thesis

Ι
SCHOOL OF INFORMATION SCIENCES
& TECHNOLOGY
DEPARTMENT OF STATISTICS
POSTGRADUATE PROGRAM
Modelling of interest rates using the Heath-Jarro-
Morton equations with applications in portfolio
theory
By
Sotiraki Anesti Gkermani
A THESIS
Submitted to the Department of Statistics
of the Athens University of Economics and Business
in partial fulfilment of the requirements for
the degree of Master of Science in Statistics
Athens, Greece
July, 2015

Ι
ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ & ΤΕΧΝΟΛΟΓΙΑΣ
ΤΗΣ ΠΛΗΡΟΦΟΡΙΑΣ
ΤΜΗΜΑ ΣΤΑΤΙΣΤΙΚΗΣ
ΜΕΤΑΠΤΥΧΙΑΚΟ
Μοντελοποιήση των επιτοκίων χρησιμοποιώντας τις
εξισώσεις Heath-Jarrow-Morton με εφαρμογές στην
θεωρία χαρτοφυλακίου
Σωτηράκη Ανέστη Γκερμάνι
ΔΙΑΤΡΙΒΗ
Που υποβλήθηκε στο Τμήμα Στατιστικής
του Οικονομικού Πανεπιστημίου ΑΘηνών
ως μέρος των απαιτήσεων για την απόκτηση
Μεταπτυχιακού Διπλώματος Ειδίκευσης στη Στατιστική
Αθήνα
Ιούλιος, 2015

Ι
DEDICATION
To my parents and to a beloved girl

Ι
ACKNOWLEDGEMENTS
I have to thank my supervisor, Professor Athanasios Yannacopoulos for his
guidance and patience through the journey of the present thesis. Also I have to
thank my parents for their help and encouragement all these years.
Ι

Ι
VITA
I have successfully completed the undergraduate program of the department of
Mathematics at the University of Athens and during my last semester I was
trainee at the National Bank of Greece. Almost immediately after I was
accepted in the Athens University of Economics and Business at the full time
postgraduate program of Statistics.
III

Ι
ABSTRACT
Sotiraki Gkermani
Modelling of interest rates using the Heath-Jarro-Morton
equations with applications in portfolio theory
July, 2015
In the present thesis we model the interest rates through the Heath-
Jarrow-Morton equations, and we model the yield of a zero-coupon bond in
infinite dimensional Hilbert spaces as a Itō process. Moreover, we use the
meaning of duality in Hilbert spaces in order to introduce a bond portfolio.
Finally we give necessary and sufficient conditions on the existence of a
optimal portfolio.
V

Ι
ΠΕΡΙΛΗΨΗ
Σωτηράκη Γκερμάνι
Μοντελοποιήση των επιτοκίων χρησιμοποιώντας τις εξισώσεις
Heath-Jarrow-Morton με εφαρμογές στην θεωρία
χαρτοφυλακίου
Ιούλιος, 2015
Στην παρούσα διπλωματική εργασία μοντελοποιούμε τα επιτόκια
χρησιμοποιώντας τις εξισώσεις Heath-Jarrow-Morton και μοντελοποιούμε τις
αποδόσεις των ομολόγων μηδενικού κουπονιού στους απειροδιάστατους
χώρους Hilbert ως διαδικασίες Itō. Επιπλέον, χρησιμοποιούμε την έννοια της
δυικότητας στους χώρους Ηilbert ώστε να εισάγουμε την έννοια του
χαρτοφυλακίου ομολόγων. Τέλος, δίδουμε επαρκείς και αναγκαίες συνθήκες
για την ύπαρξη ένος βέλτιστου χαρτοφυλακίου ομολόγων.
VII

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TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ι
VITA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ΙΙΙ
ABSTRACT. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . IV
ΠΕΡΙΛΗΨΗ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... V
1 Introduction to bond markets 1
1.1 Description of bond markets . . . . . . . . . . . . . . . . . . .. 1
1.2 A view of bonds characteristics and major types of bond
markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Different types of bonds . . . . . . . . . . . . . . . . . . . 5
1.2.2 Zero-coupon bonds. . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Pricing formula using discount . . . . . . .. . . . . . . . . . . . 8
1.3.1 Pricing in a "moving frame" . . . . . . . . . . . . . . . . 9
1.4 Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.1 Yield on a zero-coupon bond . . . . . . . . . . . . . . . 13
1.5 Yield curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5.1 The zero-coupon (or spot) yield curve . . . . . . . . 14
1.5.2 The forward yield curve . . . . . . . . . . . . . . .. . . . . 16
1.6 Bond portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 Curves versus numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Mathematical concepts of the Bond Markets 19
2.1 Random Variables in a function space . . . . . . . . . . . . . . . 19
2.1.1 Random Variables in Banach space . . . . .. . . . . . 19
2.1.2 Random Variable In Hilbert Space . . . . . . . . . . . . 22
2.2 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 Definition of Sobolev Spaces and their uses . . . . . 25
IXIX

Ι
Page
(Continued)
2.2.2 Sobolev spaces
,n p
W , m N and
,
, s Rs p n
W 
and their duals . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Semigroups and the Cauchy Problem . . . . . . . . . . .. . . . . . 36
2.3.1 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.2 The translation semigroup . . . . . . . . . . . . . . . . . . . 39
2.4 Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.1 The Homogeneous Cauchy Problem . . . . . . . . . . . 40
2.4.2 The Inhomogeneous Cauchy Problem . . . .. . . . . . 42
2.5 The Stochastic Cauchy Problem . . . . . . . . . . . . . . . . . . . . . 46
2.5.1 Itō integral and Itō formula . . . . . . . . . . . . . . . . . 47
2.5.2 SDES in Hilbert spaces . . . . . . . . . . . . . . . . . .. . . 48
3 Bond pricing in the Heath-Jarrow-Morton framework 51
4 Portfolio selection 57
4.1 The optimal portfolio problem . . . . . . . . . . . . . . . . . . . . . 62
4.1.1 The case of finite number of random sources . . . . . . . . . 65
4.1.2 The case of infinite number of random sources . . . . .. . . 70
Appendices 75
A Aspects in Functional Analysis 77
B Aspects in Stochastic Calculus in infinite dimensional spaces 83
X

Ι
LIST OF FIGURES
Figure Page
1.1 Absence of fluctuations 10
1.2 Yield of a bond with fluctuations 10
1.3 Greek Yield Curves by Neely (2012 11
XIII

Chapter 1
Introduction to bond markets
1.1 Description of bond markets
"On May 1, 1941, President Franklin D. Roosevelt bought the first of the so-called war bonds
also known as "victory bonds "... By the time the program ended and the last proceeds from the
sale were deposited into the U.S. Treasury on January 3, 1946, $185.7 billion of war bonds had
been sold and over 85 million Americans had invested in them", Crescenzi and el Erian (2002).
In that era the total population of United States of America was 130 million people, that means
65% of the citizen had invested in war bonds. The liquidity of government bonds of United
States had an impact on the result of World War II.
In our days the bond markets are bigger, stronger and even more complex. The origin of
the spectacular increase in the size of bond markets was the rise in oil prices in the early 1970s.
Higher oil prices stimulated the development of a sophisticated international banking system,
as they resulted in large capital inflows to developed country banks from the oil-producing
countries. A significant proportion of this capital was placed in Eurodollar deposits in major
banks. The growing trade deficit and level of public borrowing in United States also contributed
to that. As a result the world bond markets in 1998 was fifteen times greater than they were
in 1970s and at the end of 1998 the outstanding volume stood at over $26 trillion, Choudhry
(2003). The growth of bond markets continued in the next decade and U.S bond markets, at the
end of the third quarter of 2009, had a value of $34.644 trillion, Crescenzi and el Erian (2002).
Bond markets had an extremely rapid growth over the years, the economic growth of world-
wide economy, demands for credit availability because companies need to increase borrowing
to finance their growth and by issuing new bonds. Also many new economies had been formed
during this era and their need for capital led them to the bond markets. Borrowing needs of
government and public sector also increase as the economy grows. The globalization and the
introduction of new financial products were the ones that influenced the most the growth of
financial markets.
In order to understand how the bond markets works we should define what bonds are.
Definition 1.1.1. A bond is a debt capital markets instrument issued by a borrower, who is then
required to repay to the lender/investor the amount borrowed plus coupons, over a specified
period of time.
1

Practically everyone can buy a bond, thus effectively lending capital to the issuer. The role
of lenders in bond markets is extremely important for the whole economy and the monetary
policy. This led the central banks to a series of restrictions to the role of dealers in bond markets.
In the United States there are a small number of dealers that are called primary dealers and they
play an important role in the liquidity of bond markets and also in the monetary policy of the
Federal Reserve of New York (FED). The role of primary dealers led FED to establish very
stringent requirements for obtaining the primary dealer recognition and at the list of current
primary dealers there are banks and social security corporations.
In the procedure of trading bonds the investor/lender and issuers would had a series of
difficulties which made bond markets so elusive. Some of these difficulties in the past were: (i)
the absence of a centralized market place, (ii) the enormous plethora of bonds in the markets,
(iii) the lack of publicity, (iv) the mathematical complicity. The globalization and the extreme
growth of Internet solved the above problems and many more, but the mathematical complexity
still remains as a problem which the economic entities need to deal with.
Bonds are nothing else but contracts that describe debt and they have many similarities but
also significant differences with bank loans. Is the bond markets rather that the equity market
better for investors? The European Council at 2002 had a quite impressive answer to the above
query. They support that bond financing can combine some of the features of equity markets
and bank loans. For investors a bond represents an asset whose yield typically exceeds the
bank deposit rate and whose value, unlike the dividend on equity, is largely independent of
the issuer’s financial performance. Moreover, bonds are exposed only to two risks which are
inflation and default; in contradiction with equity which are subject to more risks. The latter
risk of bonds is asymmetric, as there is a high probability that the issuer will pay back and a
small probability of total loss if the issuer fails.
1.2 A view of bonds characteristics and major types of bond
markets
In order to understand the bond markets we are obliged to look closer at bonds and their key
features. For example the type of issuer, the term to maturity, the principal and coupon rate or
the currency can cluster the bond markets into different fragmentation with unique characteris-
tics on each one.
Let us now give a brief review of the above features of bonds.
• Type of issuer: The nature of the issuer will affect the way the bond is viewed in the
markets, there are four issuers of bonds: sovereign governments and their agencies, local
government authorities, supranational bond issuers such as the World Bank and corpora-
tions.
• Term to maturity: The term to maturity of a bond is the number of years after which
the issuer will repay the obligation. During the term the issuer will also make periodic
interest payments on the debt. The maturity of a bond refers to the date that the debts
cease to exist at which time the issuer will redeem the bond by paining the principal. In
Musiela (1993), ,Ekeland et al. (2005) it is suggested to consider the time to maturity
of a bond which is the time remaining until the maturity date. This view has a lot of
advantages which will be explained in detail in the present thesis.
2

