CFRM 405: Mathematical Methods for Quantitative Finance
Solve the exercises by hand.
1. Let K, T, σ, and r be positive constants and let
Homework 2
where . Compute g0(x).
Z x
2. Letso that Φ(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 lim Φ(x) = 1, use the properties of the integral as well as a substitution
x→∞ to show that Φ(−x) = 1 − Φ(x) (again, assuming x > 0).
3. (a) Under what condition does the following hold?
(b) Evaluate the double integral
ZZ 2
ey dA
D
where D = {(x,y) : 0 ≤ y ≤ 1, 0 ≤ x ≤ y}.
4. (a) Transform the double integral
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 ≤ x2 + y2 ≤ 9, y ≥ 0}. Compute the integral
ZZ px2 + y2 dxdy
D
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
where
