Modeling Interest-Rate Derivatives

2 MODELING INTEREST-RATE DERIVATIVES
of Osaka. This market is usually considered the distant ancestor of modern
futures markets, which are generally considered to have originated with the
formation of the Chicago Board of Trade (CBOT), in 1848.
In the 1840s, Chicago was becoming the grain transportation and distribu-
tion center of the Midwest of the USA. Farmers shipped their grain from the
farm belt to Chicago for sale and subsequent distribution eastward along rail
lines and through the Great Lakes. However, due to the seasonal nature of
grain production, large quantities of grain were shipped to Chicago in the late
summer and fall. The city’s storage facilities were inadequate and prices fell
drastically as supplies increased, but rose steadily as supplies were consumed.
In 1848, a group of businessmen took the first step towards alleviating
this problem by forming the CBOT. The CBOT was initially organized for
the purpose of standardizing the quantities and qualities of the grains. A
few years later, in 1865, the first forward contract on grain was developed. It
implied that a farmer could agree to deliver the grain at a future date at a price
determined in advance. Soon after, the exchange established a set of rules and
regulations to govern these transactions. In the 1920s, a clearinghouse was
established. By that time, most of the essential ingredients of future contracts
were in place. In 1874, the Chicago Produce Exchange was formed. This later
became the Chicago Butter and Egg Board, and in 1898 it was reorganized as
the Chicago Mercantile Exchange (CME).
In Europe, the first organized exchange for commodity futures was the
London Metal Exchange (LME), which was established in 1877, while the
Agrarische Termijnmarkt in Amsterdam opened its doors in 1888.
For the first 120 years, since the formation of CBOT, futures exchanges of-
fered trading in contracts on commodities. Then, in 1971, the major Western
countries began to allow their currency exchange rates to fluctuate as a conse-
quence of the breakdown of the Bretton Woods Agreement. This opened the
way for the formation in 1972 of the International Monetary Market (IMM), a
subsidiary of the CME that specialized in the trading of futures contracts on
foreign currencies. These were the first futures contracts that could be called
financial futures. The CBOT responded by introducing the first interest-rate
futures contract in 1975. Over the years, many futures contracts on various
financial quantities have been developed. The first financial futures exchange
to be established in Europe was the London International Financial Futures
Exchange (LIFFE), in 1982. This proved to be a great success and led to the
establishment of many derivatives exchanges across Europe during the late
1980s and the early 1990s. For example, in 1986, the Paris-based exchange
Marche´ A Terme d’Instruments Financiers (MATIF) opened, which became
Marche´ A Terme International de France in 1988, followed by the Frankfurt-
based exchange Deutsche TerminBo¨rse (DTB), which opened in 1990. The
Milan-based exchanges Mercato degli strumenti Derivati Azionari (IDEM) and
Mercato degli strumenti Derivati sui tassi di interesse (MIF) were established
Chapter 1 - INTRODUCTION 3
in 1994.
Alongside the early activity in derivatives exchanges, the 1970s witnessed
the growth of the OTC (over-the-counter) forward market, especially for for-
eign exchange, called the interbank market. This ‘market’ consists of hundreds
of banks worldwide who make forward and spot commitments with each other,
representing either themselves or their clients. These transactions are private
and unregulated, their sizes are quite large, and they are tailored to the spe-
cific needs of the two parties involved. They are subject to credit risk, as
either party may default. In the decade of the 1980s, OTC market growth
pace further increased, the primary stimulant being the development of swaps
in 1981.
Options have existed since ancient times, but the current system of option
markets traces its origins to the nineteenth century. At the beginning of the
1900s, a group of firms calling itself the Put and Call Brokers and Dealers
Association created an option market in the US. Although this OTC market
was viable, it did not allow its participants to sell their contracts before expiry.
Also the holder of such a contract was exposed to the potential bankruptcy of
the writer. Because of these deficiencies, the costs of transacting were relatively
high.
In 1973, a revolutionary change occurred in the options world. The CBOT
organized an exchange exclusively for trading options on stocks. This exchange
was named the Chicago Board Options Exchange (CBOE). It opened its doors
for call option trading on April 26
th
, 1973, and the first put options were added
in June 1977.
The CBOE created a central marketplace where option contracts were stan-
dardized to improve liquidity (i.e., the possibility for an investor with an open
position to close it before the option expiry by taking an offsetting position in
an identical contract). Most importantly, however, the CBOE added a clear-
inghouse that guaranteed to the buyer that they would not suffer if the writer
did not fulfill their part of the contract.