• Principal and coupon rate The principal of a bond is the amount that the issuer agrees
to repay the bondholder on the maturity date. This amount is also referred as redemption
value, maturity value, par value, nominal value or face amount or simply par. The coupon
rate or nominal rate s the interest rate that the issuer agrees to pay each year. The annual
amount of the interest payment made is called coupon.
• Currency Bonds can be issued in virtually any currency. As we will see the largest
volume of bonds in the global markets are denominated in US dollars, the years that
followed the nomination of the single currency in the Eurozone managed to make the
euro a respectable opponent of U.S.A dollar, also major bond markets are denominated
in Japanese yen and sterling, and liquid markets also exist in Australian, New Zealand and
Canadian dollars, Swiss francs1
and other major currencies. The currency of issue may
impact on a bond’s attractiveness and liquidity which is why borrowers in developing
countries often elect to issue in a currency other that their home currency, for example
dollars (or euro 2
) as this will make it easier to place the bond with investors. On the
other hand, if a bond is aimed solely at a country’s domestic investors it is more likely
that the borrower will issue in the home currency.
These key features of bonds can help us to cluster the bond markets for better understanding.
Domestic bonds are issued by borrowers domiciled in the country of issue and in the cur-
rency of the country of issue. On the other side there is another type of bond which can be issued
across national boundaries and can be in any currency, this type of bonds is called Eurobond or
international bonds to avoid confusion with "euro bonds", which are bonds dominated in euros.
As an issue of Eurobonds is not restricted in terms of currency or country, the borrower is not
restricted as to its nationality either. There are also foreign bonds, which are domestic bond
issued by foreign borrowers. An example of foreign bond is a Bulldog, which is a sterling bond
issued for trading in the United Kingdom (UK) markets by a foreign borrower. We can ﬁnd
also foreign bonds in United States (Yankee bonds), in Japan (Samurai bonds), in Switzerland
(Alpine bonds) and in Spain (Matador bonds).
The foundation of the entire domestic bond markets is formed by government bonds. As
their name suggests government bonds are issued by a government or sovereign. The gov-
ernment bond markets is the largest in relation to the markets as whole and that is because
government bonds represent the best credit risk in any markets as people do not expect the
government to go bankrupt, of course this is not always true. We can now take a look on some
of the most important government bond markets.
• United States The United States government bond markets is the largest among other
government bond markets worldwide and we can see that its size is more $18 trillion,
in detail the debt that is held by the Public is $12.922 trillion and the Intergovernmental
Holdings are $5.082 trillion [3]. The United States government bonds usually are called
Treasuries and they are so prominent that their interest rates are used as a benchmark for
rate markets thought the world. Treasury securities are being issued by the U.S. Treasury
Department to meet the funding requirements of the U.S. government. The Treasury
Department issues three different categories of Treasury securities: discount, coupon and
inﬂation linked. Their term of maturity starts from 4 weeks and occasional numbers of
days (the so called oddball) and can go till 30 year bonds. In general U.S government
1
Until January 14, 2015 the Swiss franc was connected with euro, traded for 1.20 per euro.
2
for example Mexico issued this year i.e. 2015 a 1.5 billion euro, maturing at 2115 with a yield of 4.2%
3

bond market is liquid enough and for the majority of people it is thought as the biggest
bond market, although it is not.
• United Kingdom The UK government bond market issues bonds known as "gilt-edged
securities gilts". The gilt market like the Treasury market is a very liquid and transparent
market with prices being very competitive at the beginning of euro. The gilt market ad
removed many of the more esoteric features of gilts such as "tic" pricing and special
exdivedend trading in order to harmonize the market with euro government bonds. The
maximum of the term of maturity of the gilts is 50 years with the first gilt of that maturity
have been issued in May 2005! Gilts still pay coupon on a semi-annually basis. The UK
government also issues bonds known as index-linked gilts whose interest and redemption
payments are linked to the rate of inflation. There are also older gilts with peculiar
features such as no redemption date and quarterly-paid coupons.
• Eurozone The introduction of euro in member states of Eurozone brought fundamental
changes to the Euro area bond markets. The single currency and the implementation of
the Financial Sector Assessment Program (FSAP) have worked toward the emergence of
an integrated framework for investors and issuers alike, instigating growth and develop-
ment on a pan-European level in each segment of the market. Especially the bond market
of Eurozone has become larger, deeper and more integrated than ever before. Although
the connectivity of financial market inside the Eurozone the article 103 of the EU Treaty-
the so-called “no bailout ”clause- states that each EU member is responsible for its own
debt and prohibits member stat from being liable for other member states. That means
that although the decreased interest rates to the whole of Eurozone members, Eurozone
is not a single state but a alliance of different nationalities, each of these is responsible
for its own national government bond market and so the description of the Eurozone
government bond market as a unity fails, and one can just describe each state separately.
For that purpose I will describe in a few words the basic aspects of the biggest economy
inside the Eurozone, the German government bond market. Government bonds in Ger-
many are known as bunds, BOBls or Schatze. These terms refer to the original maturity
of the paper and have little effect on trading patterns. Bunds pay coupon on an annual
basis and are of course now denominated in euros.
Government bond market is one of the largest markets, but it is not the largest. Also it is
accepted that government bonds usually have better rates3
in the financial market, but occasion-
ally one may come across a corporate entity, such as Gasprom in Russia, which was better rated
that Russian government bonds at the beginning of 2000. But the main misconception which
is related with government bond market is the idea that it is the largest market of the national
bond market, the mortgage-backed securities form the largest bond market in the United States
with value $13,296,703 millions at the end of the second quarter of 2014 [see of Governors
of the Federal Reserve System (2015)], at the same period U.S. Treasury bond market has pub-
lic debt at the size close to $12.922 trillion, i.e. the mortgage-backed securities bond market
is the biggest bond market worldwide. Although mortgage-backed securities form the biggest
bond market worldwide, it is not so active, in contradiction with the government bond market..
Average daily trading volume in mortgage-backed securities in 2008 was about $350 billion.
3
Where rating is done by credit rating agencies such as Moody’s, Standard&Poors and Fitch Group which are
the three biggest credit agencies in the world, as of 2013 they hold a collective global market share of "roughly 95
percent". with Moody’s and Standard& Poor’s having approximately 40% each, and Fitch around 15%.
4

The liquidity is vital for the normal and optimal functioning of the market that was generally
accepted by economists, but the last financial crises has proved that liquidity is the most im-
portant part of economic growth. In the United States where the financial markets are both
vast and mature liquidity has historically been quite good, in EU the liquidity defers among the
countries than construct the European Union.
Another important part of bond market is the so called corporate bond market which is
essential for the growth in a capitalistic environment. Corporate bonds are widely held by im-
portant institutions in the financial markets such as insurance companies, pension funds and
foreign entities. Households are also large holders of corporate bonds. The investors in the
corporate bond market are buy-and-hold investors and speculators do not dabble in corporate
bonds much. This keeps volume levels relatively low. For instance in the U.S corporate bond
market the average daily trading volume in 2009, for investment-grade corporate bonds was
around $12.0 billion and for high-yield bond it was around $5.5 billion. The corporate bond
market is quite admissible and understandable for the majority of people because of the popu-
larity of corporations that issues almost in every day basis new bonds. In the United States cor-
porate were an exception in the liquidity of the corporate bond market because of the collapse
of the Lehman Brothers at 2008, and the result was for all of 2008, corporate bond issuance
was around $800 billion, well bellow than the previews year’s record of about $1.2 trillion
(Crescenzi and el Erian (2002)). On the other side of Atlantic Ocean the corporate bond market
before “euro ”was underdeveloped as a result of the banking economy that Europe had. How-
ever the new financial environment that “euro”brought had an impact to the corporate bond
market. As argued by London Economics (2002) in their report to ECOFIN, the euro had a
series of benefits for resident and nonresident investors, which were:
• lower transaction costs, which allow investors to re-balance their portfolios more effec-
tively;
• wider possibilities for risk diversification, which help increase the risk-adjusted rate of
return of a given portfolio;
• increased price transparency, which reduces the perceived risk of asset holdings; and
• Financial innovation stemming from intense competition, which may create more highly
tailored and attractive financial products....
All these brought to the EU a new era of liquid financial market where the investors were
much more for bonds in Eurozone and as a result the corporate bonds had a success in such
period.
1.2.1 Different types of bonds
The Definition 1.1.1 in which is given the description of a bond seems like it is describing a
conventional or plain vanilla bond. In bond market in our days exist a lot of non-conventional
bonds, a lot of variation on vanilla bonds. One of them will concern this thesis. Here we list a
few of the non-conventional bonds [see Choudhry (2003)]
• Floating Rate Notes The bond markets are often called fixed income markets of fixed
interest markets in the UK. Floating rate notes (FRNs) do not have a fixed coupon at all,
but instead link their interest payments to an external reference.
5

• Index-linked bonds Index-linked bonds look like FRNs because its coupon payment
is linked to a specified index. But in contrast of FRNs index-linked bonds may have also
redemption payment linked to a specified index. When governments issues Index-linked
bonds the cash flows are linked to a price index such as consumer or commodity price.
Corporates have issued-linked bonds that are connected to inflation or a stock market
index.
• Amortised bonds This kind of bonds have a common characteristics with bank loans.
Bank loans give to the borrower the option, sometimes the borrower is obligated, to
repay the capital in stages. Issuers of amortised bonds repay portions of the borrowing in
stages during bond’s life, in contradiction of the vanilla bonds which repay on maturity
the entire nominal sum initially borrowed on issue.
• Bonds with embedded options Some bonds include a provision in their offer par-
ticulars that gives either the bondholder and/or the issuer an option to enforce early re-
demption of the bond. The most common type of option embedded in a bond is a call
feature. A call provision grants the issuer the right to redeem all or part of the debt before
the specified maturity date. These kind of bonds usually pay greater interest because the
issuer have the right to change the maturity date of a bond, something that is considered
harmful to the bondholder’s interests. A bond issue may also include a provision that
allows the investor to change the maturity of the bond. This is known as a put feature
and gives the bondholder the right to sell the bond back to the issuer at par on specified
dates. The advantage to the bondholder is that is interest rates rise after the issue date,
thus depressing the bond’s value, the investor can realize par value by putting the bond
back to the issuer. A convertible bond is an issue giving the bondholder the right to
exchange the bond for a specified amount of shares (equity) in the issuing company. This
feature allows the investor to take advantage of favourable movements in the price of
the issuer’s shares. The presence of embedded options in a bond makes valuation more
complex compared to plain vanilla bonds, and will be considered separately.
• Bond warrants A bond may be issued with a warrant attached to it, which entitles the
bond holder to buy more of the bond under specifies terms and conditions at a later date.
An issuer may include a warrant in order to make the bond more attractive to investors.
Warrants are often detached from their host bond and traded separately.
Finally there is another class of bonds known as asset-backed securities. These bonds are
formed from pooling together a set of loans such as mortgages or car loans and issuing bonds
against them. The interest payments on the original loans serve to back the interest payable on
the asset-backed bond.
1.2.2 Zero-coupon Bonds
Zero-coupon bonds are notably used by governments, especially in Europe and corporations.
In the past the government of Denmark was issuing zero-coupon bond with maturities ranging
from 3 months to 10 years. Also Belgium was using zero-coupon bills denominated in euro,
with a maturity of maximum 1 year. Germany is also one of the countries that issue zero-coupon
bonds.
Zero-coupon bonds are useful non-conventional bonds that do not have any coupon pay-
ment at all and are called also as strip. Zero-coupon bonds have only cash flow and the re-
6

demption payment on maturity. This seems very useful financial instruments for economic or-
ganization, such as governments and corporations, because of that they do not have any coupon
payments to invest during the bond’s life. Such bonds are also known as discount bonds be-
cause of the price that is paid on issue and the redemption payment; there is a difference on
these two prices because of the coupons of the bond which is paid totally at the maturity date.
Conventional coupon-bearing bonds can be stripped into a series on individualâ ˘A´Zs cash flows,
which would be traded as separate zero-coupon bonds. This is a common practice in govern-
ment bond markets such as Treasuries or gilts where the borrowing authority does not actually
issue strips, and they have to be created via the stripping process.
For a better understanding of zero-coupon bonds an example would help. Let us sup-
pose that an investor who purchased $1 million dollars, and at the issue date would have paid
$133.600, we can calculate with simple mathematics that the coupon is 6.94% annually with
term of maturity 30 years. The total increase in value in dollar terms is 648%. We can see a
huge difference between face value and the redemption payment.
That difference between face value and redemption payment is the coupons that the issuer
has to pay. If we look closer to the absence of coupon we can see that it has a negative aspect
because the risk of default of issuer lead to the total loss of the face value which in conventional
bonds does not exist since the investor will at least get some of the coupon during the bond life,
that could have been reinvested. Also in United States for example a zero-coupon bondholder
is obligated to pay taxes for the interest during the life of the bond even though these payments
are not received until maturity, that is the regard of Internal Revenue Service. For that rea-
son, in zero-coupon bonds usually invest pension funds, individual retirement accounts (IRAs),
Keoghs and SEP accounts, which are not subject to taxes.
It is important to see that there are some factors that influence the price of a zero-coupon
bond. Some of them are the quality of the bond, the length of time to maturity, the call provi-
sion, market rates of interest, and the yield:
• The issuer’s ability to redeem them at maturity.
• The investor ’s ability to sell them before maturity at a higher price than their purchase
price.
Obviously a zero-coupon bond with good quality has less risk than a speculative, low qual-
ity zero-coupon bond. Usually investors are willing to pay more for a good quality bond. Thus
a positive relationship between quality and price exists. Also that a low quality bond offers a
higher yield than a good quality zero-coupon bond. Investors pay less for a low quality zero-
coupon bond than a high quality zero-coupon bond. Therefore, price is inversely related to
yield. Moreover, the length of time to maturity is related to the yield. The longer the time
to maturity, the lower the price, and the higher the yield. This is because the zero-coupon
bondholder gets the interest payment at maturity.
But if we suppose that the investors want to sell the zero-coupon bond before maturity, then
they always face the risk of a loss in principal due to the extreme volatility of zero-coupon
bonds. They are the most volatile of all bonds. In addition, markups in the pricing of zero-
coupon bonds are high and also vary from dealer to dealer. This makes zero-coupon bonds
expensive to buy and sell.
7

