Trading Math - Part I

[abattia]

Quote from intradaybill:

Statement 1:

For any trading system, with any parameters, if there is no target quit time or target profit after which it quits, eventually there will be total ruin.

I invite all the brains here to either prove or disprove Statement 1.

...and learn something...

Thanks for posing the question. Ruin Theory is certainly something we ought to be pretty good at here on ET!! LOL!!

How about the following?

Strategy's equity at time t,
X(t)

Strategy's unrealised profit at time t,
Y(t)

So,
X(t) = X(0) + Y(t) [Equation A]

Assume that generating the set of possible values of Y(t), {Y(t)}t>=0, is a stochastic process, so that {Y(t)}t>=0 is described by some probability function.

Assume there is a finite probability Y(t) can take any value in range -infinity < Y < +infinity, so that
P(-infinity < Y <= X(0)) > 0 [Equation B]

Assume "eventually" means "enough time for all possible outcomes to occur at least once".

Therefore, Equation B applied to Equation A implies "eventually" there will come a time when X(t) <= 0, i.e. "ruin"

With a stop and/or target, or an exit time, Equation B would cease to be valid.

 

[braincell]

Ok I think abattia gave the clearest approximation to an answer to a somewhat unclear question.

Quote from kut2k2:

But I do have adaptive MAs better than anything you'll find in the public domain, for what it's worth. And they're not black boxes I bought from some vendor(s).

Adaptive MAs you say? Funny name. I'd like to know how they're "adaptive"? By definition an MA is supposed to be one thing and nothing else - a moving average. I've made several of my own also, I'm not sure if they qualify as competition to your "adaptive MAs" but they give weight to values with logarithmic, exponential or sqrt decay, and on top of that adjust for volatility (also with various weighing curves available). Is that something similar to what you seem to consider very valuable? Math discussion is much easier when everyone is using the same definitions - hence why definitions are so strictly taught in schools.

 

[DontMissTheBus]

I think the average math illiterate ET'ers are waaay to far beneath you to even get the humor of your statement....

If the strategy returns are, say, iid normal (I don't want to hear from the peanut gallery that nothing is normal; it doesn't matter - normality captures the first order nature of returns; we'll worry about the rest below) with no drift, then yes, you are of course right - your statement is just a rephrasing of the usual proof that rules out arbitrage in equilibrium inter-temporal models.

This is true even if there's a positive drift (that's alpha, for the rest of you)... because the probability of a run of bad realizations goes to 1 as sampling periods go to infinity.

But.... the probability of your NAV being greater than the number of atoms in the universe ALSO approaches 1 as sampling periods go to infinity.

Quote from intradaybill:

Statement 1:

For any trading system, with any parameters, if there is no target quit time or target profit after which it quits, eventually there will be total ruin.

I invite all the brains here to either prove or disprove Statement 1.

...and learn something...

 

[Lucrum]

Quote from kut2k2:

... I do have adaptive MAs better than anything you'll find in the public domain, for what it's worth. And they're not black boxes I bought from some vendor(s).

LOL You paid money...for moving averages?


Hey, you interested in buying a bridge?

 

[intradaybill]

Quote from DontMissTheBus:

I think the average math illiterate ET'ers are waaay to far beneath you to even get the humor of your statement....

If the strategy returns are, say, iid normal (I don't want to hear from the peanut gallery that nothing is normal; it doesn't matter - normality captures the first order nature of returns; we'll worry about the rest below) with no drift, then yes, you are of course right - your statement is just a rephrasing of the usual proof that rules out arbitrage in equilibrium inter-temporal models.

This is true even if there's a positive drift (that's alpha, for the rest of you)... because the probability of a run of bad realizations goes to 1 as sampling periods go to infinity.

But.... the probability of your NAV being greater than the number of atoms in the universe ALSO approaches 1 as sampling periods go to infinity.

Thanks. As far as NAV getting greater than number of atoms in the universe I think we can limit that via liquidity contraints so that doesn't imply the ruin result is also absurd.

Do you agree then that even if someone has an edge if he trades long enough the probability of ruin is inreasing?

 

[intradaybill]

Quote from abattia:

Thanks for posing the question. Ruin Theory is certainly something we ought to be pretty good at here on ET!! LOL!!

How about the following?

Strategy's equity at time t,
X(t)

Strategy's unrealised profit at time t,
Y(t)

So,
X(t) = X(0) + Y(t) [Equation A]

Assume that generating the set of possible values of Y(t), {Y(t)}t>=0, is a stochastic process, so that {Y(t)}t>=0 is described by some probability function.

Assume there is a finite probability Y(t) can take any value in range -infinity < Y < +infinity, so that
P(-infinity < Y <= X(0)) > 0 [Equation B]

Assume "eventually" means "enough time for all possible outcomes to occur at least once".

Therefore, Equation B applied to Equation A implies "eventually" there will come a time when X(t) <= 0, i.e. "ruin"

With a stop and/or target, or an exit time, Equation B would cease to be valid.

Thanks. Does this imply that if I trade a system I must quit at some time T1 otherwise face ruin at some time T2, T2 > T1 hopefully, regardless of edge?

 

[abattia]

Quote from intradaybill:

Thanks. Does this imply that if I trade a system I must quit at some time T1 otherwise face ruin at some time T2, T2 > T1 hopefully, regardless of edge?

For 0 < t1 < t2, for continuous t
P.of.Ruin(t1) < P.of.Ruin(t2)

But obviously, “higher risk of ruin” does not mean “certainty of ruin”. Working against “certainty of ruin” is risk management, etc.

------------
And related … ?
HFT: lower risk trades > intraday: higher risk trades > swing: still higher risk trades
… partly (but not only) because of increased “risk of ruin” the longer the trade

 

[intradaybill]

Quote from abattia:

For 0 < t1 < t2, for continuous t
P.of.Ruin(t1) < P.of.Ruin(t2)

But obviously, “higher risk of ruin” does not mean “certainty of ruin”. Working against “certainty of ruin” is risk management, etc.

Well, I was interested in the certainty of ruin, meaning P of Ruin = 1.

 

[abattia]

Quote from intradaybill:
Well, I was interested in the certainty of ruin, meaning P of Ruin = 1.

OK. If you trade a system (or systems) with no target or time of exit ("even if it has an edge")... then yes, you're better off getting out earlier rather than later.

 

[DontMissTheBus]

Yes.... but only if each trade is in the same unit of risk as all previous trades (for example, a single contract rather than a fixed % of NAV).

But this is a theoretical consequence of the 'trading until infinity' part, no?

Practically, it's only problems with a fixed stopping time (how about 50 years, if we want to think about long-run characteristics) that are interesting. Otherwise, you get entangled in all those asymptotic limits.

Quote from intradaybill:

Do you agree then that even if someone has an edge if he trades long enough the probability of ruin is inreasing?