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Part 3
The Use of Options Pricing for
Estimating Distance to Default
(DD) and Probability of Default(PD)
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Default Risk
Default risk is the uncertainty surrounding a firm’s
ability to service its debts and obligations.
Firms pay a spread over the default-risk free rate of
interest that is proportional to their default probability
to compensate the lenders for this uncertainty.
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Options Pricing
Myron S. Scholes, 1941Fisher Black
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Options Pricing
}{}{)( 21 d N Xed SN t C rt
The expected stock
price at time period t
The expected present value of
the exercise (buying) priceat time period t
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http://www.moodys.com/
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Call Options and Distance to Default
The holder of a call option on the assets has a
claim on the assets after meeting the strike price
of the option.
The strike price = book value of liabilities.
The merge between Option pricing and bankruptcy
ensures that if the value of the assets is
insufficient to meet the liabilities of the firm, then
the shareholders (holders of the call option), will
not exercise their option and will leave the firm to
its creditors.
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Call Options and Distance to Default
}{}{ 21 d N Xed N V V rt
A E
The expected Firm Value at timeperiod t
The expected present value of firmliabilities at time period t
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}{}{ 21 d N e X d N V V
rt
A E
The implied market
value of the
firm’s assets
The book value of
debt
}{}{
1
2
d N
d N e X V V
rt
E
A
Implied Market Value of Firm’s Assets
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}{
}{
1
2
d N
d N e X V V
rt
E
A
= the value of the Standard Normal Cumulative Distribution
= NORMSDIST, or NORMDIST (x,0,1) d N
= market value of equity E
V
= Book value of debt X
= time horizon
= risk-free rate of return = minimum rate the firm is to achieve
t
r
Implied Market Value of Firm’s Assets
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Call Options and Distance to Default
t d
t
t r
X
V
d
t
t r X
V
d
A
A
A
A
A
1
2
2
2
1
5.0ln
5.0ln
= Value of firm’s assets = market value of equity + total debt
= Observed volatility of firm’s assets
AV
2
A
Distance to default (DD)
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How to Calculate the Observed Volatility
of Firm’s Assets?
= Observed volatility of firm’s assets. It can be obtained from
the relationship between equity and asset volatility that follows.
2
A
A
E E
A V
V
= market value of equity
= Observed volatility of firm’s equity =
E V
E MV %
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Call Options and Distance to Default:
Exercise
Market Capitalization (billion) 3
Equity Volatility (per annum) 40%
Total Liabilities (billion) 10
= 3
= 40%
= 10
= 3 + 10 = 13
E V
E
X
AV
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Call Options and Distance to Default
3.3378109231.04301.3
3.3378109231.0
100852.05.0%510
13ln
3.4301109231.0
100852.05.0%5
10
13ln
2
1
d
d
= 13
= 10
=1 = one year
AV
X
t %231.913
%403
A
E E
A
V
V
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}3378.3{}4301.3{
N
N
The values for the Standard Normal Cumulative Distribution can
be either obtained from the statistical tables or the Excel
= NORMSDIST, or = NORMDIST (x,0,1)
0.9996}1,0,3378.3{
0.9997}1,0,4301.3{
N
N
Implied Market Value of Firm’s Assets
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12.5121
0.9997
0.999671828.2103
}{
}{
1%51
2
A
rt
E
A
V
d N
d N e X V V
Implied Market Value of Firm’s Assets
The implied market value of the firm’s assets calculates as follows
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The implied market value of the firm’s assets is used for
calculating the implied asset volatility as follows.
9.6% 12.5121
%403
A
E E
AV
V
What is the relationship between the observed and the
implied asset volatility?
Implied Asset Volatility
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Probability of Default (PD)
The “Probability of Default” can be obtained either from the
statistical tables or the Excel
Using tables, The PD = 1- Standard Normal Cumulative
Distribution.
PD = 1- N {d2}
Using Excel, the PD = NORMSDIST (-d2) ,
= NORMDIST (-d2 ,0 ,1)
0.000420.9996-1}3378.3{1 N DP
The PD = 0.00042 basis points = 0.042% risk premium
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Distance to default and Financial Ratios
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