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 Center 8-61
Phone: (212) 998-0476
Email: plakner@stern.nyu.

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 stock, and the discounted
time one price in one is big

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-period model with
interest rate r = 1/9:
a) Compute the

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 = 1
(2)
Express q3 from (2) and substitute it into (1)

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 the
following three inequalities, corresponding to each

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 q2 = .305 (with rounding).
b) The price of this put is
.

APPLIED STOCHASTIC PROCESSES FOR FINANCE
SOLUTION TO EXERCISE 16.10
Peter Lakner
a.
d1,2 =
1
0.5
log
1
+ 0.2 0.125
1.5
This gives d1 = 0.161 and d2 = 0.661.
V0 = N (0.161) 1.5e0.2 N (0.661) = 0.436 0.312 = .1237
b.
The number of shares the investor will h

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 .8
q=
q=
V0 = P [U 36]
.6733 1.1
= .7392
1.002
1
P [U

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 probability of up rst will be denoted by q, so the
probab

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 2X3 + X4 = 0. In this case X1 + 2X2 2X3 + X4 =
40 + 60

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 model is incomplete.
(b) The risk-neutral probabilities