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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).

−∞

• For x > 0, show that φ(−x) = φ(x).
• 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

e^y 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.

• 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.
• 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.

• Compute the integral

where