Approximating Maximum Drawdown for Drifted Brownian Motion

Summary

The document considers the expected maximum drawdown of a Brownian motion with constant drift and volatility over a fixed horizon. Drawdown is defined as the gap between the running peak and the process value, and the maximum drawdown is the largest such gap during the period. The answer points to theoretical work that approximates the distribution through a series approximation for its density.

It also identifies an R package with functions for the maximum-drawdown density, cumulative distribution, random generation, and summary statistics, which can be used to obtain an expected value. The discussion gives no derivation, approximation error, numerical example, or comparison with simulation. Its guidance is therefore a pointer to a model-specific analytical approximation and software; applicability depends on whether the constant-drift Brownian assumptions are suitable for the process being studied.

Key ideas

• Maximum drawdown is the largest distance from the running maximum over the chosen horizon.
• The model assumes Brownian motion with constant drift and volatility.
• A series approximation to the maximum-drawdown density is cited as a way to estimate its distribution.
• The referenced R package includes distribution and summary-statistic functions.
• The note provides no accuracy assessment or extension to more realistic price dynamics.

Tags

# Expectation of maximum draw down in the Brownian motion case


# Expectation of maximum draw down in the Brownian motion case












Let $$ X_t = \mu t + \sigma B_t $$ be a linear Brownian motion with drift. Let $$ S_t = \max(X_u, u \le t) $$ denote the process of the running max, then the draw down is given by $$ DD_t = S_t - X_t, $$ and the maximum draw down over a period $[0,T]$ is $$max_{u \in [0,T]} DD_u.$$ What can we say about $$E[ max_{u \in [0,T]} DD_u ] ?$$ How can we calculate the expected maximum draw down? Are there analytical formulas, approximations, available (R) packages?

## Answer by vanguard2k (score 3, accepted)

https://quant.stackexchange.com/a/17598

as I mentioned here, this paper provides some theoretical insight (and a way to approximate the true value).

The authors end up with an approximative series for the density. It is implemented in the function maxdd of the R-package fBasics. There are convenient functions dmaxdd, pmaxdd and rmaxdd. Calculating the Expected Drawdown should be easy. (to be honest, I found the paper as a reference provided on the help page of the functions mentioned above)

The function you are asking for would be maxddStats:

```
require(fBasics)
maxddStats(mu,sigma,t)
```

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.