Aﬃne Interest Rate Models - Theory and Practicecuchiero/Christa... · Abstract The aim of this...

D I P L O M A R B E I T

Affine Interest Rate Models - Theoryand Practice

Ausgefuhrt am

Institut fur Wirtschaftsmathematikder Technischen Universitat Wien

unter der Anleitung von

ao. Univ.-Prof. Dr. Josef Teichmann

durch

Christa Cuchiero

Hainzenbachstraße 254060 Leonding

Datum Unterschrift

Abstract

The aim of this diploma thesis is to present the theory as well as the practicalapplications of affine interest rate models. On the basis of the general theoryestablished by Duffie and Kan, we put emphasis on affine models whose statevariables have - in contrast to their theoretical abstract definition - a rea-sonable economic interpretation. Starting from the very first term structuremodels, namely the Vasicek and the Cox-Ingersoll-Ross model, we describein sequel two- and more-factor models that have appeared in literature. Bymeans of the Vasicek model we exemplify the calibration to market yields aswell as to market cap volatilities.However, our main focus are affine yield factor models developed by Duffieand Kan, which allow to relate the state variables to yields with differentmaturities. We show how to calibrate a two-factor version of this model tomarket data. The results are promising since the model fits the market yieldsfrom different dates very well while the parameters remain nearly constant.

Contents

1 Introduction to Interest Rate Theory 1

1.1 Definitions and Notations . . . . . . . . . . . . . . . . . . . . 11.1.1 Short-Term Interest Rate . . . . . . . . . . . . . . . . . 11.1.2 Zero-Coupon Bonds and Spot Interest Rates . . . . . . 21.1.3 Forward Rates . . . . . . . . . . . . . . . . . . . . . . . 31.1.4 Interest Rate Swaps . . . . . . . . . . . . . . . . . . . 4

1.2 No-Arbitrage Pricing . . . . . . . . . . . . . . . . . . . . . . . 61.3 Factor Models of the Term Structure . . . . . . . . . . . . . . 7

1.3.1 Dynamics under P∗ . . . . . . . . . . . . . . . . . . . . 9

1.3.2 The Bond Price as Solution of a PDE . . . . . . . . . . 12

2 Affine Models 13

2.1 Theory of Affine Factor Models . . . . . . . . . . . . . . . . . 132.1.1 Specification of the State Variable Process . . . . . . . 142.1.2 Affine Stochastic Differential Equations . . . . . . . . . 172.1.3 Ricatti Equations . . . . . . . . . . . . . . . . . . . . . 18

2.2 Types of Affine Models . . . . . . . . . . . . . . . . . . . . . . 192.2.1 Gaussian Affine Models . . . . . . . . . . . . . . . . . . 192.2.2 CIR Affine Models . . . . . . . . . . . . . . . . . . . . 202.2.3 The Three-Factor Affine family . . . . . . . . . . . . . 21

2.3 Classification of Affine Models . . . . . . . . . . . . . . . . . . 222.3.1 A Canonical Representation . . . . . . . . . . . . . . . 222.3.2 Invariant Transformations and Equivalent Models . . . 24

3 Examples of Affine Models 26

3.1 Examples of One-Factor Affine Models . . . . . . . . . . . . . 263.1.1 The Extended Vasicek Model . . . . . . . . . . . . . . 273.1.2 The Extended CIR Model . . . . . . . . . . . . . . . . 29

3.2 Examples of Multi-Factor Affine Models . . . . . . . . . . . . 313.2.1 The Longstaff and Schwartz Two-Factor Model . . . . 323.2.2 The Central Tendency as Second Factor . . . . . . . . 33

ii

CONTENTS iii

3.3 Economic Models . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.1 The General Framework . . . . . . . . . . . . . . . . . 363.3.2 IS - LM Framework . . . . . . . . . . . . . . . . . . . . 37

3.4 Non-Affine Models - Consol Models . . . . . . . . . . . . . . . 383.5 Criteria for Model Selection . . . . . . . . . . . . . . . . . . . 40

4 Calibration and Estimation 42

4.1 Obtaining a Data Set . . . . . . . . . . . . . . . . . . . . . . . 434.1.1 Market Data for the Current Yield Curve . . . . . . . . 434.1.2 Market Data for Bond Options . . . . . . . . . . . . . 444.1.3 Which Market Rate should be used for the

Short-Term Rate? . . . . . . . . . . . . . . . . . . . . . 474.2 Calibration to Current Market Data . . . . . . . . . . . . . . . 47

4.2.1 Calibrating the Vasicek Model to the CurrentTerm Structure . . . . . . . . . . . . . . . . . . . . . . 47

4.2.2 Calibrating the Vasicek Model to Cap Volatilities . . . 494.2.3 Calibrating the Hull-White Extended

Vasicek Model . . . . . . . . . . . . . . . . . . . . . . . 524.3 Historical Estimation . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.1 Maximum Likelihood Method . . . . . . . . . . . . . . 544.3.2 General Method of Moments . . . . . . . . . . . . . . . 56

5 Affine Yield-Factor Models 58

5.1 General Affine Yield-Factor Model . . . . . . . . . . . . . . . 585.2 A Two-Factor Affine Model of the

“Long”- and the Short-Term Rate . . . . . . . . . . . . . . . . 605.2.1 Deterministic Volatility . . . . . . . . . . . . . . . . . . 615.2.2 Calibrating the Deterministic Volatility Model

to the Current Term Structure . . . . . . . . . . . . . . 615.2.3 Stochastic Volatility . . . . . . . . . . . . . . . . . . . 645.2.4 Calibrating the Stochastic Volatility Model

to the Current Term Structure . . . . . . . . . . . . . . 675.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 71

A Numerical Methods for Calibration 72

A.1 Trust-Region Methods for NonlinearMinimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 72A.1.1 Box Constraints . . . . . . . . . . . . . . . . . . . . . . 74A.1.2 Nonlinear Least-Squares . . . . . . . . . . . . . . . . . 74

List of Figures

4.1 Market- vs. Vasicek Model Yields . . . . . . . . . . . . . . . . 494.2 Vasicek Model Parameters over Time . . . . . . . . . . . . . . 504.3 Market- vs. Vasicek Model Caps . . . . . . . . . . . . . . . . . 514.4 Hull-White Model Calibration . . . . . . . . . . . . . . . . . . 53

5.1 Market- vs. Model Yields, 2-Factor Deterministic VolatilityModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Parameters over Time, 2-Factor Deterministic Volatility Model 665.3 Market- vs. Model Yields, March 2006, 2-Factor Deterministic

Volatility Model . . . . . . . . . . . . . . . . . . . . . . . . . . 675.4 Parameters over Time, 2-Factor Stochastic Volatility Model . . 705.5 Market- vs. Model Yields, 2-Factor Stochastic Volatility Model 70

iv

List of Tables

4.1 Parameters for the Vasicek Model, Calibration to Yields . . . 484.2 ATM Cap Volatilities . . . . . . . . . . . . . . . . . . . . . . . 514.3 Parameters for the Vasicek Model, Calibration to Caps . . . . 514.4 Parameters for the Hull-White Model, Calibration to Caps . . 534.5 Maximum Likelihood Parameters for the Vasicek Model . . . . 55

5.1 Parameters for the 2-Factor Deterministic Volatility Model . . 635.2 Residuals for the 2-Factor Deterministic Volatility Model . . . 635.3 Monthly Parameters for the 2-Factor Deterministic Volatility

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.4 Parameters for the 2-Factor Stochastic Volatility Model . . . . 685.5 Residuals for the 2-Factor Stochastic Volatility Model . . . . . 695.6 Monthly Parameters for the 2-Factor Stochastic Volatility Model 69

v

Chapter 1

Introduction to Interest Rate

Theory

Although the concept of interest rates seems to be something natural thateverybody knows to deal with, the management of interest rate risk, i.e. thecontrol of changes in future cash flows due to fluctuations in interest ratesis an issue of great complexity. In particular, the pricing and hedging ofproducts depending in large part on interest rates create the necessity formathematical models.

This chapter covers the basic definitions and concepts of interest rate the-ory. The first part focuses on the different kinds of interest rates, whereasin the other sections mathematical basics for interest rate modeling are pre-sented. The approach is similar to Brigo and Mercurio [3] including supple-ments of Musiela and Rutkowski [14] and Bjork [1].

1.1 Definitions and Notations

1.1.1 Short-Term Interest Rate

The first concept that is introduced is the notion of a bank (savings) accountrepresenting a risk-free security which continuously compounds in value at arisk-free rate, namely the instantaneous interest rate (also referred as short-term interest rate).

Definition 1.1. Short-term rate, Bank account. Let r(t) denote theshort-term rate for risk-free borrowing or lending at time t over the infinites-imal time interval [t, t + dt]. r(t) is assumed to be an adapted process on a

1

CHAPTER 1. INTRODUCTION TO INTEREST RATE THEORY 2

filtered probability space (Ω,F ,P, (Ft)0≤t≤T ∗)1 for some T ∗ > 02 with almostall sample paths integrable on [0, T ∗]. B(t) = B(t, ω) is defined to be thevalue of the bank account at t ≥ 0 that evolves for almost all ω ∈ Ω accordingto the differential equation

dB(t) = r(t)B(t)dt with B(0) = 1. (1.1)

Consequently

B(t) = exp( ∫ t

0

r(s)ds)

for all t ∈ [0, T ∗]. (1.2)

By means ofB(t), two amounts of currency which are available at differenttimes can be related. In fact, in order to have one unit of cash at time Tone has to invest the amount 1/B(T ) at the beginning. At time t > 0 thevalue of this initial investment constitutes B(t)(1/B(T )), which leads to thefollowing definition.

Definition 1.2. Stochastic discount factor. The stochastic discount fac-tor D(t, T ) is the value at time t of one unit of cash payable at time T > tand is given by

D(t, T ) =B(t)

B(T )= exp

(−

∫ T

t

r(s)ds). (1.3)

1.1.2 Zero-Coupon Bonds and Spot Interest Rates

Definition 1.3. Zero-coupon bond. A zero-coupon bond of maturity Tis a financial security paying one unit of cash at a prespecified date T in thefuture without intermediate payments. The price at time t ≤ T is denotedby P (t, T ). Obviously, P (T, T ) = 1 for all T ≤ T ∗.3

Remark. It is assumed that the price process P (t, T ) follows a strictly posi-tive and adapted process on a filtered probability space (Ω,F ,P, (Ft)0≤t≤T ∗),where the filtration Ft is again the P-completed version of the filtration gen-erated by the underlying Brownian motion.

Note that there is a close relationship between the zero-coupon bondprice P (t, T ) and the stochastic discount factor D(t, T ). Actually, P (t, T )corresponds to the expectation of D(t, T ) under the risk-neutral probabilitymeasure, as we will see in the next section (equation 1.21). If r is determin-istic, then D is deterministic as well and necessarily D(t, T ) = P (t, T ).

1Ft is the P-completed (i.e. it contains all sets of null probability with respect to P)filtration generated by a standard Brownian motion in R

n.2T ∗ is the fixed horizon date for all market activities.3P (t, T ) is the discount factor at time t for cashflows occurring at time T .


Definition 1.4. Continuously-compounded spot interest rate or

yield on a zero-coupon bond. The continuously-compounded spot in-terest rate R(t, T ), also referred as yield on the zero-coupon bond P (t, T ),is the constant rate at which an investment of P (t, T ) units of cash attime t accrues continuously to yield one unit of cash at maturity T , i.e.exp(R(t, T )(T − t))P (t, T ) = 1. Hence

R(t, T ) = − lnP (t, T )

T − t. (1.4)

Remark. The short-term rate r(t) is obtained as limit of R(t, T ), that is

r(t) = limT→t+

R(t, T ) = limT→t+

− lnP (t, T )

T − t. (1.5)

Definition 1.5. Simply-compounded spot interest rate. The simply-compounded spot interest rate L(t, T ) is the constant rate at which an in-vestment has to be made to produce one unit of cash at maturity T , startingfrom P (t, T ) units of cash at time t, when accruing is proportional to theinvestment time. In formulas,

L(t, T ) =1 − P (t, T )

(T − t)P (t, T ). (1.6)

Remark. The market LIBOR and EURIBOR rates are simply-compoundedrates, whose day-count convention is ”Actual”/360. This means that theyear is assumed to be 360 days long and the corresponding year fraction isthe actual number of days between two dates divided by 360.

