Homework 1.
Your grade for the homework will be based on the best 3 solutions.
Problem 1. Consider nancial model on exchange rates. Assume that spot
exchange rate S0 = 4, domestic annualized monthly r
Solutions to homework 1.
Solution to Problem 1. The replication strategy for the payo is to
1. buy forward at time 0
2. sell forward at time N/2
The initial capital of the strategy is zero, and, hence
Solutions to Homework 10.
Solution to Problem 1. From the dynamic for interest rates we deduce
n
1 + rn = (1 + r0 )
n = 0, 1, . . . , N 1.
k,
k=1
Let D(n) denote the discount factor computed at 0 for
Homework 4.
Your grade for the homework will be based on the best 3 solutions.
Problem 1. Consider the one-period nancial model with yearly interest
rate r = 7% and maturity T = 1 year, where the stoc
Answers to test 2.
1. V0 =
4
9
= 0.4444, 0 =
1
9
= 0.1111.
2. We have the following system of equations for the functions fn , n =
0, 1, . . . , N :
1. At maturity N
fN (x) = 1cfw_xU F.
2. At times n
Solutions to homework 2.
Solution to Problem 1. We start with an auxiliary problem of evaluation
of the derivative security paying Sm at maturity N for m = 0, . . . , N . Denote
by Xn (m) the arbitrag
Solutions to homework 3.
Solution to Problem 1. The following table lists the terminal values of
traded securities as functions of the terminal stock price:
stock call put
95
0
10
120
5
0
145
30
0
The
Solutions to homework 4.
Solution to Problem 1. The sample space of the model consists of 3
points, which we denote by i , i = 1, 2, 3. The following table lists the
terminal values of the relevant se
Solutions to homework 5.
Solution to Problem 1. The one step risk-neutral probabilities in the
model are given by
p=
3
1+rd
=,
ud
4
1
q =1p= .
4
We compute the arbitrage-free prices for the barrier op
Solutions to homework 6.
Solution to Problem 1. The functions (fn ) and (gn ) satisfy the boundary
conditions:
fN (x) = max(x K, 0),
gN (x) = 1cfw_xL fN (x),
and the equations of backward induction, f
Solutions to homework 7.
Solution to Problem 1. The one-period risk-neutral probabilities are given
by
1
1
1+rd
= , q =1p= .
p=
ud
2
2
Let vn (x) be the price of the option at time n under the conditi
Solutions to homework 8.
Solution to Problem 1. Consider a self-nancing strategy, where at time
0 we invest one unit of foreign currency at foreign interest rate q . The capital
of the strategy at tim
Solutions to homework 9.
Solution to Problem 1. It is convenient to write down the price process
of the stock and the payment process of the American option in discounted
terms:
Sn
Gn
Sn =
, Gn =
.
n
Steven E. Shreve
Stochastic Calculus for Finance I
Students Manual: Solutions to Selected
Exercises
December 14, 2004
Springer
Berlin Heidelberg NewYork
Hong Kong London
Milan Paris Tokyo
Contents
1
2
Homework 10.
Your grade for the homework will be based on the best 3 solutions.
Problem 1. Consider a complete model for interest rates, where
rn+1 = (1 + rn )
n+1
1,
r0 =
1
(= 25%),
4
1
and the rand
Homework 9.
Your grade for the homework will be based on the best 3 solutions.
Problem 1. Consider the N = 5-period binomial model with interest rate
r = 1 = 100% and the price of the stock following
Final Exam for the course Discrete-Time Finance .
Your grade for the exam will be based on the best 7 solutions.
Problem 1. Consider the N = 6-period model on exchange rates1 with spot
81
exchange rat
Solutions to Final Exam.
Solution to Problem 1. The strategy where we enter a long position in
forward at 0 and a short position at n results in the payo F (n, N ) F (0, N )
at N . Hence, the value of
Solutions to test 1.
Solution to Problem 1. To replicate the payo SN/2 at N we
1. At time 0 invest
1
(1+q )N/2 (1+r)N/2
units of foreign currency up to N/2
1
2. At time N/2 convert foreign currency in
Solutions to test 2.
Solution to Problem 1. The forward exchange rate computed at t = 1 for
maturity t = 2 is given by
1+r
.
F1 = S1
1+q
Hence, the payo of the option is
V2 = max(S2 F1 , 0) = max(S2 S
Test 1.
Your grade for the test will be based on the best 2 solutions.
Problem 1. Assume spot exchange rate1 S0 = 15, domestic yearly rate
r = 20%, and foreign yearly rate q = 25%. Denote by Sn the ex
Test 2.
Your grade for the test will be based on the best 2 solutions.
Problem 1. Spot exchange rate (Sn ) evolves according to the binomial
3
model with N = 2, u = 2 , d = 1 , and S0 = 4. Assume the
Homework 2.
Your grade for the homework will be based on the best 3 solutions.
Problem 1. Consider a stock paying dividends annually in the amount q =
2
66.67% of the stock price year ago. More preci
Homework 3.
Your grade for the homework will be based on the best 3 solutions.
Problem 1. Consider a single-period nancial model with interest rate r =
5%, where one can trade
1. call option on a stoc
Homework 5.
Your grade for the homework will be based on the best 3 solutions.
Problem 1. Consider the N = 2-period binomial model with S0 = 100,
u = 1.2, d = 0.8 and take r = 0.1. For the derivative
Homework 6.
Your grade for the homework will be based on the best 3 solutions.
Problem 1. In the framework of N -period binomial model with the parameters u, d, and r consider the down-and-in call opt
Homework 7.
Your grade for the homework will be based on the best 3 solutions.
Problem 1. Consider the N = 2-period binomial model with the parameters
S0 = 4, u = 2, d = 0.5 and take the interest rate
Homework 8.
Your grade for the homework will be based on the best 3 solutions.
Problem 1. Consider a complete N -period foreign exchange model. The
evolution of exchange rate1 is given by the followin
Answers to Final Exam.
1. V0 = $ 1 .
9
2. The model is arbitrage-free and complete. C0 = $ 8 = $1.6.
5
3. p1 = 0.0144, p2 = 0.6962, p3 = 0.2894.
4. We have
V0 =
N n
3
5
F0 ,
11
0 =
F0 2
4
5
N
.
5. We