Stat 100C Linear Models Homework 2 Instructions (a) Just following instructions gives you 5% of the homework grade.
J. Sanchez UCLA Department of Statistics
(b) IN ALL OF THIS HOMEWORK, YOU MUST SHOW WORK. THERE IS ONLY ONE PROBLEM WHERE YOU HAVE TO USE
Stat 100 C. Homework 5
NAME, ID, DATE and HOMEWORK # REQUIRED ON THIS CORNER
INSTRUCTIONS: Homework must be stapled. Just following instructions gives you 5% of the homework grade. The writing must be legible and easy to follow. Turn in a clean version of
Homework 5 YOUR NAME ARE REQUIRED. YOUR ID : YOUR TA SESSION (time or session #)
Homework must be stapled. Just following instructions gives you 5% of the homework grade. The writing must be legible and easy to follow. Turn in a clean version of your answ
Mathematics ofFinance for Mathematics/Economics Students
MATH 174

Spring 2011
I ntroduction
In the last 25 years derivatives have become increasingly important in the world of
finance. Futures and options are now traded actively on many exchanges throughout
the world. Many different types of forward contracts, swaps, options, and o
1. Let the linear regression model be and the standard assumptions (zero mean, homoscedastic and noncorrelated errors) hold. A researcher takes 37 random sample observations to estimate this function. The relevant data (i.e., no transformations, no devia
Stat 100 C. Homework 6
N AME, ID, DATE and HOMEWORK # REQUIRED ON
THIS CORNER
INSTRUCTIONS:
Homework must be stapled.
Just following instructions gives you 5% of the homework grad e.
The writing must be legible and easy to follow. Turn in a clean version
Solution 3
Sec2.1 2.1.10 Suppose that T: R 2 R 2 is linear, T(1,0) = (1,4), and T(1,1) = (2,5). What is T(2,3)? Is T onetoone? Ans.: (A) (2,3) = (1)(1,0)+3(1,1) T(2,3) = (1)T(1,0)+3T(1,1) = (1)(1,4)+3(2,5) = (5,11) (B) The two vectors (1, 0) and (1,
Stat 100C Linear Models Homework 2 Instructions (a) Just following instructions gives you 5% of the homework grade.
J. Sanchez UCLA Department of Statistics
(b) IN ALL OF THIS HOMEWORK, YOU MUST SHOW WORK. THERE IS ONLY ONE PROBLEM WHERE YOU HAVE TO USE
Homework 4 YOUR NAME ARE REQUIRED. YOUR ID : YOUR TA SESSION (time or session #)
Homework must be stapled. Just following instructions gives you 5% of the homework grade. The writing must be legible and easy to follow. Turn in a clean version of your answ
Mathematics ofFinance for Mathematics/Economics Students
MATH 174

Spring 2011
Value': at Risk
In Chapter 15 we examined measures such as delta, gamma, and vega for describing
different aspects of the risk in a portfolio of derivatives. A financial institution usually
calculates each of these measures each day for every market varia
Mathematics ofFinance for Mathematics/Economics Students
MATH 174

Spring 2011
Wiener Processes
and Ito's etntna
Any variable whose value changes over time in an uncertain way is said to follow a
stochastic process. Stochastic processes can be classified as discrete time or continuous
time. A discretetime stochastic process is one
Solution 8
Sec. 6.2 6.2.2. In each part, apply the GramSchmidt process to the given subset S of the inner product space V to obtain an orthogonal basis for space(S). Then normalize the vectors in this basis to obtain an orthonormal basis for space(S), an
Mathematics ofFinance for Mathematics/Economics Students
MATH 174

Spring 2011
T rading Str"ategies
Involving Options
We discussed the profit pattern from an investment in a single stock option in
Chapter 8. In tllis chapter we cover more fully the range of profit patterns obtainable
using options. We assume that the underlying asse
Mathematics ofFinance for Mathematics/Economics Students
MATH 174

