# jumps by xiagong0815

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```									Here we look at comparative statics of the Merton Model for Corporate Debt.

Example:
Firm Value                                  \$12,500,000                  \$12,500,000                  \$12,500,000
Face Value of Debt:                         \$10,000,000                  \$10,000,000                  \$10,000,000
s-Asset                                              0.2                          0.2                          0.2
T                                                    0.1                        0.25                           0.5
r:                                                 0.05                         0.05                         0.05

Equity Value                          #NAME?                       #NAME?                       #NAME?

Value of Debt:                        #NAME?                       #NAME?                       #NAME?
PV(F) (at r):                             \$9,950,125                   \$9,875,778                   \$9,753,099
Expected Loss:                        #NAME?                       #NAME?                       #NAME?
Probability of Default:               #NAME?                       #NAME?                       #NAME?
Yield Spread (basis pts)              #NAME?                       #NAME?                       #NAME?

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1
1
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0
0
0
0
0
0                5                10                15                20                25

Here we see a case where the credit spread is 100 basis points for 3-year debt. But, we also see that debt that matures in 36 d
spread. This is a characteristic feature of the Merton model that stands at odds with the data.

A natural extension to the assumptions of Merton's original model is to allow the value of the underlying firm to jump. In the orig
the dynamic process of the firm value is a continuous process. So if the company's value exceeds the bond's par value, th
firm value to drift downward. Jumps would preclude that.

Firm Value                                  \$12,500,000                  \$12,500,000                  \$12,500,000
Face Value of Debt:                         \$10,000,000                  \$10,000,000                  \$10,000,000
s-Asset                                            0.15               0.15                   0.15
T                                                   0.1               0.25                    0.5
r:                                                 0.05               0.05                   0.05
# Jumps per Year                                    0.1                0.1                    0.1
Mean Jump Size                                     -0.3               -0.3                   -0.3
Std Dev of Jump Size                               0.15               0.15                   0.15
Equity Value                          #NAME?                #NAME?                #NAME?
Limt for infinite sum:                             150                 150                   150

Value of Debt:                        #NAME?                #NAME?                #NAME?
PV(F) (at r):                             \$9,950,125            \$9,875,778            \$9,753,099
Expected Loss:                        #NAME?                #NAME?                #NAME?
Probability of Default:               #NAME?                #NAME?                #NAME?
Yield Spread (basis pts)              #NAME?                #NAME?                #NAME?

Jumps in the Asset Value

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0
0
0
0
0
0            5            10               15         20
Time to Maturity
\$12,500,000         \$12,500,000      \$12,500,000    \$12,500,000    \$12,500,000    \$12,500,000
\$10,000,000         \$10,000,000      \$10,000,000    \$10,000,000    \$10,000,000    \$10,000,000
0.2                 0.2              0.2            0.2            0.2            0.2
0.75                    1                3              5              8             10
0.05                0.05             0.05           0.05           0.05           0.05

#NAME?                   #NAME?             #NAME?        #NAME?         #NAME?         #NAME?

#NAME?                   #NAME?             #NAME?        #NAME?         #NAME?         #NAME?
\$9,631,944             \$9,512,294        \$8,607,080    \$7,788,008     \$6,703,200     \$6,065,307
#NAME?                   #NAME?             #NAME?        #NAME?         #NAME?         #NAME?
#NAME?                   #NAME?             #NAME?        #NAME?         #NAME?         #NAME?
#NAME?                   #NAME?             #NAME?        #NAME?         #NAME?         #NAME?

at debt that matures in 36 days (1/10 of a year) has 0 credit

ing firm to jump. In the original specification (as in Black-Scholes),
eds the bond's par value, then time is required to allow the

\$12,500,000         \$12,500,000      \$12,500,000    \$12,500,000    \$12,500,000    \$12,500,000
\$10,000,000         \$10,000,000      \$10,000,000    \$10,000,000    \$10,000,000    \$10,000,000
0.15            0.15        0.15          0.15          0.15          0.15
0.75               1            3             5             8            10
0.05            0.05        0.05          0.05          0.05          0.05
0.1             0.1          0.1           0.1           0.1           0.1
-0.3            -0.3        -0.3          -0.3          -0.3          -0.3
0.15            0.15        0.15          0.15          0.15          0.15
#NAME?           #NAME?          #NAME?        #NAME?        #NAME?        #NAME?
150            150          150           150           150           150

#NAME?           #NAME?          #NAME?        #NAME?        #NAME?        #NAME?
\$9,631,944     \$9,512,294     \$8,607,080    \$7,788,008    \$6,703,200    \$6,065,307
#NAME?           #NAME?          #NAME?        #NAME?        #NAME?        #NAME?
#NAME?           #NAME?          #NAME?        #NAME?        #NAME?        #NAME?
#NAME?           #NAME?          #NAME?        #NAME?        #NAME?        #NAME?

et Value

Continuous Path
Jumps

25
\$12,500,000    \$12,500,000
\$10,000,000    \$10,000,000
0.2            0.2
15             20
0.05           0.05

#NAME?         #NAME?

#NAME?         #NAME?
\$4,723,666     \$3,678,794
#NAME?         #NAME?
#NAME?         #NAME?
#NAME?         #NAME?

\$12,500,000    \$12,500,000
\$10,000,000    \$10,000,000
0.15          0.15
15            20
0.05          0.05
0.1           0.1
-0.3          -0.3
0.15          0.15
#NAME?        #NAME?
150           150

#NAME?        #NAME?
\$4,723,666    \$3,678,794
#NAME?        #NAME?
#NAME?        #NAME?
#NAME?        #NAME?

```
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