1.3 Pricing formula using discount
We saw above that zero-coupon bonds are expensive and their volatility influences their price.
In this thesis we will see that there is a way to model the volatility, but before adding new
concepts in the pricing formula we will see how zero-coupon and in generally bonds are being
priced in the bond market. It is essential to understand the principles of present and future
value, compound interest and discounting, because they are all connected with bond pricing.
The principles of pricing in the bonds market are exactly the same as those in other financial
markets, which states that the price of any financial instrument is equal to the present value
today of all the future cash flows from the instrument. All the features of bonds that are de-
scribed above influence the interest rate at which a bond’s cash flows are discounted. In this
approach we will see the "traditional" approach of bond pricing, which is made under certain
assumptions in order to keep the analysis simple. But in the next chapters we will see more
sophisticated methods of bonds pricing.
In oder to continue to the "traditional" pricing formula we need to know the bond’s cash
flows before determining the appropriate interest rate at which to discount cash flows.
• Bond cash flows A vanilla bond’s cash flows are the interest payments o coupons that
are paid during the life of the bond, together with the final redemption payment. It is
possible to determine the cash flows with certainty only for conventional bonds of a fixed
maturity.
• The discount rate The interest rate that is used to discount a bond’s cash flows is the rate
required by the bondholder, it is therefore known as the bond’s yield. The yield on the
bond will be determined by the market and is the price demanded by investors for buying
it, which is why it is sometimes called the bond’s return. The required yield for any bond
is the product of economic and political factors, including that yield is being earned by
the bonds of the same class. Yield is always quoted as an annualized interest rate, so that
for a semi-annually paying bond exactly half of the annual rate is used to discount the
cash flows.
The fair price of a bond is the present value of all its cash flows. That is why we need
to calculate the present value of all the coupon interest payments and the present value of the
redemption payment, and the sum of these. In Choudhry (2003) we can see that the price of a
conventional bond that pays annual coupon can be given by:
(1.1) P =
C
(1 + r)
+
C
(1 + r)2
+ ... +
C
(1 + r)N
+
M
(1 + r)N
=
N
n=1
C
(1 + r)n
+
M
(1 + r)N
where:
P is the price
C is the annual coupon payment
r is the discount rate (therefore, the required yield)
N is the number of years to maturity (therefore, the number of interest in an
annually-paying bond; for a semi-annual bond the number of interest periods is
N × 2
8

M is the maturity payment or par value (usually 100 per cent of currency).
We can write 1.1 in a different, but equivalent, way as
(1.2) C
1 − 1
(1+r)N
r
or
C
r
1 −
1
(1 + r)N
At this moment we can understand better the difference between conventional bonds and
zero-coupon bonds. As we mentioned before the zero-coupon bonds do not have any coupon
payments during their life, that means that in the above formula we have C=0 and the price of
the zero-coupon bonds can be calculated by the following formula:
(1.3) P =
M
(1 + r)N
where M and r are as before and N is the number of years to maturity.
As we saw r is associated with the yield of the bond. If we wanted to comment on this
formula with respect to the yield, then we would say that the yield is influencing the price in
the opposite direction, i.e. when the price is raising the yield is decreasing, and vice versa.
1.3.1 Pricing in a "moving frame"
The time to maturity is a variable that influences the bond pricing, since N, which is exactly
the time to maturity, is a variable of equation (1.1). Also the "present" time influences equation
(1.1). With these assumptions we get:
P(t, N) = C
N
i=t
1
(1 + r)i
+
M
(1 + r)N−t
,
Where t is the present time, and P, C, r, N, M, as described in (1.1). Also we can derive
P(t + 1, T + 1) = C
T+1
i=t+1
1
(1 + r)i
+
M
(1 + r)T+1−t−1
= C
T+1
i=t+1
1
(1 + r)i
+
M
(1 + r)T−t
.
We can understand that the price of a bond although it is influenced by time, at the same
moment it is stays that the price does not change. Moreover, in Musiela (1993) was introduced
a parametrization of bond pricing in which the price of a zero-coupon bond depends on the
time to maturity. In Ekeland et al. (2005) made a step forward and described in details this
parametrization, according to them at each time t, there will be a curve (if we suppose that our
time is continuous) S → pt(S), S ≥ 0, where pt(S) is the price of a standard zero-coupon
bond maturing at time T = t + S.
Although time is influencing the bond pricing, at the same time the price of a bond stays
unaffected. That is true because we made a very strong assumption, that was simple and crucial,
the bond markets do not have fluctuations.
In the ideal case we will have the same price curve shifted on the right as it is in the follow-
ing graph.
9

Figure 1.1: Absence of ﬂuctuations
This deterministic point of view lacks of reality, since he price of a bond may evolve as we
illustrate by the following graph:
Figure 1.2: Yield of a bond with ﬂuctuations
We can see here the affect of time in the price curve which cause the changing of the price
curve at each time. In some cases the price curve of the bond may change completely, as we
can see in the case of the Greek 10 years bonds during the period of crisis in the years 2011 and
2012, before and after PSI (Private sector Involvement) at which occurred a "haircut" of Greek
bonds that were held by the Private sector (see. Neely (2012)) .
10

Figure 1.3: Greek Yield Curves by Neely (2012)
The fluctuations of the markets is missing from equation (1.1). In this thesis we will model
the volatility of bond pricing by independent Brownian motions, in order describe more care-
fully the bond pricing.
1.4 Yield
If we are looking to evaluate how much a bond is worth, then the concept of a yield or expected
return is both the most common and useful measure, especially when considered along with
its related measures such as duration and convexity. They can be categorized as: current yield,
simple yield to maturity and redemption yield, although especially in the case of redemption
yields there are many variations to the calculations. We can see them in detail as follows:
• The current yield measures the income an investor would receive on a bond, if it con-
tinues to pay interest at the current rate, as a percentage of the bond. The current yield of
a bond is given by the formula:
(1.4) CY =
g ∗ 100
CP
where:
CY is the current yield
g is the annual coupon rate
CP is the clean price (i.e. not including any accrued interest)
Current yield is also known as a flat, running or interest yield. Current yield ignores any
capital gain or loss that might arise from holding and trading a bond and does not consider
the time value of money. The current yield is useful as a “rough-and-ready”interest rate
calculation; it is often used to estimate the cost of or profit from a short-term holding of
a bond.
11

• The simple yield to maturity is also known as a Japanese yield. It takes into account
the effect of the capital gain or loss on maturity of the bond as well as the current yield.
Unlike the redemption yield calculations any capital appreciation/depreciation is deemed
to occur uniformly over the bond’s life. The simple yield to maturity of a bond is given
by:
(1.5) SY =
g + (C − CP)/L
CP
where
SY is the simple yield to maturity (SY=0.08 for a yield of 8%)
g is the annual coupon rate
CP is the clean price
C is the redemption value
L is the life of maturity in years. This is calculated by taking the
number of days to maturity, excluding any 29 February and dividing by
365.
Simple yield to maturity improves on the current yield calculation as it takes into account
the effect of any gain or loss on redemption. However, it does this in a fairly simplistic
way, which does not allow for the effect of compound interest. The capital gain of loss is
assumed to be the same each year from now until the bond is redeemed.
• The Redemption yield calculation removes the limitations of the current and simple
yield to maturity calculations. It allows for all the expected future cash flows from the
bond. The cash flows are usually from coupon payments and repayments of capital. The
redemption yield of a bond is that discount rate that would make the sum of the present
values of all assumed future cash flows equal to the gross price of the bond. The gross
price is the quoted clean price plus any associated accrued interest. In other words the
redemption yield y is given by the solving an equation of the form below:
(1.6) P =
n
i=1
CFi ∗ vLi
where
P is the gross price (i.e. clean price plus accrued interest)
n is the number of future cash flows
CFi is the ith cash flow
Li is the time in periods to the ith cash flow, taking into account the
market conventions for calculating the fraction of a period (e.g. does the
year have 360 or 365)
v is the discounting factor (i.e v = 1
1+ y
h
)
Redemption yield is clearly much more informative than current yield or simple yield to
maturity but also is has one very important characteristic which is that, often exists some un-
certainty to the exact future cash flow stream. This uncertainty can arise due to the terms of the
bond, or the issuer in unable to fulfill the terms of the issue.
12

We can see that it would be more useful to calculate present values of future cash ﬂows
using the discount rate that is equal to the markets view on where interest rates will be at that
point, known as the forward interest rate. We will deal with forward rates in the following,
in which we will talk also about the forward yield curve, and how it is connected with the
zero-coupon yield curve.
1.4.1 Yield on a zero-coupon bond
We introduced zero-coupon bonds as bonds that have only one cash ﬂow, the redemption pay-
ment on maturity. Hence the name: zero-coupon bond pay no coupon during their life. In
virtually all cases zero-coupon bonds make one payment on redemption, and this payment will
be par. Therefore, a zero-coupon bond is sold at a discount to par and trades at a discount
during its life. In (1.3) we gave a pricing formula for zero-coupon bonds in which the price P
is related with the yield r. The equation (1.3) still uses N for the number of interest periods in
the bond’s life. Because no interest is actually paid by a zero-coupon bond, the interest peri-
ods are known as quasi-interest periods. A quasi-interest period is an assumed interest period,
where the assumption is that the bond pays interest. It is important to remember this because
zero-coupon bonds in markets that use a semi-annual convention will have N equal to double
the number of years to maturity. For annual coupon bond markets, like the Germany’s strips,
N will be equal to the number of years to redemption. But we can rearrange (1.3) for the yield
r:
(1.7) r =
P
M
1
N
− 1
where we use the same notation as in (1.3). Also the redemption yield formula (1.6) for a
zero-coupon bond reduces to:
(1.8) P = C ∗ vN+f1−1
if one treats it, for the purpose of calculating f1 and N, as a bond which pays zero coupon one
a year on an anniversary of the redemption date. This automatically implies that f2 = 0 and
the clean price CP is the same as the gross price P. N + f1 − 1 is just the life L of the bond in
years; hence equation (1.8) may be rewritten as:
(1.9) y =
C
CP
1
L
− 1
1.5 Yield curve
Until now we have seen the main measure of return associated with holding bonds, which is
the yield to maturity or redemption yield. But in bond market we have a large number of bonds
trading at one time, at different yields and with varying terms to maturity. Investors and traders
frequently examine the relationship between the yields on bonds that are in the same class;
plotting yields of bonds that differ only in their term to maturity produces what is known as a
yield curve, which is an important indicator and knowledge source of the state of a debt capital
market.
13

