Negative Implied Asset Volatility in the Merton Default Model
Summary
The document considers negative annual asset-volatility estimates produced while fitting the Merton structural default model to firm data. Since volatility is nonnegative by definition, such estimates point to an estimation issue or a model fit that cannot represent the observed market information under the imposed setup. The responses emphasize that sparse balance-sheet observations and market pricing can make implied asset value and volatility difficult to identify reliably.
One suggested remedy is to estimate the logarithm of volatility and transform it back, which constrains the fitted volatility to remain positive. A fitting routine that enforces positivity is another option, though the cited implementation avoids numerical methods described as unstable. The responses also caution that a negative numerical output should not automatically be treated as a meaningful economic volatility: it may signal a parameterization that allows invalid values, or a case in which the model cannot produce a plausible implied volatility. The discussion offers no code diagnosis or systematic evidence about how frequently this occurs.
Key ideas
- Unconstrained numerical estimation can return negative values for parameters that should be positive.
- Estimating log volatility and transforming back enforces a positive volatility estimate.
- Market prices and infrequent balance-sheet data can make Merton model fits problematic.
- A failed or implausible estimate may reflect model limitations as well as an implementation issue.
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# How to interpret negative asset volatility numerical results in Merton model? # How to interpret negative asset volatility numerical results in Merton model? I am currently working on my thesis where I discuss the Merton default probability model. I have a huge sample of US firms for the period 1990-2010. I use both numerical and complex iterative approach to estimate asset volatility and asset value. I have a problem with the numerical approach because when I estimate asset value and asset volatility (in statistical software R with this code) for some firms in the sample I get a negative annual asset volatility. This does not make sense as something which is result of square root can't be negative, but it could be due estimation in numerical approach. Has anyone come across something like this or what are your thoughts regarding this phenomenon. ## Answer by CodeJockey (score 1) https://quant.stackexchange.com/a/11304 Although I, admittedly, did not go hunting through your code for an error, I have seen this phenomenon before using this model. This model (like all other models) isn't perfect. This is especially true when you can only observe those parameters that come from the balance sheet quarterly. There are scenarios where no asset vol can imply the current market prices. Usually, the is an explanation for this, such as a pending LBO, but sometimes, it's just that investors like the credit and hate the equity so much, that no reasonable vol can be implied. ## Answer by Benjamin Christoffersen (score 0) https://quant.stackexchange.com/a/39491 The estimation method you use places no restriction on the parameters. One solution would be to use the $\log$ of the volatility and backtranform in the estimation function. Alternatively, you can use the R package I have made. See the function `BS_fit`. All methods guarantee a positive volatility. A caveat is that I do not implement the numerical methods as it is unstable.
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