC/​USD, with data kindly pro­vided by Bi­nance.

That is ex­actly what we are look­ing for: each one of these bars is a bet about fu­ture or­ders, and by ap­ply­ing our strat­egy from above, we may find the prob­a­bil­ity for any price to be the next or­der.

The last step, de­vel­op­ing the bot who will au­to­mat­i­cally in­vest op­ti­mally, is left to you! How­ever, keep in mind that this ar­ti­cle is merely a math­e­mat­i­cal mus­ing, and not fi­nan­cial ad­vice.