CFRM 405: Mathematical Methods for Quantitative Finance Homework 1 solution

Description

1. Compute the following limits.
(a) lim
h→0
4(x + h − 3)2 − 4(x − 3)2
h
(b) limx→∞
1
√
4x
2 − 2x − 10 + 2x
2. Compute the derivatives of the following functions.
(a) f(x) = 1
1 − x
(b) f(x) = X
7
k=1
ke−akx
3
(the {ak} are constants)
(c) f(x) =
log
x
K


+

r − q +
σ
2
2

T − t


σ
√
T − t
(K > 0, r, q, σ > 0, and T > t constant)
(d) f(x) =
log
S
K


+

r − q +
x
2
2

T − t


x
√
T − t
(S > 0, K > 0, r, q, and T > t constant)
(e) f(x) =
log
S
K


+

x − q +
σ
2
2

T − t


σ
√
T − t
(S > 0, K > 0, q, σ > 0, and T > t constant)
3. Recall that
d+(·) =
log
S
K


+

r − q +
σ
2
2

T − t


σ
√
T − t
(a) Parts (c), (d), and (e) of Problem 2 correspond to partial derivatives of d+. What
partial derivative does each correspond to?
(b) Compute the partial derivative of d+ with respe