The logical interest of participants in the debt capital market for information about the
future, led them to the yield curve. In which they can see the current shape and level of the
yield curve, as well as what this information implies for the future. For a better understanding
of yield curve we are giving the main uses of it, which are summarized below.
• Setting the yield for all debt market instruments. We can say that the yield curve
fixes the cost of money over the maturity term structure. The yields of government bonds
from the shortest-maturity instrument to the longest set the benchmark for yields for all
other debt instruments in the market. We must be careful with benchmark interest yields
which are not necessary to be synonymous with government yields. In principle, even
corporate bonds could serve as benchmark instruments. In general issuers of debt use the
yield curve to price bonds and all other debt instruments.
• Acting as an indicator of future yield levels. The yield curve assumes certain shapes
in response to market expectations of the future interest rates. For example we can see
the yield curve of Greek government bonds, which is given in Neely (2012) , where they
analyze the yields of Greek government bonds with different maturities. This is the key
role of yield curves, i.e. the analysis of the present shape of the yield curve in an effort
to determine the implications regarding the future direction of market interest rates.
• Measuring and comparing returns across the maturity spectrum. Portfolio managers
use the yield curve to assess the relative value of investments across the maturity spec-
trum. The yield curve indicates the returns that are available at different maturity points
and is therefore very important to fixes-income managers, who can use it to assess which
point of the curve offers the best return relative to other points.
• Indicating relative value between different bonds of similar maturity. The yield
curve can be analyzed to indicate which bonds are cheap or dear to the curve. Placing
bonds relative to the zero-coupon yield curve helps to highlight which bonds should be
bought or sold either outright or as part of a bond spread trade.
• Pricing interest-rate derivative securities. The price of derivative securities revolves
around the yield curve. At the short-end, products such as Forward Rate Agreements
are prices off the futures curves, but the futures rates reflect the market’s view on for-
ward three month cash deposit rates. At the longer end, interest rate swaps are priced
off the yield curve, while hybrid instruments that incorporate an option feature such as
convertibles and callable bonds also reflect current yield curve levels.
1.5.1 The zero-coupon (or spot) yield curve
The zero-coupon (or spot) yield curve plots zero-coupon yields (or spot yields) against term
to maturity. A zero-coupon yield is the yield prevailing on a bond that has no coupons. If
there is a liquid zero-coupon bond market we can plot the yields from these bonds if we wish
to construct this curve, but it is now necessary to have a set of zero-coupon bonds in order to
construct the curve, as we can derive it from a coupon or par yield curve. We can see that in
many markets that do not use zero-coupon bonds, as we mentioned the U.S Treasuries does
not issue zero-coupon bonds, they derive the spot yield from the conventional yield to maturity
yield curve, and of course this is a theoretical zero-coupon (spot) yield curve.
14

We can see some basic concepts about zero-coupon yield curve. We can see that spot
yields must comply with equation 1.10, which assumes annual coupon payments and that the
calculation is carried out on a coupon date so that accrued interest is zero.
(1.10) Pd =
N
n=1
C
(1 + rn)n
+
M
(1 + r)N
=
N
n=1
C ∗ dn + M ∗ dN
where
rn is the spot or zero-coupon yield on a bond with n years to maturity
dn ≡ 1
(1+rn)n = the corresponding discount factor.
In (1.10) r1 is the current one-year spot yield, r2 the current two-year spot yield, and so on.
Theoretically the spot yield for a particular term to maturity is the same as the yield on a zero-
coupon bond of the same maturity, which is why spot yields are also known as zero-coupon
yields.
This last is an important result, as spot yields can be derived from redemption yields that
have been observed in the market. We can see that the spot yield curve, as redemption yield
curve, is used a lot in the market, and that is because of it is viewed as the true term structure of
interest rates because there is no reinvestment risk involved. Also, the yield on a zero-coupon
bond of n years maturity is regarded as the true n-year interest rate. Because the observed
government bond redemption yield curve is not considered to be the true interest rate, analyst
often constructs a theoretical spot yield curve.
We can now see some very interesting results about the discount factors which are derived
from the equation (1.10), in which we could set Pd = M = 100, C = rN and then we would
have
(1.11) 100 = rN ∗
N
n=1
dn + 100 ∗ dN = rN ∗ AN + 100 ∗ dN
where rN is the par yield from a term to maturity of N years, where the discount factor dN is
the fair price of a zero-coupon bond with a par value of $1 and a term to maturity of N years,
and where
(1.12) AN =
n=1
dn = AN−1 + dN
is the fair price of an annuity of $1 per year for N years (with A0 = 0 by convention) we can
see that after simple calculation we can derive from (1.11) and (1.12) the expression for the
N-year discount factor
(1.13) dN =
1 − rN AN−1
1 + rN
The above results give us useful formulae with which we can calculate the discount factor of
a zero-coupon bond, but it also has further uses because we can use them in order to calculate
the interest of a N-period zero-coupon interest rate, which is the true interest rate for an N-year
bond. And that is true because (1.11) discounts the n-year cash flow by the corresponding n-
year spot yield. In other words rn is the time-weighted rate of return on a n-year bond, and as
we said before the spot yield curve is the correct method for pricing or valuing any cash flow,
including an irregular cash flow. Because is uses the appropriate discount factors.
15

1.5.2 The forward yield curve
Most transactions in the market are for immediate delivery, which is known as cash or spot
market, also there is a large market in forward transactions in whose trades are carried out for
a forward settlement date. In the second category of transactions the two parties agree today
a security for cash at a future date, but at a price agreed today. So the forward rate applicable
to a bond is the spot bond yield as at the forward date. That is, it is the yield of a zero-coupon
bond that is purchased for settlement at the forward date. It is delivered today using data from a
present-day yield curve, so is not correct to consider forward rates to be a prediction of the spot
rates as at the forward date. Forward rates use the knowledge that spot interest rates deliver
for the present and imply it for the future behavior of interest rates. Thereafter the forward
yield curve is created by plotting forward rates against term to maturity. Forward rates satisfy
expression:
Pd =
C
(1 + r(0,1))
+
C
(1 + r(0,1))(1 + r(1,2))
+ · · · +
M
(1 + r(0,1)) · · · (1 + r(N−1,N))
=
=
N
n=1
C
N
i=1(1 + r(i−1,i))
+
M
N
i=1(1 + r(i−1,i))
.
(1.14)
where r(n−1,n) is the implicit forward rate (of forward-forward rate) on a one-year bond maturity
in year N. The connection between spot rates and forward yield is illustrated by the following
formula, which is a comparison of (1.10) and (1.14) :
(1 + sn)n
= (1 + r(0,1))(1 + r(1,2))...(1 + r(n−1,n)) ⇒
⇒ (1 + r(n−1,n)) =
(1 + sn)n
(1 + sn−1
=
dn−1
dn
(1.15)
after a simple rearrangement the left formula can be written as:
(1.16) sn = ((1 + r(1,1))(1 + r(2,1)) · · · (1 + r(n,1)))
1
n − 1
where r(1,1), r2,1, r3,1 are the one-period versus two-period, two-period versus three-period
forward rates up to the (n − 1) period versus n-period forward rates.
Forward rates that exist at any one time reﬂect everything that is known in the market up
to that point. In the unbiased expectations hypothesis exists such an implication but that is not
always the case, in fact there is no direct relationship between the forward rate curve and the
spot rate curve. To view the forward rate curve as a predictor of rates is a misuse of it. Assuming
that all developed country markets are at least semi-strong form, to preserve market equilibrium
there can only be one set of forward rates from a given spot rate curve. Nevertheless this does
not mean that such rates are a prediction because the instant after they have been calculated,
new market knowledge may become available that alters the markets view of future interest
rates.
1.6 Bond portfolio
A bond portfolio is the collection of two or more bonds. Although this description of bond port-
folio sounds simple, or naive, that is not the case because of the difﬁculties that are presented
during the process of evaluating it, in which all the characteristics of bonds are now combined
16

and sometimes the one intercepts to the other. The goal that the portfolio manager, or for that
matter any investor has to achieve is to liquidate it at a specified date, the horizon date, in the
future. Needless to say, the portfolio manager would like to have a certain amount of confi-
dence in the amount of money that would be received at this date. As mentioned by Markowitz
(1952), the process of selecting a portfolio may be divided into two stages. The first stage starts
with observation and experience and ends with beliefs about the future performance of avail-
able securities. The second stage starts with the relevant beliefs about future performances and
ends with the choice of portfolio. Moreover these two important observations of Markowitz
can be divided in four questions that any portfolio manager will be challenged to answer:
• Would any coupon received before the horizon date be able to be reinvested at a satisfac-
tory rate?
• Would the proceeds of any redemption amounts received before the horizon date be able
to be
• reinvested at a satisfactory rate?
• What will be the value of the bond portfolio at the horizon date?
• Would it be possible to redeem any outstanding bonds at a â ˘AIJfairâ ˘A˙I value? This can
sometimes be a problem as the market makers may widen the spreads considerably for
all but the most liquid bonds.
All these questions form what is known as the portfolio optimization which was introduced by
Markowitz (1952) and has experienced a long history of controversial discussion. In that paper
Markowitz had approached the optimization problem by writing the return on the portfolio as
a weighted sum of random variables, and his introduction was the expected return which was
a weighted mean and a weighted variance. For that day till now have been seen a lot of dif-
ferent approaches on that matter, all of them had something to add on portfolio management
theory. The most important of all was the introduction of description of bonds with random
variables, i.e. bonds became a mathematical entity which could be evaluated by well-known
mathematical results. Nevertheless, the financial belief about the market did not extinct con-
trary it is supplementing the mathematical formulation for a better description of portfolio and
for a better evaluation of it.
1.7 Curves versus numbers
As we mentioned in 1.3.1 someone can model the bond price following the parametrization
which is given in Musiela (1993), where is suggested to take in account the time to maturity
of a bond, which is different than the time of maturity. In Ekeland et al. (2005) is made a step
forward in that view, working in a "moving frame" where the time to maturity were the basic
variable on which the zero-coupon depends on time, instead of the time of maturity. Formally,
let T be the time of maturity and S time to maturity, i.e. T = S + t and then the bond price
would be the curve S → pt(S). This curve depends on S but at the same time it depends on t.
That means that at each t we will have a curve which is valid until T. It is natural to assume that
we can see that at t ≥ T + =⇒ pt = 0, > 0. That means whenever t changes, the bond
price also changes because for each t we have another curve pt. In that view the bond price
is simply a curve in which we can see the price of a bond at each t until the time of maturity
17

T, also that means bond portfolio is simply a linear functional operating on the space of such
curves. From the financial point of view this can be seen in different perspectives:
1. The static point of view is to consider the portfolio at time t as a linear combination of
standard zero-coupons, each of which has a fixed time of maturity T > t. Such a portfolio
has to be rebalanced each time a zero-coupon in the portfolio comes to maturity.
2. The dynamic point of view is to consider the portfolio at time t as a linear combination
of self-financing instruments, each one with a fixed time to maturity S ≥ 0. Such instru-
ments were introduced in Rutkowski (1999) under the name "rolling-horizon bond", in
Ekeland et al. (2005) is used the term Roll-Overs. Those Roll-Overs behave like stocks,
in the sense that their time to maturity is constant through time, so that their price de-
pends only on the risk they carry, one can then envision a program where portfolios
are expressed as combinations of stocks and Roll-Overs, which are treated in a uniform
fashion.
Nevertheless, this program although it seems easy to understand has some very important
mathematical difficulties. The first one is that the equations of bond prices must be rewritten
in terms of the unbounded operator ∂
∂S
acting on a space H of curves pt and also the space H
must be contained in the space of all continuous functions. In the following we will explain the
necessary mathematical instruments that will describe the bond price (or the curve) pt and also
it will describe the space where these prices belong.
18