1.1.3 Forward Rates

Forward rates are interest rates that can be locked in today for an investmentin a future time period. Their values can be derived directly from zero-coupon bond prices. Define f(t, T, S) to be the continuously-compoundedforward rate at time t for the expiry time T and maturity time S. We musthave

exp(R(t, S)(S − t)) = exp(R(t, T )(T − t)) exp(f(t, T, S)(S − T )), (1.7)

so that

f(t, T, S) =1

(S − T )lnP (t, T )

P (t, S), (1.8)

since otherwise arbitrage would be possible.


Analogous to the instantaneous short-term rate the instantaneous forwardrate f(t, T ) at time t for the maturity T is defined by

f(t, T ) = limS→T+

f(t, T, S) = −∂ lnP (t, T )

∂T, (1.9)

so that we also have

P (t, T ) = exp(−

∫ T

t

f(t, u)du). (1.10)

Beside the continuously compounded forward rate, a simply-compoundedforward rate can be defined as well.

Definition 1.6. Simply-compounded forward interest rate. The sim-ply compounded forward interest rate at time t for the expiry T and maturityS is denoted by F (t, T, S) and is defined by

F (t, T, S) =1

(S − T )

(P (t, T )

P (t, S)− 1

). (1.11)

Remark. Expression (1.11) can be derived from a forward rate agreement.This is a contract where at maturity S, a fixed payment based on a fixedrate K is exchanged against a floating payment based on the rate L(T, S).Formally, at time S one receives (S−T )K units of cash and pays the amount(S − T )L(T, S). The value of the contract at time S is therefore

(S − T )(K − L(T, S)).

Discounting this value to time t leads to

P (t, S)(S − T )K − P (t, T ) + P (t, S). (1.12)

The simply-compounded forward rate F is now the fixed rate that must beinserted for K to render the contract fair at time t, so that the contract value(1.12) is 0 at time t.

1.1.4 Interest Rate Swaps

An interest rate swap (IRS) is a contract that exchanges interest paymentsbetween two differently indexed legs, of which one is usually fixed whereasthe other one is floating. When the fixed leg is paid and the floating leg isreceived the interest rate swap is termed payer IRS and in the other casereceiver IRS.


The present value at time t = T04 of borrowing one unit of cash at a fixed

rate K with coupons paid at times Ti, i = 1, . . . , n and with τi = Ti −Ti−1 is

PV (fixed leg) =n∑

i=1

P (t, Ti)τiK + P (t, Tn). (1.13)

and the present value at time t = T0 of a stream of floating rate cashflows is

PV (floating leg) =n∑

i=1

P (t, Ti)τiL(Ti−1, Ti) + P (t, Tn). (1.14)

Thus, the present value at time t = T0 of a payer IRS is given by

PV (Payer IRS) =n∑

i=1

P (t, Ti)τi(L(Ti−1, Ti) −K). (1.15)

For simplicity we have assumed that the tenors of the floating and fixed legsare the same.5

Definition 1.7. Swap rate. The swap rate is the rate S that must beinserted for K in equation (1.13) in order to have

PV (fixed leg) = PV (floating leg). (1.16)

By simplifying the value of the floating leg to

PV (floating leg) =n∑

i=1

P (t, Ti)τiL(Ti−1, Ti) + P (t, Tn)

=n∑

i=1

P (t, Ti)τi1 − P (Ti−1, Ti)

τiP (Ti−1, Ti)+ P (t, Tn)

=n∑

i=1

( P (t, Ti)

P (Ti−1, Ti)− P (t, Ti)

)+ P (t, Tn)

=n∑

i=1

(P (t, Ti−1) − P (t, Ti)) + P (t, Tn) = 1, (1.17)

and by using equation (1.13) we can express the swap rate S(t, Tn) in termsof bonds prices

S(t, Tn) =1 − P (t, Tn)∑n

i=1 τiP (t, Ti). (1.18)

4T0 is the first reset date.5Indeed, a typical interest rate swap in the market has a fixed leg with annual payments

and a floating leg with quarterly or semiannual payments.


We have now only regarded swaps starting at time t = T0. If, however, t <T0 cashflows are exchanged starting at a future time, rather than immediatelyand we have a forward start swap. So the value of the floating leg must bediscounted and is therefore P (t, T0). Consequently the value of the payerforward start swap is

P (t, T0) −n∑

i=1

P (t, Ti)τiK − P (t, Tn).

This is 0 when K is the forward start swap rate S(t, T0, Tn),

S(t, T0, Tn) =P (t, T0) − P (t, Tn)∑n

i=1 τiP (t, Ti). (1.19)

1.2 No-Arbitrage Pricing

The absence of arbitrage opportunities between all bonds with different matu-rities and the bank account is the fundamental economic assumption whichwill be introduced in this section and which all further considerations arebased on.

Definition 1.8. Arbitrage-free family of bond prices. A familyP (t, T ), t ≤ T ≤ T ∗, of adapted processes is called an arbitrage-free familyof bond prices relative to r if the following conditions hold:

i) P (T, T ) = 1 for all T ∈ [0, T ∗] and

ii) there exists a probability measure P∗ on (Ω,FT ∗) equivalent6 to P, such

that for all t ∈ [0, T ] the discounted bond price

P (t, T ) = D(0, t)P (t, T ) =B(0)

B(t)P (t, T ) =

P (t, T )

B(t)(1.20)

is a martingale under P∗.

Any probability measure P∗ that satisfies the required conditions of def-

inition 1.8 is named martingale measure for the family P (t, T ). Actually,definition 1.8 is based on the general result that the existence of an equiva-lent martingale measure implies the absence of arbitrage opportunities in astandard market model. As P (t, T ) follows a martingale under P

∗, we have:

P (t, T ) = EP∗(P (T, T )|Ft) for t ≤ T.

6P and P

∗ are equivalent measures, if P(A) = 0 ⇔ P∗(A) = 0 for every A ∈ FT∗ .


Therefore,

D(0, t)P (t, T ) = EP∗(D(0, T )P (T, T )|Ft) = EP∗(D(0, T )|Ft),

which leads to the following expression for the bond price

P (t, T ) = D(0, t)−1EP∗(D(0, T )|Ft)

= exp(∫ t

0

r(s)ds)EP∗

(exp

(−

∫ T

0

r(s)ds)∣∣∣Ft

)

= EP∗

(exp

(−

∫ T

t

r(s)ds)∣∣∣Ft

)= EP∗(D(t, T )|Ft). (1.21)

Thus, P (t, T ) corresponds to the expectation of the stochastic discount factorD(t, T ) under P

∗. So we directly obtained the unique no-arbitrage pricefor bonds, which is again a special case of the general no-arbitrage priceassociated with an attainable contingent claim H given by

πt = EP∗(D(t, T )H|Ft). (1.22)

1.3 Factor Models of the Term Structure

As we have seen in the previous section, the zero-coupon bond price is givenby the following expression:

P (t, T ) = EP∗

(exp

(−

∫ T

t

r(s)ds)∣∣∣Ft

). (1.23)

So, whenever we can characterize the distribution of exp(−∫ T

tr(s)ds), we are

able to compute bond prices. The general idea of a factor model for the yieldcurve is to suppose that there is a Markov process X valued in some opensubset D ⊂ R

n such that, for any times t and T , the market value PM(t, T )of a zero-coupon bond at time t is given by g(Xt, τ), where τ = T − t andg ∈ C2,1(D × R≥0).

To start with we assume that X follows an n-dimensional Ito processunder the actual probability P, i.e.

dXt = µ(Xt, t)dt+ σ(Xt, t)dWt, (1.24)

where Wt is a standard P-Brownian motion in Rn. In order to guarantee the

existence of a unique solution, µ : D×[0, T ∗] → Rn and σ : D×[0, T ∗] → R

n×n

must be measurable functions that satisfy the following conditions,

|µ(x, t)|+|σ(x, t)| ≤ C1(1 + |x|) x ∈ D, t ∈ [0, T ∗],

|µ(x, t) − µ(y, t)|+|σ(x, t) − σ(y, t)| ≤ C2|x− y| x, y ∈ D, t ∈ [0, T ∗],

(1.25)


for some constants C1 and C2, where |σ|2=∑

|σij|2 (see Øksendal [15]).

Remark. Note that (1.24) is a short form of the following integral represen-tation

Xt = X0 +

∫ t

0

µ(Xs, s)ds+

∫ t

0

σ(Xs, s)dWs. (1.26)

We will now consider time-homogeneous Ito processes, meaning that thefunctions µ and σ only depend on X and not on t, i.e.

dXt = µ(Xt)dt+ σ(Xt)dWt, (1.27)

where Wt is again a standard P-Brownian motion in Rn and where µ and σ

satisfy the conditions of (1.25), which in this case can be simplified to

|µ(x) − µ(y)|+|σ(x) − σ(y)|≤ C|x− y| x, y ∈ D. (1.28)

In the following we will briefly explain an important property of theseprocesses, namely the Markov property.

Definition 1.9. Markov process. The stochastic process (Xt, t ∈ T ∗) iscalled Markov if for every n and t1 < t2 < . . . < tn,

P(Xtn|Xtn−1, . . . , Xt1) = P(Xtn|Xtn−1

). (1.29)

Theorem 1.1. The Markov property for Ito processes. Let Xxt be a

time-homogeneous Ito process of the form

dXxt = µ(Xx

t )dt+ σ(Xxt )dWt, Xx

0 = x, (1.30)

where µ and σ satisfy the conditions of (1.28) and let f be a bounded Borelfunction from R

n to R. Then, for t, s ≥ 0

E(f(Xxt+s)|Fs) = E(f(Xy

t ))|y=Xxs. (1.31)

This Markov property can now be used for calculating the price of thezero-coupon bond P (t, T ), given by (1.23).

Proposition 1.2. Let Xxt be of form (1.30) and the short rate process r

defined as r(t) := R(Xxt ) where R : D → R, then there exists a measurable

function g : D × R≥0 → R, such that

P (t, T ) = g(Xxt , T − t) = EP∗

(exp

(−

∫ T−t

0

R(Xys )ds

))∣∣∣y=Xx

t

. (1.32)


Proof. The result follows directly from the fact that P (t, T ) is given by

EP∗

(exp

(−

∫ T

t

R(Xxs )ds

)∣∣∣Ft

)

and by applying theorem (1.1) to this expression.

Remarks. To be consistent with (1.5) the function R defining the short rateprocess r must be of the form

R(x) = limτ→0+

− ln g(x, τ)

τ, x ∈ D, (1.33)

where τ = T − t.We are interested in choices for (g, µ, σ) that are compatible, in the sensethat (1.32) with (1.33) is fulfilled. One important class of compatible modelsare affine models whose main properties will be described in section 2.1.

1.3.1 Dynamics under P∗

We are now interested in the dynamics of (1.24) under an equivalent mar-tingale measure P

∗ for the bond market. Note that for a fixed maturitydate T , the zero-coupon bond price P (t, T ) is a function of Xt and t, i.e.P (t, T ) = G(Xt, t)(≡ g(Xt, T − t)), t ≤ T . By applying Ito’s formula we get

dP (t, T ) = α(Xt, t)dt+ β(Xt, t)dWt, (1.34)

α(Xt, t) =∂G(Xt, t)

∂t+∂G(Xt, t)

∂Xµ(Xt, t) +

+1

2tr

(∂2G(Xt, t)

∂X2σ(Xt, t)σ

′(Xt, t)),

β(Xt, t) =∂G(Xt, t)

∂Xσ(Xt, t),

where ∂G∂X

= ( ∂G∂X1

, . . . , ∂G∂Xn

). Furthermore we consider a self-financing portfo-

lio consisting of φ1 units of the zero-coupon bond P 1 and φ2 units of anotherzero-coupon bond P 2 (i.e. with another maturity date T ), whose value pro-cess is given by

Vt(φ) = φ1tP

1(t, T 1) + φ2tP

2(t, T 2). (1.35)

As φ is self-financing strategy, we have

dVt(φ) = φ1tdP

1(t, T 1) + φ2tdP

2(t, T 2) (1.36)


and consequently

dVt(φ) = (φ1tα

1(Xt, t) + φ2tα

2(Xt, t))dt+ (φ1tβ

1(Xt, t) + φ2tβ

2(Xt, t))dWt.

