Robert Merton used the option pricing framework to create a model for credit risk. Consider a...

 

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Robert Merton used the option pricing framework to create a model for credit risk. Consider a firm that finances itself with debt and equity. The firm has X amount of debt which comes due at time T. If the firm's net worth V is less than X when the debt is due, the company is liquidated and the shareholders receive nothing. If the firm's net worth V is more than X, the shareholders receive V-X as a payout. This payout structure is identical to the payments that the holders of a call option with strike price X receive. Merton's great insight thus was that the equity in a firm could be valued as a call option where the strike price was the amount of debt the firm had and the expiration date was the maturity date of the debt. This framework is particularly useful to model credit risk. What does it mean if the option has a positive payoff? The firm did well and could pay back its debt. What does it mean if the option has no payoff? The firm did not do well and could not pay back its debt in full - it defaulted! Therefore, to estimate the default risk, we can estimate the likelihood of the payoff of a call option. In our equation for the value of a call option where we plug in V as the firm's net worth, c=VxN(d,) − Xe-rT x N(d2), N(d₁) is the probability of payoff, and thus 1 - N(d₁) is the probability of no payoff. Consider two firms, ABC and XYZ. Today, both firms have debt in the amount of $0.9m. The debt is due in 3 years. The risk-free rate is 2% per year. The volatility of the net worth of firm ABC is 20% per year, and the volatility of the net worth of firm XYZ is 40% per year. a) Compute the probability of default, that is, the probability of no payoff for both firms today, one year from today and two years from today. Assume that the value of the assets for both firms is $1m today, $1.1m one year from today, and $0.8m two years from today. b) Which firm is riskier today, one year from today, and two years from today given the asset values in a)? Explain. Robert Merton used the option pricing framework to create a model for credit risk. Consider a firm that finances itself with debt and equity. The firm has X amount of debt which comes due at time T. If the firm's net worth V is less than X when the debt is due, the company is liquidated and the shareholders receive nothing. If the firm's net worth V is more than X, the shareholders receive V-X as a payout. This payout structure is identical to the payments that the holders of a call option with strike price X receive. Merton's great insight thus was that the equity in a firm could be valued as a call option where the strike price was the amount of debt the firm had and the expiration date was the maturity date of the debt. This framework is particularly useful to model credit risk. What does it mean if the option has a positive payoff? The firm did well and could pay back its debt. What does it mean if the option has no payoff? The firm did not do well and could not pay back its debt in full - it defaulted! Therefore, to estimate the default risk, we can estimate the likelihood of the payoff of a call option. In our equation for the value of a call option where we plug in V as the firm's net worth, c=VxN(d,) − Xe-rT x N(d2), N(d₁) is the probability of payoff, and thus 1 - N(d₁) is the probability of no payoff. Consider two firms, ABC and XYZ. Today, both firms have debt in the amount of $0.9m. The debt is due in 3 years. The risk-free rate is 2% per year. The volatility of the net worth of firm ABC is 20% per year, and the volatility of the net worth of firm XYZ is 40% per year. a) Compute the probability of default, that is, the probability of no payoff for both firms today, one year from today and two years from today. Assume that the value of the assets for both firms is $1m today, $1.1m one year from today, and $0.8m two years from today. b) Which firm is riskier today, one year from today, and two years from today given the asset values in a)? Explain.

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