Chapter 2
Mathematical concepts of the Bond
Markets
2.1 Random Variables in a function space
2.1.1 Random Variables in Banach space
Under the hypothesis that we have a continuous time bond market, we concluded that the best
way to describe the bond price is as a curve that changes continuously. In order to calculate the
risk that a bond carries and the finding of the best procedure that we have to follow in order to
construct the best portfolio we need to develop probability theory in function spaces.
In order to describe random variables a measurable space (Ω, F) is needed, where Ω is a
set and F is a σ -field of subsets of Ω .
Definition 2.1.1. If (Ω, F) and (H, G) are two measurable spaces, then a mapping X : Ω → H
such that the set {ω ∈ Ω : X(ω) ∈ A} = {X ∈ A} belongs to F for arbitrary A ∈ G is called
a measurable mapping or a random variable from (Ω, F) into (E, G).
A random variable is called simple if it takes only a finite number of values. If H is a metric
space, then the Borel σ-field of H is the smallest σ-field containing all closed (or open) subsets
of H; will be denoted as B(H). An H-valued random variable is a mapping X : Ω → H which
is measurable from (Ω, F) into (E, B(E)).
Remark 2.1.2. We assume separability of H to simplify the subtle issues concerning different
types of measurability in infinite-dimensional spaces (according to the Pettis theorem), Roach
et al. (2012).
Definition 2.1.3. We say that a metric space H is separable if there exists a subset D ⊂ H that
is countable and dense.
19

The definition of separability is quite wide and many important spaces in analysis are sep-
arable. Clearly, finite-dimensional spaces are separable, also Lp
(and lp
) spaces are separable
for 1 ≤ p < ∞, also the space C(K) of continuous functions on a compact metric space K is
separable. However, L∞
and l∞
are not separable.
If H is a separable Banach space we shall denote its norm by • and its topological
dual by H∗
. Given x ∈ H and r > 0 we set B(x, r) = {α ∈ H : x − α < r} ,
B(x, r) = {α ∈ H : x − α ≤ r} In Da Prato and Zabczyk (2014) someone can find a series
of Lemmata and Propositions which lead to the construction of the general random variable
and its distribution function, we refer the reader to the excellent monograph of Da Prato and
Zabczyk (2014) for the proofs.
Lemma 2.1.4. Let H be a separable Banach space with norm · and let X be an H-valued
random variable. Then there exists a sequence {Xm} of a simple H-valued random variables
such that, for arbitrary ω ∈ Ω the sequence X(ω) − Xm(ω) is monotonically decreasing
to 0.
Let H be a collection of subsets of Ω. The smallest σ − field on Ω which contains H is
denoted by σ(H) and is called the σ − field generated by H. Analogously, {Xi}i∈I which
is supposed to be a family of mappings from Ω into H, can generate the smallest σ − field
σ(Xi : i ∈ I) on Ω such that; all functions Xi are measurable from (Ω, σ(Xi : i ∈ I)) into
(H, G). A collection H of subsets of Ω is said to be a π − system if ∅ ∈ H and if A, B ∈ H
then A ∩ B ∈ H. In order to prove that a given mapping or a given set is measurable we can
use the following proposition.
Proposition 2.1.5. Assume that H is a π − system and let G be the smallest family of subsets
of Ω such that
i). H ⊂ G
ii). if A ∈ G then Ac
∈ G
iii). if A1, A2, ... ∈ G and An ⊂ Am = ∅ for n = m, then ∪∞
n=1An ∈ G
Then G = σ(H)
Proposition 2.1.6. Let H be a separable Banach space. Then B(H) is the smallest σ − field
of subsets of H containing all sets of the form
(2.1) {x ∈ H : ϕ(x) ≤ α}, ϕ ∈ H∗
, α ∈ R
By Proposition 2.1.6 it follows that, if H is a separable Banach space, then a mapping X :
Ω → H is an H-valued random variable if and only if, for arbitrary ϕ ∈ H∗
, ϕ(X) : Ω → R1
is an R1
-valued random variable. On the measurable space (Ω, F) we can define a σ-additive
function P from F into [0,1] such that P(Ω) = 1. Then the triplet (Ω, F, P) is called a
probability space. A measure on (Ω, F) is determined by its values on an arbitrary π-system H
which generates F. We have in fact the following result.
20

Proposition 2.1.7. Let P1 and P2 be probability on (Ω, F), and let H be a π-system such that
σ(K) = F. If P1=P2 on H, then P1=P2 on F.
We can now introduce the distribution of a random variable X from (Ω, F) into (H, G) with
probability measure P on Ω, as the image L(X) of P under the mapping P:
(2.2) L(X)(A) = P(ω ∈ Ω : X(ω) ∈ A), ∀A ∈ G
We can define the integral of a simple H-valued random variable X on (Ω, F)), where H is an
separable Banach space and: X = N
i=1 xiχAi
, Ai ∈ F, xi ∈ H Then for the integral of X we
set
(2.3)
B
X(ω)P(dω) =
B
XdP =
N
i=1
xiP(Ai ∩ B)
for all B∈ F, and χAi
denotes the indicator function of Ai. We can now define the integral of
a general random variable by the help of the following lemma.
Lemma 2.1.8. Let H be a separable Banach space and let X be an H-valued random variable
on (Ω, F). Then the real valued function X(·) .
Then X is said to be Bochner integrable or integrable if
(2.4)
Ω
X(ω) P(dω) < ∞
By Lemma 2.1.4 we get a sequence {xm} of simple random variables that the sequence {
X(ω) − Xm(ω) } decreases to 0 for all ω ∈ Ω and by Lemma 2.1.8 we have that the real
valued function X(·) is measurable, thus we have the following result:
Ω
Xm(ω)P(dω) −
Ω
Xn(ω)P(dω) ≤
≤
Ω
X(ω) − Xm(ω) P(dω) +
Ω
X(ω) − Xn(ω) P(dω)
as m,n→∞
−−−−−−→ 0
Therefore the integral of X can be defined by
Ω
X(ω)P(dω) = lim
n→∞ Ω
Xn(ω)P(dω).
The integral Ω
XdP will be denoted by E(X) or EP (X) and will be called the expectation of
X. The Bochner integral has many common properties with the Lebesgue integral.
For sequences of random variables we have the following concepts of convergence
1. Xn → X, P − a.s. if P({ω : Xn(ω) X(ω)}) = 0
2. Xn → X in L2
(Ω, F, P; H) if E( Xn − X 2
) → 0
The next proposition gives an important result which is generally used in probability theory.
21

Proposition 2.1.9. Assume that H is a separable Banach space. Let X be a Bochner integrable
H-valued random variable defined on (Ω, F, P) and let G be a σ-field contained in F. There
exists a unique, up to a set of P-probability zero, integrable H-valued random Z, measurable
with respect to G such that
(2.5)
A
XdP =
A
ZdP, ∀A ∈ G
The random variable Z will be denoted as E(X|G) and called the conditional expectation of X
given G
Finally, the last concept that will be needed is the concept of independence. Let {F}i∈I
be a family of sub σ-fields of F. These σ-fields are said to be independent if, for every finite
subset J ⊂ I and every family {Ai}i∈J such that Ai ∈ Fi, i ∈ J,
(2.6) P(∩i∈J Ai) =
i∈J
P(Ai)
The concept of independence affects a variety of elements in probability theory and in gen-
eral measure theory, from events to σ-algebras and from that to random variables, through the
concept of the σ-algebra generated by a random variable. From the equation 2.10 we can see
that:
1. If X1, X2 are independent then E((X1, X2)) = (E(X1), E(X2)), or
equivalently cov(X1, X2) = 0
2. If X is independent of the σ-algebra G, then E(X|G) = E(X)
2.1.2 Random Variable In Hilbert Space
To go beyond the first moment, and to consider generalization of operators such as the covari-
ance or correlation we need to restrict our viewpoint and consider the special case of a random
variable in Hilbert spaces. Let U and H be separable Hilbert spaces and let L(U,H) be the space
of all linear bounded operators from U into H. However if both spaces are infinite dimensional,
then L is not a separable space. The non-separability of L has several consequences, where the
most important turns out to be the non-measurability, because the corresponding Borel σ-field
is very rich to the extent that very simple L-valued functions can not be measurable. At this
point, the aforementioned (see Remark 2.1.2) Pettis theorem in which is introduced a weaker
concept of measurability, is needed.
If we constrain ourselves to smaller spaces, such as the space L1(U, H) of all nuclear op-
erators from U into H or the space L2(U, H) of all Hilbert-Schmidt operators from U into H,
then we avoid the problem of non-measurability. In that perspective we are going to describe
the above mentioned classes of operators, which are going to help us to define the covariance
of a random variable and correlation of two random variables.
Let l : U → H be a bounded linear operator in the space L(U,H). The adjoint operator l∗
is
an element of L(U,H) such that:
(2.7) (lx, y) = (x, l∗
y), ∀x ∈ U, y ∈ H.
22

Definition 2.1.10. 1. An operator l ∈ L(U, H) is called a nuclear operator if there exists a
sequence vn ∈ H and a sequence un ∈ U such that
(2.8) lx =
∞
n=1
vn(un, x)U ∀x ∈ U,
∞
n=1
vn H un U < ∞
2. Let U=H. A nuclear operator l that is non-negative (i.e. (lu, u) ≥ 0 ∀u ∈ U) and
symetric (i.e. (lu, v) = (u, lv) ∀u, v ∈ U) is called a trace class operator.
The following proposition is a very useful property of nuclear operators.
Proposition 2.1.11. Let l : U → U be a nuclear operator and let en be an orthonormal basis
of U. Define the trace of the operator l as the infinite series Tr(l) := ∞
n=1(len, en). Then
Tr(l) is a well-defined finite quantity and independent of the choice of the orthonormal basis
en.
Trace class operators can be thought as the generalization of the covariance matrix in infinite
dimensions. The solution of the eigenvalue problem for trace class operators provides us with
an orthonormal basis for the Hilbert space U.
An interesting subclass of nuclear operators consists of the Hilbert-Schmidt operator.
Definition 2.1.12. A bounded linear operator l : U → H is called a Hilbert-Schmidt operator
if ∞
n=1 len
2
< ∞, where en is an orthonormal basis of U.
If we define the inner product, as it follows, the space of Hilbert-Schmidt operators can be
turned into a separable Hilbert space.
(l1, l2)L2(U,H) =
∞
n=1
(l1en, l2en)
.
The definition of the "square root" of a trace class operator can be achieved by the following
proposition.
Proposition 2.1.13. If l : U → U is a trace class operator, then there exists a unique Hilbert-
Schmidt operator R such that R ◦ R = l. We will use the notarion R = l
1
2 . Furthermore,
l 2
L2(U)= Tr(l).
The operator l
1
2 has the useful property that L ◦ l
1
2 ∈ L2(U, H) for any L ∈ L(U, H). We
will need the following result on measurable decomposition of an L1(U, U) = L1(U) valued
random variable.
Proposition 2.1.14. Let U be a separable Hilbert space and assume that Φ is an (L1(U), B(L1(U)))
random variable on (Ω, F) such that Φ(ω) is a nonnegative operator for all ω ∈ Ω. Then there
23

exists a deceasing sequence {λn} of nonnegative random variables and a sequence {gn} of
U-valued random variables such that 1
(2.9) Φ(ω) =
∞
n=1
λn(ω)gn(ω) ⊗ gn(ω), ω ∈ Ω.
Moreover the sequences λn and gn can be chosen in such a way that:
(gn)(ω) =



1 if λn(ω) > 0,
0 if λn(ω) = 0,
and
(2.10) < gn(ω), gm(ω) >= 0 ∀ n = m and ∀ω ∈ Ω
The proof of the proposition is based on the following classical result of Rutkowski (1999).
Lemma 2.1.15. Let E be a compact metric space and let ψ : E × Ω → R1
be a mapping
such that ψ(x, ·) is measurable for arbitrary x ∈ E and ψ(·, ω) is a continuous mapping for
arbitrary ω ∈ Ω. Then there exists an E-valued random variable X:Ω → E such that
(2.11) ψ(X(ω), ω) = sup
x∈E
ψ(x, ω), ω ∈ Ω
If X,Y ∈ L2
(Ω, F, P:H) and H is a Hilbert space, with inner product < ·, · >, we deﬁne
the covariance operator of X and the correlation operator of (X,Y) by the formulae
Cov(X) = E(X − E(X)) ⊗ (X − E(X)),
and
Cor(X, Y ) = E(X − E(X)) ⊗ (Y − E(Y ))
Cov(X) is a symmetric positive and nuclear operator and
TrCov(X) = E(|X − E(X)|2
)
In fact if {ek} is a complete orthonormal basis in H and, for simplicity E(X)=0 we have
TrCov(X) =
∞
h=1
< Cov(X)eh, eh >=
=
∞
h=1 Ω
| < X(ω), ek > |2
P(dω) = E|X|2
1
Fro arbitrary a, b ∈ H we denote by a ⊗ b the linear operator deﬁnes by (a ⊗ b)h = a < b, h >,h ∈ H
24