In order to have a risk-less portfolio we chose φ1tβ

1i (Xt, t) + φ2

tβ2i (Xt, t) = 0

for all i. Due to the absence of arbitrage it has to satisfy the conditiondVt = r(t)Vtdt, which leads to the following equality

dVt(φ) = (φ1tα

1(Xt, t) + φ2tα

2(Xt, t))dt = r(t)(φ1tP

1(t, T 1) + φ2tP

2(t, T 2))dt.

Thus, we get a linear system of equations whose unknowns are φ1t and φ2

t ,

φ1t (α

1(Xt, t) − r(t)P 1(t, T 1)) + φ2t (α

2(Xt, t) − r(t)P 2(t, T 2)) = 0,

φ1tβ

1i (Xt, t) + φ2

tβ2i (Xt, t) = 0 for all i. (1.37)

In order to have a solution for φ1t and φ2

t that is different from 0 it is necessarythat the following equality holds for each component i of the vector β:

α1(Xt, t) − r(t)P 1(t, T 1)

nβ1i (Xt, t)

=α2(Xt, t) − r(t)P 2(t, T 2)

nβ2i (Xt, t)

. (1.38)

Hence, the term

λi(t) =α(Xt, t) − r(t)P (t, T )

nβi(Xt, t)=

α(Xt,t)P (t,T )

− r(t)

nβi(Xt,t)P (t,T )

(1.39)

is invariant for every considered bond P (t, T ), i.e. it does not depend on T .

Remark. Equation (1.34) expresses the bond-price dynamics in terms of theshort-term rate r, with α/P being the return and β/P the volatility of thebond. For this reason λ = (λ1, . . . , λn) can be interpreted as follows: Thedifference α(Xt, t)/P (t, T ) − r(t) represents the difference in returns withrespect to the risk-less bank account. By dividing by (β(Xt, t)/P (t, T )), wedivide by the riskiness of the zero-coupon bond. That is why λ is referred toas risk premium, market price of risk or as proposed by Brigo and Mercurio[3] as “excess return with respect to a risk-free investment per unit of risk”.

If λ(t) satisfies Novikov’s condition, i.e.

EP

(exp

(1

2

∫ T

0

λ(s)λ′(s)ds))

<∞, (1.40)


we are able to define a probability measure P∗ equivalent to P by the Radon

Nikodym derivative

dP∗

dP

∣∣∣Ft

= exp(−

∫ t

0

λ(s)dWs −1

2

∫ t

0

λ(s)λ′(s)ds), P-a.s. (1.41)

Then, in view of Girsanov’s theorem7, the process

W ∗t = Wt +

∫ t

0

λ(s)′ds for all t ∈ [0, T ∗] (1.42)

follows a standard Brownian motion under P∗.

Remark. Due to the construction of λ, where we used the no-arbitrage ar-gument, P

∗ satisfies the conditions of definition 1.8 and is therefore an equiv-alent martingale measure. More specifically, the dynamics of the discountedbond price under P are given by

dP (t, T ) = d(

exp(−

∫ t

0

r(s)ds)P (t, T )

)= (1.43)

= exp(−

∫ t

0

r(s)ds)β(Xt, t)(λ(t)′dt+ dWt).

By moving from P to P∗ using (1.42) we have dW ∗

t = dWt + λ(t)′dt, whichleads to

dP (t, T ) = exp(−

∫ t

0

r(s)ds)β(Xt, t)dW

∗t . (1.44)

Thus, as P (t, T ) can be represented as a stochastic integral under P∗, it is a

martingale and P∗ therefore an equivalent martingale measure.

Now we have all necessary tools to state the following result that showsthe behavior of X under P

∗.

Proposition 1.3. Let P (t, T ) be an arbitrage-free family of bond prices andassume that X follows an Ito process under the actual probability P, as spec-ified by (1.24). Then for any equivalent martingale measure P

∗ of definition1.8, whose Radon-Nikodym derivative is given by (1.41) the process X satis-fies under P

∗

dXt = µ∗(Xt, t)dt+ σ(Xt, t)dW∗t with (1.45)

µ∗(Xt, t) = µ(Xt, t) − σ(Xt, t)λ(t)′. (1.46)

7See Øksendal [15] for details.


Proof. From (1.42) we know that dWt = dW ∗t − λ(t)′dt holds. Combining

this expression with (1.24) we immediately get

dXt = µ(Xt, t)dt+ σ(Xt, t)(dW∗t − λ(t)′dt).

Remark. It is essential to assume that the function λ is sufficiently regular,so that the SDE (1.45) admits a unique global strong solution.

1.3.2 The Bond Price as Solution of a PDE

If we replace in equation (1.39) α and β by their definition we obtain

∂G(Xt, t)

∂t+∂G(Xt, t)

∂X(µ(Xt, t) − σ(Xt, t)λ(t)′) +

+1

2tr

(∂2G(Xt, t)

∂X2σ(Xt, t)σ

′(Xt, t))− rG(Xt, t) = 0.

By using (1.46) the above expression becomes

∂G(Xt, t)

∂t+∂G(Xt, t)

∂Xµ∗(Xt, t) +

+1

2tr

(∂2G(Xt, t)

∂X2σ(Xt, t)σ

′(Xt, t))− rG(Xt, t) = 0 (1.47)

with the terminal condition P (T, T ) = G(XT , T ) = 1. It follows fromthe Feynman-Kac8 formula that under mild technical assumptions the risk-neutral valuation formula for bond prices EP∗(exp(−

∫ T

tr(s)ds)|Ft) solves

PDE (1.47).

8See Øksendal [15] for details.

Chapter 2

Affine Models

In this chapter we study the theory of affine models which rank among themost popular models in theory and practice and of which many exampleshave been investigated. For instance, the very first term structure model,the Vasicek model, is an affine model. Other popular models such as Cox,Ingersoll, Ross (CIR), Hull and White or Longstaff and Schwartz are also ofthis type. Their popularity is due to their tractability and to their flexibility,because there are often explicit solution for bond prices and bond optionprices.

Affine models were investigated amongst others by Duffie and Kan [8]who developed a general theory. Dai and Singelton [6] have provided a clas-sification and have established the most general representative example ofeach class of affine models.

In the following sections we explain the general theory and properties,whereas important examples and practical applications are examined in chap-ter 3. Furthermore we give an overview of the three main families of affinemodels that have appeared in the literature. Complementary to this, wedescribe the Dai and Singelton [6] classification. We follow the approach ofJames & Webber [11] and Cairns [4] considering in particular the papers ofDuffie and Kan [8] and Dai and Singelton[6] .

2.1 Theory of Affine Factor Models

To start with we consider a model with n state variables

X(t) = (X1(t), . . . , Xn(t))′,

which are valued in some open subset D ⊂ R and follow a n-dimensional Itoprocess described by (1.45) (i.e under P

∗), whose exact form will be specified

13

CHAPTER 2. AFFINE MODELS 14

by theorem 2.2. Affine multi-factor models are characterized by the fact thatthe zero-coupon bond prices can be written in the form

P (t, T ) = exp(A(t, T ) +

n∑

i=1

Bi(t, T )Xi(t))

= exp(A(t, T ) +B(t, T )′X(t)) (2.1)

withB(t, T ) = (B1(t, T ), . . . , Bn(t, T ))′.

Concerning the yields R(t, T ), it is obvious that they have to be of thefollowing form

R(t, T ) = −A(t, T )

T − t− B(t, T )′

T − tX(t). (2.2)

Considering remark (1.33) the short-term rate r(t) must be the limit ofR(t, T ) for T going to t. Thus, we can express r(t) by means of (2.2), i.e.

r(t) = R(X(t)) = g + h′X(t), (2.3)

where g and h are constants:

g = limT→t+

−A(t, T )

T − t,

h′ = limT→t+

−B(t, T )′

T − t.

Remarks. The model is time-homogeneous if X(t) is time-homogeneous andthe functions A(t, T ) and B(t, T ) are functions of τ = T − t. In the followingwe will restrict our considerations to this time-homogeneous case.The function g(X(t), T−t) specified by proposition (1.2) is here consequentlygiven by

g(X(t), τ) = exp(A(τ) +B(τ)′X(t)). (2.4)

2.1.1 Specification of the State Variable Process

The next proposition and the following theorem specify the conditions thatthe functions µ∗ and σ of SDE (1.45)1 must satisfy under the assumptionthat (g, µ∗, σ) is a compatible model in the sense of remark (1.33), where ggiven by (2.4).

1In contrast to equation (1.45) µ∗ and σ only depend on X and not on t, as we onlyconsider the time-homogeneous case.


Proposition 2.1. Suppose that (g, µ∗, σ) is a compatible term structure fac-tor model with functions µ∗ and σ of SDE (1.45). If g is of form (2.4) andthere exist maturities τ1, . . . , τN for N = 2n+(n2−n)/2 such that the N×Nmatrix C(τ1, . . . , τN), whose ith row is of the form2

c(τi)′ = (c1(τi), . . . , cn(τi), cn+1(τi), cn+2(τi), cn+3(τi), . . . , cN(τi))

=(B1(τi), . . . , Bn(τi),

B1(τi)2

2, B1(τi)B2(τi), B1(τi)B3(τi) . . . ,

Bn(τi)2

2

),

(2.5)

is non-singular, then µ∗, σσ′ and r are affine.

Proof. From (2.3) we already know that r is affine. In order to show that µ∗

and σσ′ are affine functions we have to calculate the derivatives of g(X(t), τ)and insert them in PDE (1.47). The partial derivatives of G(X(t), t) ≡g(X(t), τ) are given by

∂G

∂t= −∂g

∂τ= g(X(t), τ)

(− ∂A(τ)

∂τ− ∂B(τ)′

∂τX(t)

),

∂G

∂Xi

=∂g

∂Xi

= g(X(t), τ)Bi(τ),

∂2G

∂Xi∂Xj

=∂2g

∂Xi∂Xj

= g(X(t), τ)Bi(τ)Bj(τ).

Consequently, by (1.47) and (2.3)

g(X(t), τ)(− ∂A(τ)

∂τ− ∂B(τ)′

∂τX(t) +B(τ)′µ∗(X(t)) +

+1

2

n∑

i=1

n∑

j=1

Bi(τ)Bj(τ)σi(X(t))σj(X(t))′ − (g + h′X(t)))

= 0, (2.6)

where σi denotes the ith row of the matrix σ. Since g(X(t), τ) is strictlypositive valued, (2.6) is equivalent to

n∑

i=1

Bi(τ)µ∗i (X(t)) +

1

2

n∑

i=1

n∑

j=1

Bi(τ)Bj(τ)κij(X(t)) =

∂A(τ)

∂τ+∂B(τ)′

∂τX(t) + g + h′X(t)

︸︷︷︸:=a(X(t),τ)

, (2.7)

2Regard the proof for the construction of this matrix.


where κij(X(t)) = σi(X(t))σj(X(t))′. Under a mild non-degeneracy condi-tion this differential equation implies that µ∗ and σσ′ are affine functions.In order to see this, note at first that the right side of (2.7), which we willdenote a(., τ), is affine for each fixed τ . Define now a function H : D → R

N

for N = 2n+ (n2 − n)/2

H(x) := (µ∗1(x), µ

∗2(x), . . . , κ11(x), κ12(x), . . . , κnn(x))′,

where only the κij(x) with i ≤ j are included. Now the aim is to show thateach component of H is affine in x. Equation (2.7) can be written as a systemof equation in τ and x of the form

c(τ)′H(x) = a(x, τ), (2.8)

where c : R≥0 → RN . For example, c1(τ) = B1(τ)(the coefficient of H1(x)),

while cn+1(τ) = B1(τ)2/2 (the coefficient of κ11(x)). By extending (2.8) to

N maturities τ1, . . . , τN we get

C(τ1, . . . , τN)H(x) =

a(x, τ1)a(x, τ2)

...a(x, τN)

, (2.9)

where C(τ1, . . . , τN) is the N × N matrix whose i-th row is c(τi)′. So, if

τ1, . . . , τN can be chosen, such that C(τ1, . . . , τN) is non-singular, then thereis a unique solution H(.) to (2.9), which is a linear combination of affinefunctions (since the right side is a vector of affine functions), and is thereforeaffine (compare with Duffie and Kan [8]).

Remark. Of course, for arbitrary distinct non-zero maturity times τ1, . . . , τNthe matrix C(τ1, . . . , τN) is non-singular except for (B(τ1), . . . , B(τN)) in aclosed subset of measure zero of R

Nn.

