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 rate r = 12% and foreign
annualized monthly rate q = 24%
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, the arbitrage-free price
is given by
V0 = 0.
Solution
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 stock price at maturity
takes values according to probabili
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 maturity n. We have
n1
D(n) = E[
=
n1
1
1
1
]=
]
E[
nk
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 = 1, . . . , N 1
fn (x) = 1cfw_xU
1
[pfn+1 (ux) + q f
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 arbitrage-free price of such a security at n m. We clearly
have
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 risk-neutral probabilities pi , i = 1, 2, 3, are the s
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 securities:
event stock long forward call
1
4
-4
0
2
8
0
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 option using the algorithm of backward induction.
Time 2:
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, for n = 0, . . . , N 1:
1
(fn+1 (ux) + q fn+1 (dx),
p
1+
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 conditions that
no barriers were crossed before n and Sn = x.
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 time n expressed in domestic currency is given by
Vn = (1
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
(1 + r)
(1 + r)n
We have
Sn+1 = Sn + a
n+1 ,
Gn = exp(S
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
3
The Binomial No-Arbitrage Pricing Model . . . . . .
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 random variables n take values u = 2 and d = 2 with the sin
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 the recurrent equation:
Sn+1 = (1 + r)(Sn + a
n+1 ),
wh
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 rate S0 = $ 64 , domestic single period rate r = 1 = 12.5%
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 F (n, N ) paid at N is the same as the value of
F (0,
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 into (1+r)N/2 SN/2 units of domestic currency and invest
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 S1
1+r
, 0).
1+q
A portfolio with Xn , total wealth, and
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 exchange rate
at year n.
Compute V0 , the arbitrage-free
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 domestic in2
1
terest rate r = 2 and the foreign intere
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 precisely, the dividend paid at
3
year n is given by
Dn = qS
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 stock with strike K1 = $115 at premium C1 = $12,
2. put opt
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 security expiring at
N = 2 and having the payo
100,
0,
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 option, which becomes the
standard call option with strike
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 r = 0.25. Consider the
up-or-down-and-rebate option wi
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 following equation:
Sn+1 = Sn (1 +
n+1 ).
Here ( n )n=1,.,N are
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 have the following system of equations for the functio