[Solved] CFRM405 Homework 2

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CFRM 405: Mathematical Methods for Quantitative FinanceSolve the exercises by hand.1. Let K, T, σ, and r be positive constants and let Homework 2

where. Compute g0(x).

Z x

  1. 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).

  1. (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}.

  1. (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).
  1. (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 (that means r on one axis and θ on the other) planes.

  • Compute the integral

where

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[Solved] CFRM405 Homework 2
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