STERN SCHOOL OF BUSINESS
NEW YORK UNIVERSITY
COURSE SUPPLEMENT
STAT-GB.2219.89 and STAT-UB.0008.01
APPLIED STOCHASTIC PROCESSES
FOR FINANCIAL MODELS
Professor Peter Lakner
Oce: Kaufman Management Cent
APPLIED STOCHASTIC PROCESSES FOR FINANCE
PRACTICE MIDTERM EXAMINATION, 2015 FALL
Peter Lakner
This is an open book and notes exam.
1. The following tree shows the non-discounted stock prices in a one-
STOCHASTIC MODELS FOR FINANCE
MIDTERM EXAM, SOLUTIONS, 2015 FALL, VERSION B
Peter Lakner
1.
a. The martingale measure is ( 1 ,
2
1
).
2
b. There is no arbitrage in this model because there is only one
STOCHASTIC MODELS FOR FINANCE
MIDTERM EXAM, SOLUTIONS, 2015 FALL, VERSION A
Peter Lakner
1.
a. We have the following equations for the risk-neutral probabilities:
11q1 + 7q2 + 2q3 = 5
(1)
q1 + q2 + q3
APPLIED STOCHASTIC PROCESSES FOR FINANCE
HOMEWORK 1, SOLUTION
Peter Lakner
(a) Below is the tree for the discounted prices:
We need to satisfy G () 0 for every and G () > 0 for at least one . We get t
APPLIED STOCHASTIC PROCESSES FOR FINANCE
PRACTICE MIDTERM EXAM, 2015 FALL, SOLUTIONS
Peter Lakner
1.
a) Here are the discounted prices:
We must have
6.48q1 + 1.62q2 = 5
q1 + q2 = 1,
so q1 = .695 and q
APPLIED STOCHASTIC PROCESSES FOR FINANCE
Exercise 16.7, SOLUTION
Peter Lakner
a.
50
1
.8
log 1.1
.8
log
n
=
4.8455
= 35.035
.1383
Since n must be an integer, we select n = 36.
1 + .002 .8
= .6733
1.1
APPLIED STOCHASTIC PROCESSES FOR FINANCE
EXERCISE 16.5, SOLUTION
Peter Lakner
(a) We denote the probabilities for up, up, up, down, down, up, and down, down
by q1 , q2 , q3 , q4 , respectively. The pr
APPLIED STOCHASTIC PROCESSES FOR FINANCE
EXERCISE 16.3, Solution
Peter Lakner
(a) We already know from Exercise 15.2 that a contingent claim (X1 , X2 , X3 , X4 ) is attainable if and only if X1 + 2X2
APPLIED STOCHASTIC PROCESSES FOR FINANCE
EXERCISE 16.2, SOLUTION
Peter Lakner
Here is the tree for the discounted prices:
The interest rate is r = 1/9.
(a) We have N = 2, K = 4. Since N + 1 < K, the m