FRE6083, Midterm Examination, Monday October 21 2013
2:00pm-4:30pm
1. Number of pages including this one: 2
2. For this examination, you may only use a 2-page cheat sheet. No other notes, books or electronic
devices may be used. Cell phones may not be use

FRE 6083 Homework 5 Question 1 Is the mid-range, 1/2( + ) a statistic?
Question 2 Show that the average of the rst two observations in a random sample is unbiased but not consistent, in estimating a population mean. Question 3 Show that if is an unbiased

Solutions Question 1 Similar to the example in the notes, diering only in the form of the starting pdf. ( )= ( )= (
2
)= 1 2 ()
(
)=
( ) 1/ (1 + )
(
)
The pdf is obtained from the distribution function by dierentiation, ( )= ( )= () ( ) 1 = for 0
2
()

FRE 6083 Homework 4 Question 1 Scaling Normal VaR with independent and with autocorrelated returns. A portfolio has daily returns, discounted to today, that are normally with zero expectation and standard deviation 1.5%. (a) Calculate the 1% 1-day VaR. (b

Continuous Optimization Problems Mathematical optimization techniques are crucial in the design and construction of portfolios of financial assets. It is assumed that you have taken an introductory course in linear and nonlinear programming such as MA 614

FRE 6083 Homework 2
Problem 2.1
We let the random variables Xn be a binomial process,
Xn =
n
X
Bi ,
i=1
where each Bi is independent and is distributed according to
(
1 with probability p
Bi =
0 with probability 1 p
1. Calculate the probability P (X4 > k)

Midterm Review
1. Let
fX|Y (x|y) = yexy f or x > 0, 0 < y < 1
and assume that the distribution of Y is uniform on (0, 1). Calculate
1.The marginal density of X, fX (x),
2.P [XY > 1],
3.E[X|Y ],
4.V ar[X|Y ].
2. Consider a standard normal random variable X

Quantitative methods
Lecture 4-5
A basic introduction to stochastic processes
October 3, 2016
1
Introduction
Stochastic processes are used for modeling the time evolution of financial
assets. In the course of the past 3 lectures, we have already encounter

Quantitative methods
Lecture 6
The arithmetic random walk and the
geometric random walk
October 3, 2016
1
Introduction
In this lecture, we draw heavily from the textbook in Stochastic Calculus by
S. Shreve, Stochastic Calculus for Finance. In particular y

Quantitative Methods
Week 2
Convergence concepts, Law of large numbers,
Central Limit Theorem, Markov sequences,
the Martingale property
September 16, 2016
For the concepts of convergence, the Law of large numbers and the central
limit Theorem, we draw he

FRE 6083 Final Review
Problem 1
Compute the characteristic function E[eiW (t) ] for any R, and W (t) a Wiener process.
1
2
Problem 2
A stock is currently $100. Over the next two six-month periods is expected to go up by
10% or down 10%. The risk free rate

Quantitative methods, Homework Assignment for Lecture 1
Where appropriate, show the space or universe, the field of subsets of the
space, and the probability measure for outcomes. Clearly state the events
for which probabilities are to be calculated.
Prob

Quantitative Methods
Lecture 3
Markov chains
September 19, 2016
1
1.1
Markov chains
Introduction
This week, we study Markov chains. This study is motivated by the numerous applications to Insurance and Finance, in particular to credit risk
but also to the

Tandon School of Engineering
FRE 6083
Final Review
(1) We construct a Binomial tree with 3 periods for the parameters
r = 1/4; d = 1/2; u = 2
and for the current stock price S0 = 8, for computing the price C0 of a call option
at time 0 with payoff
C3 = S3

Quantitative methods, Homework Assignment 6
Due December 10th
In the following exercises B(t), t 0 is a standard Brownian motion process and Ta denotes the time it takes this process to hit a.
1. What is the distribution of B(s) + B(t), s t?
2. Compute th

Quantitative methods, Homework Assignment 3
Problem 3.1
Let X1 , X2 , ., X10 be a sample of size 10 from the standardized normal distri = X1 +X2 +.+X10 .
bution N (0, 1). Determine probability P (X 1) where X
10
Problem 3.2
Let the random variables X and

Quantitative methods,
Homework Assignment 4
Problem 4.1
Consider the process X(t), t 0 defined by
1, t Y ;
X(t) =
0, t > Y .
where Y is a uniformly distributed random variable on the interval (0, 1).
1. Compute, for t (0, 1), the first-order probability m

Sum of Normally Distributed Random Variables This appendix provides a series of derivations for the density function of the sum of independent random variables. We begin with the simplest case, namely, the sum of two independent standard normal random var