CFRM 405: Mathematical Methods for Quantitative Finance Homework 2
Solve the exercises by hand.
1. Let K, T, σ, and r be positive constants and let
g(x) = 1
√
2π
Z b(x)
0
e
−
y
2
2 dy
where b(x) = 1
σ
√
T
h
log
x
K
+
r +
σ
2
2
T
i
. Compute g
0
(x).
2. Let φ(u) = 1
√
2π
e
−u
2/2
so that Φ(x) = Z x
−∞
φ(u) du (i.e., the Φ(x) in Black-Scholes).
(a) For x > 0, show that φ(−x) = φ(x).
(b) Given that limx→∞
Φ(x) = 1, use the properties of the integral as well as a substitution
to show that Φ(−x) = 1 − Φ(x) (again, assuming x > 0).
3. (a) Under what condition does the following hold?
Z Z
D
f(x, y) dA =
Z Z
D
f(x, y) dy dx =
Z Z
D
f(x, y) dx dy
(b) Evaluate the double integral
Z Z
D
e
y
2
dA
where D = {(x, y) : 0 ≤ y ≤ 1, 0 ≤ x ≤ y}
4. (a) Transform the double integral
Z Z
D
e
x+y
x−y dA
into an integral of u and v using the change of variables
u = x + y v = x − y
and call the domain in the uv plane S.
(b) Let D be the trapezoidal region with vertices (1, 0), (2, 0), (0, −2) and (0, −1). Find
the corresponding region S in the uv plane by evaluating the transformation at the
vertices of D and connecting the dots. Sketch both regions.
(c) Compute the integral found in part (a) over the domain S from part (b).
5. (a) Let D = {(x, y) : 1 ≤ x
2 + y
2 ≤ 9, y ≥ 0}. Compute the integral
Z Z
D
p
x
2 + y
2 dx dy
by changing to polar coordinates. Sketch the domains of integration in both the xy
and rθ (that means r on one axis and θ on the other) planes.
(b) Compute the integral
Z Z
D
sin(p
x
2 + y
2
) dx dy
where D = {(x, y) : π
2 ≤ x
2 + y
2 ≤ 4π
2}.
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