Theorem 2.2. Duffie and Kan, 1996 [8]. If P (t, T ) = P (t, t + τ) =g(X(t), τ) can be written in the form (2.4) then the process for X(t) mustsolve the following SDE

dX(t) = (α+ βX(t))dt+ SD(X(t))dW ∗t , (2.10)

where W ∗t is a n-dimensional Brownian motion under P

∗, α = (α1, . . . , αn)′

is a constant vector, β = (βij) and S = (sij) are constant n × n matricesand, finally, D(X(t)) is a diagonal matrix of the form

D(X(t)) =

√γ′1X(t) + δ1 0

. . .

0√γ′nX(t) + δn

, (2.11)


where δi ∈ R and γi = (γi1, . . . , γin)′ ∈ Rn.

Proof. The proof of this theorem is based on proposition (2.1). If σσ′ isaffine in x, then under non-degeneracy conditions and a possible re-orderingof indices σ(X(t)) = SD(X(t)), where D(X(t)) is of form (2.11).See Duffie and Kan [8] for details.

2.1.2 Affine Stochastic Differential Equations

As indicated by the last theorem, the affine class of term structure modelsseems to be well behaved. Now we formulate the conditions on the coefficientsof equation (2.10) under which there is indeed a unique strong solution to theSDE. In order to assure this, there are two problems to cope with: Firstly,the diffusion function σ(X(t)) = SD(X(t)) is not Lipschitz continuous andsecondly γ′iX(t) + δi must be non-negative for all i and t. The open domainG with non-negative volatilities is

G = x ∈ Rn : γ′ix+ δi > 0, i ∈ 1, . . . , n. (2.12)

In effect, to guarantee the existence of a solution X, it is necessary to assumethat for each i the volatility process γ′iX(t) + δi has as sufficiently strongpositive drift on the i-th boundary segment ∂Gi = x ∈ G : γ′ix + δi = 0.The following theorem states the conditions which guarantee the existenceof a unique solution to (2.10) that remains in G.

Theorem 2.3. If the following conditions hold for all i,

i) if γ′ix+ δi=0, then γ′i(βx+ α) >(γ′

iSS′γi)

2for all x ∈ R

n,

ii) if (γ′iS)j 6= 0, then γ′ix+ δi = γ′jx+ δj for all j,

then there exists a unique (strong) solution X in G to the stochastic differ-ential equation (2.10) with (2.11) and (2.12). Moreover, for this solution Xand for all i, we have γ′iX(t) + δi > 0 for all t almost surely.

Proof. See Duffie and Kan [8]

Remarks. Both conditions of theorem (2.3) are designed to ensure strictlypositive volatility. As they are not generally satisfied, they are a significantrestriction on the model.For a state process X(t) satisfying the conditions of this theorem, there isalways a strictly positive non constant short rate process r(t) given by (2.3).


2.1.3 Ricatti Equations

In this section we show how to obtain two differential equations for A(τ) andB(τ), which are known as Ricatti equations. At first, we can now write (2.7)in the form

−∂A(τ)

∂τ− ∂B(τ)′

∂τX(t) +B(τ)′(α+ βX(t)) +

+1

2

n∑

i=1

n∑

j=1

Bi(τ)Bj(τ)σiσ′j − (g + h′X(t)) = 0,

where σi denotes the ith row of the matrix SD(X(t)), i.e

σi = (si1

√γ′1X(t) + δ1, . . . , sin

√γ′nX(t) + δn).

This equation is affine in x and if an affine relationship of the form a+bx = 0for all x in some non-empty open set, then a = 0 and b = 0. Therefore wehave

−∂A(τ)

∂τ+B(τ)′α+

1

2B(τ)′Sdiag(δ)S ′B(τ) − g = 0, (2.13)

−∂Bk(τ)

∂τ+B(τ)′βk +

1

2

n∑

i=1

n∑

j=1

n∑

l=1

Bi(τ)Bj(τ)silsjlγlk − h′ = 0, (2.14)

for k = 1, . . . , n with boundary conditions

A(0) = 0 and B(0) = 0. (2.15)

βk is the kth column of the matrix β and γlk is the kth component of thevector γl. The boundary conditions follow from the fact that P (T, T ) =g(X(T ), 0) = 1, which implies that A(0) + B(0)′X(T ) = 0. Since T isarbitrary, A(0) +B(0)′x = 0 must hold for all x in D, which yields (2.15).

Remarks. There is a non-trivial issue of the existence of finite solutions toRicatti equations, since the coefficients are not Lipschitz continuous. Solu-tions exist on the whole time domain for special cases. For any particulargiven case they exist up to some given time T > 0, which is due to the localLipschitz continuity of the coefficients.In general, the Ricatti equations need to be solved numerically, although inthe case of Vasicek and CIR they are explicitly solvable.


2.2 Types of Affine Models

Commonly used affine models can be conveniently separated into three maintypes:

• Gaussian affine models

• CIR affine models

• Three factor affine family

Remark. This categorization is distinct from the classification provided byDai and Singelton, which is a complete canonical mathematical classification(see section 2.3).

We consider now each of the three categories in turn. Note that thedynamics of X are always specified under the martingale measure P

∗.

2.2.1 Gaussian Affine Models

All time-homogeneous Gaussian models are based on the following generalmodel. Let X(t) = (X1(t), . . . , Xn(t))′ evolve according to the following SDE

dX(t) = (α+ βX(t))dt+ ΣdW ∗t , (2.16)

where α ∈ Rn, β, Σ ∈ R

n×n and W ∗t is a n-dimensional Brownian motion

under P∗. As specified by (2.3) the short rate r(t) is a function

r(t) = g + h′X(t) (2.17)

of the state variables X(t), where g ∈ R and h ∈ Rn.

Remark. If all hi are non-zero, it is possible to rescale the Xi(t) and assumethat the hi = 1 for all i without loss of generality.

In order to solve SDE (2.16) we apply the following theorem.

Theorem 2.4. Variation of Constants. Let B be some real valued n×nmatrix, a, σ1, . . . , σd ∈ R

n and (Wt)t≥0 a d-dimensional Brownian motion on(Ω,F ,P, (Ft)0≤t≤T ∗). Then the solution of the SDE

dX(t) = (a+BX(t)dt+d∑

i=1

σidWi(t), X(0) = x ∈ Rn (2.18)

is given by

X(t) = eBtx+

∫ t

0

eB(t−s)ads+d∑

i=1

∫ t

0

eB(t−s)σidWi(t). (2.19)


Remark. For any real-valued matrix C, eC =∞∑

i=1

Ci

i!.

So, in our case X(t) = eβtX(0) +∫ t

0eβ(t−s)αds +

∫ t

0eβ(t−s)ΣdW ∗

s , is theunique solution of SDE (2.16) and r(t) is therefore given by

r(t) = g + h′eβtX(0) + h′∫ t

0

eβ(t−s)αds+ h′∫ t

0

eβ(t−s)ΣdW ∗s . (2.20)

Remark. The matrix β has a spectral decomposition β = βRΛβL, whereΛ =diag(λ1, . . . , λn) is the diagonal matrix of eigenvalues of β. βL and βR

are the matrices of left and right eigenvectors respectively. The columns ofβR can be scaled in a way which ensures that βRβL = I, so that βi can beeasily calculated by βRΛiβL. Concerning the eigenvalues we have to requirethat the real parts of them are negative, so that X(t) is stationary. Thisensures that exp(βt) tends to zero as t tends to infinity.

Equation (2.20) implies that r(t) is normally distributed with mean andvariance given respectively by

EP∗(r(t)) = g + h′eβtX(0) + h′∫ t

0

eβ(t−s)αds, (2.21)

varP∗(r(t)) =

∫ t

0

(h′eβ(t−s)ΣΣ′eβ′(t−s)h)ds. (2.22)

Remark. One major drawback of gaussian models is the positive probabilityof negative interest rates, which is incompatible with no-arbitrage and thepresence of cash.

The bond prices P (t, T ) can now be derived by computing the expecta-

tion EP∗(exp(−∫ T

tr(s)ds)|Ft). We will calculate this precisely for the one-

dimensional Vasicek Model in section 3.1.1. Another way would be solvingthe PDEs (2.13) and (2.14), where S = Σ and γi = 0 and δi = 1.

2.2.2 CIR Affine Models

A model is CIR (Cox, Ingersoll, Ross) affine if all state variables X(t) =(X1(t), . . . , Xn(t))′ are independent processes of the one factor CIR-type.Thus for i = 1, . . . , n,

dXi(t) = (αi − βiXi(t))dt+ σi

√Xi(t)dW

∗i (t), (2.23)


where W ∗1 (t), . . . ,W ∗

n(t) are independent standard Brownian motions. Forsimplicity we assume here that the short-term rate is then defined as

r(t) =n∑

i=1

Xi(t), (2.24)

i.e. g and h of equation (2.3) are assumed to be 0 and hi = 1 for all irespectively. As the model is affine in every factor, the bond price Pi(t, T )for the ith one-factor CIR-process can be written in the form exp(Ai(τ) +Bi(τ)Xi(t)). Due to the independence of factors it is therefore immediate toderive the following formula. The price P (t, T ) at time t of a zero couponbond with maturity T − t is explicitly given by

P (t, T ) = EP∗

(exp

(−

∫ T

t

r(s)ds)∣∣∣Ft

)

= EP∗

(exp

(−

∫ T

t

n∑

i=1

Xi(s)ds)∣∣∣Ft

)

=n∏

i=1

EP∗

(exp

(−

∫ T

t

Xi(s)ds)∣∣∣Ft

)=

n∏

i=1

Pi(t, T )

= exp( n∑

i=1

(Ai(τ) +Bi(τ)Xi(t))). (2.25)

2.2.3 The Three-Factor Affine family

This family represents models that mix Gaussian and CIR type state vari-ables. The motivation for that stems from the desire to have a stochasticmean and a stochastic volatility. The three processes are:

1. The short rate process:

dr(t) = κ(µ(t) − r(t))dt+√σ(t)dW ∗

r (t) (2.26)

2. The drift process:

dµ(t) = β(γ − µ(t))dt+ ηµφ(t)dW ∗µ(t), (2.27)

where φ = 0, 12.

3. The volatility process:

dσ(t) = δ(α− σ(t))dt+ λ√σ(t)dW ∗

σ (t) (2.28)

As an example we will mention the BDFS three-factor model in section 3.2.2.


2.3 Classification of Affine Models

In order to enable affine models to be compared and classified Dai and Sin-gelton [6] have provided a general framework where affine models can beclassified according to

1. The number of state variables, and

2. How many of the state variables appear in the volatility matrix.

Since a model could be represented by a transformation of variables in twoapparently different ways, it is necessary to find a general representative ofeach class, so that every affine model can be seen as special case of a generaltype. This section follows the treatment of Dai and Singelton.

2.3.1 A Canonical Representation

Dai and Singelton base their classification on processes for X(t) that arespecified in the form 3

dX(t) = κ∗(θ∗ −X(t))dt+ SD(X(t))dW ∗t , (2.29)

under P∗, where κ∗ is a constant n × n matrix and θ∗ is a constant n-

dimensional vector. S and D(X(t)) have the same form as in equation (2.10).In order to obtain also an affine model under the measure P it is assumedthat the market price of risk is given by 4

Λ(t) = D(X(t))λ, (2.30)

where λ ∈ Rn. Then, the process (2.29) evolves under P according to

dX(t) = κ(θ −X(t))dt+ SD(X(t))dWt, (2.31)

where, after proposition 1.3, κ = κ∗ − SΦ and θ = κ−1(κ∗θ∗ + Sψ). Theith row of the matrix Φ is (λiγi1, . . . , λiγin) and the ith component of thevector ψ is given by λiδi. As specified by (2.3) the short rate is set to ber(t) = g+ h′X(t). The model under P is therefore specified by the structure

A = (g, h, κ, θ, S,Γ, δ, λ),

where Γ = (γ1, . . . , γn) ∈ Rn×n is the matrix of coefficients on X(t) in D and

δ the vector whose components are δi.

3The coefficients α and β in (2.10) are now given by α = κ∗θ∗ and β = −κ∗.4Λ(t) corresponds to λ(t)′ of equation (1.46).