2.2 Sobolev Spaces
If we take account on previous considerations about the bond market, and the mathematical
representation of bonds we conclude that the state-space of bonds (precisely the state-space
of zero-coupon bonds) must be an infinite-dimensional topological vector space. In order to
choose the proper state space we quote the considerations of Ekeland et al. (2005):
(a) H is a space of continuous functions going to zero at infinity, because zero-coupon
bond prices are continuous with respect to time to maturity and they tend to zero as time
to maturity tends to infinity.
(b) H should be a Hilbert space, because it is the simplest possible infinite-dimensional
topological vector space.
(c) pt(S) must be differentiable with respect to S at S = 0, so that the spot interest rate is
well defined.
(d) pt(S) should be positive for all S > 0 and pt(0) = 1.
(e) pt(S) should be decreasing with respect to S.
From conditions (a) and (b) we can derive that the proper state space H should be a Sobolev
space such as Hs
((0, ∞)) with s > 1
2
. The next three conditions are required by the realistic
view of bond, which will be achieved by the proposal of a model that illustrates the risks that
bonds carries in the portfolio creation. In that model condition (c) and (d) will be satisfied. In
contradiction with the condition (e) which wont be, because of the necessity to include simple
Gaussian interest rate models. Also the state space of portfolio at each time H∗
, which is the
dual of the zero-coupon bond state space, will contain measures as it shall. If wanted, we can
now choose H such that portfolios have certain regularity properties, for example, such that
derivatives of measures are not elements of H∗
An introduction to Sobolev spaces is needed in order to understand their necessity in the
description of bond pricing.
2.2.1 Definition of Sobolev Spaces and their uses
Our first hypothesis that the observed yield do not present sudden jumps, implies that the yield
should be modeled as a continuous function of time. However the continuity may not be enough
for the purpose of bond portfolio selection and we have to consider functions of a certain degree
of smoothness. Stronger notion of smoothness is that of differentiabillity, and yet a stronger
notion of smoothness is that the derivative also be continuous, i.e the function must be C1
.
Nonetheless in the twentieth century, it was observed that the space C1
is not the proper space
to study solutions of differential equations. The modern view in this problem is the replacement
of these spaces in which to look for solutions of partial differential equations, more precisely
Sobolev spaces are related to functions that are differentiable but in a weak sense, and they
provide a solution on that manner.
In order to understand Sobolev spaces why are so useful in the description of bond pricing
as well as the bond market, we should take a look on what functions belong into Sobolev spaces.
Definition 2.2.1. The Sobolev Space W1,p
(I) is defined to be
25

W1,p
(I) = {f ∈ Lp
(I); ∃g ∈ Lp
(I) such that I
fϕ = − I
gϕ, ∀ϕ ∈ C1
c (I)}.
We set
H1
(I) = W1,2
(I)
.
For
u ∈ W1,2
(I)
we denote f = g.
Remark 2.2.2. In the above definition φ is called a testfunction, and it could equally be a
C∞
c (I) function instead of a C1
c (I). That equality is clear if ϕ ∈ C1
c (I), then ρ ϕ ∈ C∞
c (I) 2
for n large enough and ρ ϕ → ϕ in C1
.
Remark 2.2.3. It is clear that if f ∈ C1
(I) ∩ Lp
(I) and if f ∈ Lp
(I) (at this point f is
the classical derivative of f) then f ∈ W1,p
(I). Moreover, the classical derivative of f and
its W1,p
sense derivative coincide, so the notation is consistent! In particular if I is bounded,
C1
(I) ⊂ W1,p
(I), ∀ 1 ≤ p ≤ ∞.
Remark 2.2.4. Someone can use the language of distributions in order to define the W1,p
.
By doing that, all functions f ∈ Lp
(I) admit a derivative in the sense of distributions; this
derivative is an element of the huge space of distributions D (I). We say that f ∈ W1,p
(I) if is
distributional derivative happens to lie in Lp
, which is a subspace of D (I). When I = R and
p = 2, Sobolev spaces can also be defined using the Fourier transforms.
In the previews section we quoted a series of Propositions and Lemmas where became
obvious the necessity of working in Banach spaces and more precisely in separable Hilbert
spaces in order to have nuclear and Hilbert-Schmidt operators. For that manner the following
proposition proves that the above defined Sobolev space is separable Banach space and H1
is
separable Hilbert space.
Proposition 2.2.5. The space W1,p
, is a Banach space for 1 < p < ∞ equipped with the
norm.
f W1,p = f Lp + f LP .
or sometimes, if 1 < p < ∞, with the equivalent norm
( f p
Lp + f p
Lp )
1
p .
2
See Theorem A.0.17 for the definition of the convolution product .
26

It is reflexive3
for 1 < p < ∞ and separable for 1 ≤ p < ∞. The space H1
is a separable
Hilbert space equipped with the scalar product.
(f, g)H1 = (f, g)L2 + (f , g )L2 =
b
a
(fg + f g )
and with the associated norm
f H1 = ( f 2
L2 + f 2
L2 )
1
2
Proof. (a) Let (un) be a Cauchy sequence in W1,p
; then (un) and (un) are Cauchy se-
quences in Lp
. It follows that un converges to some limit u in Lp
and un converges to
some limit h in Lp
. We have
I
unϕ = −
I
unϕ ∀ϕ ∈ C1
c (I),
and in the limit
I
uϕ = −
I
gϕ ∀ϕ ∈ C1
c (I),
Thus u ∈ W1,p
, u = g and un − u W1,p → 0.
(b) W1,p
is reflexive for 1 < p < ∞. Clearly, the product space E = Lp
(I) × Lp
(I)
is reflexive. The operator T : W1,p
→ E defined by Tu = [u, u ] is an isometry from
W1,p
into E. Since W1,p
is a Banach space, T(W1,p
) is a closed subspace of E. It follows
that T(W1,p
) is reflexive (Proposition 3.20, Brezis (2010) ) . Consequently W1,p
is also
reflexive.
(c) W1,p
is separable for 1 < p < ∞. Clearly, the product space E = Lp
(I) ×
Lp
(I) is separable. thus T(W1,p
) is also separable (Proposition 3.25 in Brezis (2010))
Consequently W1,p
is separable.
A conclusion of the Definition 2.2.1 of W1,p
may be that if a function u belongs to W1,p
then
all functions v such that v = u a.e. onI also belong to W1,p
. The following theorem proves
that the class of functions in W1,p
is much wider because every function u ∈ W1,p
admits one
(and only one) continuous representative on I, i.e. there exists a continuous function on I that
belongs to the equivalence class of u (v u if v = u a.e.)
Theorem 2.2.6. Let u ∈ W1,p
(I) with 1 ≤ p ≤ ∞, and I bounded or unbounded then there
exists a function û ∈ C(I) such that
u = û a.e. on I
3
see A.0.16 for the definition of a reflexive space.
27

and
û(x) − û(y) =
x
y
u (t)dt ∀x, y ∈ I.
Remark 2.2.7. It follows from Theorem 2.2.6 that if u ∈ W1,p
and if u ∈ C(I)(i.e. u admits
a continuous representative on I), then u ∈ C1
(I); more precisely û ∈ C(
I), but in order to
simply the notation we also write u for its continuous representative.
Proof. of Theorem 2.2.6 Fix y0 ∈ I and set u(x) =
x
y0
u (t)dt. By LemmaA.0.21 we have
I
uϕ = −
I
u ϕ ∀ϕ ∈ C1
c (I).
Thus I
(u − u)ϕ = 0 ∀ϕ ∈ C1
c (I). It follows from Lemma A.0.20 that u − u = C a.e. on I.
The function û = u(x) + C has the desired properties.
The following proposition defines the functions u ∈ Lp
where 1 < p < ∞ that belong to
W1,p
.
Proposition 2.2.8. Let u ∈ Lp
with 1 < p < ∞. The following propeerties are equivalent:
(i) u ∈ W1,p
,
(ii) there is a constant C such that
|
I
uϕ |≤ C ϕ Lp (I) ∀ϕ ∈ C1
c (I).
Furthermore, we can take C = u Lp(I) in (ii)
Proof. (i) ⇒ (ii). This is obvious.
(ii) ⇒ (i). The linear functional
ϕ ∈ C1
c (I) →
I
uϕ
is defined on a dense subspace of Lp
(since p < ∞) and it is continuous for the Lp
norm.
Therefore it extends to a bounded linear functional F defined on all of Lp
(applying the Hahn-
Banach theorem, or simply extension by continuity). By the Riesz representation theorems
(Theorems 4.11 and 4.14 Brezis (2010)) There exists g ∈ Lp
such that
< F, ϕ >=
I
gϕ ∀ϕ ∈ Lp
.
In particular
I
uϕ =
I
gϕ ∀ ∈ C1
c
and thus u ∈ W1,p
.
28

Remark 2.2.9. (absolutely continuous functions and functions of bounded variation).
In Proposition2.2.8 if p = 1 then the functions u that satisfy (i) are called the absolutely
continuous functions and we can see that these functions satisfy (ii). But the converse (i.e.
((i) ⇒ (ii))) it is not always true. The functions u that satisfy (ii) with p = 1 are the so called
functions of bounded variation, and they need not to have a continuous representative i.e.
by Theorem 2.2.6 they do not belong to W1,1
.
Proposition 2.2.10. A function u in L∞
(I) belongs to W1,∞
if and only if there exists a con-
stant C such that
| u(x) − u(y) |≤ C | x − y | for a.e. x, y ∈ I.
Proof. If u ∈ W1,∞
(I) we may apply Theorem 2.2.6 to deduce that
| u(x) − u(y) |≤ u L∞ | x − y | for a.e. x, y ∈ I.
Conversely, let ϕ ∈ C1
c (I). For h ∈ R, with | H | small enough, we have
I
[u(x + h) − u(x)] ϕ(x)dx =
I
u(x) [ϕ(x − h) − ϕ(x)] dx
(these integrals make sense for h small, since ϕ is supported in a compact subset of I). Using
the assumption on u we obtain
|
I
u(x) [[ϕ(x − h) − ϕ(x)]] dx |≤ C | H | ϕ L1 .
Dividing by | h | and letting h → 0, we are led to
I
uϕ |≤ C ϕ L1 ∀ϕ ∈ C1
c (I).
We may now apply Proposition 2.2.8 and conclude that u ∈ W1,∞
.
In Remark 2.2.4 we mentioned that Sobolev spaces can be deﬁned through the Fourier
transformation. In that point of view we must extend a function u : I → R to a function
u : R → R in order for the Fourier transformation to be well deﬁned4
using the obvious
notation W1,p
(R). For that purpose we give the following Theorem.
Theorem 2.2.11. (extension operator) Let 1 ≤ p ≤ ∞. There exists a bounded linear
operator P : W1,p
(I) → W1,p
(R), called an extension operator, satisfying the following prop-
erties:
4
If u is extended as 0 outside I then the resulting function will not, in general, be in W1,p
(R).
29

(i) Pu|I = u ∀u ∈ W1,p
(I),
(ii) Pu Lp(R)≤ C u Lp(I) ∀u ∈ W1,p
(I),
(iii) Pu W1,p(R)≤ C u W1,p(I) ∀u ∈ W1,p
(I),
where C depends only on | I |≤ ∞. 5
Proof. Beginning with the case I = (0, ∞) we show that extension by reﬂexion
(Pu)(x) = u∗
(x) =



u(x) if x ≥ 0,
u(−x) if x < 0,
works. Clearly we have
u∗
Lp(R)≤ 2 u Lp(I)
Setting
v(x) =



u (x) if x > 0,
−u (−x) if x < 0,
we easily check that v ∈ Lp
(R) and
u∗
(x) − u∗
(0) =
x
0
v(t)dt ∀x ∈ R.
It follows that u∗
∈ W1,p
(R) and u∗
W1,p(R)≤ 2 u W1,p(I).
Now consider the case of a bounded interval I; without loss of generality we can take
I = (0, 1). Fix a function η ∈ C1
(R) , 0 ≤ η ≤ 1, such that
η(x) =



