AMS 513 Stony Brook University Instructor: Pinezich
Exercise 1.1 Using the properties of Denition 1.1.2 for a probability measure P show the
following.
i. If A F , B F and A B , then P(A) P(B ).
ii. If A F and cfw_An is a sequence of sets in F with limn
Quantitative Finance Qualifying Exam
2013 Summer
INSTRUCTIONS
You have 4 hours to do this exam.
Reminder: This exam is closed notes and closed books. No electronic devices are permitted.
Phones must be turned completely o for the duration of the exam.
Do
Feynman-Kac and Zero Coupon Bonds
AMS 513: Financial Derivatives and Stochastic Calculus
J.D. Pinezich
Department of Applied Mathematics and Statistics
Stony Brook University
Fall 2012
J.D. Pinezich
Feynman-Kac and Zero Coupon Bonds
1 / 21
Deriving PDEs f
Lecture 1
MatLab Basics
Prof. Dr. Svetlozar Rachev
Anna Serbinenko
Institute for Statistics and Mathematical Economics
University of Karlsruhe
MatLab for Financial Econometrics, Summer Semester 2009
Prof. Dr. Svetlozar Rachev Anna Serbinenko Institute for
Completion Problem
Let (, F, P) be a probability space, and A . A need not be in F. In order to define a
measure on A, if there exists a set B F with A B and P(B) = 0, then P(A) = 0. The
family of all subsets of type C A where C F and P(A) = 0 is called t
Path Dependent Options
AMS 513: Financial Derivatives and Stochastic Calculus
J.D. Pinezich
Department of Applied Mathematics and Statistics
Stony Brook University
Fall 2012
J.D. Pinezich
Path Dependent Options
1 / 32
Path Dependent Options
1
Mostly we ha
Stochastic Integration
AMS 513: Financial Derivatives and Stochastic Calculus
J.D. Pinezich
Department of Applied Mathematics and Statistics
Stony Brook University
J.D.Pinezich
Stochastic Integration
1 / 29
Ito Integral
1
Expressions of the form
T
(t)dW (
AMS 513 QUIZ 4 FALL 2013
NAME:
ID:
CLOSED BOOK TURN OFF ALL ELECTRONIC DEVICES DURING EXAM
1. Let W (t) be Brownian motion, t [0, 2]. Compute
2
2
sin2 (s)dW (s)
E
.
0
Solution By the Ito isometry, and since sin(s) is deterministic,
2
2
2
2
0
2
sin4 (s)ds
Shreve Chapter 8: American
Derivative Securities
From Stochastic Calculus for Finance II: Continuous-Time Models
Stony Brook University, Applied Mathematics and
Statistics
Program in Quantitative Finance
Instructor: J.D. Pinezich
pinezich@ams.sunysb.edu
F
AMS-510 Solution for Midterm 2
October 29, 2012
Problem 1
Proof:
1
1
n
When n = 1, (2n1)(2n+1) = 3 = 2n+1 the equation is proved;
Assume the equation holds for n k, then when n = k + 1
1
1
1
1
+
+ +
+
=
13 35
(2k 1) (2k + 1) (2 (k + 1) 1) (2 (k + 1) + 1)
AMS-510 Analytical Method for AMS
Midterm Exam III
(1). Find the inverse of the following matrices if exist:
2 0 0 0 0
1 2 0 0 0
1 1 2
A = 0 1 2 0 0 , B = 1 2 5
0 0 1 2 0
1 3 7
0 0 0 1 2
1
(2). Find the dimension and a basis of the general solution o
AMS-510 Solution for Midterm 3
December 5, 2012
Problem 1
1
z
= x2
y
1+ y
x
2z
=
xy
x
then we have
x3
x2 + y 2
=
2
x3
1
= 2
x
x + y2
3x2 x2 + y 2 x3 2x
2
(x2 + y 2 )
=
x4 + 3x2 y 2
2
(x2 + y 2 )
2z
x4 + 3x2 y 2
1+3
|(1,1) =
|(1,1) =
=1
2 + y 2 )2
xy
22
(x
AMS 513 QUIZ 5 FALL 2013
NAME:
ID:
CLOSED BOOK TURN OFF ALL ELECTRONIC DEVICES DURING EXAM
1. Consider the one-period binomial asset pricing model for a stock S with S0 = 20,
S1 (H) = 25, S1 (T ) = 16, and risk-free interest rate r = 1 . Let V represent t
AMS 513 QUIZ 6 FALL 2013
NAME:
ID:
CLOSED BOOK TURN OFF ALL ELECTRONIC DEVICES DURING EXAM
1. Let V (T ) be a derivative payo in a space (, F, P) with risk-neutral measure
P, and let R(t) be a positive adapted risk-free interest rate process.
i. What is t
AMS 513 Stony Brook University Instructor: Pinezich
Definition Let (, F, P) be a probability space such that F has a finite number of
sets in it. Let A F satisfy
1. A 6= ,
2. If B A then either B = A or B = .
Then A is said to be an atom of (, F, P).
Ques
AMS 513 2009 Stony Brook University
Instructor: John D. Pinezich
Shreve Vol. II Ex. 4.6 Let S(t) = S0 exp W (t) + 12 2 t be a geometric Brownian motion. Let p be a positive constant. Compute d(S p (t).
Let f (x) = ex and X(t) = pW (t) + p( 21 2 )t. The
AMS 513 Instructor: Pinezich
Shreve Volume II
Ex. 4.5 Solving the generalized geometric Brownian motion stochastic differential equation:
dS(t) = (t)S(t)dt + (t)S(t)dW (t)
(SDE)
(i) Compute d log S(t). (ii) Derive the solution to the SDE.
Solution
(i) Let
AMS 513 Instructor: Pinezich
Shreve Volume II
Ex. 4.4 (Stratonovich Integral) Let W (t) be a Brownian motion, = cfw_t0 , ., tn be a
partition of [0, T ], and for j = 0, ., n 1 define
tj + tj+1
tj =
.
2
(i) Define the half-sample quadratic variation:
Q/2
Ex. 2.1 Let (, F, P ) be a probability space and X a random variable measurable
with respect to the trivial -algebra. Prove that X is constant.
Proof
1. Since is not empty, there exists 0 , and we let x0 = X(0 ).
2. Now consider X 1 (x0 ).
3. Given that X
AMS 513 Instructor: Pinezich
Shreve Volume II
Ex. 4.1 Suppose M (t), 0 t T is a martingale with respect to a filtration F and (t)
is a simple process adapted to F(t). For t [tk , tk+1 ) define the stochastic integral
I(t) =
k1
X
(tj )(M (tj+1 ) M (tj ) +
Stochastic Calculus for Finance, Volume I and II
Solution of Exercise Problems
Yan Zeng
August 20, 2007
Contents
1 Stochastic Calculus for Finance I: The Binomial Asset
1.1 The Binomial No-Arbitrage Pricing Model . . . .
1.2 Probability Theory on Coin Tos
Name:
ID:
Quiz 1, AMS 513, September 17, 2012
1. Consider a probability space (, F, P), and let A, B be subsets of such that A F,
B F. Suppose P(A) = 1 , P(B) = 3 . Find the largest real a, and the smallest real b
3
4
such that
a P(A B) b.
Solution: Since
NAME:
ID:
Quiz 5, November 26, 2012
AMS 513
1. Denition: An arbitrage is a portfolio value process X(t) satisfying X(0) = 0 and
for some T > 0:
P(X(T ) 0) = 1
P(X(T ) > 0) > 0.
Prove the First Fundamental Theorem of Asset Pricing: If a market model has a