Let m = rank(Γ), i.e. m is the number of state variables that appear inthe matrix D. Using this index each n-factor model can be classified intoone of n + 1 subfamilies based on its value of m, since m can range from 0to n. Am(n) denotes the class of n-factor affine models with index m.

Definition 2.1. Canonical Representation. Let m = rank(Γ), then foreach m, we partition X(t) as X ′ = (X ′

1, X′2), where X1 is a m-dimensional

vector and X2 is a (n−m)-dimensional vector. The canonical representationof Am(n) is defined as the special case of equation (2.31) with

κ =

(κ1,1 0κ2,1 κ2,2

), (2.32)

for κ1,1 ∈ Rm×m, κ2,1 ∈ R

(n−m)×m, κ2,2 ∈ R(n−m)×(n−m) if m > 0 and for

m = 0, κ is either upper or lower triangular, and

θ =

(θ1

0

), (2.33)

S = In×n, (2.34)

δ =

(01

), (2.35)

Γ =

(Im×m Γ1,2

0 0

), (2.36)

where I is the identity matrix, θ1 ∈ Rm, 1 ∈ R

(n−m) and Γ1,2 ∈ Rm×(n−m).

Remark. Consistently with equation (2.12) parametric restrictions must beimposed. Basically γiX(t)+ δi must be strictly positive for all i and t. Thus,it is the same requirement as that of Duffie and Kan. However as the processis regarded under P, D also affects the drift parameters. Consequently it isalso necessary to constrain κ and θ (see Dai and Singleton [6]).

Definition 2.2. Equivalence class Am(n). Am(n) is defined as the set ofall affine models that are nested special cases of the canonical model or ofany equivalent model obtained by invariant transformations of the canonicalmodel.5

Remarks. For instance, there are obviously two classes of one-factor models.On the one hand A0(1), the class of the Vasicek model, where the statevariable does not appear in the volatility matrix and on the other handA1(1), the class of the CIR model, whose volatility is modeled by a square root

5These invariant transformations are formally defined in the next section.


process of the state variable. Both models are described in detail in chapter 3.This classification allows to place existing models that have appeared in theliterature in a general context, enabling us to determine when a given modelis over-specified, which means, for instance, that the original formulation ofthe model imposes too many restrictions on parameters. Such facts may notbe apparent in the original specification. We dwell on that in section 3.2.1when dealing with the Longstaff and Schwartz model.

2.3.2 Invariant Transformations and Equivalent Mod-

els

Invariant transformation are transformations of the state variables and pa-rameter vectors in ways that leave the bond prices unchanged. More pre-cisely, two models are equivalent in the sense that they generate identicalbond prices for all interest rate instruments if they can be transformed intoone another by a sequence of operations enumerated in the following list. Weconsider the model, specified by A = (g, h, κ, θ, S, γi, δi, i ∈ 1, . . . , n, λ),to which equivalent models can be obtained by means of the subsequentinvariant transformations.

1. Permutation TP : If π is a permutation of 1, . . . , n and if AP is themodel obtained by permuting the elements of X(t) and the componentsof the parameters accordingly, then AP is equivalent to A.

2. Brownian motion rotation TO: TO rotates a vector of independentBrownian motions into another vector of independent Brownian mo-tions by using a n × n orthogonal matrix O, i.e. O′ = O−1, thatcommutates with D. Then TOWt = OWt and

AO = (g, h, κ, θ, SO′, γi, δi, i ∈ 1, . . . , n, Oλ)

are the Brownian motions and the transformed equivalent model, re-spectively. Note that the state vector is not affected.

3. Diffusion rescaling TR: This transformation rescales the parameters ofD and λ by the same constant. So, for any n× n non-singular matrixR,

AR = (g, h, κ, θ, SR−1, R2iiγi, R

2iiδi, i ∈ 1, . . . , n, Rλ)

is the transformed model. The state vector and the Brownian motionsare not modified. Such rescaling is possible because only the combina-tions SDDS ′ and SDDλ enter the zero-coupon bond price equation.


4. Invariant affine transformation TA: It is defined by a non-singular n×nmatrix L and a n-dimensional vector ϑ, such that the state vector andthe equivalent model are given by

TAX(t) = LX(t) + ϑ,

AA = (g − h′L−1ϑ, L′−1h,

LκL−1, ϑ+ Lθ, LS, γi, δi, i ∈ 1, . . . , n, λ),

where γi = L′−1γi and δi = δi − γ′iL−1ϑ . The Brownian motions are

not changed. These transformation are possible because of the affinestructure of the models.

Chapter 3

Examples of Affine Models

Beyond doubt the best-known examples of affine models are the Vasicek andthe Cox, Ingersoll Ross (CIR) one-factor models. Beside these we also inves-tigate examples of multi-factor models, namely the Longstaff and Schwartzand the Balduzzi, Das, Foresi and Sundaram (BDFS) models. We also de-scribe an economic model of affine form that is based on the general macroe-conomic IS-LM relationship. The reason why we have chosen to present theseexamples is on the one hand their popularity and their historic importance,and on the other hand the fact that all of them allow an economic interpre-tation. At the end of the chapter we will conclude that affine models rangeamong the most appropriate models for practical purposes because of theirtractability and flexibility as well as the possibility of an useful economicinterpretation. We compare them with consol-rate-models, which were de-veloped to take economic principles into account and which, however, entailsome difficulties.

The dynamics are always specified under the martingale measure P∗ if

the opposite is not mentioned explicitly.

3.1 Examples of One-Factor Affine Models

A number of widely used one-factor models have the general (Hull-White)form

dr(t) = (α(t) − β(t)r(t))dt+ σ(t)r(t)γdW ∗t , r(0) = r0, (3.1)

for some constant γ ≥ 0 and where α, β, σ: R+ → R are locally boundedfunctions. α/β can be interpreted as the mean reversion level, whereas β is

26

CHAPTER 3. EXAMPLES OF AFFINE MODELS 27

the mean reversion rate.1

3.1.1 The Extended Vasicek Model

By setting γ = 0 in (3.1) we obtain the generalized Vasicek model, in whichthe dynamics of r are

dr(t) = (α(t) − β(t)r(t))dt+ σ(t)dW ∗t , r(0) = r0. (3.2)

To explicitly solve this equation, let us denote l(t) =∫ t

0β(u)du. By applying

a slightly different version of theorem (2.4) for the time-dependent functionsα, β and σ, we find the following explicit expression for r(t),

r(t) = e−l(t)(r0 +

∫ t

0

el(s)α(s)ds+

∫ t

0

el(s)σ(s)dW ∗s

). (3.3)

Properties of the Classical Vasicek Model

If all parameters in (3.2) are constants we get the classical Vasicek model.More precisely, r is defined as the unique strong solution of the SDE

dr(t) = (α− βr(t))dt+ σdW ∗t , r(0) = r0, (3.4)

where α, β and σ are strictly positive constants. This SDE is know as mean-reverting Ornstein-Uhlenbeck process. In this case r(t) is given by the follow-ing equation

r(t) =α

β+ (r0 −

α

β)e−βt + σe−βt

∫ t

0

eβsdW ∗s . (3.5)

(3.5) implies that r(t) is normally distributed with mean and variance givenrespectively by

EP∗(r(t)) =α

β+ (r0 −

α

β)e−βt (3.6)

varP∗(r(t)) = σ2e−2βt

∫ t

0

e2βsds =σ2

2β(1 − e−2βt) (3.7)

Remarks. Considering (3.6), it is obvious that r is mean reverting, sincethe expected rate tends, for t going to infinity to α

β, which can be regarded

as a long term average rate.

1The model is often written in the form dr(t) = β(t)(µ(t) − r(t))dt + σ(t)r(t)γdW ∗

t ,where µ is the mean reversion level.


The bond prices P (t, T ) can now be derived by computing the expecta-

tion EP∗(exp(−∫ T

tr(s)ds)|Ft). Using the Markov property for Ito processes

(Theorem 1.1) this expression is equivalent to

EP∗

(exp

(−

∫ T−t

0

r(s)yds))∣∣∣

y=r(t). (3.8)

Hence, at first we calculate (∫ T−t

0r(s)yds)|y=r(t):

( ∫ T−t

0

r(s)yds)∣∣∣

y=r(t)=

=α

β(T − t) +

(r(t) − α

β

)1 − e−β(T−t)

β+ σ

∫ T−t

0

∫ s

0

eβ(u−s)dW ∗uds

=α

β(T − t) +

(r(t) − α

β

)1 − e−β(T−t)

β+ σ

∫ T−t

0

( ∫ T−t

u

eβ(u−s)ds)dW ∗

u

=α

β(T − t) +

(r(t) − α

β

)1 − e−β(T−t)

β+ σ

∫ T−t

0

1 − eβ(u−(T−t))

βdW ∗

u .

As (∫ T−t

0r(s)yds)|y=r(t) is obviously a gaussian variable we can now easily

calculate the expectation

P (t, T ) = EP∗

(exp

(−

∫ T−t

0

r(s)yds))∣∣∣

y=r(t)

= exp[EP∗

(−

∫ T−t

0

r(s)yds)∣∣∣

y=r(t)

+1

2varP∗

(−

∫ T−t

0

r(s)ds)∣∣∣

y=r(t)

]

with

EP∗

(−

∫ T−t

0

r(s)yds)∣∣∣

y=r(t)= −

(αβ

(T − t) + (r(t) − α

β)1 − e−β(T−t)

β

),

varP∗

(−

∫ T−t

0

r(s)ds)∣∣∣

y=r(t)=

σ2

β2

∫ T−t

0

(1 − eβ(u−(T−t)))2du. (3.9)

So, the bond price can be expressed in the affine form

P (t, T ) = eA(T−t)−B(T−t)r(t),


where

B(T − t) =1 − e−β(T−t)

β,

A(T − t) = −αβ

(T − t− 1 − e−β(T−t)

β

)+

σ2

2β2

∫ T−t

0

(1 − eβ(u−(T−t)))2du

= (B(t, T ) − (T − t))(αβ− σ2

2β2

)− σ2

4βB(t, T )2, (3.10)

which could also be found by solving (2.13) and (2.14).

3.1.2 The Extended CIR Model

Extended CIR models have γ = 1/2 in equation (3.1), so that the short rateprocess is

dr(t) = (α(t) − β(t)r(t))dt+ σ(t)√r(t)dW ∗

t . (3.11)

When α, β and σ are strictly positive constants we have the classicalCIR model, which was developed by Cox, Ingersoll and Ross. Due to thesquare-root term in the diffusion coefficient, r can only take positive valueswhich is a major advantage over the Vasicek model.

Distribution of the Classical CIR Model

For deducing the distribution of r we take the following approach whichallows to obtain the dynamics of form (3.11) with constant coefficients bya transformation of a d-dimensional Gaussian process. Let (W ∗

i )i=1,...,d be ad-dimensional Brownian motion and for i = 1, . . . , d we define Xi as solutionof the SDE

dXi(t) = −1

2βXi(t)dt+

1

2σdW ∗

i (t). (3.12)

Applying Theorem (2.4) we have

Xi(t) = Xi(0)e−1

2βt +

1

2σe−

1

2βt

∫ t

0

e1

2βsdW ∗

i (s).


We now define r(t) asd∑

i=1

X2i (t). Hence the dynamics of r equal

dr(t) =d∑

i=1

2Xi(t)dXi(t) +1

2

d∑

i=1

2σ2

4dt

=(− β

d∑

i=1

X2i (t) + d

σ2

4

)dt+ σ

d∑

i=1

Xi(t)dW∗i (t)

=(dσ2

4− βr(t)

)dt+ σ

d∑

i=1

Xi(t)dW∗i (t). (3.13)

Setting 2

dW ∗t =

d∑

i=1

Xi(t)√r(t)

dW ∗i (t)

yields immediately to the CIR equation (3.11) with α = dσ2

4. In order to

guarantee positive short-term rates (a.s), the dimension d of the underly-ing Brownian motion must be greater than 2. For this reason we have thecondition α > 2σ2

4= σ2

2, which coincides with condition (i) of theorem 2.3.

Proposition 3.1. r(t)/ρ has a non-central χ2 distribution under P∗ with

d = 4α/σ2 degrees of freedom and non-centrality parameter λ, where

ρ = varP∗(Xi(t)) =σ2

4β(1 − e−βt),

λ =4βr0

σ2(eβt−1). (3.14)

Proof. In order to see this, we write rρ

=d∑

i=1

(Xi√ρ

)2

in the following form:

r

ρ=

d∑

i=1

(zi + δi)2,

where z1, z2, . . . , zd are independent and identically distributed standard nor-mal random variables and δi = mi/

√ρ, where

mi = EP∗(Xi(t)) = Xi(0)e−1

2βt.

