Credit Analysis Part 3 Options DD(4)

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Spring 2012 ©Prof.Tarek Eldomiaty 1

Part 3

The Use of Options Pricing for

Estimating Distance to Default

(DD) and Probability of Default(PD)




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.




Options Pricing

Myron S. Scholes, 1941Fisher Black




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




http://www.moodys.com/




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.





}{}{ 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




}{}{ 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




}{

}{

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






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)




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 %




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



Spring 2012 14 ©Prof.Tarek Eldomiaty


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




}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





12.5121

0.9997

0.999671828.2103

}{

}{

1%51

2

A

rt

E

A

V

d N

d N e X V V


The implied market value of the firm’s assets calculates as follows




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




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




Distance to default and Financial Ratios

Credit Analysis Part 3 Options DD(4)

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