Math 294, Winter 2014
Homework 3 Solutions
1. Show that Xt = Wt3 3tWt is a martingale. [Hint: One way to do this is to compute the
conditional expectation E[Xt |Fs ] (0 < s < t) by using the trick of writing Wt = Ws + (Wt Ws ).
t
2
Another is to use Its f
Math 294, Winter 2014
Homework 3 Due March 7
In problems 1 and 2, (Wt )t0 is a (standard) Brownian motion dened on some ltered
probability space (, F, (Ft ), P).
1.
Show that Xt = Wt3 3tWt is a martingale. [Hint: One way to do this is to
compute the condi
Math 294, Winter 2014
Homework 2 m Solutions
Section 2.4, Exercise 6. Observe that p* = .4.
(a) The payoff process for this option is given by
Y0 = 20.
Y1 (d) = 70 11(5) = 0,
Y2 (dd) = 95, Y2(du) = Y2(ud) = 20, Y2 (uu) = O.
Nowusing the backward recursion
Math 294
Mathematics of Finance
Winter 2014
This course is an introduction to the mathematics of nancial models, especially hedging and arbitrage pricing. The course begins with the development of the basic ideas of
hedging and arbitrage pricing in the di
Math 294, Winter 2014
Homework 4 Due March 14
In problems 1 and 2, W = (Wt )0tT is a (standard) Brownian motion dened on
some ltered probability space (, F, (Ft )0tT , P), and F = FT .
1. Let and be real numbers. With W as described above, the process Xt
Math 294, Winter 2014
An Integral
The following integral arises in the calculation of the Laplace transform of the density function of the
rst passage time Tb .
Proposition. For > 0 and 0,
et/t t1/2 dt =
(1)
0
2
e
.
Proof. Fix > 0 and view the left side