ECONOMIC VALUATION OF LIABILITIES: CALIBRATING A TWO-FACTOR SHORT-RATE MODEL.

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ECONOMIC VALUATION OF LIABILITIES:

CALIBRATING A TWO-FACTOR SHORT-RATE MODEL.

Douwe-Klaas Bijl

s1451472

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE AT THE

UNIVERSITY OF GRONINGEN GRONINGEN, THE NETHERLANDS

APRIL 2006

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UNIVERSITY OF GRONINGEN DEPARTMENT OF

ECONOMETRICS

The undersigned hereby certify that they have read and recommend to the Faculty of Economics for acceptance a thesis entitled “Economic Valuation of Liabilities:

Calibrating a Two-Factor Short-rate model.” by Douwe-Klaas Bijl in partial

fulfillment of the requirements for the degree of Master of Science.

Dated: April 2006

Supervisors:

Dr. J.W. Nieuwenhuis

Jhr. Dr. P.W. van Foreest

Reader:

Prof. Dr. P.A. Bekker

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UNIVERSITY OF GRONINGEN

Date: April 2006

Author: Douwe-Klaas Bijl

Studentnumber: s1451472

Title: Economic Valuation of Liabilities:

Calibrating a Two-Factor Short-rate model.

Department: Econometrics

Degree: M.Sc.

Permission is herewith granted to University of Groningen to circulate and to have copied for non-commercial purposes, at its discretion, the above title upon the request of individuals or institutions.

Douwe-Klaas Bijl

THE AUTHOR RESERVES OTHER PUBLICATION RIGHTS, AND NEITHER THE THESIS NOR EXTENSIVE EXTRACTS FROM IT MAY BE PRINTED OR OTHERWISE REPRODUCED WITHOUT THE AUTHOR’S WRITTEN PERMISSION.

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The Hague - April 6, 2006

This thesis has been submitted for the Degree of Master of Science in Econometrics at the University of Groningen. Research for this thesis

took place during an internship at the Group Risk department of AEGON N.V. from the 1st of August 2005 till

the 15th of March 2006.

The internship at AEGON was under guidance of Jhr. Dr. P.W. van Foreest. The supervisor of the thesis was

Dr. J.W. Nieuwenhuis, University of Groningen.

With great pleasure I hereby wish to thank all those who invested in this thesis. I am especially grateful to the committee members,

Dr. J.W. Nieuwenhuis and Jhr. Dr. P.W. van Foreest and feel indebted to the entire staff of the Group Risk

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Abstract

The main objective of our internship was to build a calibration tool that enables one to efficiently cali-brate short-rate models. Below we will describe why the need for such a tool exists and in broad outline what will be discussed in this thesis.

There is a variety of reasons that drive the urge of life insurance companies to value their insurance liabilities in accordance with current market circumstances. For a great deal of the insurance liabilities, there exists a closed-form approximation of the market-consistent value. However, the market-consistent valuation of the more complex liabilities - especially the ones resulting from products with imbedded options - requires a stochastic valuation. For the stochastic valuation we require:

- a model that describes the liability cashflows in all possible future states;

- a model that is capable of valuing financial securities with characteristics similar to the insurance liability cashflows and that is calibrated to these securities.

Barrie and Hibbert will provide AEGON with an Economic Scenario Generator that can run Monte-Carlo simulations based on these two models. We will restrict ourselves to insurance liabilities that are subject to interest rate fluctuations and we will focus on the second model. The interest rate model we are going to use is from the class of short-rate models. We are mainly interested in the short-rate model for which the ESG can run simulations.

To set the stage, we begin with examining some theory about pricing1_{. We describe a number of}

short-rate models. Next we discuss how these models can be approximated with a tree. Furthermore we describe how to find the price of discount bonds and European swaptions with the tree approximation. Subsequently we will discuss the calibration procedure and the calibration tool. The calibration tool enables us to systematically search for the parameters that make the tree approximation of the short-rate model replicate the prices of the discount bonds and European swaptions as closely as possible. We will conclude with some remarks on difficulties related to the calibration procedure.

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Introduction

In recent years there has been a trend towards market-consistent valuation in life insurance companies. AEGON is no exception. AEGON’s aim is to develop an Economic Capital Model that can be consis-tently applied across the organization. It should provide a comprehensive economic risk assessment that will define the risk capital required to support all products. The framework will affect the economic valuation of both assets and liabilities.

In this thesis we will focus on the market-consistent valuation of liabilities. The question we want to an-swer is: ”What price would insurance liabilities trade at, if they were sold in the market?”. The valuation of the liabilities should therefore be done in a manner that is consistent with the way the market prices the risks of similar financial securities. Because, as we will indicate in chapter 1, we will mainly deal with interest rate risk related to insurance liabilities, we will make use of an interest rate model. To be more precise: we will make use of a short-rate model, which is defined by one or more diffusion processes. Chapter 2 will discuss a variety of available short-rate models. An alternative to using a short-rate model is to use a forward-rate model. Chapter 3 will discuss the relationship between short-rate models and forward-rate models.

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This leads to the main objective our internship. To facilitate the calibration of the short-rate model, the need for a practical and user-friendly tool arises. The main objective is therefore to:

Build a tool that enables one to efficiently calibrate short-rate models.

The absence of more-sophisticated software packages like Matlab forced us to build the tool in Microsoft Excel and Visual Basic. Chapter 7 discusses the calibration tool and can be best regarded as a manual for the tool. Moreover it contains an example of the results obtained by applying the tool to market data.

The calibration results, i.e. the values of the parameters of the short-rate model that makes it repli-cate the market prices of the calibration securities as closely as possible, will subsequently be used in a software package that enables one to run Monte-Carlo simulations. This software package will be provided by Barrie and Hibbert [B&H]2, and is commonly referred to as an Economic Scenario Generator [ESG]. Unfortunately the ESG only has the ability to run Monte-Carlo simulations for one particular short-rate model: a parametric version of the two-factor Black-Karasinski short-rate model. Therefore in part of this thesis we fix our attention on this model.

Finally, the simulation results can be used to price an insurance liability. This is however out of the scope of our this thesis.

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Liabilities One-factor Two-factor

short-rate model short-rate model The Liabilities

should have

roughly similar Choose one

properties as the

calibration Assume an initial set

securities of model

parameter-values

Choose appropriate Market Data Compare Calculate Prices of - Discount bond prices Discount Bonds and - Swaption Implied Vols Swaption Implied Vols

if difference is tolerable Repeat the set of parameters will be until used in Monte Carlo simulations difference

is tolerable else

Assume a set of model parameter-values

Run simulation using B&H's Economic Scenario Generator

Market consistent price of liabilities

A broad outline of the procedure that can be used to find the market-consistent price of insurance liabilities.

Figure 1: Economic Valuation

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Chapter 1 Economic Valuation Model

1.1 Introduction

The approaches to liability valuation within insurance companies are undergoing a radical change. While currently most of the large insurance companies still employ quite traditional actuarial approaches, in the near future they are expected to develop a framework that enables them to make a more market-consistent valuation. In essence market-market-consistent valuation of an insurance liability is nothing more than attaching a value to a set of future, potentially uncertain, cashflows that is consistent with the prices at which securities with similar cashflow profiles trade in the market. The market-consistent value of liabilities can provide a measure of economic capital requirements for in-force business: ”Is there a third party on the market that is willing to take over the risk for the capital that is available?” Furthermore it can give some important information about the appropriate price for new products.

There are also external reasons that drive the development of such a framework. The first of these reasons is the development on the regulatory front. A good example is Solvency II, that is currently developed by the EU. One of the key objectives of Solvency II is to establish a solvency system that is better matched to the true risks of an insurance company. It aims to ensure that insurance companies have sufficient capital to fund the poorest outcomes that can arise from the risks taken. In other words: An insurance company should have a solvency position that is sufficient to fulfill its obligations to poli-cyholders and other parties. Other external reasons include demands from rating agencies, investors and institutional clients for a consistent internal framework for measuring risk.

Now that the need for a proper economic1 _{valuation of insurance liabilities has been briefly touched}

upon, we will discuss how this valuation can be done in practice.

