The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Spring 2015
MATH 4512 Fundamentals of Mathematical Finance
Solution to Homework One
Course Instructor: Prof. Y.K. Kwok
2
1. (a) The portfolio variance P is given by
2
2
2
P = 2 A + (1 )2 B + 2(1 )A B .
2
Dierentiating P with respect to , we have
2
dP
2
2
= 2A 2(1 )B
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH 4512
Fundamentals of Mathematical Finance
Midterm test Spring 2013
Time allowed: 80 minutes
Course instructor: Prof. Y. K. Kwok
[points]
1. Suppose there are only 2 fully negatively correlated risky assets in the portfolio, whose
2
expected rate of
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH 362 Fundamentals of Mathematical Finance
Test One
Fall, 2006 Time allowed: 60 minutes [points] 1. Suppose the financial market provides one riskfree asset and one risky asset. The risky asset either doubles with probability or halves with probability
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH 362 Fundamentals of Mathematical Finance
Solution to Homework Four
Fall, 2006
Course Instructor: Prof. Y.K. Kwok
1. (Capital budgeting)
Project
1
2
BenetCost Ratio
2
5/3
3
3/2
4
4/3
5
5/3
The approximate method based on costbenet ratios implies Pro
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Spring 2013
MATH362 Fundamentals of Mathematical Finance
Topic 2 Capital asset pricing model and factor models
2.1 Capital asset pricing model and beta values
2.2 Interpretation and uses of the capital asset pricing model
2.3 Arbitrage pricing theory and factor model
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Spring 2015
MATH 4512 Fundamentals of Mathematical Finance
Homework Four
Course instructor: Prof. Y.K. Kwok
1. Let c be the coupon rate per period and y be the yield per period. There are m periods
per year (say, m = 4 for quarterly coupon payments), and let n be the
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Spring 2015
MATH 4512 Fundamentals of Mathematical Finance
Homework One
Course instructor: Prof. Y.K. Kwok
1. (Two correlated assets) The correlation coecient between the random rates of return
of assets A and B is 0.1, and other data are given in the following table
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH362 Fundamentals of Mathematical Finance
Homework Three
Course instructor: Prof. Y.K. Kwok
1. In a betting game with m possible outcomes, the gain from a unit bet on i if outcome j
occurs is given by
di if j = i
gij =
, j = 1, 2, , m,
1 if j = i
where
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH 362 Fundamentals of Mathematical Finance
Homework One
Course instructor: Prof. Y.K. Kwok
1. (Two correlated assets) The correlation coecient between the random rates of return
of assets A and B is 0.1, and other data are given in the following table.
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH362 Fundamentals of Mathematical Finance
Homework Two
Course instructor: Prof. Y.K. Kwok
2
1. Suppose there are n mutually uncorrelated assets. The return on asset i has variance i .
The expected rates of return are unspecied at this point. The dollar
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH 4512
Fundamentals of Mathematical Finance
Final Examination Spring 2013
Time allowed: 120 minutes
Course instructor: Prof. Y. K. Kwok
[points]
1. Consider the following onefactor model, where the rate of return ri of the risky asset
i, i = 1, 2, is
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
4. Capital budgeting and optimal management
(4.1) Capital budgeting as linear programming formulation
(4.2) Cash matching problem
(4.3) Dynamic cash ow processes
(4.4) Harmony theorem maximum present value criterion and maximum return criterion
(4.5) Valu
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH 362 Fundamentals of Mathematical Finance
Solution to Homework 5
Fall, 2006
Course Instructor: Prof. Y. K. Kwok
1. () The trading strategy with V0 < 0 and V1 () 0, , dominates the noninvesting trading strategy H = (0 0 0)T as in this case V1 () = V0 <
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH362 Fundamentals of Mathematical Finance
Homework Four
Course instructor: Prof. Y.K. Kwok
1. Show that a dominant trading strategy exists if and only if there exists a trading strategy
satisfying V0 < 0 and V1 () 0 for all .
Hint: Consider the dominan
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH 362 Fundamentals of Mathematical Finance
Homework One
Fall 2008
Course instructor: Prof. Y.K. Kwok
1. (Two correlated assets) The correlation between assets A and B is 0.1, and other data
are given in the following table. [Note: = AB /(A B ).]
r
10.0
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH 362
Fundamentals of Mathematical Finance
Test Two
Course Instructor: Prof. Y. K. Kwok
Fall, 2007
Time allowed: 60 minutes
[points]
1. Consider the meanvariance model, where the minimum variance funds lie on the left boundary
(parabolic curve) of the
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH362 Fundamentals of Mathematical Finance
Homework Two
Fall 2007
Course instructor: Prof. Y.K. Kwok
1. Suppose there are n mutually uncorrelated assets. The return on asset i has variance
2
i . The expected rates of return are unspecied at this point.
