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
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• 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
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• 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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