1 if x < 1
4
,
0 if x > 3
4
.
given a function η on (0, 1) set
η(x) =



η(x) if 0 < x < 1,
0 if x > 1.
At this point we shall need Lemma 2.2.12 Given u ∈ W1,p
(I), write
u = ηu + (1 − η)u.
5
One can take C = 4 in (ii) and C = 4(1 + 1
|I| ) in (iii)
30

The function ηu is first extended to (0, ∞) by ηu (in view of Lemma 2.2.12) an then to R
by reflection. In this way we obtain a function v1 ∈ W1,p
(R) that extends ηu and such that
v1 Lp(R)≤ 2 u Lp(I), v1 W1,p(R)≤ C u W1,p(I)
(where C depends on η L∞ ).
Proceed in the same way with (1 − η)u , that is, first extend (1 − η)u to (−∞, 1) by 0 on
(−∞, 0) and then extend to R by reflection (this time about the point 1, not 0). In this way we
obtain a function v2 ∈ W1,p
(R) that extends (1 − η)u and satisfies
v2 Lp(R)≤ 2 u Lp(I), v2 W1,p(R)≤ C u W1,p(I) .
Then Pu = v1 + v2 satisfies the condition of the theorem.
Lemma 2.2.12. Let u ∈ W1,p
(I). Then
ηu ∈ W1,p
(o, ∞) and (ηu) = η u + ηu .
Proof. Let ϕ ∈ C1
c ((0, ∞)); then
∞
0
ηuϕ =
1
0
ηuϕ =
1
0
u [(ηϕ) − η ϕ]
= −
1
0
u ηϕ −
1
0
uη ϕ since ηϕ ∈ C1
c ((0, 1))
= −
∞
0
(u η + uη )ϕ
In 1938, the original Sobolev inequality (an embedding theorem) was published in the cele-
brated paper by S.L. Sobolev ”On a theorem of functional analysis”, see. Sobolev (1938).
The next result is an important prototype of a Sobolev inequality, among the numerous versions
that have appeared since 1938.
Theorem 2.2.13. There exists a constant C (depending only on | I |≤ ∞) such that
(2.12) u L∞(I)≤ C u W1,p(I) ∀u ∈ W1,p
(I), ∀ 1 ≤ p ≤ ∞.
In other words, W1,p
(I) ⊂ L∞
(I) with continuous injection for all 1 ≤ p ≤ ∞.
Further, if I is bounded then
(2.13) the injection W1,p
(I) ⊂ C(I) is compact for all 1 < p ≤ ∞,
(2.14) the injection W1,1
(I) ⊂ Lq
(I) is compact for all 1 ≤ q < ∞.
31

Proof. We start by proving 2.12 for I = R; the general case then follows from this by the
extension theorem (Theorem 2.2.11). Let v ∈ C1
c (R); if 1 ≤ p < ∞ set G(s) =| s |p−1
s. The
funstion w = G(v) belongs to C1
c (R) and
w = G (v)v = p | v |p−1
v .
thus, for x ∈ R, We have
G(c(x)) =
x
∞
p | (t) |p−1
v (t)dt,
and by Hölder’s inequality
| v(x) |p
≤ p v p−1
p v v p,
from which we conclude that
(2.15) v ∞≤ C v W1,p ∀v ∈ C1
c (R),
where C is a universal constant (independent of p).6
Argue now by density. Let u ∈ W1,p
(R); there exists a sequence (u)n ⊂ C1
c (R) such
that un → u in W1,p
(R) (by TheoremA.0.22). Applying (2.15), we see that (u)n is a Cauchy
sequence in L∞
(R). Thus un → u in L∞
(R) and we obtain (2.12).
We now give a proof of (2.13) Let H be the unit ball in W1,p
(I) with 1 < p ≤ ∞. For
u ∈ H we have
| u(x) − u(y) |=|
x
y
u (t)dt |≤ u p| x − y |1/p
≤| x − y |1/p
∀x, y ∈ I.
It follows then from the Ascoli-Arzela theorem (Theorem 4.24 in Brezis (2010) ) that H has a
compact closure in C(I)
We ﬁnally prove 2.14. Let H be the unit ball in W1,1
(I). Let P be the extension operator
of Theorem2.2.11 and let F = P(H), so that H = F|I. We prove that H has a compact closure
in Lq
(I) (for all 1 ≤ q < ∞) by applying Theorem 4.26 of Brezis (2010) . Clearly, F is
bounded in W1,1
R; therefore F is also bounded in Lq
(R), since it is bounded both in L1
(R)
and in L∞
(R). We now check Condition 22 of Chapter 4 in Brezis (2010), i.e.,
lim
h→0
τh
7
f − f q= 0 uniformly in f ∈ F
6
Noting that p1/p
≤ e1/e
∀p ≥ 1.
7
(τhu)(x) = u(x + h)
32

By Proposition 8.5 in Brezis (2010) we have, for every f ∈ F,
τhf − f L1(R)≤| h | f L1(R)≤ C | h |,
since F is a bounded subset of W1,1
(R). Thus
τhf − f q
Lq(R)≤ (2 f L∞
(R))q−1
τhf − f L1(R)≤ C | h |
and consequently
τhf − f Lq(R)≤ C | h |1/q
,
where C is independent of f . The desired conclusion follows since q = ∞.
Remark 2.2.14. The injection W1,1
(I) ⊂ C(I) is continuous but it is never compact even if I
is a bounded interval. Nevertheless, if (un) is a bounded sequence in W1,1
(I) (with I bounded
or unbounded) there exists a subsequence (unk
) such that unk
(x) converges for all x ∈ I (this is
Helly’s selection theorem). When I is unbounded and 1 < p ≤ ∞ we know that the injection
W1,p
(I) ⊂ L∞
(I) is continuous; this injection is never compact. However, if (un) is bounded
in W1,p
(I) with 1 < p ≤ ∞ there exist a subsequence (unk
) and some u ∈ W1,p
(I) such that
unk
→ u in L∞
(J) for every bounded subset j of I.
2.2.2 Sobolev spaces Wm,p
, m ∈ N+ and Ws,p
, s ∈ Rn
and their duals
Someone could define the Sobolev Spaces Wm,p
, m ∈ N+ naturally through the Definition
2.2.1 with the use of partial derivatives i.e.
I
uDj
ϕ = (−1)j
I
gjϕ ∀ϕ ∈ C∞
c (I), ∀j = 1, 2, ..., m,
where Dj
ϕ denotes the jth derivative of ϕ, u ∈ Wm,p
(I) and gm ∈ Lp
(I), which by the
above equation can derived that u = g1, (u ) = g2, ..., up to order m. They are denoted by
Du, D2
u, ..., Dm
u. The space Wm,p
(I) is equipped with the norm
(2.16) u Wm,p = u p +
m
a=1
Da
U p,
and the space Hm
(I) is equipped with the scalar product
(u, v)Hm = (u, v)L2 +
m
a=1
(Da
u, Da
v)L2 =
I
uv +
m
a=1 I
Da
uDa
v.
Someone can extend to the space Wm,p
all the properties shown for W1,p
. We can extend
the definition of Sobolev spaces Wm,p
, m ∈ N+ into Ws,p
, s ∈ R and more precisely we are
33

interested in the Hilbert−Sobolev spaces Hs
= Ws,2
. We will give a Fourier Characterization
of these spaces in the following.
For φ ∈ C∞
c (R) let the Fourier transform be defined by,
TF [φ] = (2π)−1/2
R
φ(x)e−i·x·z
dx = ˆφ(z)
In Prasad and Iyengar (1997) is proved that the Fourier transform is an isometric iso-
morphism on L2
(R), from which we can derive that it is also an isometric isomorphism on
L2
= C∞
c (R) L2
(R), in particular we have
(a) ∀u ∈ L2
u L2 = û L2 (Parseval relation)
(b) ∀u, v ∈ L2
(u, v)L2 = (û, ˆv)L2 (Plancherel relation)
The results (a) and (b) hold for every test function, and since the test functions are dense in
L2
(Rn
), they extend to L2
(Rn
) by continuity. Then the Fourier transform can be extended to
L2
(Rn
) by continuity as well. In addition,
If u, Dα
u ∈ L2
then TF [Dα
u] = (iz)α
û(z) ∈ L2
(2.17)
If u, xα
u(x) ∈ L2
then TF [xα
u(x)] = (iDz)α
û(z) ∈ L2
(2.18)
i.e.
TF [Dα
u] =
Rn
Dα
u(x)e−ixz
dx = (−1)|α|
Rn
u(x)Dα
x (eixz
)dˆx
= (−1)|α|
Rn
u(x)(−iz)α
eixz
dˆx = (iz)α
)û(z);
TF [xα
u(x)] = iα
Rn
u(x)(−ix)α
eixz
dˆx = iα
Rn
u(x)Dα
z (e−ixz
)dˆx = (iDz)α
û(z).
The result (i) asserts that when u is smooth, û decays rapidly at infinity and result (ii) asserts
the converse. This suggests the definition,
Definition 2.2.15.
for s ≥ 0, Hs
= {u ∈ L2
: (1+ | z |2
)s/2
û(z) ∈ L2
}
with u 2
Hs =
Rn
(1+ | z |2
)s
û(z)2
dz,
and (u, v)Hs =
Rn
(1+ | z |2
)s
û(z)ˆv(z)dz.
Then H0
= L2
, and s ≥ t ≥ 0 implies Hs
⊂ Ht
⊂ H0
= L2
It is easy to prove that
∃c1, c2 ∈ R such that c1(1+ | z |2
)m
≤
|α|≤m
| zα
|2
≤ c2(1+ | z |2
)m
34

, then
c1
Rn
(1+ | z |2
)m
| û(z) |2
dz ≤ u Hm(Rn)≤ c2
Rn
(1+ | z |2
)m
| û(z) |2
dz.8
Thus, the norm Rn (1+ | z |2
)m
| û(z) |2
dz is equivalent to the standard norm in
W(m,2)
(Rn
) as it can be derived from (2.16) and someone can prove that the space Hs
(Rn
)
is the closure of C∞
0 (Rn
) with respect to the aforementioned norm.
In the beginning of this account of Sobolev spaces we mentioned that the state space of
portfolios at each time H∗
, would be the dual of the zero-coupon bond state space. We will
define the bond portfolio as a continuous linear functional from H into C therefore its state
space should be the H∗
. Thereby, the dual spaces are that much important in the description of
bond market.
We recall here that the algebraic and topological dual H∗
of a Banach space H is the
linear space of all bounded linear functionals on H which is itself a Banach space with respect
to the norm
l H∗ := sup
v=0
| l(v) |
v H
.
Definition 2.2.16. Let Ω ⊂ Rn
be a bounded domain, let n ∈ N be a negative integer and sup-
pose p ∈ [1, ∞). Then, the Sobolev space W−n,q
(Ω) is defined as the dual space of (Wn,q
(Ω))∗
,
where q is conjugate to p, i.e. 1
q
+ 1
p
= 1.
Correspondingly the dual spaces of Sobolev spaces Hs
, as they are defined in Definition
2.2.15, are H−s
something that it is proved in Theorem 2.2.18 .
Remark 2.2.17. The Dirac δ-function The Sobolev spaces Wn,p
(Ω), n < 0, are proper sub-
spaces of Lp
(Ω). For instance, for n < −d + d
p
, if p < ∞, and n ≤ −d,if p = ∞, they contain
the Dirac δ-function considered as linear functional
(2.19) δ : W−n,p
(Ω) → R , u → δx(u),
where x is some given point in Ω
The following theorem proves that H−s
(Rn
) is dual to Hs
(Rn
) with respect to L2
−duality.
Theorem 2.2.18. Let l(u) be a linear continuous functional on Hs
(Rn
). Then there exists
unique element v ∈ Hs
(Rn
), such that
(2.20) l(u) =
Rn
uvdx, ∀u ∈ Hs
(Rn
),
and
(2.21) l = v Hs .
8
By 2.16 and using the Fourier transformation we have u Hm(Rn)= |α|≤m Rn | Dα
u |2
dx =
Rn ( |α|≤m | z |2
) | û(z) |2
dz
35

