Math 489/889
Stochastic Processes and
Advanced Mathematical Finance
Homework 6
Steve Dunbar
Due Monday, October 13, 2010
Problem 1
(1.1)
Part a
For $ T_0 = 10 $ and $ a = 20 $ , draw a graph of theduration as a function of
the probability $ q $ .
100
90
80
70
60
50
40
30
20
10
0
1
q
Part b
For $ a = 20 $ and $ q = 0.55 $ draw a graph of the expected duration as a
function of $ T_0 $ .
80
70
60
50
40
30
20
10
0
0
5
10
T0
15
20
Part c
For $ a = 20 $ and $ q = 0.45 $ draw a graph of the expected duration as a
function of $ T_0 $ .
80
70
60
50
40
30
20
10
0
0
5
10
T0
15
20
Problem 2
The boundary condition at state 26 says that the duration of the "game" from 26
down to 18 is the sum of hte duration of the subsequent games from 26 down to
25, and then from 25 to 18. We can compute the duration of the first sub-game
using Corollary 1. Then the set of 9 first-step equations in 9 unknowns is:
(2.1)
(2.2)
513
Problem 3
Insert the trial solution
into the difference equation.
For s < k the substitution yields
(2.3)
so the equation is satisfied by the trial particular solution.
For
the difference equation becomes
and again the
equation is satisfied identically.
For
, the difference equation becomes
Therefore, the particular solution is
=
.
Problem 4
(4.1)
simplify symbolic
(4.2)
(4.3)
simplify symbolic
(4.4)
(4.5)
simplify symbolic
(4.6)
(4.7)
simplify symbolic
(4.8)
(4.9)
simplify symbolic
(4.10)
Problem 5
(5.1)
Part a
(5.1.1)
Part b
(5.2.1)
Part c
(5.3.1)
(5.3.2)
Inspect carefully, one solution set with is real, the other terms have complex
factors
and
.
We can also solve the partial derivative eqations "by hand".
Set the derivative
equal to 0, then clear denominators.
(5.3.3)
isolate for K
(5.3.4)
Substitute this into the derivative with respect to .
(5.3.5)
simplify symbolic
(5.3.6)
Now this derivative is
when
. Note that
and
are not realistic
values of , that is, the values
and
are not in the domain so the
denominator is never in the domain. Likewise, and are not . Then
substitute
back into the expression for
(5.3.7)
isolate for S
(5.3.8)
(5.3.9)