Consider 2 related aspects of a process for prices in a financial market:
- time &
Say I've identified an exponential distribution that reasonably models the occurrence of the lengths of price trends (how long in time the price moves up or down) in this financial market. Given the length of a given trend up to some point in time (i.e., now), one can readily calculate the mean residual life of the trend (the average residual lifetime of the trend given that the trend has survived up to a given time point)-- essentially conditional expectation.
I can then take the next step and calculate the conditional probability of reaching that mean residual life.
Separately, I have another distribution which models returns (a bit wonkier than the exponential distribution for the trends, but maybe useful).
I can do a similar set of calculations for returns as I've done for the length of the trend. Given the current return within the current trend I can calculate the mean residual life for the return and the conditional probability that the current move will reach that target.
What to do with the above?
Lots of extenuating issues and questions (i.i.d?, stable distribution?,...etc.) but just keeping it simple for now…
Take this probability of reaching the return's mean residual life to scale a bet similar to the way black jack players do:
Think of "payoff" as what one stands to make if you reach the mean residual life from where you currently stand. So basically you do the following:
(probability * payoff - (1- probability)) / payoff
Something like the above (derived via various paths from John Kelly's "A New Interpretation of Information Rate"; I think I've got the basic formula right but if anyone has a better take on this please advise) tells you how much of your capital (or bankroll) to bet on the opportunity to optimize the geometric return on your bets (investments) over the long run.
What about the "time" information?
But, I also have this information about "time" and the probability that a trend will last a certain amount of time.
Intuitively, longer trends ought to have greater returns than shorter tends. Trend length and return give different views of the same process or perhaps describe different but related aspects of the same process. Again, intuitively the two distributions should have some relationship.
Can I use this time information together with the return information to better calculate risk and bet size?
How can I define or identify the relationship between time and returns (would this be useful)?
Can I combine the probabilities of reaching the mean residual lives of both returns and trends to get something more reliable or more robust than just scaling bets with returns information?
How would one go about doing this or even thinking about it?
The time analysis seems a bit tricky to use because while related to returns, its not measured in the same units.
Just seems like an interesting question. Looking forward to the community's thoughts.
Just a follow up, it occurred to me that joriki's answer in A Question about Poker (and probability in general) might have some bearing on my question. Just a gut feeling.
A quick footnote, from wikipedia's entry for Volatility:
For a financial instrument whose price follows a Gaussian random walk, or Wiener process, the width of the distribution increases as time increases. This is because there is an increasing probability that the instrument's price will be farther away from the initial price as time increases. However, rather than increase linearly, the volatility increases with the square-root of time as time increases, because some fluctuations are expected to cancel each other out, so the most likely deviation after twice the time will not be twice the distance from zero.
But prices don't follow a Gaussian or Wiener process.
So, how do I establish the relationship for my distributions?