Solutions
Question 1. True: 1,2
Question 2. Let W1 = X and W2 = X + Y . X, Y are independent standard
normal random variables. The probability of having losses at both 1 and 2
is
P (300X < 0, 300X + 100
√
3× Y < 0) = P (X < 0,
√
3X + Y < 0) =
5
12
Question 3.
(a) Equity holders: (AT − F )+. Debt holders: F − (F − AT )+
(b) E0 = C(A0, F ), D0 = Fe
−rT − P (A0, F ), D0 + E0 − A0 = 0
Question 4. Taking
Yt =
Xt
t+ 2
,
we know that Yt has mean 1. using Ito’s lemma, it satisfies the SDE
dYt = dWt.
So
Yt = Y0 +Wt = 1 +Wt.
So
Xt = (t+ 2) + (t+ 2)Wt.
Question 5. We would like E[S1 logS1] under the risk-neutral measure.
E[S1 logS1] = E[S1(logS0 + logS1/S0)] = S0 logS0eX + S0XeX ,
where X = logS1/S0 ∼ N(0.06− 120.22, 0.22).
E[S1 logS1] = 8 log 8e0.06 + 8(0.04 + 0.22)e0.06 = 8e0.06(3 log 2 + 0.08) = 18.34
The price of the derivative security is e−0.08(18.34) = 16.93.
Question 6. I(0, t) is normal distributed. Solving the SDE,
Xt = xe
−at +
∫ t
0
e−a(t−s)σdWs.
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Then ∫ t
0
Xsds =
∫ t
0
[xe−as +
∫ s
0
e−a(s−u)σdWu]ds
=
∫ t
0
xe−asds+
∫ t
0
∫ s
0
e−a(s−u)σdWuds
The mean is
E
∫ t
0
Xsds =
∫ t
0
xe−asds =
x(1− e−at)
a
.
The variance is
Var
[ ∫ t
0
Xsds
]
= σ2Var
[ ∫ t
0
∫ s
0
e−a(s−u)dWuds
]
.
Then,
Var
[ ∫ t
0
∫ s
0
e−a(s−u)dWuds
]
= Var
[ ∫ t
0
1− e−a(t−s)
a
dWs
]
=
1
a2
[
t+
2e−at
a
− e
−2at
2a
− 3
2a
] .
Hence, the variance is
Var
[ ∫ t
0
Xsds
]
=
σ2
a2
[
t+
2e−at
a
− e
−2at
2a
− 3
2a
]
.
Question 7.
(a) logG(4) = 1
4
[logS1 + logS2 + logS3 + logS4]
Var[logG(4)]
=
1
16
Var[4(W1 −W0) + 3(W2 −W1) + 2(W3 −W2) + (W4 −W3)]
=
30
16
0.42
= 0.3
(b) Plugging Sk
′
t into BS equation,
k′r +
σ2
2
k′(k′ − 1) = r.
Then k′ = − 2r
σ2
= −0.5 or 1. Hence, k = 0.5.
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Question 8.
(a) The bond price
Z(0, T ) = E
[
e−
∫ T
0 rtdt
]
∫ T
0
rtdt = r0T +
1
2
θT 2 + σ
∫ T
0
W (t)dt.
Since Var
[ ∫ T
0
Wtdt
]
= 1
3
T 3,
Z(0, T ) = e−r0T−
1
2
θT 2+ 1
6
σ2T 3
(b) The instantaneous forward rate f(t, T ) = − ∂
∂T
logZ(t, T ). Then
f(t, T ) = rt + θ(T − t)− 1
2
σ2(T − t)2.
Hence,
df(t, T ) = σ2(T − t)dt+ σdWt.
(c) No mean reversion; yield curve fitting not available; constant volatility;
negative interest rate.
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