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Figure 12.4: Equity Value as Function of Asset Value when Face Value of Debt is $50

Figure 12.4: Equity Value as Function of Asset Value when Face Value of Debt is $50

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Equity is a Call Option on the Assets of

the Firm



15



• The BSM formula can be used to value the equity in the firm

in the Merton model

• Assuming that asset volatility, σA, and the risk-free rate, rf ,

are constant, and assuming that the log asset value is

normally distributed we get the current value of the equity to

be



• where



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



Equity is a Call Option on the Assets

of the Firm



16



• Note that the risk-free rate, rf , is not the rate earned on the

company’s debt; it is instead the rate earned on risk-free

debt that can be obtained from the price of a government

bond

• Investors who are long options are long volatility

• The Merton model therefore provides the additional insight

that equity holders are long asset volatility

• The option value is particularly large when the option is atthe-money; that is, when the asset value is close to the face

value of debt

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



Equity is a Call Option on the Assets

of the Firm



17



• In this case if the manager holds equity he or she has an

incentive to increase the asset value volatility (perhaps by

taking on more risky projects) so as to increase the option

value of equity

• This action is not in the interest of the debt holders as we

shall see now



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



Corporate Debt is a Put Option Sold



18



• The simple accounting identity states that the asset value must

equal the sum of debt and equity at any point in time and so we

have



• where we have used the option payoff on equity described

earlier.

• We use Dt+T to denote the market value of the debt at time

t+T.

• Solving for the value of company debt, we get

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



Figure 12.5: Market Value of Debt as a Function of

Asset Value when Face Value of Debt is $50



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



19



20



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



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Figure 12.4: Equity Value as Function of Asset Value when Face Value of Debt is $50

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