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Financial Risk Management
Lecture 15: Interest Rate Derivatives
An Introduction to Derivative Securities, Financial Markets, and Risk Management
Jarrow and Chatterjea 2013
This chapter discusses yields and forward rates, which
form the foundation of our
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Financial Risk Management
MBA & EMBA
An Introduction to Derivative Securities, Financial Markets, and Risk Management
Jarrow and Chatterjea 2013
In this chapter:
We introduce the simplest and the most derivative, a
forward contract.
A forward has a lin
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Financial Risk Management
MBA & EMBA
An Introduction to Derivative Securities, Financial Markets, and Risk Management
Jarrow and Chatterjea 2013
This chapter introduces exchangetraded
options.
An option is the most basic derivative with a
nonlinear pay
National University of Sciences & Technology, Islamabad
Financial management
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Financial Risk Management
MBA & EMBA
An Introduction to Derivative Securities, Financial Markets, and Risk Management
Jarrow and Chatterjea 2013
This chapter
explores the reasons for hedging
discusses its cost and benefits
introduces futures hedging s
National University of Sciences & Technology, Islamabad
Financial management
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Financial Risk Management
Lecture 13
An Introduction to Derivative Securities, Financial Markets, and Risk Management
Jarrow and Chatterjea 2013
This chapter studies some popular options trading
strategies.
We present profit diagrams for the basic optio
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Liquidity and Leverage
Financial Risk ManagementFRM
Lecture 8
Liquidity
Transactions liquidity,
Property of an asset being easy to exchange for other assets.
Funding liquidity
Ability to finance assets continuously at an acceptable borrowing rate.
Tr
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1.7. Symmetric and asymmetric ciphers
45
fascinating subject in Sections 2.1 and 8.2. For now, we content ourselves with
a few brief remarks.
Although no one has yet conclusively proven that pseudorandom number
generators exist, many candidates have been
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Exercises
57
(d) Explain how any simple substitution cipher that involves a permutation of the
alphabet can be thought of as a special case of a Hill cipher.
1.43. Let N be a large integer and let K = M = C = Z/N Z. For each of the
functions
e : K M C
lis
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Exercises
(b) The following message was encrypted using this transposition cipher. Decrypt
it.
WNOOA HTUFN EHRHE NESUV ICEME
(c) There are many variations on this type of cipher. We can form the letters into a
rectangle instead of a square, and we can
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2. Discrete Logarithms and DieHellman
In practice, unless one were to build a specialpurpose machine, the process
of computing g x should not be counted as a single basic operation. Using the
fast exponentiation method described in Section 1.3.2, it t
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2.5. An overview of the theory of groups
73
Example 2.12. Groups are ubiquitous in mathematics and in the physical
sciences. Here are a few examples, the first two repeating those mentioned
earlier:
(a) G = Fp and = multiplication. The identity element is
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2.9. The PohligHellman algorithm
89
of powers of small primes. More generally, g x = h is easy to solve if the
order of the element g is a product of powers of small primes. This applies, in
particular, to the discrete logarithm problem in Fp if p 1 facto
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1. An Introduction to Cryptography
c c = (k m) (k m ) = m m
to extract information about m or m . Its not obvious how Eve would proceed
to find k, m, or m , but simply the fact that the key k can be removed so
easily, revealing the potentially less ran
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Exercises
47
a b c d e f g h i j k l m n o p q r s t u v w x y z
S C J A X U F B Q K T P R W E Z H V L I G Y D N M O
Table 1.11: Simple substitution encryption table for exercise 1.3
and Hellmans key exchange method in Section 2.3 and then go on to discus
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Exercises
49
(ii) at least one letter fixed?
(iii) exactly one letter fixed?
(iv) at least two letters fixed?
(Part (b) is quite challenging! You might try doing the problem first with an alphabet
of four or five letters to get an idea of what is going on
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66
2. Discrete Logarithms and DieHellman
knows them, too. For various reasons to be discussed later, it is best if they
choose g such that its order in Fp is a large prime. (See Exercise 1.31 for a
way of finding such a g.)
The next step is for Alice to p
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Exercises
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1.27. Without using the fact that every integer has a unique factorization into
primes, prove that if gcd(a, b) = 1 and if a  bc, then a  c. (Hint. Use the fact that
it is possible to find a solution to au + bv = 1.)
1.28. Compute the follo
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Exercises
and a list of the most common bigrams that appear in the ciphertext. (If you do not
want to recopy the ciphertexts by hand, they can be downloaded or printed from
the web site listed in the preface.)
(a) A Piratical Treasure
JNRZR BNIGI BJRGZ
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96
2. Discrete Logarithms and DieHellman
Definition. Let R be a ring and let m R with m = 0. For any a R,
we write a for the set of all a R such that a a (mod m). The set a is
called the congruence class of a, and we denote the collection of all congruenc
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Exercises
55
Section 1.6. Cryptography by hand
1.37. Write a 2 to 5 page paper on one of the following topics, including both
cryptographic information and placing events in their historical context:
(a) Cryptography in the Arab world to the 15th century.
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2.4. The ElGamal public key cryptosystem
71
Proposition 2.10. Fix a prime p and base g to use for ElGamal encryption.
Suppose that Eve has access to an oracle that decrypts arbitrary ElGamal
ciphertexts encrypted using arbitrary ElGamal public keys. Then
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2. Discrete Logarithms and DieHellman
to get a second square root of 197 modulo 437. If the modulus were prime,
there would be only these two square roots (Exercise 1.34(a). However,
since 437 = 19 23 is composite, there are two others. In order to fin
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2.10. Rings, quotients, polynomials, and finite fields
95
and if u R is an element that has a multiplicative inverse u1 R, then we
can factor any element a R by writing it as a = u1 (ua). Elements that
have multiplicative inverses and elements that have o
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2.6. How hard is the discrete logarithm problem?
75
Proof. We give a simple proof in the case that G is commutative. For a proof
in the general case, see any basic algebra textbook, for example [37, 3.2]
or [42, 2.3].
Since G is finite, we can list its el
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2.10. Rings, quotients, polynomials, and finite fields
93
mental properties of these operations and use them to formulate the following
fundamental definition.
Definition. A ring is a set R that has two operations, which we denote by +
and , satisfying th
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2.9. The PohligHellman algorithm
87
Theorem 2.32 (PohligHellman Algorithm). Let G be a group, and suppose
that we have an algorithm to solve the discrete logarithm problem in G for
any element whose order is a power of a prime. To be concrete, if g G has
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1.7. Symmetric and asymmetric ciphers
ek (m) m + k
(mod p)
and
dk (c) c k
43
(mod p),
which is nothing other than the shift or Caesar cipher that we studied in
Section 1.1. Another variant, called an ane cipher, is a combination of the
shift cipher and th
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54
Exercises
(f) Use your program from (e) to find all primes less than 100 for which 2 is a
primitive root.
(g) Repeat the previous exercise to find all primes less than 100 for which 3 is a
primitive root. Ditto to find the primes for which 4 is a primi
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2.4
2. Discrete Logarithms and DieHellman
The ElGamal public key cryptosystem
Although the DieHellman key exchange algorithm provides a method of
publicly sharing a random secret key, it does not achieve the full goal of being
a public key cryptosystem