This made options more attractive to the general public, and their trading
grew tremendously until the stock market crash of 1987. Meanwhile, options
started to be traded in OTC markets as well, between corporations and finan-
cial institutions.
In the early 1990s, the derivatives market went through an accelerated
period of growth, though it has slowed somewhat in recent years. This is
illustrated for the OTC market in Figure 1.1.
A key factor that played a primary role in this frantic growth was the
explosion of activity in the OTC market, especially as regards interest-rate
derivatives. This is clearly illustrated in Table 1.1.
Another factor was the steady creation of new and useful derivative secu-
rities to meet the diverse needs of investors.
4 MODELING INTEREST-RATE DERIVATIVES
Figure 1.1: total over-the-counter derivatives activity.
The Role of Derivative Markets and Their Players Usually the players
in the derivative markets are divided into three main categories: hedgers,
speculators and arbitrageurs. We will make use of this partition in order to
describe why derivatives are useful and the objectives that can be achieved by
means of them.
Interest-rate Currency Interest-rate
Years swaps swaps options
1990 2,311.5 577.5 561.3
1991 3,065.1 807.2 577.2
1992 3,850.8 860.4 634.5
1993 6,177.3 899.6 1,397.6
1994 8,815.6 914.8 1,572.8
1995 12,810.7 1,197.4 3,704.5
1996 19,170.9 1,559.6 4,722.6
1997 22,291.3 1,823.6 4,920.1
Source: ISDA market survey
Table 1.1: breakdown of outstandings in $bn.
Risk Management Because derivative prices are related to the prices of
the underlying spot market securities, they can be used to reduce or increase
risk associated with owning the spot items. For example, buying the spot item
Chapter 1 - INTRODUCTION 5
and selling a futures contract or buying a put option reduces the investor’s
risk. This type of position is known as a hedge and the person who owns this
position is called the hedger.
However, investors have different risk preferences. Some are more tolerant
of risk than others. Derivative markets enable those wishing to reduce their risk
to transfer it to those wishing to increase it, the latter being called speculators.
So, the other side of hedging is speculation. Unless a hedger can find another
hedger with opposite needs, the hedger’s risk must be assumed by a speculator.
Therefore, the presence of speculators increases market liquidity.
Price Discovery Futures and forwards are an important means of ob-
taining information about investors’ expectations of future spot prices. Op-
tions markets do not directly provide forecasts of future spot prices. They do,
however, provide valuable information about the volatility and hence the risk
of the underlying cash security.
Operational Advantages First, derivative markets allow lower trans-
action costs. Second, they have greater liquidity than the corresponding spot
markets because they can accommodate high-volume trades more easily than
cash markets. The reason for these benefits is partly due to the smaller amount
of capital required for participation in derivative markets. Third, derivative
markets allow investors to sell short more easily.
Market Efficiency and Arbitrage The existence of derivative mar-
kets increases overall market efficiency. In fact, there usually exist profitable
short-term arbitrage opportunities in the markets. The presence of these op-
portunities means that the prices of some securities are temporarily out of line
with what they should be. as a result, investors can earn returns that exceed
what the market considers fair for the given risk level.
Arbitrage is a kind of transaction whereby profits are derived from pricing
anomalies between identical or very similar instruments. The individual en-
gaging in the arbitrage, called the arbitrageur, buys the mispriced security at
the lower price and immediately sells it at the higher price. By doing so, a
risk-less profit formed from the difference between the two prices is locked in.
This can happen in the same market, or between two different markets where
the same or very similar securities are listed and price differences may arise.
These differences may be due to either a different interaction between supply
and demand in the two markets or the fact that the prices of these securities
are expressed in different currencies in the two markets.
Arbitrage plays a crucial role in the pricing of all futures contracts. Such
contracts have a ‘fair value’ price at which they should be trading. But, some-
times, deviations occur so that arbitrage opportunities arise because deriva-
tives are linked to the corresponding cash instruments. Arbitrageurs exploit
6 MODELING INTEREST-RATE DERIVATIVES
these opportunities, thus contributing to keeping cash and futures markets
aligned. In effect, the presence of arbitrageurs in the market allows us to as-
sume that any arbitrage opportunities are removed as soon as they arise, and
hence that there are, effectively, no arbitrage opportunities available.
