21-270
Introduction to Mathematical Finance
D. Handron
Exam #1 Review
The exam will be closed book and notes; only the following calculators will be permitted:
TI-30X IIS, TI-30X IIB, TI-30Xa.
1. (25 points)
Consider a simple nancial model with two time t
21-270 Introduction to Mathematical Finance D. Handron
Exam #2 , \
March 19, 2014 - Name: Sahib ans.
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Closed book and notes; only the following calculators will be permitted: TI30X IIS,
TI-SOXS MultiView, TI30Xa. '
Write your solutions in the space
21270 Introduction to Mathematical Finance D. Handron
Exam #3 A
April 18, 2014 Name: 3447»;
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Closed book and notes; only the following calculators will be permitted: TI30X IIS,
TI30XS MultiVieW, TI-30Xa.
Write your solutions in the space below each probl
63
Part II Solutions
SOLUTIONS FOR PART II
5. COMBINATORIAL REASONING
5.1. When rolling n dice, the probability is 1/2 that the sum of the numbers
obtained is even. There are 6n equally likely outcomes; we show that in
half of them the sum is even. For ea
1
Part I Solutions
SOLUTIONS FOR PART I
1. NUMBERS, SETS, AND FUNCTIONS
1.1. We have at least four times as many chairs as tables. The number of
chairs (c) is at least () four times the number of tables (t ). Hence c 4t .
1.2. Fill in the blanks. The equa
to the average of the spot rates R*(1) and R* (2), which were computed
in part Therefore, we assume
RR (1.5) = 12*(1) 312,42)
= 4.56%.
Under this assumption about R*(1.5), determine the price Z at t =' 0 of
a zerocoupon bond with maturity 1.5 and face val
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Chapter 2
Fixed-Income Securities and
Interest Rates
We now begin a systematic study of xed-income securities and interest rates. By a
xed-income security we mean a nancial instrument that promises xed (or denite)
payments at prescribed future dates. In s
Course Notes for Introduction to Mathematical
Finance (21-270)
William J. Hrusa & Dmitry Kramkov
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
January 10, 2011
2
Part I
Introduction to Financial Markets,
Replication a
Recitation 21
21-127 Concepts of Math
11.08.2011
Problems and (some) solutions
Congruence sense: Let x, y, a, b Z and n N. Assume x y mod n. Prove that ax + b ay + b
mod n.
Solution: We can use the denition of congruence modulo n, and then modify terms a
Recitation 21
21-127 Concepts of Math
11.08.2011
Problems and (some) solutions
Congruence sense: Let x, y, a, b Z and n N. Assume x y mod n. Prove that ax + b ay + b
mod n.
Solution: We can use the denition of congruence modulo n, and then modify terms a
Recitation 21
21-127 Concepts of Math
11.08.2011
Lecture material Divisibility, factorization, and modular arithmetic:
Throughout these denitions and theorems, assume all variables are integers, unless otherwise specied.
Denition: We say a divides b and w
183
Chapter 13: The Real Numbers
SOLUTIONS FOR PART IV
13. THE REAL NUMBERS
13.1. Sequences.
a) x dened by x n = n is monotone but not bounded. Each succeeding
term is larger than the previous term, so the sequence is increasing and
hence monotone. The se
183
Chapter 13: The Real Numbers
SOLUTIONS FOR PART IV
13. THE REAL NUMBERS
13.1. Sequences.
a) x dened by x n = n is monotone but not bounded. Each succeeding
term is larger than the previous term, so the sequence is increasing and
hence monotone. The se
125
Chapter 9: Probability
SOLUTIONS FOR PART III
9. PROBABILITY
9.1. If A B , then P ( A) P ( B )TRUE. P ( B ) is the sum of the probabilities assigned to points in A plus the sum of the proabilities assigned to
points in B A.
9.2. If P ( A) and P ( B )