The Hong Kong University of Science and Technology
Quantitative Methods for Fixed Income Securities
MATH 4511

Fall 2013
Homework 1. Quantitative Methods for fixed Income Securities
CHAPTER 1 Bond Prices, Discount Factors, and Arbitrage (Tuckman)
For the following problems, we assume that today is May 15, 2001.
1.1. Wri
The Hong Kong University of Science and Technology
MATH 4511

Fall 2015
Swaptions
A swaption is an option to enter into a swap
for a prespecified swap rate in the future.
Let the
T0 maturity of the option
TN T0 life of the underlying swap
k the strike rate
Payoff of s
The Hong Kong University of Science and Technology
MATH 4511

Fall 2015
Amodelfortermrates
rt
t
Letbetheannualizedrateforterm,we
haveasimilarmodel
rt t t t B
where
1, with probability 1/2
B
1, with probability 1/2
11/4/2016
L.Wu
1
MeanandVaraince
rt
Themeanandvariance
The Hong Kong University of Science and Technology
Quantitative Methods for Fixed Income Securities
MATH 4511

Fall 2013
Homework 4. Quantitative Methods for fixed Income Securities
CHAPTER 3 Yield to Maturity (Tuckman)
3.13. In October 15, 2016, the forwardrate curve for quarterly compounding is
f ( i ) 0.0175 0.00125
The Hong Kong University of Science and Technology
Quantitative Methods for Fixed Income Securities
MATH 4511

Fall 2013
Homework 6. Quantitative Methods for fixed Income Securities
Chapter 13 and 31, Hull
1. Prove that the solutions to the equations
S dt (1 r t ) Cdt
S ut (1 r t ) Cut
are
Cut Cdt
, and
S ut S dt
S ut
The Hong Kong University of Science and Technology
Quantitative Methods for Fixed Income Securities
MATH 4511

Fall 2013
Homework 3. Quantitative Methods for fixed Income Securities
CHAPTER 3 YieldtoMaturity
3.9. In October 15, 2015, the spotrate curve for quarterly compounding is
r( i ) 0.0175 0.00125 (i 1) , i 1, 2
The Hong Kong University of Science and Technology
Quantitative Methods for Fixed Income Securities
MATH 4511

Fall 2013
Homework 2 Solution. Quantitative Methods for fixed Income Securities
CHAPTER 24 (Tuckman)
4.1 There are 89 days between February 15, 2001, and May 15, 2001. There are 181 days between
February 15, 2
The Hong Kong University of Science and Technology
Quantitative Methods for Fixed Income Securities
MATH 4511

Fall 2013
Homework 5 Solution. Quantitative Methods for fixed Income Securities
8.2 Since the yield of the 6.5s of August 15, 2004, is assumed to move .9619 basis points for every basis point
move of the 6s of
The Hong Kong University of Science and Technology
Quantitative Methods for Fixed Income Securities
MATH 4511

Fall 2013
Homework 1 Solution. Quantitative Methods for fixed Income Securities
CHAPTER 1 Bond Prices, Discount Factors, and Arbitrage (Tuckman)
For the following problems, we assume that today is May 15, 2001.
The Hong Kong University of Science and Technology
Quantitative Methods for Fixed Income Securities
MATH 4511

Fall 2013
Forward Rate Agreements (FRA)
An financial contract to pay/receive the
difference between a fixed interest rate and
the realized interest rate, LIBOR in particular,
applied to certain notional value.
The Hong Kong University of Science and Technology
Quantitative Methods for Fixed Income Securities
MATH 4511

Fall 2013
2. Using the riskneutral tree above, price $100 face amount of the following 1.5 year collared floater.
Payments are made every six months according to this rule: If the short rate on date i is ri ,
The Hong Kong University of Science and Technology
Quantitative Methods for Fixed Income Securities
MATH 4511

