Reconciling Drawdown Probability Models for Gaussian Returns

Summary

The document compares a drawdown survival formula attributed to John Wolberg with results from two research papers on drawdowns. Wolberg’s expression gives the probability of a drawdown at least as large as a specified percentage, using the drift and variance of a Gaussian return process. The questioner notes that the book provides little derivation beyond a reference to a private communication.

The document also cites a drawdown density from Bouchaud and coauthors and an exponential density from Zhong and Maslov. Integrating the latter yields a survival expression that appears different from Wolberg’s formula, prompting questions about how the results relate and how Wolberg’s equation was derived. No reconciliation or derivation is supplied. The source is therefore useful as a focused theoretical question, but it does not establish that the formulas describe identical assumptions or processes; resolving that requires examining the papers’ definitions and model conditions.

Key ideas

• Wolberg’s cited formula expresses the probability of a drawdown exceeding a threshold using drift and return variance.
• The document compares that formula with drawdown distributions reported in two research papers.
• Integrating the exponential density cited from Zhong and Maslov appears to produce a different survival form.
• The document asks how the formulas can be reconciled and how Wolberg’s expression was derived.
• It provides no resolution, and equivalence would depend on the assumptions and definitions used by each source.

Tags

# Drawdown distribution mentioned in Expert Trading Systems (John Wolberg)


# Drawdown distribution mentioned in Expert Trading Systems (John Wolberg)












In equation 2.13 of Chapter 2 (pg. 41), in his book "Expert Trading Systems: modeling financial markets with kernel regression," John Wolberg writes the probability of drawdown of $P$ percentage units or greater as $(1-\frac{P}{100})^\frac{2\mu}{\sigma^2}$, where $\mu$ and $\sigma^2$ are the first and second central moments of a Gaussian process for returns.

There is no run-up to this statement, only a reference (in the footnotes) to a private communication by P. Feigin, which I am unable to find online.

Bouchaud et al. obtain a density function for the drawdown, in equation 13 of "You are in a drawdown. When should you start worrying?" (link https://arxiv.org/pdf/1707.01457.pdf). However, this is expressed as a fearsome marginal distribution, integrating out the drawdown length, and I wonder if it will lead to the Wolberg survival function. I am unable to coax it there.

Zhong and Maslov in "Probability distribution of drawdowns in risky investments" (link https://arxiv.org/pdf/cond-mat/9808295.pdf) as far as I can understand, show that the density function for the same process is $f(x)=\frac{2\mu}{\sigma^2}e^{\frac{-2\mu}{\sigma^2}x}$. Again, I cannot relate this to Wolberg's result,since $\int_\frac{P}{100}^\infty f(x).dx=e^{-\frac{2\mu}{\sigma^2}(\frac{P}{100})}$.

My questions are: (1) can anyone help me reconcile all three statements, please? and (2) how does Wolberg get there? the other two articles are adequately clear in development.

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