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Calibrating a Two-Factor Short-rate model. Economic Valuation Model - Insurance Liabilities

1.2 Insurance Liabilities

To be able to make a reasonable approximation of the value of an insurance liability, one has to be aware of the risks or rather uncertainties underlying the insurance liability. A natural approach is to classify these risks. Of course, how these risks are classified is not a critical issue. What is critical is that all risks are considered. For internal use AEGON made a mutually exclusive and exhaustive risk-classification. For our ends, it is however not necessary, to go into a similar level of detail and therefore we will confine ourselves to using the rough classification made by Babbel and Merrill [1997]. Babbel and Merrill divide uncertainty into three categories: actuarial risk, market risk, and nonmarket systematic risk. The cate-gories will be discussed successively.

Actuarial risks include, but is not limited to, mortality and morbidity risk. Mortality rate is the an-nual number of deaths per 1000 people. Morbidity rate refers to the number of people who have a disease compared to the total number of people in a population. Whereas mortality risk is the risk of the actual mortality differing from the expected mortality, morbidity risk is risk of the actual morbidity differing from the expected. Mortality risk is directly related to life insurance, while morbidity risk is inherent to income protection plans (disability insurance) and health insurance. Exposure to actuarial risks can theoretically be lowered to a certain suitable level through diversification and writing large numbers of similar policies.

Market risks include fluctuating interest rates, inflation rates and exchange rates. Insurance compa-nies can measure and value their exposure to these risks, because of the existence of an active market for securities that are exposed to the same sources of uncertainty. Unlike, mortality risk, the majority of market risks is inherently non-diversifiable. This category, and especially interest rate changes, will be of our main concern. Therefore, in the remaining, that is from section 1.3 onwards, we will mainly focus on interest rates.

Nonmarket systematic risks consequently holds all remaining risks, such as changes in the legal envi-ronment, tax laws, or regulatory requirements. Compared to market risks, they generally have a less significant effect on the value of insurance liabilities [Babbel and Merrill, 1997].

1.3 Valuing Liabilities I

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GR Economic Valuation Model - Valuing Liabilities I

as closely as possible. This definition is, although less precise, similar to Babbel and Merrill’s [1997]. Babbel and Merrill interpret the economic value of an insurance liability as the amount of money an insurer would need today to satisfy, on a probabilistic basis across various economic states of the world, the obligations imposed on it through the insurance it has sold.

In any valuation approach there will be cashflows, with their associated probabilities of occurrence, and discount rates. The most well-known method to account for risk is to adjust either the discount rate or the probabilities. The former is usually referred to as the deflator approach, the latter as the risk-neutral approach. Pelsser [2003] gives a nice overview of the relationship between the two approaches.

In the deflator approach the discount rate will be adjusted so that it is consistent with the return that is demanded by investors. Because of additional risk they have to bear, the return they demand naturally exceeds the risk-free return. In risk-neutral valuation approach the probability distribution is adjusted to compensate for the additional risk so that the cashflows can be discounted at risk-free rate as if investors were risk-neutral. Harrison and Pliska [1981] show that if the probability distribution can be adjusted in such a way, or equivalently if a risk-neutral measure (which is commonly denoted as Q, while the ’real world’-measure is denoted as P) exists, the market must be free of arbitrage opportunities. Thus, in spite of possible differences in the level of risk aversion, investors will agree upon which value is correct. In this thesis we will adopt the risk-neutral valuation approach. Pelsser [2003] shows that, correctly applied, both approaches give the same value.

There are still a number of valuation approaches conceivable. To be of any use for, market-consistent valuation purposes, it certainly has to be able to fulfill the requirement of calibrating closely to prices observable on the market. Economic value is equivalent to market value when the financial security in question is tradable on the market. For basic insurance liabilities it is enough when the model closely matches observed discount bond prices. For insurance liabilities which have future cashflows that are, possibly indirectly, subject to future prices of a certain security that is tradable on the market, it is important to find a model that is also capable of matching more than just observed discount bond prices. We will return to this later. One thing that should be clear though, is that we need a general approach that features both stochastic cashflows and interest rates.

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Calibrating a Two-Factor Short-rate model. Economic Valuation Model - Valuing Liabilities II

1.4 Valuing Liabilities II

We will start with giving a time horizon, a probability space, a filtration and a number of definitions2_.

Let’s fix a time horizon [0, T ]. It is common practice to model the uncertainty through a d-dimensional Brownian motion w defined on its probability space (Ω, F , P). The filtration F = {F(t) : 0 ≤ t ≤ T } is the augmentation of the natural filtration generated by the Brownian motion w, i.e. F (t) is the σ-algebra generated by σ(w1(s), . . . , wd(s) : 0 ≤ s ≤ t) and the null-sets of F .

Assume a market M where there are N + 1 non-dividend paying securities.

Define r(t) as the short-rate. The short-rate should be regarded as the spot-rate that applies to the shortest time increment possible. Chapter 2 will elaborate on possible short-rate processes, (and hence the probabilistic nature of r(t)) and in particular r(t) will be defined in mathematical terms in section 2.2.

Next, define B0(t) to be the value of a bank account at time t, 0 ≤ t ≤ T . A bank account

repre-sents an investment, where profit is accrued continuously at the short-rate r(t) prevailing in the market at every instant. We assume B0(0) = b0,0 = 1 and that the bank account evolves according to the

following differential equation:

dB0(t) = r(t)B0(t)dt. (1.4.1)

As a consequence:

B0(t) = e− Rt

0r(s)ds.

The rate of return of B0 is locally deterministic because it is equal to short rate r(t) prevailing in the

market at time t:

dB0(t)

B0(t)

= r(t)dt.

Therefore the bank account is said to be a locally riskless investment.

The discount factor D(t, T ) between two time instants t and T is the amount at time t that is equivalent to one unit payable at time T and is given by:

D(t, T ) = B0(t) B0(T )

= e−RtTr(s)ds. (1.4.2)

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GR Economic Valuation Model - Valuing Liabilities II

A bank account can be viewed as being one of the N + 1 non-dividend paying securities. The prices of the other securities, Bi, i = 1, . . . , N , are positive and evolve according to similar equations:

dBi(t) = µiBi(t)dt + σiBi(t)dw(t) = µiBi(t)dt + Bi(t) d X j=1 σi,jdwj(t), Bi(0) = b0,i, i = 1, . . . , N.

The rate of return on the securities Bi, i = 1, . . . , N is can be written as3:

dBi(t)

Bi(t)dt

= µi+ σi

dw(t)

dt , i = 1, . . . , N.

and is not observable at time t, as it contains the random dw(t)_dt term. Contrary to B0, the securities

Bi, i = 1, . . . , N have a stochastic rate of return, even on the infinitesimal scale [Bj¨ork, 1998].

Some other terminology will be introduced as the discussion progresses.

Arbitrage opportunities

A simple example of an arbitrage opportunity would be a self-financing trading strategy, such that the evolution process of the value of the corresponding portfolio starts out with zero value and terminates at some definite date T with a positive value. In mathematical terms a trading strategy is a N + 1-dimensional process φ = {φ(t) : 0 ≤ t ≤ T }, whose components φ0, . . . , φN are F-previsible. The value

process associated with a strategy φ is defined by:

Vφ(t) =

N

X

i=0

φi(t)Bi(t), 0 ≤ t ≤ T.

A portfolio φ holds an amount φi(t) of security i at time t, 0 ≤ t ≤ T , i = 0, . . . , N . Short-selling

security Bi leads to a negative value of φi. The previsibility condition on each φi i = 0, . . . , N , means

that the value φi(t) is known immediately before t, or, in mathematical terms, that φi(t) is measurable

with respect to F (t).

The gains associated with a strategy φ is defined by:

Gφ(t) = N X i=0 Z t 0 φi(t)dBi(t), 0 ≤ t ≤ T.

Self-financing trading strategy

Mathematically, under some technical conditions, we can say that a portfolio is self-financing if:

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Calibrating a Two-Factor Short-rate model. Economic Valuation Model - Valuing Liabilities II

A trading strategy is self-financing if the change in its value only depends on the change of the prices of the securities it contains, the gains (no money is added or subtracted). To prohibit portfolios with doubling strategies we have to impose the condition that Vφ(t) is bounded from below:

Vφ(t) ≥ −K, _{K ∈ R.}

for all 0 ≤ t ≤ T . Doubling strategies make almost sure profits starting with zero value [Øksendal, 1992].