2W ∗

t =

d∑

i=1

∫ t

0

Xi(s)√r(s)

dW ∗

i (s) is a standard Brownian motion. This follows by Levy’s

Characterization Theorem, since this expression is a martingale with respect to the stan-dard Brownian filtration, whose quadratic variation equals t.


It is well known that∑d

i=1 z2i has a (central) χ2 distribution with d degrees

of freedom. Hence, r/ρ has a non-central chi-squared distribution with ddegrees of freedom and non-centrality parameter

λ =d∑

i=1

δ2i .

If we assume that Xi(0) = . . . = Xd−1(0) = 0 and Xd(0) =√r0, then

δi = 0, i ∈ 1, . . . , d− 1 and δd =

√r0e

− 1

2βt

√ρ

and therefore λ = r0e−βt/ρ.

Similar to the Vasicek’s model the bond price P (t, T ) can be derived bysolving PDEs (2.13) and (2.14). In this case the bond price is given by

P (t, T ) = eA(T−t)−B(T−t)r(t),

where

A(T − t) =2α

σ2ln

( 2γe(γ+β)(T−t)/2

(γ + β)(eγ(T−t) − 1) + 2γ

),

γ =√β2 + 2σ2,

B(T − t) =2(eγ(T−t) − 1)

(γ + β)(eγ(T−t) − 1) + 2γ. (3.15)

3.2 Examples of Multi-Factor Affine Models

A major drawback of one-factor affine models is the implication that interestrates with different maturities are perfectly correlated. Bearing in mind thatR(t, T ) can be expressed by (2.2) it is obvious that the correlation betweentwo yields R(t, T1) and R(t, T2),

Corr(R(t, T1), R(t, T2)) =

Corr(− A(t, T1)

T1 − t− B(t, T1)

T1 − tr(t),−A(t, T2)

T2 − t− B(t, T2)

T2 − tr(t)

),(3.16)

equals 1. This means that a shock to the interest rate curve at time t istransmitted equally through all maturities, which can easily be avoided byincluding a second factor.


In the following we will give examples of multi-factor models whose factorshave an economic interpretation. At first, we will describe the Longstaff andSchwartz model, where the second factor is identified with the volatility ofthe short rate. Longstaff and Schwartz is a well known two-factor modelwhich has a great deal of flexibility, achieving good calibration to a varietyof term structures. Then we deal with a model developed by Balduzzi, Dasand Foresi (BDF), where the mean reversion level is chosen to be the secondfactor. An extension of this approach is the three factor BDFS (Balduzzi,Das, Foresi, Sundaram) model.

3.2.1 The Longstaff and Schwartz Two-Factor Model

Longstaff and Schwartz consider an interest rate model where the short rater(t) is obtained as a linear combination of two basic processes as follows:

dY1(t) = α1(µ1 − Y1(t))dt+√Y1(t)dW

∗1 (t),

dY2(t) = α2(µ2 − Y2(t))dt+√Y2(t)dW

∗2 (t),

r(t) = c1Y1(t) + c2Y2(t), (3.17)

where all parameters have positive values and W ∗1 and W ∗

2 are independent.Our first aim is to show that the model is essentially a two-factor CIR model,as described in section 2.2.2. We set

X1(t) = c1Y1(t), X2(t) = c2Y2(t),

so thatr(t) = X1(t) +X2(t).

It is immediate to check that

dX1(t) = α1(c1µ1 −X1(t))dt+√c1

√X1(t)dW

∗1 (t),

dX2(t) = α2(c2µ2 −X2(t))dt+√c2

√X2(t)dW

∗2 (t).

Since X1 and X2 describe one factor CIR processes, we see that the Longstaffand Schwartz model can be interpreted as two-factor CIR model.

Via a change of variable it can be expressed as a stochastic-volatilitymodel, so that the state variables also have an economic interpretation. Bydefining a process V (t) = c21Y1(t) + c22Y2(t) we get the following dynamics for


r(t).

dr(t) = c1dY1(t) + c2dY2(t)

= α1c1(µ1 − Y1(t))dt+ c1√Y1(t)dW

∗1 (t)

+α2c2(µ2 − Y2(t))dt+ c2√Y2(t)dW

∗2 (t)

=(α1c1µ1 + α2c2µ2 −

(α1c2 − α2c1)r(t) + (α2 − α1)V (t)

c2 − c1

)dt

+

√c1(c2r(t) − V (t))

c2 − c1dW ∗

1 (t) +

√c2(V (t) − c1r(t))

c2 − c1dW ∗

2 (t).

The reason for formulating the model in this way is the fact that V (t)dt isthe instantaneous variance of r(t). The use of r(t) and V (t) rather thanY1(t) and Y2(t) allows to express the dynamics of r(t) by means of level andvolatility.

Remark. Referring to the Classification of Dai and Singleton, which we dis-cussed in section 2.3, it is worth mentioning that the Longstaff and Schwartzmodel belongs to the class A2(2), since two state variables appear in thevolatility matrix of process (3.17). However it is over-specified, because thereare only six parameters that could be chosen, whereas the correspondingcanonical model has nine parameters.

3.2.2 The Central Tendency as Second Factor

In the model developed by Balduzzi, Das and Foresi [16] the second factoris identified with the central tendency of the short term rate. This approachis based on the fact that the behavior of short-term rates indicates fluctua-tions around a time-varying rest level - the central tendency - that changesstochastically over time.

In other words, by regressing the future short term rate r(t + 1) on thecurrent short-term rate and current longer-term yields, they found evidencefor the fact that the future short-term rate does not only depend on the cur-rent one (as implied by one-factor models, where the short rate is implicitlyassumed to follow an autoregressive process), but also on longer-maturityyields. Generally, the negligence of this fact is one of the reasons why one-factor models are insufficient.

Indeed, they could show that β2, the coefficient of the long-term yields ofthe following regression

r(t+ 1) = β0 + β1r(t) + β2R(t, τ) + error(t+ 1),


where R(t, τ) is a vector of yields with maturities τ = 1, 2, 3, 4 years, issignificant for both, the regression in levels and first differences. On accountof these findings Balduzzi, Das and Foresi developed the following model.

The behavior of the short-term rate, the first factor, is described by theSDE

dr(t) = κ(µ(t) − r(t))dt+√σ2

0 + σ21r(t)dWr(t), (3.18)

where κ, σ0, σ1 are constants, Wr is a standard Brownian motion under theactual measure P and µ is the central tendency, toward which the short termrate reverts and which evolves according to the SDE

dµ(t) = (m0 +m1µ(t))dt+√s20 + s2

1µ(t)dWµ(t), (3.19)

where m0,m1, s0 and s1 are constants and Wµ is a standard Brownian motionunder P, independent from Wr. Under the assumption that the market priceof risk is given by3

Λ(t) = Dλ, (3.20)

where λ ∈ Rn and

D =

( √σ2

0 + σ21r(t) 0

0√s20 + s2

1µ(t)

),

so that4

DΛ(t) =

(λr0 + λr1r(t)λµ0 + λµ1µ(t)

),

we immediately get an affine model under P∗, as our processes can be written

in form (2.10),

(dr(t)dµ(t)

)=

[ (−λr0

m0 − λµ0

)+

(−κ− λr1 κ

0 m1 − λµ1

) (r(t)µ(t)

) ]dt

+

( √σ2

0 + σ21r(t) 0

0√s20 + s2

1µ(t)

) (dWr(t)

∗

dWµ(t)∗

). (3.21)

In the following we show how a linear combination of two yields can beused to approximate the central tendency, which is obviously an unobservablefactor. From (2.2) we know that yields can be expressed by

R(t, τi) = −A(t, τi)

τi− B1(t, τi)r(t)

τi− B2(t, τi)µ(t)

τii = 1, 2.

3Λ(t) corresponds to λ(t)′ of equation (1.46).4i.e. σ(X, t)λ(t)′ in the notation of formula (1.46)


Solving for r from the first yield, substituting into the second and rearrangingleads to

τ1B1(t, τ2)R(t, τ1) − τ2B1(t, τ1)R(t, τ2)

= B1(t, τ2)[A(t, τ1) +B2(t, τ1)µ(t)] −B1(t, τ1)[A(t, τ2) +B2(t, τ2)µ(t)].

Note that this quantity does not depend on r. Therefore µ can be written inthe form

µ(t) =B1(t, τ2)[τ1R(t, τ1) − A(t, τ1)] −B1(t, τ1)[τ2R(t, τ2) − A(t, τ2)]

B1(t, τ2)B2(t, τ1) −B1(t, τ1)B2(t, τ2).

(3.22)This equation justifies the following approximation µ for µ

ˆµ(t) = a0 + a1[B1(t, τ2)τ1R(t, τ1) −B1(t, τ1)τ2R(t, τ2)]. (3.23)

Using this approximation the model can be estimated by applying the methodof Maximum Likelihood.

The BDFS Three-Factor Model

An extension of the two-factor model described in the last section is theBalduzzi, Das, Foresi and Sundaram model, which belongs to the three-factor affine family (see equations (2.26) (2.27) and (2.28)). BDFS termstructures are highly flexible, giving rise to hump- and spoon-shaped yieldcurves. However the lack of explicit formulas makes it awkward to calibratethe parameters to market data, which is a major disadvantage. The Fongand Vasicek model, where µ is constant, is a special case of the BDFS model.

3.3 Economic Models

Understanding the links between output, prices, interest rates and moneysupply has been a major objective of economists for decades. The standardmacroeconomic model is the IS-LM5 model that presumes an equilibriumbetween output and income as well as an equilibrium between money supplyand money demand. In contrast to this approach financial models of interestrates emphasize no-arbitrage at the expense of an economic context. Thus,this section is concerned with understanding how the economic concepts canbe applied to financial models of interest.

The economic model that we regard is directly based on the IS-LM frame-work and was formulated by Tice and Webber [12]. As already mentioned the

5LM stands for “Liquidity and Money”, while IS signifies “Investment and Savings”.


LM relationship comes from an assumption of equilibrium in the money mar-ket. In particular, it is supposed that the demand for real money increasesin income and decreases in the short-term rate r(t). The expenditure, mean-ing the IS-Line also increases in income and decreases in the short rate, butbesides that it depends on the public spending that is modeled as constant.In our case these relationships are specified by the following equations6:

md(t) = ky(t) − ur(t) (LM), (3.24)

e(t) = a− br(t) + cy(t) (IS), (3.25)

where md denotes the (log-) demand for real money and e(t) the expenditure.Both, money demand and expenditure are affine functions of (log-) incomey(t) and the short-term rate r(t). k, u, a, b, c are constants. It is supposedthat the equilibrium dynamics are given by

dmd(t) = α(ms −md(t))dt+ σmdWm(t),

dy(t) = β(e(t) − y(t))dt+ σydWy(t), (3.26)

so that money demand reverts to the level of money supply ms, which isassumed to be constant, and income to the level of expenditure. Wm(t) andWy(t) are independent standard Brownian motions under P.

3.3.1 The General Framework

In a general framework we suppose to have n state variables X1(t), . . . , Xn(t)and n economic variables mi(t) and mi(t), where

mi(t) = mi(X1(t), . . . , Xn(t)) i = 1, . . . n, (3.27)

mi(t) = mi(X1(t), . . . , Xn(t)) i = 1, . . . n. (3.28)

We suppose that the variable mi(t) reverts to the level mi(t), so that mi(t)is interpreted as the equilibrium level of mi(t). In particular, the dynamicsof the equilibrium relationship are assumed to evolve according to the SDE

dmi(t) = αi(mi(t) −mi(t))dt+ SidWt i = 1, . . . , n, (3.29)

where α = (α1, . . . , αn)′ is a constant vector, S is a constant n × n ma-trix (Si denotes the ith row) and Wt is a n-dimensional standard Brow-nian motion under P. If the functions M(t) = (m1(t), . . . ,mn(t))′ andM(t) = (m1(t), . . . ,mn(t))′ are sufficiently regular we can invert (3.29) to

6Instead of r(t), the difference between the long-term interest rate and the long runfuture inflation is often used to model this relationship.


find the process followed by the state variables X(t) = (X1(t), . . . , Xn(t))′.Under the assumption that the dynamics of X(t) are given by (1.27), thus

dX(t) = µ(X(t))dt+ σ(X(t))dWt,

we obtain by means of Ito’s formula

dM(t) = (Mxµ+1

2h)dt+MxσdWt,

where Mx = ∂mi

∂Xji,j=1,...,n and h = hii=1,...,n is defined as

hi =n∑

k,l,j=1

∂2mi

∂Xk∂Xl

σkjσlj.

