Maximum Drawdown (Binomial tree)

[kk007]

Imagine there is a game and a gambler has a prob. of P1 in winning one unit of capital in a trial and 1 - P1 in lossing one unit.

He wants to know the prob. of HAVING EVER lost more than or equal to a threshold no. of units (drawdown threshold) at or before the end of a number of trials.

For example, if his prob. winning 1 unit is 0.6 and lossing 1 unit is 0.4, what is his probability of having ever lost more than or equal to 10 units at or before the end of 30 trials?

To illustrate the "having ever" concept a bit more, imagine the gambler has 10 golden coins, at any time-step, he uses 1 golden coin for gambling, if he has lost all of them before the end of the 30 trials, his attempt is over.

Thanks in advance!

 

[kk007]

Problem solved!

Imagine there is a game and a gambler has a prob. of P1 in winning one unit of capital in a trial and 1 - P1 in lossing one unit.

He wants to know the prob. of HAVING EVER lost more than or equal to a threshold no. of units (drawdown threshold) at or before the end of a number of trials.

For example, if his prob. winning 1 unit is 0.6 and lossing 1 unit is 0.4, what is his probability of having ever lost more than or equal to 10 units at or before the end of 30 trials?

To illustrate the "having ever" concept a bit more, imagine the gambler has 10 golden coins, at any time-step, he uses 1 golden coin for gambling, if he has lost all of them before the end of the 30 trials, his attempt is over.

Thanks in advance!

 

[techmologist]

Awesome. Did you solve it using a generating function or with the Inclusion/Exclusion principle? That seems like a hard problem.

 

[techmologist]

I just now learned of the technique for counting random walk paths called the reflection principle, or method of images. It simplifies things a lot. Perhaps that's how you solved it so quickly.

 

[blue_raver22]

The concept of Maximum Drawdown in a binomial tree scenario is a useful tool for evaluating the risk associated with a particular game or gambling strategy. In this case, we are looking at a game where the gambler has a probability of P1 in winning one unit of capital in a trial and a probability of 1 - P1 in losing one unit.

The question posed is what is the probability of the gambler ever losing more than or equal to a certain threshold number of units at or before the end of a given number of trials. This can be interpreted as the likelihood of the gambler reaching a point where they have lost a significant amount of their initial capital, and potentially having to end their attempts at the game.

To calculate this probability, we can use the binomial distribution formula, where n is the number of trials and p is the probability of success (winning one unit) in each trial. In this case, n=30 and p=0.6. We then need to consider the different outcomes that can lead to the gambler reaching the threshold of 10 units lost.

One possible outcome is that the gambler loses 10 units in the first 10 trials, and then wins the remaining 20 trials. This can be calculated using the binomial distribution formula as:

P(X=10) = (30 choose 10) * (0.6)^10 * (0.4)^20 = 0.00149

However, there are also other possible outcomes that can lead to the gambler reaching the threshold. For example, they could lose 9 units in the first 9 trials, win the 10th trial, lose 1 unit in the 11th trial, and then win the remaining 19 trials. This can be calculated as:

P(X=9) * P(X=1) = (30 choose 9) * (0.6)^9 * (0.4)^21 * (0.6)^1 * (0.4)^0 = 0.00406 * 1 * 1 = 0.00406

We would need to consider all possible combinations of trial outcomes that can lead to the gambler reaching the threshold of 10 units lost. This can be a complex calculation, but it ultimately depends on the specific rules and outcomes of the game being played.

In summary, the probability of the gambler ever losing more than or equal to 10 units at or