# Maximum Drawdown (Binomial tree)

• kk007
In summary, the gambler has a 0.6 probability of winning one unit of capital and a 0.4 probability of losing one unit of capital in 30 trials. If he loses all 10 golden coins before the end of the 30 trials, his attempt is over.

#### 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.

Problem solved!

kk007 said:
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.

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

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.

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

## What is Maximum Drawdown (Binomial tree)?

Maximum Drawdown (Binomial tree) is a risk measurement tool used in finance to calculate the maximum loss that an investment portfolio can experience from its peak value to its lowest value. It is often represented as a percentage of the portfolio's peak value.

## How is Maximum Drawdown (Binomial tree) calculated?

The Maximum Drawdown (Binomial tree) is calculated by taking the difference between the peak value of an investment portfolio and its lowest value, divided by the peak value. This is then expressed as a percentage.

## What is the significance of Maximum Drawdown (Binomial tree)?

Maximum Drawdown (Binomial tree) is a useful tool for investors and fund managers as it allows them to assess the potential risks associated with a particular investment portfolio. It can also help in determining the appropriate level of risk for a given portfolio.

## What are the limitations of Maximum Drawdown (Binomial tree)?

One limitation of Maximum Drawdown (Binomial tree) is that it only considers the historical performance of an investment portfolio and does not take into account future market conditions. It also does not factor in the timing of the losses, which can be important for investors with a specific time horizon.

## How can Maximum Drawdown (Binomial tree) be used in portfolio management?

Maximum Drawdown (Binomial tree) can be used as a risk management tool in portfolio management. It can help investors and fund managers to identify potential risks and adjust the portfolio accordingly to minimize losses. It can also be used to compare the performance of different investment portfolios.