From (3.29) we immediately get

σ(X(t)) = M−1x S, (3.30)

µ(X(t)) = M−1x diag(α)(M(t) −M(t)) − 1

2M−1

x h. (3.31)

3.3.2 IS - LM Framework

We can now apply the results of the previous section to our IS-LM relation-ship defined by (3.24)-(3.26). The economic variables are md(t) and e(t),whereas r(t) and y(t) are the underlying state variables, on which the valuesof md(t) and e(t) depend. Hence, we invert (3.26) to obtain

dr(t) = αr(βr + γry(t) − r(t))dt+ σrdWr(t), (3.32)

dy(t) = αy(βy − γyr(t) − y(t))dt+ σydWy(t), (3.33)

where

αr =αu+ βbk

u,

βr =βak − αms

αu+ βbk,

γr =k(α− β(1 − c))

αu+ βbk,

αy = β(1 − c)

βy =a

1 − c,

γy =b

1 − c.


Furthermore σr =

√k2σ2

y+σ2m

u, so that Wr(t), defined by

dWr(t) =−σm

uσr

dWm(t) +kσy

uσr

dWy(t),

is a (with Wy corraleted) Brownian motion. Obviously this is a two-factoraffine model under P that can easily be transformed into a two-factor affinemodel under P

∗ by assuming an appropriate market price of risk (see equation(2.30)).We can re-parameterize the system in terms of r(t) and x(t) = βr +γry(t) to get the following equations,

dr(t) = αr(x(t) − r(t))dt+ σrdWr(t), (3.34)

dx(t) = αx(βxr(t) − (1 − βx)µ− x(t))dt+ σxdWx(t), (3.35)

for appropriate constants αx, βx, µ, σx and Wx(t) = Wy(t).

Remarks. This model is a generalized Vasicek model, where r(t) reverts tox(t), and x(t) to a weighted sum of r(t) and µ. In a way, it gives an economicjustification for interest rate models with a second stochastic factor identi-fied with the drift function. We have already encountered this idea in section(3.2.2), where we described the BDF model which also has a stochastic drift,but whose dynamics are specified by two generalized CIR processes.µ can be interpreted as a variable that is controlled by the monetary author-ities via ms and the public spending a. Hence µ could reasonably be time-dependent, which provides some economic justification for time-dependentvariables that in general need to be treated with caution.

3.4 Non-Affine Models - Consol Models

We present here a non-affine model with the aim to put the advantages ofaffine models across. We will see in chapter 5 that affine yield factor modelscan be constructed which allow to relate yields of long maturities to thestate variables. Another (earlier) approach to include a long term rate intoan interest rate model in order to have a reasonable economic interpretationare consol models. They are two factor models whose state variables are theconsol rate and the short rate. The consol rate can be defined as the yield ona bond that has a continuous coupon paid at a constant rate c and infinitematurity, i.e. we have to consider an economy with an infinite horizon date,T ∗ = ∞. The price of the consol at time t is then given by

C(t) = EP∗

( ∫ ∞

t

c exp(−

∫ s

t

r(u)du)ds

∣∣∣Ft

)=

∫ ∞

t

cP (t, s)ds. (3.36)


We suppose without loss of generality that c = 1 and substitute r(u) by theFt-measurable random variable l(t), which gives

C(t) =

∫ ∞

t

exp(−

∫ s

t

l(t)du)ds =

∫ ∞

t

exp(−l(t)(s− t))ds =1

l(t)(3.37)

So the consol rate is simply defined as the reciprocal of its price l(t) = C(t)−1

and can be seen as an approximation of a long-term rate of interest. Brennanand Schwartz used it to extend the short rate model to a two-factor model,in which the short rate r and the long-term rate l are intertwined. SinceC(t) = l(t)−1, we may work directly with the price of the consol. Then thetwo-dimensional process (r(t), C(t)) evolves according to the following pairof SDEs

dr(t) = µr(r, C)dt+ σr(r, C)dW ∗r (t), (3.38)

dC(t) = µC(r, C)dt+ σC(r, C)dW ∗l (t) (3.39)

under the martingale measure P∗. The Application of Ito’s formula to (3.36)

with c = 1 implies thatµC = r(t)C(t) − 1.

For any choice of the short-term rate coefficients µr and σr, σC can be chosenconsistently, which was conjectured by Black and confirmed by Duffie, Maand Yong [7]. They could also show that under some technical regularityconditions C(t) is necessarily of the form g(r(t)) for a function g, that is theunique solution of the following ordinary differential equation,

g′(r)µr(r, g(r)) − rg(r) +1

2σ2

r(r, g(r)) + 1 = 0.

Thus the consol model can be reduced to a model with only one statevariable r, whose dynamics are given by (3.38) and with C(t) = g(r(t)) orl(t) = 1/g(rt). This is rather surprising as the aim of this model was toprovide two state variables for the term structure, the short rate r(t) and thelong rate l(t).

Remark. Although the technical regularity conditions that are imposed tohave this result may rule out some interesting cases, modeling C(t) or l(t)is always fraught with difficulty, as the diffusion term of the consol has tobe chosen consistently with the solution of a non-trivial fix point probleminvolving the drift and the diffusion of the short rate.


3.5 Criteria for Model Selection

Generally speaking, affine models are those among the wide range of possibleinterest rate models which fulfill most of the criteria that are crucial for themodel selection. Their theoretical as well as their practical properties aredecisive for their frequent utilization.

We have now seen a number of examples of affine interest rate models (andone non-affine model) that all could be used in practice. So, the first questionthat arises is “Which model is the best one and should consequently bechosen.” Although there is no general answer to this question, it is certainlypossible to state important features which good interest rate models shouldhave and which determine the model selection. So when choosing a particularmodel the following criteria should be considered:

• Tractability: Interest rate models should have explicit or easy numericalsolutions for bond prices and other instruments such as caplets or swap-tions (see 4.1.2). As the solutions of the ricatti equations (2.13-2.14)can be quickly computed numerically in cases where explicit solutionsare not available, the tractability of affine models is beyond doubt,which is one important reason for their popularity.

• Fitting the yield curve: The idea behind fitting a model to the currentmarket yield curve is to fit it to given points and judge the goodness offit regarding the resulting residuals. To a large extend this is purely afunction of how many parameters the model contains. In other words,a model must have few enough parameters that a good fit is significant,and enough to ensure that a good fit is possible. One-factor affine mod-els, for example the Vasicek model, do not have a large range of shapesand will provide a poor fit to some initial yield curves. Multi-factoraffine models, however, offer large flexibility due to a larger number ofparameters.

• Good dynamics: Beside fitting the current market prices, a modelshould also be able to fit, to some extend, the way that prices changeover time. The dynamical features that may be desirable to matchcould be the dynamics of the short rate as well as the dynamics ofthe whole yield curve. By allowing time-dependent coefficients, whichentails the problem of the right starting time, the dynamics of themarket can sometimes be better reproduced. However, good results,for instance, of the short rate dynamics can also be attained by his-torical estimation of its mean reversion level, its mean reversion rateand its volatility. The historical estimation of these parameters that


are elements of many commonly used affine models implicates the es-timation of the market price of risk in order to be able to move from P

to P∗.

• Economic interpretation: The chosen factors, i.e. state variables shouldhave an economic interpretation. In most of the examples that exist inliterature the first factor is identified with the short rate, whereas thesecond factor (in two-factor models) can be related to

– the volatility of the short-term rate (Longstaff & Schwartz: section3.2.1, Vasicek & Fong),

– the mean level of the short-term rate (Balduzzi, Das & Foresi:section 3.2.2),

– the inflation (Heston, Pearson & Sun),

– the spread between long and short-term rates(Schaefer & Schwartz: consol rate model) and

– the long-term rate (Brennan & Schwartz: consol rate model, Duffie& Kan [8]: affine yield-factor model, chapter 5)

Both, the Brennan and Schwartz and the Schaefer and Schwartz modelbelong to the class of consol models, whose use could raise a numberof difficulties as we examined in section 3.4. Some of the other men-tioned affine models include factors which have indeed an economicrelevance, however which cannot be observed directly (e.g. the volatil-ity of the short-term rate). So the affine yield factor model developedby Duffie and Kan, which we examine in detail in chapter 5, seems tobe a promising approach, as the state variables are identified with theyields which are of course observable.

• Two-factor model: The choice of a two factor model seems to be a goodcompromise between flexibility and complexity. The calibration and so-lution of three (or more) factor models which provide high flexibilityare already rather complicated, whereas single factor models are hardlyadequate to describe the behavior of long rates, which play an impor-tant role, for example when pricing interest rate dependent productswith long maturities.

Chapter 4

Calibration and Estimation

The process of fitting an interest rate model to market data is generallyknown as estimation. Depending on the kind of data, there are two notionsthat must be distinguished: calibration and historical estimation. Calibrationmeans fitting the parameters of the model to current market data, whereashistorical estimation of the parameters is based on statistical methods whichare used to filter information out of historical time series. As the valuationof financial products should be market consistent, the calibration to currentmarket prices, is in most cases preferable. Nevertheless it is still important tolook at historical data. If the parameter estimates from historical and currentdata are systematically out of line, then one may be inclined to investigatethe cause.

The next question that arise is, which aspects of the complete set ofavailable market data should be fitted. For example, one could try to fit

• the current yield curve (which is usually the first objective),

• current bond option (cap, swaption) prices or

• the current volatility structure of bond options.

In practice it is usually only possible to fit a few aspects as the models are notadequate to match all market prices of all available interest rate products.In effect, interest rate models can be regarded as methods of interpolationand extrapolation. On the one hand, the parameters of the model have to bechosen so that at least one aspect of the market data is well reproduced bythe model. On the other hand, using the found parameters other instrumentsmay now be priced consistently with the market data.

In the following sections we describe different methods for both, calibra-tion and historical estimation. In order to test and compare these approaches,

42

CHAPTER 4. CALIBRATION AND ESTIMATION 43

we apply them to the Vasicek model using real market data. We have chosenthe Vasicek model because of its simplicity, with the aim to illustrate eachmethod clearly, however being aware of the fact that it is not an appropriatemodel to fit the data well. So we do not focus on the absolute results, sincethere are many other models which are more adequate.

Before starting to explain various ways of calibration and estimation wecomment on different types of data and their preparation.

4.1 Obtaining a Data Set

4.1.1 Market Data for the Current Yield Curve

Calibrating a model to the current yield curve means trying to reproducethe curve of the today’s continuously-compounded spot interest rate T →RM(0, T )1 (see (1.4)) for different maturities T . However, these marketyields RM(0, T ) are not available in this form and have to be calculated fromEURIBOR- and swap rates.2 So we use the 1-12 month EURIBOR rates tocalculate RM(0, n

12), n = 1, . . . , 12 and for the longer maturities 1, 2, . . . , 30

years the corresponding swap rates. From equation (1.18) we know that theswap rates S(0, Tn), Tn = 1, 2, . . . , 30 are given by

S(0, Tn) =1 − P (0, Tn)∑n

i=1 τiP (0, Ti). (4.1)

We can rearrange (4.1) to express P (0, Tn) in terms of bond prices withshorter maturity times Ti, i < n and the swap rate S(0, Tn),

P (0, Tn) =1 − S(0, Tn)

∑n−1i=1 τiP (0, Ti)

1 + S(0, Tn)τn. (4.2)

On the basis of this equation the zero bond prices can be calculated by thebootstrapping method. The yields are then easily deduced from the bondprices.

Remarks. When converting the EURIBOR rates to yields the day-countconvention must be taken into consideration.It is worth remarking that RM(0, 1) can be derived from the EURIBOR rateas well as from the swap rate. However, L(0, 1) and S(0, 1) which should beequal (having already considered the day-count convention) do not coincide.This difference must therefore be incorporated into the EURIBOR rates.

1We write RM for the market yields.2They are denoted by L and S respectively.


