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.

  • 1
    $\begingroup$ I think it would be more helpful if you were to post your code and link to the data. Without knowing what calculations you did, help is limited. $\endgroup$
    – rocinante
    May 15, 2014 at 8:55
  • $\begingroup$ here is the link R code $\endgroup$ May 15, 2014 at 8:57

2 Answers 2


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.

  • $\begingroup$ I totally agree with you the thing is that many paper reports summary statistics and they get appropriately defined quantiles of asset volatility distribution i.e. no negative asset volatilities. Do you think in my cases it could be directly related to sample I use $\endgroup$ May 15, 2014 at 12:48
  • $\begingroup$ You may also want to check to make sure you are expressing data in the right terms. Balance sheet data is usually in M$, CDS Spreads are in bps Notional, CDS Upfront quotes are in pct Notional etc. $\endgroup$
    – CodeJockey
    May 15, 2014 at 15:09

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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