MATH530a
(FALL 2014)
HOMEWORK #1
Due Friday, September 12, 2014
2
1) Recall that the Variance of a random variable X is dened by X := E[X E(X)]2 . Show that
2
(i) X = E[X 2 ] (E[X])2 ;
2
(ii) X = Ecfw_X(X 1) + X 2 , where X = E[X].
X
Argue that 2 E[X 2 ],
Homework 5 Key Points
December 9, 2014
1
1.1
Problem 1(Textbook Exercises)
Exercise 8.1
By the Meta-theorem 8.3.1 we have M = 1 and R = 2, so
(a) The model is arbitrage free;
(b) The model is not complete;
(c) Since the model is not complete, there is no
MATH530a
(FALL 2014)
Homework #5
1) Exercise 8.1, 8.2, 8.3 (of Bjrks book).
o
2) Let p(t, s) = C(t, s, K, r, , T ) be the price function of the European call option. Namely
p(t, s) = sN (d1 (t, s) er(T t) KN (d2 (t, s),
where
s
1
1
ln
+ r + 2 (T t) ;
K
2
MA530a
HOMEWORK #4
(Fall 2014)
Due Friday, November 28, 2014
1) Exercise 6.1 (of Bjrks book).
o
2) Consider a market that has one bond and one stock, with the following price dynamics:
dB(t) = B(t)r(t)dt;
dS(t) = S(t)[(t)dt + (t)dW (t)].
Let h(t) = (x(t),
Homework 3 Key Points
1
Assigned Problems
Every problem in this section was graded out of ten points.
1) Jensens Inequality We only need to show that they are convex functions.
i) Let f (x) = |x|.
f (x) + (1 )f (y) = |x| + (1 )|y| |x + (1 )y|
= f (x + (1
MA530a
(FALL 2014)
HOMEWORK #3
Due Friday, October 31, 2014
1) Using Jensens Inequality to argue that
i) |Ecfw_X| E|X|;
ii) |Ecfw_X|2 Ecfw_|X|2 ,
provided that all the expectations involved exist. (A challenge: if you claim that a function is
convex, you
Math 530 HW2 Solution
October 25, 2014
Book Problems
Assigned Problems
2 Hint
market is complete Ah = X has a solution for any X Rk rank(A) = K
nullity(AT ) = 0 AT Q = S has unique solution, if exists.
3 Let h = (h0 , h1 , ., hN ), then:
V1h
B1
= h0 +
=
MA530a
(Fall 2014)
HOMEWORK #2
Due Friday, October 3, 2014
1) Exercise 2.2-2.4.
2) Consider a market with N risky assets and 1 riskless
the matrix
1
1 + R S1 (1 )
1
1 + R S1 (2 )
A=
1
1 + R S1 (K )
asset. Assume that = (1 , , K ), and recall
N
S1 (1 )
N