In mathematical terms an arbitrage opportunity can be, briefly and to the point, represented by a self-financing strategy φ such that Vφ_{(0) = 0, but V}φ_{(T ) ≥ 0 almost surely, with a positive probability}

of the value being strictly positive at time T , i.e. P(Vφ(T ) > 0) > 0. A market M is arbitrage-free if there are no arbitrage opportunities.

Absence of arbitrage opportunities

Harrison and Pliska [1981] describe the relationship between the concept of absence of arbitrage oppor-tunities and the existence of an alternative probability measure Q. They proved that if an equivalent martingale measure exists the market is free of arbitrage. Related results can be found in [Delbaen and Schachermayer, 1994; 1995].

Equivalent martingale measure

An equivalent martingale measure Q is a probability measure on the space (Ω, F) such that [Brigo and Mercurio, 2001, 25]:

- P and Q are equivalent if they operate on the same sample space and agree on what is possible. Formally, if A ∈ F :

P(A) > 0 ⇔ Q(A) > 0,

- The discounted security price process D(0, ·)Bi, i = 0, . . . , N is an (F , Q)-martingale,

so that:

EQ_{(D(0, t)B}

i(t)|F (u)) = D(0, u)Bi(u), (1.4.3)

for all i = 0, . . . , N and all 0 ≤ u ≤ t ≤ T .

Thus the conversion from P to Q represent a redistribution of probability mass that causes every security to earn (in expected value) at the riskless rate zero without changing the set of events that receive positive probability [Harrison and Kreps, 1979].

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GR Economic Valuation Model - Risk-neutral valuation

Risk-neutral approach

The measure Q is commonly referred to as a risk-neutral probability measure. To explain this properly, we assume a T -claim H. Intuitively H is a contract which specifies that an, at time t, unknown amount, conveniently also denoted by H, is to be paid out to the holder of the contract at time T . Mathematically, H is a random variable on (Ω, F , P) and EP_{(|H|) < ∞.}

A T -claim H is attainable if there exists some φ such that Vφ_{(T ) = H. Such a φ is said to hedge}

against T -claim H. For every T -claim, we will assume attainability to hold. This implies that we assume the market M to be complete [Brigo and Mercurio, 2001, 26]. Harrison and Pliska [1983] show that a market is only complete if there exists a unique equivalent martingale measure. If a unique martingale measure exists, for each time t, 0 ≤ t ≤ T , there exists an unique price π(t), associated with any T -claim H, i.e:

π(t) = EQ_{(D(t, T )H|F (t)).} _(1.4.4)

T -claim H is said to be priced by arbitrage, and the unique price is occasionally referred to as the arbi-trage value of H.

In (1.4.4) the unique no-arbitrage price of an attainable T -claim H is written in terms of the expec-tation of the T -claim payoff under Q. However, in practice Q might not be the most convenient measure for pricing the T -claim H. In section 1.5 we will show how to transfer to a measure with respect to a different numeraire. We will start with explaining the concept of a numeraire and subsequently we will discuss a certain measure that will prove to be of use later in this thesis.

1.5 Risk-neutral valuation

A change in measure requires a different numeraire.

Numeraire

The prices of the securities in the market M are denoted in a fixed pricing unit. However they may be expressed in terms of their relative value to any security which has a positive value at all times. The security relative to which the value of other securities can be judged is called a numeraire.

More formally, a numeraire Z is the value process of a portfolio such that Z(t) > 0 for all t, 0 ≤ t ≤ T almost surely. A numeraire Z is indentifiable with a self-financing strategy φ such that Z(t) = Vφ(t) for all t, 0 ≤ t ≤ T .

So, any of the other4 _{(or some combinations of) securities B}

i, i = 1, . . . , N could serve as a numeraire.

If Z is a numeraire, then the securities B0/Z, . . . , BN/Z make up a market (M

0

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Calibrating a Two-Factor Short-rate model. Economic Valuation Model - Risk-neutral valuation

securities are denoted in units of the numeraire Z. This transformation is called change of numeraire.

Geman et al. [1995]/ Duffie [1996] show not only that, under some technical conditions, after a nu-meraire change self-financing strategies remain self-financing, but also that the attainability assumption continues to hold. This means that if a T -claim is attainable in the market M it is also in M0.

Now, let Z just be any of the securities Bi, i = 1, . . . , N . Geman et al. demonstrate that there exists a

probability measure QZ, equivalent to P, such that the price of any attainable T -claim H normalized by Z is a martingale under QZ_{, i.e:}

π(t) Z(t) = E QZ _H Z(T ) F (t) , 0 ≤ t ≤ T.

A commonly used numeraire is the discount bond5_{whose maturity T coincides with that of the T -claim}

to price. In such a case:

Z(t) = P (t, T ),

where P (t, T ) denotes the price at time t of a discount bond maturing at time T, t ≤ T , with unit ma-turity value, i.e. Z(T ) = P (T, T ) = 1. The probability measure associated with the bond maturing at time T is called the T -forward probability measure and will be denoted by QT_{. The relationship between}

the forward probability measure QT

and the risk-neutral probability measure Q can be expressed by a Radon-Nikodym derivative.

With a Radon-Nikodym derivative it is possible to express two equivalent probability measures P and Q in terms of each other. The Radon-Nikodym derivative of Q with respect to P restricted to F is a random variable and is normally denoted as dQ_dP. Both Brigo and Mercurio [2001, 483] and Williams [1991, 146] show that the following equation holds:

Q(A) = Z

A

dQ

dPdP, A ∈ F .

The choice of notation reflects the fact that the function is analogous to a derivative in calculus in the sense that it describes the rate of change in probability density of one probability measure with respect to another. When one wants to calculate the conditional expectation of a random variable H, it may be useful to switch from probability measure (see for instance [Jamshidian, 1989]). It can be shown that:

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GR Economic Valuation Model - Risk-neutral valuation

The Radon-Nikodym derivative of QT with respect to Q can be calculated [Brigo and Mercurio, 2001, 36] using: dQT dQ = P (T, T )B0(0) P (0, T )B0(T ) =e −RT 0 r(s)ds P (0, T ) = D(0, T ) P (0, T ). (1.5.2)

For a proof of (1.5.2) we refer to [Rutkowski and Musiela, 1997].

Using(1.5.1) and (1.5.2) results in an equation for the price of the T -claim H at time t similar to (1.4.4):

π(t) = P (t, T )EQT_{(H|F (t)),} _(1.5.3)

Notice that, almost all of the above is subject to the evolution of r under the probability measure Q in time. Chapter 2 will discuss models of the term structure of interest rates and concentrates on short-rate models. For reasons pointed out in the introduction we are highly interested in the parametric version of the two-factor Black-Karasinski model that is being used in the ESG. This model, and its relation to other more well-known models, will be discussed in chapter 2.

1_{We will use the terms ”economic value” and ”market-consistent value” interchangeably throughout this thesis} 2_{The reader interested in a more formal treatment of arbitrage theory in continuous time is referred to [Bj¨}_{ork, 1998].}

For further details on probability theory we refer to [Williams, 1991].

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Chapter 2 Models of the Term Structure of Interest Rates

2.1 Introduction

There are different approaches to model the term structure of interest rates. We can distinguish two classes: short-rate models and forward-rate models.

Because one-factor models assume there is just one cause of uncertainty in the market, represented by an one-dimensional Brownian Motion, they often imply perfect correlation of movements between short-term and long-term1 _{interest rates over time. Multi-factor models can achieve a better fit to observed prices}

and observed economic phenomena without imposing perfect correlation in movements across the term structure [Babbel and Merrill, 1997]. Hull and White [1994b] came up with a quite general two-factor model.

