Merton’s model, credit risk and volatility skews

Equations (8) and (9) define an implicit relationship between the implied volatil- i t y o f a n o p t i o n a n d t h e m o n e y n e s s , κ , f o r a s e t o f m o d e l p a r a m e t e r v a l u e s L , σ A and T, and the option maturity, τ. For different values of κ different implied volatilities, v, will result, leading to a volatility skew. For all values of the model parameters, the implied volatilities are of the form observed in practice where an increase in the strike price leads to a reduction in the implied volatility. ,

# 2.4 An alternative implementation of Merton’s model

Section 2.3 suggests a new way of implementing Merton’s model using two implied volatilities. With one implied volatility we can solve Equations (8) and (9) for a particular value of T to obtain a relationship between the leverage ratio, L , a n d t h e a s s e t v o l a t i l i t y , σ A . W i t h t w o i m p l i e d v o l a t i l i t i e s w e h a v e t w o s u c h r e l a - t i o n s h i p s t h a t c a n b e s o l v e d f o r L a n d σ A . T h e c r e d i t s p r e a d f o r a z e r o - c o u p bond maturing at time T can then be calculated using Equation (5). o n

This implementation approach allows credit spreads to be estimated directly from implied volatility data. It is a potentially attractive alternative to the tradi- tional implementation based on Equations (1) and (2) because it avoids the need to estimate the instantaneous equity volatility and the need to map the company’s liability structure (some of which may be off balance sheet) on to a single zero- coupon bond.

In the sections that follow we will compare the results obtained from our implementation of Merton’s model with the traditional implementation. As a benchmark we will also examine the performance of a simpler model where a Poisson process generates defaults.

# 3 DATA

Our empirical tests are based on credit default swap data, implied volatility data, equity price data and balance sheet data for companies between January and December, 2002.

# 3.1 Credit default swap data

A credit default swap or CDS provides insurance against a default by a particular company or sovereign entity. The company is known as the reference entity and a default by the company is known as a credit event. The buyer of the insurance makes periodic payments to the seller and in return obtains the right to sell a bond issued by the reference entity for its face value if a credit event occurs. The amount of the payments made per year by the buyer is known as the CDS spread.^{7,8 }The credit default swap market has grown rapidly since the International Swaps

7 In a standard contract, payments by the buyer are made quarterly or semiannually in arrears. If the reference entity defaults, there is a final accrual payment and payments then stop. Contracts are sometimes settled in cash rather than by the delivery of bonds. In this case there is a calculation agent who has the responsibility of determining the (continued on next page)

## Research papers

www.journalofcreditrisk.com

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