What is your strategy?

[nonlinear5]

I need the same type of rigorous mathematical proof that I supplied to you, not just speculation.

See here: http://www.bjmath.com/bjmath/thorp/paper.htm, equation 7.4.

Ln(r) makes no sense to me. Why do it, if we already have r?

See here: http://quantivity.wordpress.com/2011/02/21/why-log-returns/

The notion of leverage implies a Kelly fraction greater than one. What are the odds of that happening?

Some of my strategies have Kelly-optimal leverage of 180.

 

[kut2k2]

See here: http://quantivity.wordpress.com/2011/02/21/why-log-returns/

There's a huge difference between log(r) and log(1+r).

And the Kelly equation is the result of first taking the logarithm of the gain before taking the differential. It looks like you're conflating a derivation with a final result.

Some of my strategies have Kelly-optimal leverage of 180.

And do you trust this number? And how is this leverage applied? To the bankroll or to the trade size?

 

[nonlinear5]

There's a huge difference between log(r) and log(1+r).

Both can be used, depending on how you calculate the return:
LN(1 + (price2 - price1) / price1) = LN(price2 / price1) = LN(price2) - LN(price1)

 

[kut2k2]

Both can be used, depending on how you calculate the return:
LN(1 + (price2 - price1) / price1) = LN(price2 / price1) = LN(price2) - LN(price1)

Yes I said that.

If r == ln(p2/p1), then what do you mean by ln(r)?

 

[nonlinear5]

Yes I said that.

If r == ln(p2/p1), then what do you mean by ln(r)?

You are becoming defensive, dude.

 

[kut2k2]

You are becoming defensive, dude.

Not at all. Here's what you posted.

One thing that I'd note is that your approximation of Kelly:
k ~ sum[Ri]_n / sum[Ri²]_n

resembles the continuous Kelly:
CK = R / (s^2)

Note that:
-- your nominator sum[Ri] is proportional to R
-- your denominator sum[Ri²] is a component in (s^2)

I think that if you started with ln(r), instead of r, you would have arrived at the true continuous Kelly, which is
CK = R / (s^2)

Nowhere in the link you provided is ln(r) or log(r) mentioned. What is talked about is log(1+r), which is vastly different from log(r), and which I already said is one step in the derivation of the Kelly equation.

The derivation of the Kelly equation is straightforward. There's nothing ambiguous about it.

I am now convinced that the CK formula is just an approximation, and not a very good one.

To be sure, all of the Kelly formulae are approximations for all but the simplest scenarios (binary outcomes, like coin flips). But some approximations are better than others, and I see nothing to recommend the CK formula, especially if it is giving results like a leverage of 180. :eek:

 

[nonlinear5]

I am now convinced that the CK formula is just an approximation, and not a very good one.

So, your approximation is a better one, then?

 

[kut2k2]

So, your approximation is a better one, then?

My approximation doesn't output leverages, especially in the three-digit range.

 

[kut2k2]

A Tale of Two Horses II

Cheeky! Give up? I have solved it, friend!

At long last, so have I.

I tried different ways but got nowhere with my unit-bet analysis. So I went back to the relationship established by Splawndarts and, with the aid of an on-line equation solver, I got the following results:

F == 5.72% and S == 8.43%

Surprisingly, at least to me, both F and S are larger than their respective single-bet Kelly fractions.

Specifically, F is 4.4% larger than k14 and S is 2% larger than k16.

This implies some sort of synergy is created by combining mutually exclusive bets. Any thoughts?

 

[kut2k2]

My Monte-Carlo simulation agrees. Here are the top 10 strategies, if betting on Red is not allowed:

	R16	R14	Score
	8.4	5.7	8.009759
	8.5	5.7	8.009752
	8.4	5.8	8.009728
	8.5	5.8	8.009715
	8.3	5.7	8.009667
	8.4	5.6	8.009662
	8.5	5.6	8.009660
	8.6	5.7	8.009647
	8.3	5.8	8.009641
	8.6	5.8	8.009606

We agree.