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GR Models of the Term Structure of Interest Rates - The Fixed Income market

2.2 The Fixed Income market

Consider a market M in which investors buy and issue default free claims on a specified sum of money to be delivered at a given future date. Such claims are generally called discount bonds. Let P (t, T ) denote the price at time t of a discount bond maturing at time T, t ≤ T , with unit maturity value, i.e P (T, T ) = 1. The difference between D(t, T ) in (1.4.2) lies in the fact that, at time t, D(t, T ) is a random quantity, while P (t, T ) is known. P (t, T ) can be viewed as the expectation of D(t, T ) under the probability measure Q:

P (t, T ) = EQ_{(D(t, T )|F (t)) = E}Q_(e−RtTr(s)ds|F (t)) (2.2.1)

The continuously compounded yield to maturity R(t, T ) is the internal rate of return at time t on a bond with maturity date T and is by definition equal to:

R(t, T ) = −ln(P (t, T ))

T − t . (2.2.2)

R(t, T ) is the constant rate at which an investment of P (t, T ) at time t accrues continuously to yield 1 at maturity T . The rates R(t, T ) considered as a function of T will be referred to as the term structure at time t.

The short-rate r(t) at time t is defined as limT >t,T →tR(t, T ), where it is assumed this limit exists for t,

0 ≤ t ≤ T . It is sometimes referred to as the instantaneous rate at time t.

We also assume the existence of a function f (t, u) such that:

R(t, T ) = 1 T − t

Z T

t

f (t, u)du, (2.2.3)

where f (t, T ) is called the instantaneous rate. In the form explicit for the instantaneous forward-rate this equation can be written as:

f (t, T ) = ∂

∂T((T − t)R(t, T − t)), or, to clarify the relation with discount bond prices, as:

f (t, T ) = − ∂

∂T ln(P (t, T )). (2.2.4)

The instantaneous forward rate as seen from the term structure at time t is equal to minus the logarithmic derivative of the time-t price of a discount bond of maturity T with respect to its maturity. We will refer to the forward-rate f (t, T ) as a function of T , as the forward-rate term structure at time t.

By definition:

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Models of the Term Structure of Interest Rates - One-Factor Interest Rate models

Specifying the dynamics of the short-rate process r under the probability measure Q is very convenient, since all fundamental quantities are, by no-arbitrage arguments, defined as the expectation of a functional of the process r [Brigo and Mercurio, 2001/ Bj¨ork, 1998, 43,253]. This is the classic approach to interest rate modeling.

Now, it is time to review models for r.

2.3 One-Factor Interest Rate models

We assume that the market M is arbitrage-free. We set a time horizon [0, T ] and consider a 1-dimensional Brownian motion w1 on a probability space (Ω, F , Q). We denote by F(t) the augmentation of the

nat-ural filtration of w. In mathematical terms, that is F (t) = σ(w1(s) : 0 ≤ s ≤ t) [Williams, 1991, 17,93].

Many different one-factor interest models have been proposed. The ones we will consider in this sec-tion can all be written in the form:

dr(t) = bdt + cdw1(t), t ≤ T, r(0) = r0,

where w1(t) is a 1-dimensional Brownian motion and b = b(t, r, σ, α) is a function of t, short-rate r, σ,

and possibly a mean-reversion parameter α. In the models we considered c = c(r) is only a function of r. Three important examples of one-factor models are listed in table 2.1.

Model Evolution Equation Functions

[HL] Ho-Lee dr(t) = θ(t : σ)dt + σdw1(t) b = θ(t : σ), c = σ

[HW] Hull-White dr(t) = (θ(t : σ, α) − αr(t))dt + σdw1(t) b = θ(t : σ, α) − αr, c = σ

[BK] Black-Karasinski d ln(r(t)) = (θ(t : σ, α) − α ln(r(t)))dt + σdw1(t) b = rθ(t : σ, α) − α ln(r)r+

rσ2/2, c = σr Table 2.1: One-Factor Interest Rate models

In all of these models we typically choose σ, and when present α, to obtain a nice volatility structure (Chapter 5 will discuss what we mean with the concept of volatility structure). Thereafter θ is chosen to fit the observed term-structure. The notation θ(t : σ, α) represents the function that makes an individual model consistent with the observed term-structure given a fixed σ and α. To simplify notation from now on the arguments σ and α will be omitted (we will write θ(t : σ, α) as θ).

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GR Models of the Term Structure of Interest Rates - One-Factor Interest Rate models

Ho and Lee

The HL-model under the risk-neutral measure Q is given by:

dr(t) = θ(t)dt + σdw1(t), r(0) = r0. (2.3.1)

In this model σ is constant and θ(t) is a function that is chosen to make the model consistent with the initial term structure. It can be proven [Bj¨ork, 1998] that θ is given by (see chapter 3):

θ(t) = ∂

∂tf (0, t) + σ

2_t.

The most interesting feature of this model is its simplicity. It has a closed-form solution for the price of a discount bond [Ho and Lee, 1986]. A disadvantage however is that it fails to capture the mean-reverting property that is observed in the movements of interest rates.

Hull and White

Hull and White’s model can be regarded as an extension of HL that incorporates mean-reversion. The short-rate r(t) follows under Q the process:

dr(t) = (θ(t) − αr(t))dt + σdw1(t), r(0) = r0. (2.3.2)

The model can be rewritten as:

dr(t) = α θ(t)

α − r(t)

dt + σdw1(t), r(0) = r0.

This shows that at any given t the short-rate is pulled toward θ(t)_α at rate α. There are two volatility parameters, α and σ. The parameter σ determines the overall level of volatility. The reversion rate parameter α determines the relative volatilities of short-term and long-term rates. A high value of α causes short-term movements to damp out quickly, so long-term volatility is reduced. A mean reverting-process is also known as an Ornstein-Uhlenbeck reverting-process. Finally θ(t) is implied from the initial yield curve, σ and α. It can be shown (see chapter 3) that:

θ(t) = ∂

∂tf (0, t) + αf (0, t) + σ2

2α(1 − e

−2αt_).

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GR

Black and Karasinski

BK [Black and Karasinski, 1991] has the desirable feature that the short-rate cannot become negative:

d ln r(t) = (θ(t) − α ln r(t))dt + σdw1(t), r(0) = r0. (2.3.3)

The probability distribution of the short-rate is lognormal at all times. Now, α is the mean-reverting rate of ln(r), while again θ(t) is a function used to fit to the initial term-structure. In general this model does not have a closed-form solution for discount bond prices.

T R(0, T ) 0 2.00% 0.25 2.14% 0.50 2.16% 1 2.35% 2 2.51% 3 2.64% 4 2.74% 5 2.83% 6 2.92% 7 3.01% 8 3.10% 9 3.18% 10 3.25% 15 3.51% 20 3.65% 30 3.74%

Table 2.2: European term structure as observed at September 30, 2005

Figures 2.2, 2.3 and 2.4 each show 5 random

sam-ple paths generated, using the same 5 Brownian

mo-tions, by the HL, HW and BK model respectively. In

all of the models θ(t) is chosen to make the model

consistent with the European term structure as

ob-served at September 30, 20052. The term structure

is tabulated in table 2.2 and depicted in figure 2.1.

The reversion rate parameter α in figure 2.3 and

fig-ure 2.4 is set equal to 0.1. For ease of

compari-son σ is set equal to 0.25 in figure 2.4, while for

the other figures it is 0.01. Figure 2.4 illustrates

that for BK the short rate is indeed always

posi-tive.

We would like to remark that on the European market interest rates R(0, T ) are only available for a certain set of maturities TM _{of T , i.e. T}M _{= {0, 0.25, 0.5, 1, 2, . . . , 10, 15, 20, 30}. In}

figure 2.1 the available rates are equipped with a marker. All rates in between these known rates are found by using a simple linear interpolation method, i.e. for all T∗_{∈ T}_/ M _we

calculated R(0, T∗) using: R(0, T∗) = _TM+_{− T}∗ TM+_{− T}M− R(0, TM−) + _T∗_{− T}M− TM+_{− T}M− R(0, TM+), where TM+= min(T ∈ TM: T > T∗) and TM−= max(T ∈ TM: T < T∗).

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GR Models of the Term Structure of Interest Rates - One-Factor Interest Rate models 0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 3.50% 4.00% 0 5 10 15 20 25 30 T R(0,T)

The data was obtained from Bloomberg. We set R(0, 0) equal to the overnight interest rate. The corresponding interest

rates are tabulated in table 2.2. All interest rates in between are calculated by using a simple linear interpolation method.

