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 *g*^{0}(*x*).

Z *x*

- 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).

- (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*}.

- (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).

- (a) Let
*D*= {(*x,y*) : 1 ≤*x*^{2 }+*y*^{2 }≤ 9*, y*≥ 0}. Compute the integral

ZZ ^{p}*x*^{2 }+ *y*^{2 }*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

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