The importance of quickly eradicating these profit opportunities will be
fully understood in the following chapters, where the absence of arbitrage is
postulated as the fundamental condition required for derivative pricing. With-
out arbitrageurs, this assumption would be incorrect and all the modern theory
of derivative pricing, which is based on this hypothesis, would receive a deadly
blow.
The State of the Art Currently, a number of derivative exchanges are in
operation worldwide. The biggest derivative markets are located in the US
(e.g., CBOT, CBOE, CME, NYMEX, COMEX, CSCE, and NYCE), Western
Europe (e.g., LIFFE, IPE, LME, EUREX, and MATIF), and Asia-Pacific (e.g.,
TIFFE, SIMEX, SFE, HKFE, and OSE). Besides these exchanges, newcomers
from emerging markets are expected to join as soon as the liquidity conditions
of the corresponding cash markets are more developed.
In the past, the only way of dealing was by open outcry, but, due to ad-
vances of information technology, a trend towards electronic trading systems
replacing the open outcry method began at the end of the 1980s. Since then,
every new exchange (e.g., EUREX) has adopted an electronic trading plat-
form, and exchanges established before that period, started studying ways to
improve their open outcry transactions by adding electronic support to the
normal trading activity. Eventually, some exchanges, such as MATIF, LIFFE,
TIFFE,SFE,SIMEX,HKFE,andOSE,switchedfromopenoutcrytoelec-
tronic trading. In spite of this, US exchanges stubbornly preserved their open
outcry dealing systems until recently, when some exchanges (e.g.,CBOT)an-
nounced their intention to transform their trading systems. In fact, there has
been a realization by these US exchanges that pure electronic exchanges, like
EUREX, are subtracting an increasing amount of business from their trad-
ing pits, thus endangering their world leadership. So, it is likely that in the
next few years open outcry trading will disappear entirely, to be replaced with
automated trading systems.
Another notable recent trend in derivative markets concerns alliances (e.g,
that among CME, MATIF, and SIMEX is called GLOBEX) and eventually
mergers (e.g., between CSCE and NYCE to form NYBOT) between different
exchanges, thus creating bigger and more liquid marketplaces able to attract
new investors. This process allows investors to trade virtually at any time,
irrespective of their time zone, and using any derivative product that can
fulfill their needs. In fact, these links are usually established among exchanges
located in different time zones. This results in the possibility of 24-hour trading
on the entire range of products of the allied exchanges since these linkages are
Chapter 1 - INTRODUCTION 7
usually characterized by mutuality, meaning that every exchange offers not
only its own products but other allied exchanges’ products as well.
These tighter linkages between exchanges can be viewed as a part of the
wider phenomenon of international market integration, also called globaliza-
tion. This creates challenges and opportunities for today’s financial markets:
no longer can a single country’s market be seen in isolation but instead the
interaction amongst markets in different countries must be considered together.
In conclusion to this section, we must stress that nowadays derivatives are
an indispensable instrument in the financial markets. However, given their
risky nature, those banks who wish to compete in this market need to employ
sound methods to price derivative securities; their failure to do so could result
in bankruptcy. In the following chapters we will focus on the issue of valuation
of derivative securities. We restrict our attention to interest-rate derivative
pricing models although the importance of having accurate valuation meth-
ods and models applies equally to other areas such as equity and commodity
derivative pricing.
1.2 Objective
Interest-rate derivative valuation is currently a ‘hot’ topic, as newer and more
innovative products are being created. These new products are also increas-
ingly more complex, and thus represent a challenge to the financial community,
who need to price and hedge them. The objective of this work is to investigate
the mechanics of interest-rate derivative valuation, to analyse the different ap-
proaches and models that can be utilized for pricing contingent claims, and
possibly to provide some useful overall comments on this subject.
1.3 Interest-Rate Derivatives
An interest-rate derivative is a derivative product that provides a pay-off deter-
mined by how an interest rate changes. While interest-rate options, forwards,
and futures contracts have proven to be quite popular, the most widely used
interest-rate derivative is the swap. We shall briefly describe forward rate
agreements (FRAs), options on interest rates (which are commonly packaged
as instruments called caps, floors, and collars), as well as interest-rate swaps,
and options on interest-rate swaps.
These instruments have grown in popularity for several reasons. Nearly
all businesses face some form of interest-rate risk; for example, a portfolio
manager holding bonds, a corporation planning to borrow money, or even a
family with a mortgage may each face interest-rate risk00.
