Merton’s model, credit risk and volatility skews

sample size to 319 firms. For this sample, the implied volatility for the two- month 50-delta put and the 25-delta put were downloaded for every trading day in the year 2002. This produced 61,544 observations.

In the Bloomberg system a two-month option is defined as an option with a maturity of between one and two months. In most circumstances the exact maturity date of the option can be calculated from knowledge of the maturity dates of options traded on the Chicago Board Option Exchange. The two-month 25-delta put implied volatility is an estimate of the implied volatility of a two- month put option that has a delta of – 0.25. This is calculated by interpolating between the implied volatility of the two two-month options whose deltas are closest to – 0.25, one greater than – 0.25 and the other less than – 0.25. The implied volatility for a two-month 50-delta put is calculated similarly. We chose the 50-delta and 25-delta implied volatilities because they are usually calculated from relatively liquid options.^{11 }

# 3.3 Data for traditional implementation

For the traditional approach to implementing Merton’s model we downloaded daily closing stock prices and quarterly balance sheet information from Bloomberg for the period between January 1, 2001, and December 31, 2002. We used the reported quarterly total liabilities divided by the quarterly reported outstanding number of shares as our measure of the debt claim, D. On any particular day the current value of the debt claim was set to the most recently reported value. The equity price and the instantaneous equity volatility used in Equations (1) and (2) were the daily closing price and a historic volatility estimated using the most recent 40 returns. The choice of 40 business days to estimate volatility is a trade-off. Using a longer (shorter) period to estimate volatility results in more (less) accurate but less (more) timely estimates. The total number of observations was 63,627.

# 3.4 Merged data

We merged the volatility data and the data used in the traditional implementation with the CDS data. This combined data set was then filtered to eliminate cases in which the volatility exceeded 90% (65 cases) and firms for which there were 10 or fewer implied volatility observations or CDS quotes. This resulted in a pool of 127 firms, each with between 11 and 178 days of data for a total of 6,220 firm days of data. In other words we ended up with 6,220 cases where, for a particular company on a particular day, we had (1) a CDS spread observation, (2) a two-month 50-delta put implied volatility, (3) a two-month 25-delta put implied volatility, (4) an equity value, (5) a debt claim, and (6) a historic equity volatility estimate. These are the data that were used for the analysis described in the following sections.

11 Bloomberg implied volatilities are sometimes criticized for the way they handle items such as dividends, bid–offer spreads and American exercise features. Our objective is to test the ability of the model to rank order credit spreads. Any consistent biases in Bloomberg’s implied volatility estimates may not therefore be important.

## Research papers

www.journalofcreditrisk.com

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