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# John C. Hull, Izzy Nelken and Alan D. White

company is a European call option on the assets of the company with maturity T and a strike price equal to the face value of the debt. The model can be used to estimate either the risk-neutral probability that the company will default or the credit spread on the debt.^{1 }

As inputs, Merton’s model requires the current value of the company’s assets, the volatility of the company’s assets, the outstanding debt and the debt maturity. One popular way of implementing his model estimates the current value of the company’s assets and the volatility of the assets from the market value of the company’s equity and the equity’s instantaneous volatility using an approach suggested by Jones, Mason and Rosenfeld (1984). A debt maturity date is chosen and debt payments are mapped on to a single payment on the debt maturity date in some way.

In this paper we develop a new way of implementing Merton’s model. This is based on use of the implied volatilities of options on the company’s stock to estimate model parameters. Our approach is interesting both because it provides an alternative to Jones, Mason and Rosenfeld (1984) and because it gives insights into the linkages between credit markets and options markets.

Under Merton’s model an option on the equity of a company is a compound option on the company’s assets. Geske (1979), who provides a valuation formula for compound options, also shows that Merton’s model is consistent with the type of volatility skew observed in equity markets.^{2 }In this paper we carry Geske’s analysis one stage further to show that the credit spread in Merton’s model can be calculated from the implied volatilities of two equity options. The options we choose are two-month at-the-money and out-of-the money put options.

To test our implementation of Merton’s model and compare it with the more traditional approach to implementation we use credit default swap (CDS) spread data. A CDS is a derivative that protects the buyer against default by a particular company. The CDS spread is the amount paid for protection and is a direct market-based measure of the company’s credit risk. Most previous researchers have used bond data to test implementations of Merton’s model. Using CDS spreads is an attractive alternative. Bond prices have the disadvantage that they are often indications rather than firm quotes. Also, the credit spread calculated from a bond price depends on the bond’s liquidity and involves an assumption about the benchmark risk-free rate.^{3 }

1 A number of authors, such as Black and Cox (1976), Geske (1977), Longstaff and Schwartz (1995), Leland and Toft (1996) and Collin-Dufresne and Goldstein (2001), have developed interesting extensions of Merton’s model, but none has emerged as clearly superior. See Eom, Helwege and Huang (2004), who compare the performance of alternative models using bond spreads. Gemmill (2002) shows that Merton’s model works well in the particular case where zero-coupon bonds are used for funding.

2 As the strike price of an equity option increases its volatility decreases. See Rubinstein (1994) and Jackwerth and Rubinstein (1996) for a discussion of this.

3 A counterargument here is that the CDS market is not as well developed as the bond market. Players sometimes come to the market seeking unidirectional execution rather than asking for a bid and an offer.

# Journal of Credit Risk

Volume 1/Number 1, Winter 2004/05