Kelly Formula Revisited

Some discussions on Kelly's formula amongst a reader Steven L:

Q:

"I am to a greater extent than than one-half agency through your majority together with am stuck at a concept that I can't seem to honour an reply inwards whatsoever other forum.

I accept read Ralph Vince's "Portfolio Management Formulas," which uses Kelly's formula to calculate an optimal "fraction" of the bankroll to bet on each trial. So a trader tin calculate a fraction of his full trading trouble concern human relationship value to jeopardy on each trade. What I am referring to is the so-called "fixed-fractional" trading. There exists an optimal fraction that volition maximize the geometric increase charge per unit of measurement of the trading equity, inwards theory anyway.

However, inwards the coin management chapter of your book, yous role Kelly's formula to derive an optimal "leverage." This seems to survive inwards conflict amongst what I learned from Ralph Vince, since leverage is commonly keen than unity together with fraction is commonly less than unity. I can't seem to brand a connecter betwixt these 2 concepts. I accept likewise seen the same optimal leverage formula inwards Lars Kestner's Strategies together with asked the same query on only about forums, but no i was able to laissez passer on me a clear satisfactory answer. It would survive greatly helpful if yous tin attention me course out the confusion."

A:

I don't accept Ralph Vince's majority amongst me, but if I think correctly, his formulation is based on discrete bets (win or lose, no intermediate outcome), much similar horse-betting or inwards a casino game. My approach, or rather, Professor Ed Thorp's approach, is based on continuous finance, assuming that every second, your P&L could fluctuatate inwards a Gaussian ("log-normal") fashion.

For discrete bets where yous could accept lost all of your equity inwards i bet, for sure i should solely bet a fraction of your full equity. For continuous finance, in that place is really footling run a jeopardy i could accept lost all of the equity inwards i fourth dimension period, due to the assumed log-normal distribution of prices. Hence i should bet to a greater extent than than your equity, i.e. role leverage.

Q:

In instance 6.2 inwards your book, the portfolio consists of solely long SPY, which has footling run a jeopardy of going to zero. So I tin run into how it is reasonable that yous role the continuous finance approach together with apply the optimal leverage to scale upwards the return.

But let's assume that the portfolio consists of a unmarried strategy that buys options. Suppose this strategy volition lose almost of the fourth dimension due to fourth dimension decay but volition brand net i time inwards a spell due to black-swan events. I don't mean value it's a skillful take in to bet the entire portfolio equity on each merchandise for this strategy. Can yous even hence apply the continuous finance approach inwards this case, since inwards reality trading is similar making discreet bets? Should nosotros hold off the hateful together with variance of this strategy automatically outcome inwards an Optimal Leverage that is less than one? So that nosotros truly demand to jeopardy a fraction of the trouble concern human relationship equity per trade?

A:

The formula I depicted inwards the majority is valid solely if the P&L distributions are Gaussian. If i expects a fat-tailed distribution due to black-swan events, a dissimilar mathematical model needs to survive used, though it tin even hence survive inside the continuous finance framework. However, for simplicity's sake, if the distribution looks multinomial (e.g. high probability of "Win a lot" five "Lose a lot"), together with then yous may model it amongst fractional betting only similar a casino game.