Figure 2.1: European term structure as observed at September 30, 2005

dr(t) = θ(t)dt + 0.01dw1(t), r0= 2%. -2.00% 0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00% 16.00% 0 5 10 15 20 25 30 t r(t)

Sample paths generated for the Ho and Lee model (2.3.1), with σ = 0.01 and θ(t)

(see also figure 2.5) so that the model is consistent with the European term structure in figure 2.1.

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GR

dr(t) = (θ(t) − 0.1r(t))dt + 0.01dw1(t), r0 = 2%. -2.00% 0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00% 16.00% 0 5 10 15 20 25 30 t r(t)

Sample paths generated for the Hull and White model (2.3.2), with α = 0.1, σ = 0.01 and θ(t)

(see also figure 2.6) so that the model is consistent with the European term structure in figure 2.1.

Figure 2.3: HW sample paths

d ln r(t) = (θ(t) − 0.01 ln r(t))dt + 0.25dw1(t), r0= 2%. -2.00% 0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00% 16.00% 0 5 10 15 20 25 30 t r(t)

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GR Models of the Term Structure of Interest Rates - One-Factor Interest Rate models θ(t) = ∂ ∂tf (0, t) + (0.10) 2 t. 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.02 0 5 10 15 20 25 30 t theta(t) 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0 5 10 15 20 25 30 t theta(t) -5.00 -4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 0 5 10 15 20 25 30 t theta(t)

The θ(t) function of the Ho and Lee model (2.3.1), with σ = 0.01, so that the model

is consistent with the European term structure in figure 2.1.

Figure 2.5: HL θ(t) θ(t) = ∂ ∂tf (0, t) + (0.1)f (0, t) + (0.01)2 2(0.01)(1 − e −2(0.01)t ). 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.02 0 5 10 15 20 25 30 t theta(t) 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0 5 10 15 20 25 30 t theta(t) -3.00 -2.00 -1.00 0.00 1.00 2.00 0 5 10 15 20 25 30 theta(t)

The θ(t) function of the Hull and White model (2.3.2), with α = 0.10 and σ = 0.01, so that the model

Figure 2.6: HW θ(t)

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Models of the Term Structure of Interest Rates - One-Factor Interest Rate models 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.02 0 5 10 15 20 25 30 t theta(t) 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0 5 10 15 20 25 30 t theta(t) -5.00 -4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 0 5 10 15 20 25 30 t theta(t)

The θ(t) function of the Black and Karasinski model (2.3.3), with α = 0.10 and σ = 0.25, so that the model

The method in which θ(t) is determined will be discussed in chapter 4 (on tree approximations).

Figure 2.7: BK θ(t)

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GR Models of the Term Structure of Interest Rates - Two-Factor Interest Rate models

2.4 Two-Factor Interest Rate models

Before discussing two-factor interest rate models, we would like to stress the weaknesses of the one-factor models in the previous section. Brigo and Mercurio [2001, 128] illustrate with a straightforward example that the correlation between two rates, say R(t, T1) and R(t, T2), equals 1. This means that, for example,

the thirty years rate R(0, 30) at a given instant is perfectly correlated with the one year rate R(0, 1) at the same instant. An upward (downward) shock to the short-rate at time t, r(t), will almost rigidly make the entire term structure shift up (down). A multi-factor model allows more flexibility. This is especially important for the valuation of claims (insurance liabilities) that depend on the joint distribution of the two rates R(t, T1) and R(t, T2).

For the two-factor interest rate models, we will assume that the market M is arbitrage-free and set a time horizon [0, T ]. But now we consider a 2-dimensional Brownian motion w on a complete probabil-ity space (Ω, F , Q). We denote the augmentation of the natural filtration of w, by F(t), which in this case can be expressed as F (t) = σ(w1(s), w2(s) : 0 ≤ s ≤ t) [Williams, 1991, 17,93].

For notational comfort we define z(t) = g(r(t)), where g(r(t)) can be either r(t) or ln(r(t)).

In the general two-factor Hull-White model [Hull and White, 1994b] the risk-neutral process for the short-rate, under the risk-neutral measure Q, can be expressed as:

dz(t) = (θ(t) + m(t) − α1z(t))dt + σ1dw1(t), z(0) = r0, or z(0) = ln(r0) (2.4.1a)

where the drift parameter m has an initial value of zero, is stochastic, and follows the process:

dm(t) = α2m(t)dt + σ2dw2(t), m(0) = 0. (2.4.1b)

As in the one-factor models considered in [Hull and White, 1994a] and in the previous section, the function θ(t) is chosen to make the model consistent with the initial term structure. The stochastic variable m is a component of the reversion level of z and itself reverts to a level of zero at rate α2. The parameters

α1, α2, σ1 and σ2 are constants and w1 and w2 are Brownian motions with instantaneous correlation ρ.

The instantaneous correlation ρ between the Brownian motions w1 and w1is modeled as:

EQ_(dw

1dw2) = ρdt, −1 ≤ ρ ≤ 1.

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Models of the Term Structure of Interest Rates - Two-Factor Interest Rate models

To implement the model, it is convenient to write (2.4.1) in the more symmetrical additive form [Brigo and Mercurio, 2001, 132]. In the additive form, z(t) is written as:

z(t) = x(t) + y(t) + φ(t), (2.4.2a)

where φ(t) is an arbitrary deterministic function and x and y follow the stochastic processes:

dx(t) = −β1x(t)dt + η1d ¯w1(t), x(0) = 0, (2.4.2b)

dy(t) = −β2y(t)dt + η2d ¯w2(t), y(0) = 0.

Again, ¯w1 and ¯w2 are Brownian motions, now with instantaneous correlation ¯ρ:

EQ_(dw

1dw2) = ¯ρdt, −1 ≤ ρ ≤ 1.

Brigo and Mercurio [2001, 149] show in detail that the two representations, (2.4.1) and (2.4.2), are analogous, i.e. they imply the same evolution equation for the short-rate r(t), if we choose:

β1 = α1, β2 = α2, η1 = s σ2 1+ σ2 2 (α1− α2)2 + 2ρ σ1σ2 α2− α1 , (2.4.3a) η2 = σ2 α1− α2 , ¯ ρ = σ1ρ − η2 η1 , and: φ(t) = z(0)e−α1t₊ Z t 0

θ(u)e−α1(t−u)_du. _(2.4.3b)

In the additive form, we see that z(t) is the sum of two Ornstein-Uhlenbeck processes plus a deterministic shift φ(t).

In the general two-factor Hull-White model (2.4.1), the function θ(t) is arbitrary. This allows the model to be exactly fitted to the initial term structure if required. Alternatively, as noted earlier, we might choose to specify a parametric form for θ(t) which allows a close (but not exact) fit. This is the approach adopted by B&H [Morrison, 2002]. Define µ(t) = g(n(t)), where g(n(t)) is either n(t) or ln(n(t)) and consider:

dz(t) = χ1(µ(t) − z(t))dt + τ1dw1(t), z(0) = r0, or z(0) = ln(r0) (2.4.4)

dµ(t) = χ2(˜µ − µ(t))dt + τ2dw2(t), µ(0) = n0, or µ(0) = ln(n0)

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GR Models of the Term Structure of Interest Rates - Two-Factor Interest Rate models

This, i.e. (2.4.4), is the specification of interest, as it is being used in the ESG. We can show that this model is analogous to the general two-factor Hull-White model if:

χ1 = α1,

χ2 = α2, (2.4.5a)

τ1 = σ1,

τ2 = σ2/α1,

and most importantly, if θ(t) in 2.4.1 can be written as:

θ(t) = α1µ(0)e−α2t+ α1µ(1 − e˜ −α2t). (2.4.5b)

This model (2.4.4) can also be rewritten into the additive form (2.4.2). To make the additive form then, equivalent to (2.4.1) with θ(t) as in (2.4.5b) we should use:

β1 = χ1, β2 = χ2, η1 = s τ2 1+ χ2 1τ 2 2 (χ1− χ2)2 + 2ρ τ1χ1τ2 χ2− χ1 , (2.4.6a) η2 = τ2 χ2− χ1 , ¯ ρ = τ1ρ − η2 η1 , and the additive function φ(t) is then:

φ(t) = ˜µ + (z(0) − ˜µ)e−χ1t₊ χ1

χ1− χ2

(µ(0) − ˜µ)(e−χ2t_{− e}−χ1t_). _(2.4.6b)

B&H assume ρ to be equal to η2

τ1, so that ¯ρ = 0. It can, if so desired, be relaxed without difficulty, and

we assume B&H only retain it to keep the calibration as simple and parsimonious as possible.

