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.