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Figure 12.5: Market Value of Debt as a Function of Asset Value when Face Value of Debt is $50

Figure 12.5: Market Value of Debt as a Function of Asset Value when Face Value of Debt is $50

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Corporate Debt is a Put Option Sold

• Figure 12.5 shows the payoff to the debt holder of the firm as

a function of the asset value At+T when the face value of debt

D is $50

• Comparing Figure 12.5 with the option payoffs we see that

the debt holders look as if they have sold a put option

although the out-of-the-money payoff has been lifted from 0

to $50 on the vertical axis corresponding to the face value of

debt in this example



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Corporate Debt is a Put Option Sold



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• Figure 12.5 suggests that we can rewrite the debt holder

payoff as



• which shows that the holder of company debt can be

viewed as being long a risk-free bond with face value D

and short a put option on the asset value of the company,

At+T , with a strike value of D

• We can therefore use the model to value corporate debt;

for example, corporate bonds

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Corporate Debt is a Put Option Sold



22



• Using the put option formula from Chapter 10 the value

today of the corporate debt with face value D is



• where d is again defined by

• The debt holder is short a put option and so is short asset

volatility

• If the manager takes actions that increase the asset

volatility of the firm, then the debt holders suffer because

the put option becomes more valuable

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Implementing the Model



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• Stock return volatility needs to be estimated for the BSM

model to be implemented

• In order to implement the Merton model we need values for

σA and At, which are not directly observable

• In practice, if the stock of the firm is publicly traded then

we do observe the number of shares outstanding and we

also observe the stock price, and we therefore do observe Et



• where NS is the number of shares outstanding

Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen



Implementing the Model

From the call option relationship earlier we know that Et is

related to σA and At via the equation



• This gives us one equation in two unknowns

• We need another equation

• The preceding equation for Et implies a dynamic for the

stock price that can be used to derive the following

relationship between the equity and asset volatilities:



• where σ is the stock price volatility

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



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Implementing the Model



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• The stock price volatility can be estimated from historical

returns or implied from stock option prices

• We therefore now have two equations in two unknowns, At

and σA

• The two equations are nonlinear and so must be solved

numerically using, for example, Solver in Excel

• Note that a crucially powerful feature of the Merton model is

that we can use it to price corporate debt on firms even

without observing the asset value as long as the stock price is

available

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



The Risk-Neutral Probability of

Default



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• The risk-neutral probability of default in the Merton model

corresponds to the probability that the put option is exercised

• It is simply



• Note that this probability of default is constructed from risk

neutral distribution of asset values and so it may well be

different from the actual physical probability

• The physical default probability could be derived in the

model but would require an estimate of the physical growth

rate of firm assets.

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



The Risk-Neutral Probability of

Default



27



• Default risk is also sometimes measured in terms of distance

to default, which is defined as



• The interpretation of dd is that it is the number of standard

deviations the asset value must move down for the firm to

default

• As expected, distance to default is increasing in the asset

value and decreasing in the face value of debt

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



The Risk-Neutral Probability of

Default

• The distance to default is also decreasing in the asset

volatility

• Note that the probability of default is



• The probability of default is therefore increasing in asset

volatility

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



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Portfolio Credit Risk



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• The Merton model gives powerful intuition about corporate

default and debt pricing

• It enables us to link the debt value to equity price and

volatility, which in the case of public companies can be

observed or estimated

• While much can be learned from the Merton model, we

have several motivations for going further

• First, we are interested in studying the portfolio implications

of credit risk

• Default is a highly nonlinear event and furthermore default

is correlated across firms and so credit risk is likely to

impose limits on the benefits to diversification

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Portfolio Credit Risk

• Second, certain credit derivatives, such as collateralized

debt obligations (CDOs), depend on the correlation of

defaults that we therefore need to model

• Third, for privately held companies we may not have

information necessary to implement the Merton model

• Fourth, even if we have the information needed, for a

portfolio of many loans, the implementation of Merton’s

model for each loan would be cumbersome

• To keep things relatively simple, we will assume a single

factor model similar to the market index model

• For simplicity, we will also assume normal distribution

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



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Figure 12.5: Market Value of Debt as a Function of Asset Value when Face Value of Debt is $50

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