Solutions for Unit 10 Questions
1. A firm’s asset value is modeled using Merton’s model. Assume
V
0
=
1
,
000
,
000, and that the return rate on the assets is
μ
V
= 0
.
1 The firm
has issued a 3year zero coupon bond with a face value of
B
= 500
,
000.
(a) Find the volatility
σ
V
of the firm’s return on its asset value if its
probability of default is 0.06.
Solution:
We need to write the probability of default under Merton’s
model and solve for
σ
V
:
0
.
06 =
P
(
V
T
< B
) =
P
(
N
(0
,
1)
<
(ln
B

ln(
V
0
)

(
μ
V

σ
2
V
/
2)
T
)
/
(
σ
V
√
T
))
because under Merton’s model, we have that ln
V
T
∼
N
(ln
V
0
+ (
μ
V

σ
2
V
/
2)
T, σ
2
V
T
), where here
V
0
= 10
6
,
μ
V
= 0
.
1,
B
= 500
,
000,
T
= 3.
Hence we need to have
z
0
.
06
= (ln
B

ln(
V
0
)

(
μ
V

σ
2
V
/
2)
T
)
/
(
σ
V
√
T
)
,
where
z
0
.
06
≈
1
.
5551. Solving for
σ
V
we find
σ
V
= 0
.
31386.
(b) Determine the expected fraction of money recovered by the lender
when the firm defaults (given that there has been default).
Use the
fact that if
X
∼
N
(
a, b
) then
E(
e
X
1
e
X
<c
) = exp(
a
+
b/
2)
φ
((ln(
c
)

a

b
)
/
√
b
)
,
where, as usual,
φ
(
x
) =
P
(
N
(0
,
1)
≤
x
).
Solution:
We need to compute
E
V
T
B

V
T
< B
where ln
V
T
∼
N
(
a, b
) with
a
= ln
V
0
+ (
μ
V

σ
2
V
/
2)
T
and
b
=
σ
2
V
T
.
Hence the above expectation is equal to
E
V
T
B
1
V
T
<B
/P
(
V
T
< B
) = (1
/B
)(exp(
a
+
b/
2)
φ
((ln(
c
)

a

b
)
/
√
b
))
/
0
.
06 = 0
.
80635
,
where
c
=
B
. Note that we take the expectation under the physical
measure here, because we are not pricing.
1
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2. A firm’s asset value is modeled using Merton’s model. Assume
V
0
=
750
,
000, that the return rate
μ
V
= 0
.
08 and
σ
V
= 0
.
25.
The firm
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 Fall '09
 ChristianeLemieux
 Zerocoupon bond, σv, Gauss copula

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