Note the diffence between (2.4.3b) and (2.4.6b). Whereas (2.4.3b) holds a virtually infinite number of parameters, (2.4.6b) is composed of ˜µ, µ0, r0, χ1, χ2, τ1 and τ2. So instead of having a deterministic

function that allows the model to be exactly fitted to the observed term structure, we will have to find the values of the parameters that make the model fit the term structure as closely as possible.

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GR

Models of the Term Structure of Interest Rates - Two-Factor Interest Rate models

Table 2.3 summarizes the models that will be of interest later in this thesis and holds the abbrevia-tions we will use, in the remaining, to refer to them. We are especially interested in BK2P, because B&H supplied AEGON with a scenario generator that is able to run simulations based on this model.

Model Evolution Equations

[HW2] Hull-White dr(t) = (θ(t) + m(t) − α1r(t))dt + σ1dw1(t) r(0) = r0 dm(t) = α2m(t)dt + σ2dw2(t) m(0) = 0 [BK2] Black-Karasinski d ln(r(t)) = (θ(t) + m(t) − α1ln(r(t)))dt + σ1dw1(t) r(0) = r0 dm(t) = α2m(t)dt + σ2dw2(t) m(0) = 0 [HW2P] Hull-White dr(t) = χ1(n(t) − r(t))dt + τ1dw1(t) r(0) = r0 Parametric dn(t) = χ2(˜µ − n(t))dt + τ2dw2(t) n(0) = n0 [BK2P] Black-Karasinksi (B&H) d ln(r(t)) = χ1(n(t) − ln(r(t)))dt + τ1dw1(t) r(0) = r0 Parametric d ln(n(t)) = χ2(˜µ − ln(n(t)))dt + τ2dw2(t) n(0) = n0

Table 2.3: Two-Factor Interest Rate models

Figures 2.8 and 2.9 each show 5 random sample paths generated, using the same 5 Brownian motions, by the HW2P and BK2P-model. The values of the parameters that have been used to generate these paths can be found in table 2.4.

HW2P BK2P (figure 2.8 ) (figure 2.9 ) χ1 0.150 0.060 χ2 0.020 0.020 τ1 0.005 0.010 τ2 0.003 0.050 r0 0.025 0.025 n0 0.045 0.060 ˜ µ 0.040 0.040

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GR Models of the Term Structure of Interest Rates - Two-Factor Interest Rate models dr(t) = 0.15(n(t) − r(t))dt + 0.01dw1(t), r0= 2.5% dn(t) = 0.02(0.04 − n(t))dt + 0.05dw2(t), n0= 4.5% HW2P 0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 0 10 20 30

Sample paths generated for HW2P (HW2P in table 2.3, with the parameter values as in table 2.4)

These paths are inconsistent with the term structure in figure 2.1.

Figure 2.8: HW2P sample paths

d ln(r(t)) = 0.06(ln(n(t)) − ln(r(t)))dt + 0.005dw1(t), r0 = 2.5% d ln(n(t)) = 0.02(0.04 − ln(n(t)))dt + 0.003dw2(t), n0= 6.0% BK2P 0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 0 10 20 30

Sample paths generated for BK2P (BK2P in table 2.3, with the parameter values as in table 2.4)

These paths are inconsistent with the term structure in figure 2.1.

Figure 2.9: BK2P sample paths

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Chapter 3 The relationship between short-rate models and

forward-rate models

3.1 Introduction

Up to this point we have only considered interest rate models where the short-rate r is the only explanatory variable. The short-rate models in chapter 2 all specify the stochastic behavior of the short-rate r, which in fact is unobservable in the market. Therefore, the calibration of these models, as we will see in chapter 6, requires a transformation of the short-rate into dynamics of observable quantities such as discount bond and European interest rate swaption prices. This shortcoming has inspired the development of another approach which uses forward-rates, which as opposed to short-rates are directly observable, as building blocks.

Heath et al. [1992] proposed a framework in which the entire forward-rate term structure, i.e. all f (t, s) for every s ∈ [0, T ], is modeled. Section 3.2 will briefly discuss their framework, whereas section 3.3 will show how the one-factor short-rate models that we discussed in section 2.3 fit in this framework. Chiarella and Kwon [1999] show that also the two-factor models can be obtained from the Heath-Jarrow-Morton framework.

3.2 The Heath-Jarrow-Morton Framework

In this section we will give an overview of the Heath-Jarrow-Morton [HJM] framework. For details we refer to [Heath et al., 1992] or [Bj¨ork, 1998].

Assume uncertainty is modeled through a d-dimensional Brownian motion w. The filtration F = {F(t) : 0 ≤ t ≤ T } is the augmentation of the natural filtration generated by the Brownian motion w. In the gen-eral HJM-framework, for all s ∈ [0, T ], the instantaneous forward-rate f (t, s) is assumed to be specified under the probability measure Q as:

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GR Short-rate models and forward-rate models - The Heath-Jarrow-Morton Framework

where σ(t, s) is d-dimensional vector of adapted processes and ˜σ(t, s) is itself an adapted process. Roughly speaking, a process is said to be adapted if it is measurable on F (t).

In the HJM framework the entire forward-rate term structure is being modeled. The index s in (3.2.1) only serves as a parameter in order to indicate which maturity is considered. To make the (3.2.1) con-sistent with the forward-rate term structure observed on the market, we will use the following initial condition:

f (0, s) = fM(0, s), (3.2.2)

where for all s ∈ [0.T ], fM(0, s) denotes the forward-rate with a maturity s as observed on the market at t = 0. In this way, all models within the HJM framework automatically fit the observed term structure.

In section 2.2, to clarify the relation with discount bond prices, we wrote f (t, T ) as:

f (t, T ) = − ∂

∂T ln(P (t, T )).

In section 2.2 we silently assumed that discount bonds are differentiable in T . This is a requirement for the HJM framework. The price at t = 0 of a discount bond maturing at time T could be written as:

P (0, T ) = e−R0Tf (0,s)ds. (3.2.3)

However, we already have an expression for the price of such a bond in term of the short rate r:

P (0, T ) = EQ_(e−RT

0 r(s)ds). (3.2.4)

These two equations, (3.2.3) and (3.2.4), and the fact that the short-rate r and the forward-rates f are tied together by r(t) = f (t, t) imposes a condition on the relation between ˜σ and σ in 3.2.1. Bj¨ork [1998] proves that under the probability measure Q the following relation must hold:

˜

σ(t, T ) = σ(t, T ) Z T

t

σ(t, s)ds, (3.2.5)

for every t and every T ≥ t.

The drift ˜σ(t, T ) is completely determined by the choice of the volatility processes σ(t, T ).

One of the most commonly used models for pricing interest rate derivatives, the LIBOR Market Model [LMM], is based on the HJM framework. It is also known as the BGM or BGM/J [Brace et al., 1997/ Jamshidian, 1997], from the name of the authors that were the first to discuss it rigorously.

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GR

Short-rate models and forward-rate models - Short-rate models within the HJM framework

3.3 Short-rate models within the HJM framework

In this section we will show that suitable specifications of the forward-rate volatility processes σ(t, T ) produce interest rate models that closely resemble the single factor HL, HW and BK discussed in section 2.3. Because we only consider one-factor models in this section we assume the Brownian motion and vector σ(t, s) in (3.2.1) to be one-dimensional (d = 1).

From (3.2.1) and (3.2.2), we can derive that in the general HJM-framework the instantaneous forward-rate f (t, T ) is assumed to satisfy the following stochastic integral equation:

f (t, T ) = f (0, T ) + Z t 0 ˜ σ(s, T )ds + Z t 0 σ(s, T )dw1(s).