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH 362 Fundamentals of Mathematical Finance
Homework 5
Fall, 2006
Course Instructor: Prof. Y. K. Kwok
1. Show that a dominant trading strategy exists if and only if there exists a
trading strategy satisfying V0 < 0 and V1 () 0 for all .
Hint: Consider t
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH362 Fundamentals of Mathematical Finance
Tutorial Note #11
Utility Function and Decision Making
In this section, we will look at several decision problems. Recall that we have three different
kinds of people (with different risk aversion): Risk averse
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH362 Fundamentals of Mathematical Finance
Tutorial Note #12
More on the Stochastic Dominance
Issue 1: Relationship between FSD and SSD
Example 1
Suppose dominates in FSD, then dominates in SSD.
Solution 1: (By using definition)
Note that dominates in F
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH 4512 Fundamentals of Mathematical Finance
Solution to Homework Three
Course Instructor: Prof. Y.K. Kwok
1. If outcome j occurs, then the corresponding gain is given by
Gj =
m
gij i ,
i=1
where i =
1
1
1 + di
m
1
i=1
cfw_
and gij =
di if j = i
.
1 if
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Fall 2015
MATH 4512 Fundamentals of Mathematical Finance
Solution to Homework Two
Course Instructor: Prof. Y.K. Kwok
1. The market portfolio consists of n uncorrelated assets with weight vector (x1 xn )T . Since the assets are uncorrelated, we obtain
2 M
= cov(x1 r
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Spring 2015
MATH4512 Fundamentals of Mathematical Finance
Homework Two
Course instructor: Prof. Y.K. Kwok
1. Suppose there are n mutually uncorrelated assets. The rate of return on asset i has
2
variance i . The expected rates of return are unspecied at this point. T
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Spring 2015
MATH 4512 Fundamentals of Mathematical Finance
Solution to Homework Four
Course instructor: Prof. Y.K. Kwok
1. Recall that
n
ci
1
i
D=
im
B i=1 (1 + y)
(cash ow ci occurs at time
i
years), where
m
B=
n
ci (1 + y)i .
i=1
Taking the derivative of B with re
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Spring 2015
MATH4512 Fundamentals of Mathematical Finance
Homework Three
Course instructor: Prof. Y.K. Kwok
1. In a betting game with m possible outcomes, the gain from a unit bet on i if outcome j
occurs is given by
cfw_
di if j = i
, j = 1, 2, , m,
gij =
1 if j = i
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Spring 2015
MATH 4512 Fundamentals of Mathematical Finance
Topic Four Bond portfolio management and immunization
4.1 Duration measures and convexity
4.2 Horizon rate of return: return from the bond investment over a time
horizon
4.3 Immunization of bond investment
1
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Spring 2015
MATH4512 Fundamentals of Mathematical Finance
Topic 3 Utility theory and utility maximization for portfolio
choices
3.1 Optimal longterm investment criterion log utility criterion
3.2 Axiomatic approach to the construction of utility functions
3.3 Maximu
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Spring 2015
MATH4512 Fundamentals of Mathematical Finance
Topic 2 Capital asset pricing model and factor models
2.1 Capital asset pricing model and beta values
2.2 Interpretation and uses of the capital asset pricing model
2.3 Arbitrage pricing theory and factor mode
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Spring 2015
MATH4512 Fundamentals of Mathematical Finance
Topic 1 Mean variance portfolio theory
1.1 Mean and variance of portfolio return
1.2 Markowitz meanvariance formulation
1.3 Twofund Theorem
1.4 Inclusion of the risk free asset: Onefund Theorem
1.5 Addition
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Spring 2013
MATH4512 Fundamentals of Mathematical Finance
Topic Three Capital asset pricing model and factor models
3.1 Capital asset pricing model and beta values
3.2 Interpretation and uses of the capital asset pricing model
3.3 Arbitrage pricing theory and factor
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Spring 2013
MATH 4512 Fundamentals of Mathematical Finance
Topic One Bond portfolio management and immunization
1.1 Duration measures and convexity
1.2 Horizon rate of return: return from the bond investment over a time
horizon
1.3 Immunization of bond investment
1.4
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Spring 2013
MATH4512 Fundamentals of Mathematical Finance
Topic Four Utility optimization and stochastic dominance
for investment decisions
4.1 Optimal longterm investment criterion log utility criterion
4.2 Axiomatic approach to the construction of utility function
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Spring 2013
MATH 4512 Fundamentals of Mathematical Finance
Solution to Homework Three
Course Instructor: Prof. Y.K. Kwok
1. The market portfolio consists of n uncorrelated assets with weight vector (x1 xn )T .
Since the assets are uncorrelated, we obtain
2
M
= cov(x1
The Hong Kong University of Science and Technology
Fundamentals of Mathematical Finance
MATH 4512

Spring 2013
MATH4512 Fundamentals of Mathematical Finance
Homework Three
Course instructor: Prof. Y.K. Kwok
1. Suppose there are n mutually uncorrelated assets. The rate of return on asset i has
variance i2 . The expected rates of return are unspecied at this point.