Fall 2013
MATH 361 L1 Quantitative Method for Interest Rate Derivatives
Tutorial 13
10May2007
1. Forward Contract
A forward contract is an agreement to buy or sell a security in the future at a price specifie
The Hong Kong University of Science and Technology
MATH 4511

Fall 2015
SolvingforS (t )
Example
Theexpressionforalognormalstockpriceis
dS (t ) rS (t )dt S (t )dW (t )
LetY(t)=ln[S(t)].AccordingtoItoslemma
dS t 1 dS t
dS t 1 2
d ln S (t )
dt
2
S t 2 S t
S t 2
2
1 2
(
The Hong Kong University of Science and Technology
MATH 4511

Fall 2015
Forward price as a random variable
Any European options can be treated as
option on forward prices.
Forward price is defined by
St
=
Ft
,
d (t , T )
for t T ,
which is also a random variable.
11/20/
The Hong Kong University of Science and Technology
MATH 4511

Fall 2015
Final Review
What Have We Learned?
I.
II.
III.
IV.
V.
Static hedging and databased hedging
Term structures of interest rates
Interestrate or fixedincome derivatives
Binomial or HoLee model
Blacks
The Hong Kong University of Science and Technology
MATH 4511

Fall 2015
A Model for Term Rates
Let rt be the annualized rate for term t , we
have a similar model
where
rt = t t + t B
1, with probability 1/2
B =
1, with probability 1/2
11/7/2017
L. Wu
1
Mean and Varianc
The Hong Kong University of Science and Technology
MATH 4511

Fall 2015
Forward Revisited
A forward contract is allows its counterparties
to buy/sell
certain asset
for certain price
in certain date in the future.
The buy/sell price is chosen such that the
price of th
The Hong Kong University of Science and Technology
MATH 4511

Fall 2015
BlackScholes call formula
The BlackScholes call formula
C0 e rT S0e rT (d1 ) K (d 2 )
=
=S0 (d1 ) Ke rT (d 2 )
Two more questions:
Can it reprise the underlying?
What should be the limit of alp
The Hong Kong University of Science and Technology
MATH 4511

Fall 2015
Two issues
For the binomial option pricing model, as
N t =T and N ,
What is the limiting distribution of S,N ?
What is the limiting value of the option
price?
11/22/2017
L. Wu
1
The Binomial Model
The Hong Kong University of Science and Technology
MATH 4511

Fall 2015
Binomial Model
Regarding bond option
10/17/2017
L. Wu
2
Why we need a model?
When pricing by static replications fails, we
need a dynamical model for the state variable.
Such a model
Describes the
The Hong Kong University of Science and Technology
MATH 4511

Fall 2015
Replication Pricing turned Expectation Pricing
Rewrite the option formula into
C0 S0 +
=
Cut Cdt
S ut Cdt S dt Cut
S0 +
=
u
d
(1 + r t )( S ut S dt )
S t S t
1 S ut S0 (1 + r t ) d S0 (1 + r0 t ) S
The Hong Kong University of Science and Technology
MATH 4511

Fall 2015
More on Swaps
CashFlow Pattern
The dashline arrow means uncertainty
The redline is the artificial principal
1
1
10/19/2017
2
4
L. Wu
8
Pricing of Swaps
The cash flows are generated from
A fixed
The Hong Kong University of Science and Technology
MATH 4511

Fall 2015
Chapter 20: Mortgages and
MortgageBacked Securities
MORTGAGE LOANS
Mortgage loans come in many different
varieties.
Fixed rate or variable rates
Residential or commercial purposes.
Residential mo
The Hong Kong University of Science and Technology
Quantitative Methods for Fixed Income Securities
MATH 4511

Fall 2013
MATH 361 L1 Quantitative Method for Interest Rate Derivatives
Tutorial 11
26Apr2007
1. Interest Rate Models with Constant Volatility
(a) Mean Reversion: The Vasicek Model (Model 4)
i. The Continuous