By realizing r(t) = f (t, t) the integral equation for the corresponding short-rate process r(t) is:

r(t) = f (0, t) + Z t 0 ˜ σ(s, t)ds + Z t 0 σ(s, t)dw1(s). (3.3.1)

Differentiating the previous equation yields:

dr(t) = ∂f (0, t) ∂t + Z t 0 ∂ ˜σ(s, t) ∂t + σ 2 (s, t) ds + Z t 0 ∂σ(s, t) ∂t dw1(s) + σ(t, t)dw1(t). (3.3.2) Unlike the one-factor models we discussed before, in (3.3.2) the evolution of the short-rate at time t depends on the entire history of shocksRt

0 ∂σ(s,t)

∂t dw1(s). By choosing an appropriate specifications of the

forward-rate volatility processes σ(t, T ) we can overcome this problem.

Volatility process of the form:

σ(s, t) = σe−α(t−s), (3.3.3)

where σ and α both are constants turn out to be appropriate. In the remaining of this section we will only consider models within the HJM framework that have a volatility process as in (3.3.3). With such a volatility process, the stochastic differential equation in (3.3.2) can [Chiarella and Kwon, 1999] be rewritten into: dr(t) = ∂f (0, t) ∂t + ∂ ∂t Z t 0 σ(s, t) Z t s σ(s, u)duds + Z t 0 ∂σ(s, t) ∂t dw1(s) dt + σ(t, t)dw1(t), = ∂f (0, t) ∂t + ( Z t 0 σ2(s, t)ds − α Z t 0 ˜ σ(s, t)ds) − α Z t 0 σ(s, t)dw1(s) dt + σdw1(t), (3.3.4)

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GR Short-rate models and forward-rate models - Short-rate models within the HJM framework

Ho and Lee

Remember that the HL-model under the risk-neutral measure Q is given by:

dr(t) = θ(t)dt + σdw1(t). (3.3.5)

In this model σ is constant and θ(t) is a function that is chosen to make the model consistent with the initial term structure.

If we choose α in (3.3.3) equal to zero, σ(s, t) will become a constant σ. Consequently, (3.3.4) can be written as: dr(t) = ∂f (0, t) ∂t + σ 2_t dt + σdw1(t). (3.3.6)

We find that if the volatility process σ(s, t) is chosen to be a constant σ, there is a model in the HJM framework that closely resembles HL. Moreover we see that to make HL consistent with the observed term structure, just as models within the HJM framework are, θ(t) in (3.3.5) should be∂f (0,t)_∂t + σ2_t_.

Hull and White

In Hull and White’s model, the short-rate r(t) follows under Q the process:

dr(t) = (θ(t) − αr(t))dt + σdw1(t). (3.3.7)

Now, instead of choosing α in (3.3.3) equal to zero, let it be a positive constant (α > 0). Now σ(s, t) is an exponentially decaying function and, by using (3.3.1), (3.3.4) can be written as:

dr(t) = ∂f (0, t) ∂t + αf (0, t) + Z t 0 σ2(s, t) − αr(t) dt + σdw1(t) = ∂f (0, t) ∂t + αf (0, t) + σ2 2α(1 − e −2αt_{) − αr(t)} dt + σdw1(t) (3.3.8)

It is easily seen that (3.3.7) and (3.3.8) are equivalent if θ(t) in the former is:

θ(t) = ∂

∂tf (0, t) + αf (0, t) + σ2 2α(1 − e

−2αt_).

The derivation of HW from the HJM framework provides an insight into the deterministic function θ(t).

Black and Karasinski

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Chapter 4 Tree Building Procedure

4.1 Introduction

Interest rate trees can be used as a computationally efficient method for the valuation of interest rate derivatives, which is, as will become clear later, the key-element of the calibration procedure. In this chapter we will describe how short-rate processes can be approximated with a recombining two or three dimensional trinomial tree. An interest rate tree is an approximation of the stochastic process for the short-rate r(t).

Section 6 will discuss the calibration procedure and how parameter values are chosen.

4.2 One-Factor Interest Rate models

This section will present the approach we adopted for constructing a recombining tree that represents risk-neutral movements in the short-rate r(t). Usually, a recombining trinomial tree is preferred to a binomial tree, since the additional flexibility provided by the trinomial tree can be used to match not only the mean, but also the variance of the process for the short-rate r(t) [Leippold and Wiener, 2004]. Although the focus of this thesis is on BK2P, for completeness this section will discuss a quite general procedure for building a recombining trinomial tree for the one-factor short rate models HW and BK. The procedure for the two-factor models is a natural extension and will be discussed in the next section.

We will follow the approach put forward by Hull and White [1994a]. Although there are analytical solutions for discount bonds and European interest rate swaptions in their model (2.3.2) (and for calibra-tion purposes it is not necessary to construct a tree) it is useful to start with their tree-building procedure. For one, the tree building procedure for HW is less complicated than the procedure for BK. Secondly the existence of the analytical solutions provided a convenient means for inspecting the correctness of the algorithm we programmed in Visual Basic. The Visual Basic script can be found in Appendix A.1. The careful reader will notice that the outline of main functions in the script is almost identical to the procedure described below.

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GR Tree Building Procedure - One-Factor Interest Rate models

have to rely on tree-approximations.

Following Hull and White our initial aim is to construct a trinomial tree that approximates the evo-lution of the short-rate r(t) according to:

dr(t) = (θ(t) − αr(t))dt + σdw1(t), r(0) = r0. (4.2.1)

Hull and White use an iterative procedure to model (4.2.1). Instead of directly modeling the discrete approximation of the stochastic process (4.2.1) by using the closed-form formula for θ:

θ(t) = ∂

∂tf (0, t) + αf (0, t) + σ2 2α(1 − e

−2αt_), _(4.2.2)

they prefer to build the trinomial tree in a two-step procedure. In the first step they construct an auxiliary tree for the process r0(t):

dr0(t) = −αr0(t)dt + σdw1(t), r

0

(0) = r0₀= 0. (4.2.3)

In the second step bθ, which is the approximation of θ, will be calculated. This will be done by means of a set of parameters that is introduced later in this section. Their two-step procedure will lead to a tree where the initial term structure is exactly matched. If, however, the value of θ is assumed to apply to the time interval between t and t + ∆t, the initial term structure is only matched in the limit as ∆t tends to 0.

We start with constructing a tree for the process of r0(t) in (4.2.3). Note that apart from the added prime, (4.2.3) is in fact (4.2.1) with θ(t) = 0. Equation (4.2.3) can be integrated so as to yield:

r0(t + ∆t) = r0(t)eα∆t+ σ Z t+∆t

t

e−α(t−u)dw(u)

Therefore, r0(t + ∆t) conditional on Ftis normally distributed with mean and variance respectively given

by:

E(r0(t + ∆t)|Ft) = r

0

(t)(e−α∆t− 1), (4.2.4)

V ar(r0(t + ∆t)|Ft) = σ2(1 − e−2α∆t)/2α.

Let us, now, first fix a time horizon [0, T ] and the times 0 = t0 < t1 < . . . < tN = T and set the time

length of the timesteps equal to ∆t = T /N = ti+1− ti for each i. From this point onwards time is being

measured in years.

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GR

Tree Building Procedure - One-Factor Interest Rate models

We set M and V equal to their exact values in (4.2.4), so that M = e−α∆t−1 and V = σ2_(1−e−2α∆t_)/2α.

In any case V thus depends on the chosen timestep ∆t. A variety of choices could be made for ∆r0/√∆t. A standard choice, of which Hull and White [Brigo and Mercurio, 2001/ Hull and White, 1994a] claim it is motivated by convergence purposes, is σ√3, so that the interest rate step ∆r0 is:

∆r0 =√3V . (4.2.5)

We will come back to the choice Hull and White made later.

