Economics 765 Assignment 1
1.3 In the one-period binomial model (as considered in the rst class), suppose we want
to determine the price at time zero of the derivative security with payo V1 = S1 . This
means that the derivative security pays the stock pri
June 2011
Final Examination
Models for Financial Economics
Economics 765
Take-home exam due on or before Friday June 21st 2013.
This exam comprises 6 pages, including the cover page
Economics 765
Page 1 of six pages
Economics 765
Page 2 of six pages
1. Us
Economics 765 Assignment 5
6.1
Consider the stochastic dierential equation
dX (u) = (a(u) + b(u)X (u) du + ( (u) + (u)X (u) dW (u),
(6.2.4)
where W (u) is a Brownian motion relative to a ltration F (u), u 0, and we allow a(u),
b(u), (u), and (u) to be pro
Economics 765 Final Exam
1. Use the probabilistic argument in Shreve section 5.2.5 to obtain an explicit
expression for the function p(t, x), which gives the value of a European put option
at time t when the price S (t) of the underlying asset is x. The o
Economics 765 Assignment 4
4.19
Let W (t) be a Brownian motion, and dene
t
sign(W (s) dW (s),
B (t) =
0
where
sign(x) =
1
1
if x 0,
if x < 0.
(i) Show that B (t) is a Brownian motion.
The dierential of B (t) is
dB (t) = sign(W (t) dW (t).
(1)
Then, since
Economics 765 Assignment 3
3.2 Let W (t), t 0, be a Brownian motion, and let F (t), t 0, be a ltration for this
Brownian motion. Show that W 2 (t) t is a martingale.
The easiest way to show this is to compute the dierential of W 2 (t)t using the It-Doebli
Economics 765 Assignment 2
2.5
Let (X, Y ) be a pair of random variables with joint density function
fX,Y (x, y ) =
2|x|+y
2
exp
(2|x|+y )2
2
if y |x|,
if y < |x|.
0
Show that X and Y are standard normal variables and that they are uncorrelated but not
i