Spring 2011
O ptions on
Stock Indices,
Currencies,
and Futures
In this chapter we tackle the problem of valuing options on stock 'indices, currencies,
and futures contracts. As a first step, we produce results for options on a stock paying a
known dividend yield. We
Chapter 21
The BlackScholes Equation
Question 21.1.
If V (S, t) = er(T t) then the partial derivatives are VS = VSS = 0 and Vt = rV . Hence Vt +
(r ) SVS + S 2 2 VSS /2 = rV .
Question 21.2.
If V (S, t) = AS a e t then Vt = V , VS = aS a 1 e t = aV /S ,
Chapter 22
Exotic Options: II
Question 22.1.
With a premium of P paid at maturity if ST > K , the COD will have the same value (which will
initially be set to zero) as a regular call minus P cash or nothing call options. That is,
0 = BSCall(S0 , K, , r, T
Chapter 20
Brownian Motion and Its Lemma
Question 20.1.
If y = ln (S) then S = ey and dy =
2
2e2y
a)
dy =
ey
b)
dy =
a
ey
c)
dy =
2
2
dt +
2
2e2y
(S,t)
S
(S,t)2
2S 2
dt +
(S,t)
S dZt ,
ey dZt .
dt +
ey dZt .
d t + dZt .
Question 20.2.
If y = S 2 then
Chapter 19
Monte Carlo Valuation
Question 19.1.
The histogram should resemble the uniform density, the mean should be close to 0.5, and the
standard deviation should be close to 1/ 12 = 0.2887.
Question 19.2.
The histogram should be similar to a standard
Chapter 18
The Lognormal Distribution
Question 18.1. The five standard normals are
15+8 15 7+8 15
= .2582,
= 1.8074.
11+8 15
= .7746,
3+8 15
= 1.291,
2+8 15
= 2.582, and
Question 18.2. If z is standard normal, + z is N , 2 hence our five standard no
Chapter 14
Exotic Options: I
Question 14.1. The geometric averages for stocks will always be lower. Question 14.2. The arithmetic average is 5 (three 5's, one 4, and one 6) and the geometric average is (5 4 5 6 5)1/5 = 4.9593. For the next sequence, the a
Chapter 12
The BlackScholes Formula
Question 12.1. You can use the NORMSDIST() function of Microsoft Excel to calculate the values for N(d1) and N (d2). NORMSDIST(z) returns the standard normal cumulative distribution evaluated at z. Here are the interme
Chapter 24
Interest Rate Models
Question 24.1.
a)
F = P (0, 2) /P (0, 1) = .8495/.9259 = .91749.
b)
Using Blacks Formula,
BSCall (.8495, .9009 .9259, .1, 0, 1, 0) = $0.0418.
c)
(1)
Using put call parity for futures options,
p = c + KP (0, 1) F P (0, 1) =
Chapter 13
MarketMaking and DeltaHedging
Question 13.1. The delta of the option is .2815. To delta hedge writing 100 options we must purchase 28.15 shares for a delta hedge. The total value of this position is 1028.9 which is the amount we will initiall
Mathematics 174E: Lecture 1
David Wihr Taylor
UCLA, Los Angeles
January 6, 2014
David Wihr Taylor
Mathematics 174E: Lecture 1
UCLA, Los Angeles
Course Description
This course is a mathematical nance course. Courses with
similar titles are oered in other d
Mathematics 174E: Lecture 7: Mechanics of
Futures Markets and No Arbitrage
David Wihr Taylor
UCLA, Los Angeles
January 17, 2014
David Wihr Taylor
Mathematics 174E: Lecture 7: Mechanics of Futures Markets and No Arbitrage
UCLA, Los Angeles
Closing Out Posi
Basics
Independence
Mathematics 174E: Lecture 3: Discrete Random
Walks
David Wihr Taylor (based on lecture notes by R. Caisch)
UCLA, Los Angeles
January 10, 2014
David Wihr Taylor (based on lecture notes by R. Caisch)
Mathematics 174E: Lecture 3: Discrete
Math 174E, Lecture 1 (Winter 2014)
Mathematics of Finance
Instructor: David W. Taylor
Time/Location: MWF 3:00pm3:50pm in MS 5127.
Text: Options, Futures, and Other Derivatives (8th edition) by John C. Hull.
Instructor: David W. Taylor
Oce Hours: Monday a
UCLA Math 174EExam #2Fall, 2013
1
1. (20 points) This problem has parts (a) and (b)
(a) (8 points) What are the two characteristic properties of a Wiener process?
(b) (12 points) A quantity Q obeys a law dQ = 3dt + 3dx, where dx is a generalized Wiener
pr