8 MODELING INTEREST-RATE DERIVATIVES
A FRA is similar to any type of forward contract, but the pay-off is based
on an interest rate, rather than a security price. More precisely, a FRA is
a contract between two parties in which one party agrees to make an inter-
est payment at a future time at an agreed fixed rate, and the other party
agrees to make an interest payment at the same time at a floating rate such
as EURIBOR.
Interest-rate options are like FRAs, but instead of being a commitment to
receive one interest rate and pay another, they provide the right to receive
one interest rate and pay another. A call option gives the holder the right
to receive an interest payment at a floating rate in exchange for making an
interest payment at a fixed rate, whereas a put option gives the holder the
right to receive an interest payment at a fixed rate in exchange for making an
interest payment at a floating rate.
An interest-rate cap is a series of European interest-rate calls that pay off
at dates corresponding to interest payment dates on a loan. Each individual
option is called a caplet. Caps are designed to limit the cost of a floating-
rate loan with multiple interest payments (such as a mortgage). The opposite
instrument is called an interest-rate floor, and is designed to protect the return
on a floating-rate loan with multiple interest payments. Floors are collections
of European interest-rate puts, each one called a floorlet. A combination of a
long position in an interest-rate cap and a short position in an interest-rate
floor is called an interest-rate collar. It is used to lower the cost of an interest-
rate cap because, while limiting the effects of interest-rate increases, it gives
up the benefits of interest-rate decreases. The net effect is that this strategy
will establish both a floor and a ceiling on the interest cost.
An interest-rate swap is an agreement between two parties in which each
party makes a series of interest payments to the other at predetermined dates
at different rates, where at least one set of payments is determined by a variable
(floating) interest rate. The most common type of interest-rate swap is the
so-called plain vanilla, in which one set of payments is fixed and the other is
variable.
A swaption is an option in which the holder acquires the right to enter into
a swap either as a fixed-rate payer, floating-rate receiver, or as a floating-rate
payer, fixed-rate receiver. The former is called a payer option while the latter
is called a receiver option.
1.4 Overview
This dissertation begins in Chapter 2 by describing the traditional term struc-
ture of interest rates models, also called equilibrium models, developed by
Vasicek [60], Cox, Ingersoll, and Ross [21], and Longstaff and Schwartz [51].
We examine the assumptions underlying these models and, more importantly,
Chapter 1 - INTRODUCTION 9
their ability to replicate the real world. We also show how these models price
various types of interest-rate derivative securities. Finally, some numerical
examples are provided.
Chapter 3 deals with the more modern no-arbitrage models. It starts with
a comparison between these models and equilibrium models, focusing on their
different approach to the absence of arbitrage: the former adopt the martingale
framework developed by Harrison and Pliska [35]; the latter use the approach of
Vasicek [60]. Subsequently, specific models are addressed, namely the Ho and
Lee [36], the Black, Derman, and Toy [4], and the Hull and White [38] models.
We chose these models amongst many others because we believe they are the
models most used by practitioners in the real world. For each, a brief review
is provided, in which we stress the corresponding benefits and limitations.
A major problem with the short-rate models considered in Chapters 2 and 3
lies in insuring that they produce a realistic volatility function for zero-coupon
bond yields. Chapter 4 considers the general approach of Heath, Jarrow, and
Morton [33], which models interest rates more realistically than the earlier
short-rate models. We present a brief history of the approach and then explore
its structure. We show how the Heath, Jarrow, and Morton (HJM) framework
is limited in practice, at least in its full generality, because of its intensive
computational requirements.
Chapter 5 illustrates how derivatives can be priced within the HJM frame-
work by means of Monte Carlo simulation. This provides a viable numerical
method for derivative pricing, especially if techniques such as martingale vari-
ance reduction, which results in a notable improvement in efficiency compared
to standard Monte Carlo simulation, are employed. In the second part of this
chapter an empirical investigation of the Italian government bond market is
undertaken, and explores the extension of the number of volatility factors in
the HJM framework from one to two or, possibly, three. This investigation
employs factor analysis of the covariance matrix of the historical changes in
zero-coupon bond yields.
Chapter 6 summarizes the dissertation and provides suggestions for future
research.
Appendixes A, B, and C review numerical pricing methods that are exten-
sively utilized throughout the dissertation. Appendix D displays the numerical
results of the MC simulations described in Chapter 5, Appendix E contains
the C++ code used to carry out the MC simulation of two particular cases of
the HJM approach, Appendix F shows the data set used to carry out factor
analysis, and Appendix G displays the correlation matrix for the changes in
zero-coupon bond yields.