Proof. Consider the mapping
As : Hs
(Rn
) → L2
(Rn
), (Asu)(z) = u∗(z) = û(z)(1+ | z |2
)s/2
.
Then u = A−1
s u∗. We define the functional ˜l(u∗) on L2
(Rn
) by the formula
˜l(u∗) = l(A−1
s u∗) = l(u).
Then ˜l is a linear continuous functional on L2
(Rn
). By the Riesz theorem A.0.23 for the func-
tional ˜l there exists unique function w ∈ L2
(R2
) such that
˜l(u∗) =
Rn
u∗(z)w(z)dz, and ˜l = w L2 .
Then l(u) = ˜l(u∗) = Rn û(z)(1+ | z |2
)s/2
w(z)dz.
We denote v(x) = F−1
(w(z)(1+ | z |2
)s/2
). Then
û(z) = w(z)(1+ | z |2
)s/2
;
R2
| û(z) |2
(1+ | z |2
)−s
dz =
Rn
| w(z) |2
dz.
So, v ∈ H−s
, and v H−s = w L2 . We have w(z) = (1+ | z |2
)−s/2
ˆv(z) = (Asv)(z),
l(u) = ˜l(u∗) = Rn (Asu)(z)(A−sv)(z)dz = Rn uvdx.
For the norm of the functional l we have:
l = sup
0=u∈Hs
| l(u) |
u Hs
= sup
0=u∗∈L2
| ˜l(u∗)
u∗ L2
= ˜l = w L2 = v H−s .
2.3 Semigroups and the Cauchy Problem
2.3.1 Semigroups
In this section we introduce the theory of semigroups which is needed in our model of the
mathematical representation of the "moving frame" as it was described in the section 1.3.1.
The analytical theory of semigroups of bounded linear operators in a Banach space (B-space)
deals with the exponential functions in infinite dimensional function spaces. The use of Fourier
characterized sobolev spaces in our model, as a useful tool for that.
The analytical theory of semigroups is concerned with the problem of determining the most
general bounded linear operator valued function T(t), t ≥ 0, which satisfies the equations
T(t + s) = T(t) · T(s),(2.22)
T(0) = I(2.23)
36

The problem was investigated by Hille (1948) and Yosida (1948) independently of each
other. They introduce the notion of the infinitesimal generator A of T(t) defined by
(2.24) A = s − limt→0+ t−1
(T(t) − I)
Where the limit is considered in the strong topology and discussed the generation ot T(t)
in terms of A and obtained a characterization of the infinitesimal generator A in terms of the
spectral properties of A.
This theory comes from the necessity to solve the Cauchy problem and it has application to
stochastic processes9
and to the integration of the evolution equations.
In the following we are going to give a series of definitions in order to define strictly the
aforementioned idea of semigroups.
Definition 2.3.1. Let H be a Banach space. A one parameter family T(t), 0 ≤ t < ∞, of
bounded linear operators from H into H is a semigroup of bounded linear operators on H if:
(i) T(0) = I, (I is the identity operator on H).
(ii) T(t + s) = T(t) ◦ T(s), ∀t, s ∈ R+
, where by "◦" we denote the composition of
operators (the semigroup property).
The linear operator A defined by
(2.25) Ax = limt→0+
T(t)x − x
t
=
d+
T(t)x
dt
|t=0, for x ∈ D(A)
is the infinitesimal generator of the semigroup T(t),
and
(2.26) D(A) = {x ∈ H : limt→0+
T(t)x − x
t
exists in H}
is the domain of A.
The definition of the strong continuity of semigroups, will be helpful afterwards.
Definition 2.3.2. A semigroup T(t), 0 ≤ t < ∞, of bounded linear operators on H is a
strongly continuous semigroup of bounded linear operators if
(2.27) lim
t→0+
T(t)x = x for every x ∈ H,
where the limit is taken in the strong topology.
A strongly continuous semigroup of bounded linear operators on H will be called a semi-
group of class C0
10
or simply a C0 semigroup.
We defined the infinitesimal generator A in (2.25) of T, and its domain D(A) in (2.26). D(A)
is non-empty; it contains at least the vector 0. Actually D(A) is larger, it is proved in Yosida
(1995) that
9
This is another reason to use semigroups in this description of bond market, were the volatilities are supposed
to come from a independent Brownian motion as it was mentioned in section 1.3.1.
10
"C0 is the abbreviation for "Cesàro summable of order 0"" Roach et al. (2012)
37

Theorem 2.3.3. D(A) is dense in H.
In order to define a special class of C0 semigroups, the so called C0 semigroup of contrac-
tion we need the following Theorem.
Theorem 2.3.4. Let T(t) be a C0 semigroup. There exist constants ω ≥ 0 and M ≥ 1 such
that
(2.28) T(t) ≤ Meωt
for 0 ≤ t < ∞.
Proof. We show first that there is an η > 0 such that T(t) is bounded for 0 ≥ t ≥ η. If
this is false then there is a sequence {tn} satisfying tn ≥ 0, limn→∞ tn = 0 and T(tn) ≥ n.
From the uniform boundedness theorem it then follows that for some x ∈ H, T(tn)x is
unbounded contrary to (2.27). Thus, T(t) ≤ M for 0 ≤ t ≤ η. Since T(0) = 1, M ≥ 1.
Let ω = η−1
log M ≥ 0. Given t ≥ 0 we have nη + δ where 0 ≥ δ < η and therefore by the
semigroup property
T(t) = T(δ)T(η)n
≤ Mn+1
≤ MMt/n
= Meωt
Corollary 2.3.5. If T(t) is a C0 semigroup then for every x ∈ H, t → T(t)x is a continuous function
from R+
0 into H.
Proof. Let t, h ≥ 0. The continuity of t → T(t)x follows from
T(t + h)x − T(t)x ≤ T(t) T(h)x − x ≤ Meωt
T(h)x − x ,
and for t ≥ h ≥ 0
T(t − h)x − T(t)x ≤ T(t) T(h)x − x
≤ Meωt
x − T(h)x .
Definition 2.3.6. Let A be a linear, not necessarily bounded, operator in H. The resolvent set
ρ(A) of A is the set of all complex numbers λ for which λI − A is invertible, i.e. (λI − A)−1
is a bounded linear operator in H. The family R(λ : A) = (λI − A)−1
, λ ∈ ρ(A) of bounded
linear operators is called the resolvent of A.
38

Let T(t) be a C0 semigroup. From Theorem 2.3.4 it follows that there are constants
ω ≥ 0 and M ≥ 1 such that T(t) ≤ MeωT
for t ≥ 0. If ω = 0 T(t) is called
uniformly bounded and if moreover M = 1 it is called a C0 semigroup of contractions. In
order to give conditions on the behavior of the resolvent of an operator A, which are necessary
and sufficient for A to be the infinitesimal generator of a C0 semigroup of contractions, we give
the famous Hille − Y osida Theorem.
Theorem 2.3.7. (Hille-Yosida) A linear (unbounded) operator A is the infinitesimal generator
of a C0 semigroup of contractions T(t), t ≥ 0 if and only if
(i) A is closed and D(A) = H.
(ii) The resolvent set p(A) of A contains R+
and for every λ > 0
(2.29) R(λ : A) ≤
1
λ
.
We refer the reader to Pazy (1983) (Theorem 3.1, Chapter 1) for the proof.
2.3.2 The translation semigroup
Definition 2.3.8. For a function f : R → C and t ≥ 0 we call
(Ttf)(s) := f(t + s), s ∈ R
the left translation (of f by t) 11
.
It is immediately clear that the operator (Ttf) satisfy the semigroup property (2.3.1). We
have only to choose appropriate function spaces to produce one-parameter operator semigroups.
Since we insisted that Sobolev spaces are the right function spaces which describe better the
bond market, we shall denote L : [0, ∞) × H → H the semigroup of left translations in H,
where H is the Sobolev space as it is defined in section 2.2:
(Laf)(s) = f(a + s), a ≥ 0, s ≥ 0 and f ∈ H.
We need to see if this shift influences the functions f that belong in the Sobolev spaces.
Since we defined the functions f as those which are weakly differentiable, it is obvious that the
weak derivatives of f are invariant under left translations, that means Sobolev spaces are also
invariant under left translations. Also it is readily verifiable that L is a strongly continuous
contraction semigroup in H. Therefore, it has an infinitesimal generator which we shall denote
11
Respectively (Ttf)(s) := f(s − t) is the right translation (of f by t).
39

by ∂. This can be characterized as follows
lim
α↓0
Lαf(z) − f(z)
α
2
Hs
from2.2.15
= lim
α↓0 R
(1+ | z |2
)s
(
((Lαf)(z) − f(z))
α
)2
dz =
lim
α↓0 R
(1+ | z |2
)s
(
(f(α + z) − ˆf(z))
α
)2
dz = lim
α↓0 R
(1+ | z |2
)s
(
(eiαz ˆf(z) − ˆf(z))
α
)2
dz =
lim
α↓0 R
(1+ | z |2
)s
( ˆf(z)(
eiαz
− 1)
α
)2
dz
from L Hpital s rule
= lim
α↓0 R
(1+ | z |2
)s ˆf2
(z)
∂(e2iαz−2eiαz+1)
∂α
∂α2
∂α
dz =
lim
α↓0 R
(1+ | z |2
)s ˆf2
(z)
ize2iαz
− izeiαz
α
dz
from L Hpital s rule
=
lim
α↓0 R
(1+ | z |2
)s ˆf2
(z)
2i2
z2
e2iαz
− i2
z2
eiαz
1
=
R
lim
α↓0
(1+ | z |2
)s ˆf2
(z)(2i2
z2
e2iαz
− i2
z2
eiαz
) =
R
(1+ | z |2
)s
(iz ˆf(z))2
dz
from 2.17
=
R
(1|z |2
)s ˆf
2
(z)
from 2.2.15
= f Hs
i.e. (Lαf)(z) − f(z)
α↓0
→ ∂f in the strong topology of Hs
. Where in the seventh equality we
use the Lebesgue dominated convergence theorem.
Therefore from (2.26) we obtain the domain
(2.30) D(∂) = {f ∈ H : ∂f := lim
↓0
−1
(L f − f) ∈ Hs
}
The space D(∂) is a Hilbert space with the Sobolev space norm or graph norm,
(2.31) f D(∂)= ( f 2
H + ∂f 2
H)1/2
We can easily see that D(∂) ⊂ H.
2.4 Cauchy Problem
2.4.1 The Homogeneous Cauchy Problem
We will now set up the well known and studied Cauchy problem beginning with the Homoge-
neous Initial Value Problem and how the semigroups are related with it In that case we suppose
that H is a Banach space and A is a linear operator problem from D(A) ⊂ H into H. Given
f ∈ H the abstract Cauchy problem for A with initial data f consists of ﬁnding a solution u(t)
to the initial value problem
(2.32)
du(t)
dt
= Au(t), t > 0
u(0) = x
where by solution we mean an H valued function u(t) such that u(t) is continuous for
t ≥ 0, continuously differentiable 12
and u(t) ∈ D(A) for t > 0 and (2.32) is satisﬁed.13
If A
12
Since our state space of bonds is the space of weakly differentiable functions, the solution of the Cauchy
problems has to be differentiable, in weak sense.
13
It is important to note that since u(t) ∈ D(A) for t > 0 and u is continuous at t = 0, (2.32) cannot have a
solution for x /∈ D(A).
40

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