We will start out with building a symmetrical tree, similar to the one in figure 4.1, where the nodes are equally spaced in r0 and t. Because r0 is assumed to start from 0, after a number of timesteps r0 is one of: r0₀+ j∆r0 = j∆r0, _{j ∈ Z,} dr0(t) = −0.10r0(t)dt + 0.01dw1(t), r 0 (0) = r00 = 0. 0.00% -1.73% -3.45% -3.43% -3.45% -1.70% -3.45% 0.04% -3.45% -1.73% -1.71% -3.43% -1.71% -1.70% -1.71% 0.04% -1.71% -1.73% 0.02% -1.70% 0.02% 0.04% 0.02% 1.77% 0.02% -1.73% 0.00% 0.00% -1.71% -3.43% -1.71% -1.70% -1.71% 0.04% -1.71% 0.00% 0.02% -1.70% 0.02% 0.04% 0.02% 1.77% 0.02% 0.00% 1.75% 0.04% 1.75% 1.77% 1.75% 3.50% 1.75% 0.00% 0.00% 1.74% 0.02% -1.70% 0.02% 0.04% 0.02% 1.77% 0.02% 1.74% 1.75% 0.04% 1.75% 1.77% 1.75% 3.50% 1.75% 1.74% 3.48% 0.04% 3.48% 1.77% 3.48% 3.50% 3.48% 1.74% 0.00% -4.00% -3.00% -2.00% -1.00% 0.00% 1.00% 2.00% 3.00% 4.00% 0.00 0.50 1.00 1.50 2.00 2.50 3.00 t r(t)

Tree approximation of the process (4.2.3), with α = 0.10 and σ = 0.01.

The length of the timestep ∆t is set equal to 1.

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The tree branching can take any of the forms shown in figure 4.2.

(a) (b) (c)

(non-standard) (standard) (non-standard)

Different forms of tree branching. The non-standard branching

is used in the edges of the tree (see figure 4.1)

Figure 4.2: Branching Processes

We will refer to figure 4.2(b) as standard or normal branching. If the standard branching applies to a certain node, r0 could either go up by ∆r0, remain unchanged or go down by ∆r0. Both figures 4.2(a) and 4.2(c) will be referred to as non-standard branching. If the branching is as in figure 4.2(a), and r0 can remain unchanged or it can go up by either ∆r0 or 2∆r0. Finally, if it is as in figure 4.2(c) r0 can remain unchanged or go down by ∆r0 or 2∆r0.

Define (i, j) as the node for which t = i∆t and r0 = j∆r0. Let r0(i, j) be the value of r0 at node (i, j). Define pu, pm, and pd as the probabilities of the upper, middle and lowest branches emanating

from a node. The probabilities at each node are chosen to match the conditional mean and the condi-tional variance of (r0(t + ∆t) − r0(t)) and to sum to one. This yields a system of three linear equations and three unknowns. If the branching from node (i, j) is standard as in figure 4.2(b) the short-rate can go up by ∆r0, remain unchanged or go down by ∆r0 and the expectation of the short-rate at time ti+1,

contingent upon r0 having value j∆r0 at time ti is given by:

E((r0(ti+1) − r 0 (ti))|r 0 (ti) = j∆r 0 ) = r0(ti)M, (4.2.6) = j∆r0M.

A similar calculation for the conditional variance gives:

V ar((r0(ti+1) − r 0 (ti))|r 0 (ti) = r 0 0+ j∆r 0 ) = 1 3(∆r 0 )2+ (j∆r0M )2. (4.2.7)

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pu∆r 0 + pm0 + pd(−∆r 0 ) = j∆r0M, pu(∆r 0 )2+ pm0 + pd(−∆r 0 )2 = 1 3(∆r 0 )2+ (j∆r0M )2, pu+ pm+ pd = 1.

So that the up, middle and down-branching probabilities are:

pu = 1 6+ j2M2+ jM 2 , (4.2.8a) pm = 2 3− j 2_M2_, _(4.2.8b) pd = 1 6+ j2_M2_{− jM} 2 . (4.2.8c)

We prevent pu, pmand pdfrom getting outside the interval (0, 1) by switching to non-standard branching.

We define jmax and jmin as the values of j where we switch to non-standard branching. The binding

constraint on j comes from (4.2.8b), the equation for pm. From (4.2.8b) we can easily derive that for pm

to be positive, j has to satisfy:

−1 −p(2/3)

M < j <

p(2/3) M

Hull and White suggest to set jmaxequal to the smallest integer greater than −

1−√(2/3)

M , jmin= −jmax.

When j is small, i.e. equal to jmin, at the bottom edge of the tree the branching will be as in figure

4.2(a), and r0 can remain unchanged or it can go up by either ∆r0 or 2∆r0. The modified probabilities become: pu = 1 6 + j2M2− jM 2 , pm = − 1 3 − j 2_M2_{− 2jM,} _(4.2.9) pd = 7 6 + j2_M2_{− 3jM} 2 .

and when j is big, equal to jmax, at the top edge of the tree where the branching will be as in figure

4.2(c). In this case r0 can remain unchanged or go down by ∆r0 or 2∆r0. The probabilities are:

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Now, we will return to our earlier remark about the choice of ∆r0.

In addition to mean and variance, we might consider to match higher order moments.

We are under the impression that this is what Hull and White did as well. Unfortunately without report-ing it in their articles. The skewness is zero for both the continuous normal process and the trinomial tree, since both are symmetric. The fourth moment, the kurtosis, in node (0, 0), where r is r0 equals:

pu(∆r 0 )4+ pm0 + pd(−∆r 0 )4= 1 6( √ 3V )4+1 6(− √ 3V4) = 3V2, which is identical of the kurtosis of the normal distribution.

We believe that this the reason that Hull and White decided to choose ∆r0 equal to√3V .

The tree in figure 4.1 is constructed for σ = 0.01, α = 0.1 and ∆t = one year. M is set to equal to −α∆t and V to σ2_{∆t. This leads to ∆t = 0.0173 and j}

max= 2.

The next stage is to introduce the time-varying drift θ(t). To do this we displace the nodes at time i∆t by an amount γi. The value of r at node (i, j) in the new tree equals the value of r

0

in the old tree at node (i, j) plus γi. The values of the γi are chosen so that the short-rate process represented by the

tree is consistent with the initial term structure observed in the market. In this stage the process that is being approximated is (4.2.1).

If we define bθ(ti) as the approximation of θ between ti and ti+1. The drift in r at time ti at the

midpoint of the tree, that is in node(i, 0), equals γi− γi−1:

(bθ(ti) − αγi)∆t = γi− γi−1, so that: b θ(ti) = γi− γi−1 ∆t + αγi. As ∆t tends to 0, bθ tends to θ.

Define Qi,j as the present value of a security that only pays off 1 if node (i, j) is reached. These Qi,j are

so-called Arrow-Debreu prices. By no-arbitrage arguments the following must hold for 0 ≤ i ≤ N :

P (0, i∆t) =

ni

X

j=−ni

Qi,j, (4.2.11)

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The γi and Qi,j are calculated using forward induction. An useful byproduct of storing Qi,j’s is that

all discount bonds prices and European interest rate Swaptions with payouts at time ti can be valued

immediately using the Qi,j, as we will see in chapter 6.

The value of Q0,0 is equal to 1. The value of γ0 is chosen to give the right price for P (0, ∆t). There is

a probability of pu that node (1, 1) will be reached and the discount rate for ∆t is γ0∆t. Q1,1 can be

calculated using:

Q1,1= pueγ0∆t.

Analogously:

Q1,0 = pmeγ0∆t,

Q1,−1 = pdeγ0∆t.

Now, with (4.2.11) we get:

P (0, ∆t) = n1 X j=−n1 Qi,j= eγ0∆t. Consequently γ0equals: γ0= − ln(P (0, ∆t)) ∆t ,

which is equal to R(0, ∆t) (see (2.2.2)).

Now γ1 can be determined. It is chosen to give the right price for P (0, 2∆t) and follows from:

P (0, 2∆t) = Q1,1e−(γ1+∆r 0 )_{+ Q} 1,0e−(γ1)+ Q1,−1e−(γ1−∆r 0 )_.

More general: Assuming that Qi,j have been determined for i ≤ m, γm can be calculated so that it

correctly prices P (0, (m + 1)∆t). The interest rate at node (m, j) is γm+ j∆r

0 so that: P (0, (m + 1)∆t) = nm X j=−nm Qm,je(−(γm+j∆r 0 )∆t)_. _(4.2.12)

The solution is:

γm=

ln(Pnm

j=−nmQm,je

−j∆r0∆t_{) − ln(P (0, (m + 1)∆t))}

∆t .

Once γm has been determined, the Qi,j for i = m + 1 can be calculated using [Hull and White, 2000]:

Qm+1,j=

X

Qm,kq((m, k) → (m + 1, j)e(−(γm+k∆r

0

ECONOMIC VALUATION OF LIABILITIES: CALIBRATING A TWO-FACTOR SHORT-RATE MODEL.