4.1.2 Market Data for Bond Options

We introduce now the two main derivative products of the interest rate mar-ket, namely caps and swaptions, to which interest rate models can be cali-brated.

Interest Rate Caps

A cap is a contract that can be viewed as a payer interest rate swap (compareequation (1.15)), where each exchange payment is executed only if it haspositive value. The cap discounted payoff at time t is therefore given by

n∑

i=1

P (t, Ti)τi(L(Ti−1, Ti) −K)+, (4.3)

where T0 is the first reset date, Ti, i = 1, . . . , n are the payment times, τi =Ti − Ti−1 and K is a fixed rate, termed the strike of the cap contract. Eachsummand of the above sum defines a contract that is called caplet. Let ci bethe value at time t of the ith caplet. It is market practice to price a caplet ciwith Black’s formula, i.e.

ci = P (t, Ti)τiBl(K,F (t, Ti−1, Ti), σi

√Ti−1), (4.4)

where, denoting by φ the standard gaussian distribution function,

Bl(K,F, σi

√Ti−1) = Fφ(d1(K,F, σi

√Ti−1)) −Kφ(d2(K,F, σi

√Ti−1)),

d1(K,F, σi

√Ti−1) =

ln(F/K) + σ2i Ti−1/2

σi

√Ti−1

,

d2(K,F, σi

√Ti−1) =

ln(F/K) − σ2i Ti−1/2

σi

√Ti−1

.

F is the simply-compounded forward rate and σi the Black’s volatility of theith caplet. As in the classical Black and Scholes option pricing setup, Black’sformula is based on the assumption that the underlying of the option, in ourcase the forward rate F , is log-normal and evolves under P

∗ according to

dF (t) = σiF (t)dW ∗t . (4.5)

So σi coincides with the actual forward rate volatility if the forward rate isthe solution of the above SDE. The cap price c is now

∑ni=1 ci(σi) and the

Black’cap volatility is σ such that c =∑n

i=1 ci(σ). In some sense σ representsan average volatility of the set of individual caplets. Caps are quoted in themarket in terms of Black’s volatility σ.


Definition 4.1. ATM Cap. A cap with payment times Ti, i = 1, . . . , n,τi = Ti − Ti−1 and strike K is said to be at-the-money (ATM) if and only if

K = KATM = S(0, Tn) =1 − P (0, Tn)∑n

i=1 τiP (0, Ti). (4.6)

The cap is instead said to be in-the-money (ITM) if K < KATM and out-of-the-money (OTM) if K > KATM .

The pricing of caps: We now show that a cap is actually equivalent to aportfolio of European zero coupon put options. This equivalence is used toderive explicit formulas for cap prices under analytically tractable short-ratemodels like the Vasicek model. The arbitrage free price at time t of the ith

caplet is according to the general pricing formula (1.22)

ci = EP∗

(exp

(−

∫ Ti

t

r(s)ds)τi(L(Ti−1, Ti) −K)+

∣∣∣Ft

)

= EP∗

[EP∗

(exp

(−

∫ Ti

t


∣∣∣FTi−1

)∣∣∣Ft

]

= EP∗

[exp

(−

∫ Ti−1

t


EP∗

(exp

(−

∫ Ti

Ti−1

r(s)ds)∣∣∣FTi−1

)∣∣∣Ft

]

= EP∗

[exp

(−

∫ Ti−1

t

r(s)ds)P (Ti−1, Ti)τi(L(Ti−1, Ti) −K)+

∣∣∣Ft

].

(4.7)

Using the definition of L(Ti−1, Ti), we obtain

ci(t) = EP∗

[exp

(−

∫ Ti−1

t

r(s)ds)(1 − (1 +Kτi)P (Ti−1, Ti))

+∣∣∣Ft

]

= (1 +Kτi)EP∗

[exp

(−

∫ Ti−1

t

r(s)ds)( 1

1 +Kτi− P (Ti−1, Ti)

)+∣∣∣Ft

],

(4.8)

which is actually an European put option price with strike 11+Kτi

and nominalvalue (1 +Kτi). Finally, cap prices are simply obtained by summing up theprices of the underlying caplets.

So, this formula is used to get explicit expressions for the model cap priceswhich we need, for instance, when fitting an interest rate model to market


ATM cap volatilities which is one possibility of calibration. However, thiskind of calibration can be fraught with difficulty, since the price produced bya short-rate model is not compatible with Black’s market formula, which weinvestigate in detail in section 4.2.2.

Swaptions

Swaptions are options on an forward start interest rate swap (IRS). Thereare two main types, a payer version and a receiver version. A European payerswaption is an option giving the right (and no obligation) to a enter a payerforward start IRS at a given future time, the swaption maturity. Usually theswaption maturity coincides with the first reset date T0 of the underlyingIRS. So the payoff at time T0 of a payer swaption on a forward start swapwith strike K is

( n∑

i=1

P (T0, Ti)τi(S(t, T0, Tn) −K))+

, (4.9)

where S(t, T0, Tn) is the forward start swap rate (1.19). It is market practiceto value swaptions with a Black-like formula. Precisely, the price of the abovepayer swaption (at time zero) is

s = Bl(K,S(0, T0, Tn), σ√T0)

n∑

i=1

τiP (0, Ti), (4.10)

where σ is now a volatility parameter that is quoted in the market that isdifferent from the corresponding σ in the caps case. Again, Black’s formulawould be “correct” if the forward start swap rates S(t, T0, Tn) were log-normalwith volatility σ. Similar to a cap a swaption is said to be ATM if and onlyif its strike is equal to the forward start swap rate corresponding to theunderlying swap.

Concerning the calibration of interest rate models to swaption volatili-ies one has to face the same problem as in the case of the calibration tocap volatilities, since the distribution of the forward start swap rate derivedby the model is not compatible with Black’s formula. Moreover one alsohas to decide which maturity the underlying swap should have because thewhole market volatility surface, as function of the swaption maturity and thematurity of the underlying swap, can never be fitted.


4.1.3 Which Market Rate should be used for the

Short-Term Rate?

The short rate r(t) is the key interest rate in all models, even though itcannot be directly observed. As the short rate is defined to have an instan-taneous holding period, one could assume that the overnight rate would bethe best approximation. However, this assumption is rebutted by the highvolatility and the low correlation with other yields. So the one- or threemonth spot rate, RM(0, 1/12) or RM(0, 1/4) respectively, is often taken tobe the best approximation. One reason for this choice is their liquidity. Forour calibration examples we always take the one month rate as surrogate forthe short-term rate.

4.2 Calibration to Current Market Data

In this section we exemplify the calibration to current market data by meansof the Vasicek and alternatively the Hull and White model. At first wecalibrate the Vasicek model to the current term structure and then to capvolatilities.

4.2.1 Calibrating the Vasicek Model to the Current

Term Structure

As already explained in section 4.1.1 we calibrate the Vasicek model to marketyields RM(0, Tn), where Tn are the different maturities

1/12, 2/12, . . . , 11/12, 1, 2, . . . , 30.

The Vasicek model yields are given by

R(0, Tn) = −A(Tn)

Tn

+B(Tn)

Tn

r(0), (4.11)

where A and B are the functions of (3.10). We denote by RM(0, T ) thevector of market yields and by R(0, T ) the vector of model yields. For thetoday’s short rate r(0) we use the one-month rate RM(0, 1/12). We now haveto choose the parameters α, β, σ of the Vasicek model (3.4), so that R(0, T )best matches RM(0, T ). Our approach is to minimize

ResR(α, β, σ) = (RM(0, T ) −R(0, T ))′(RM(0, T ) −R(0, T )), (4.12)


the sum of squared deviations.3 Consequently, our calibration problem isin fact a nonlinear least-squares problem. In order to solve this we use theMatlab function “lsqnonlin” choosing the large scale optimization algorithm4

that is based on a subspace trust region method. We describe this algorithmin detail in appendix A.1.

For the least-squares minimization it is necessary to specify the initialparameters α0, β0 and σ0, where the Matlab routine starts searching the min-imum of (4.12). Since the final result is highly sensitive to these start valueswe generate various α0, β0 and σ0 randomly and calculate some optimizationsteps. The values with minimal residual are taken as start parameters forthe final least-squares optimization which then supplies the final parameters.We use this kind of algorithm as well for all other calibration issues that wedescribe in the following sections. It is worth mentioning that we also definea set of lower and upper bounds between which the parameters must range(see appendix A.1.1).

So when fitting the Vasicek model to market yields from 31 March 2006using the above described algorithm we found the the following optimal pa-rameters.5

Date α β σ ResR

31 March 2006 0.02468 0.48995 0.06042 1.95828e-005

Table 4.1: Parameters for the Vasicek Model, Calibration to Market Yields,31 March 2006

Figure 4.1 illustrates the deviations between market yields and the yieldsproduced by the Vasicek model with the above parameters. Although thedifferences are rather small (the maximal deviation is 0.00126 for maturity7 years) it is obvious that the Vasicek model does not fit the market data inan optimal way because it lacks a large range of possible shapes.

Stability of the Parameters over Time

As all parameters of the Vasicek model are supposed to be constants, theyshould not vary much over time. In order to see if the parameters reallybehave like this, we investigate their development between January 2005and March 2006 by calibrating the model every month to the correspondingmarket data using the same method as described above. Figure 4.2 shows the

3Res stands for residuals.4For the large scale algorithm, the number of equations (the number of elements of

equation (4.12)) must be at least as many as the parameters that we want to find.5If the model was written in form dr(t) = β(µ − r(t))dt + σdW ∗

t , µ would be 0.05038.


0 5 10 15 20 25 300.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04

0.042

0.044Market Yields versus Vasicek Model Yields − 31 march 2006

Maturity

Inte

rest

rate

Model YieldsMarket Yields

Figure 4.1: Market- versus Vasicek Model Yields, 31 March 2006

parameter values for α, β and σ since January 2005. While α, σ are relativelyconstant, β, the mean reversion rate, varies extremely.

4.2.2 Calibrating the Vasicek Model to Cap Volatili-

ties

For illustration purposes we now try to calibrate the Vasicek model to marketATM cap volatilities. As we will see the Vasicek model is not appropriatefor this kind of calibration since the current term structure can not be fittedsimultaneously when calibrating to cap volatilities. In the next section we willget to know the Hull and White model, which allows a calibration to bothkinds of data, however with the disadvantage of having a time-dependentparameters.

Fitting to cap volatilities means computing the market cap price by insert-ing the market cap volatility into Black’s formula at first and then adaptingthe parameters of the model, so that the model cap prices best match themarket cap prices obtained by Black’s formula. Tabel 4.2 contains the mar-ket quoted ATM cap volatilities for different maturity times Tn to which wetry to fit the model. The underlying caplets of each cap have maturitiesTni

, which are up to one year equally three-month spaced and after one year


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Parameters over Time

January 2005 − March 2006

Para

met

ers

alphabetasigma

Figure 4.2: Vasicek Model Parameters over Time, 15 month: January 2005- March 2006

equally six-month spaced.Using the same algorithm as described in section 4.2.1, we minimize now

the sum of the square differences between market and model cap prices,

minResCap = min(∑

Tn

(CapM(Tn) − Cap(Tn))2), (4.13)

where CapM and Cap denote the market and the model cap price respec-tively. The model cap price is obtained by means of formula (4.8). With thisapproach we find the parameters of table 4.3, which do not really match theparameters that we obtained when calibrating to the current yields.

Figure 4.3 opposes the market cap prices against the model cap prices, aswell as the market cap volatilities against the model cap implied volatilities.The model cap implied volatility is the volatility parameter that must beplugged into Black’s formula (4.4) in order to obtain the model price.

Although the model implied volatility does not differ much from the mar-ket volatility in this case, there are some problems with this calibrationmethod. Firstly, the parameters which are supposed to be constant varyagain a lot over time, which we have already experienced when calibratingthe model to market yields. Secondly, as already hinted at the beginning,


Maturity Tn (years) ATM volatilities σ2 14.09%3 15.94%4 16.50%5 16.74%6 16.73%7 16.64%8 16.50%9 16.33%10 16.17%12 15.87%15 15.31%20 14.56%25 14.06%30 13.67%

Table 4.2: ATM Cap Volatilities, EUR, 31 March 2006

Date α β σ31 March 2006 0.0063 0.1274 0.0115

Table 4.3: Parameters for the Vasicek Model - Calibration to Cap Volatilities,31 March 2006

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14Market Cap Prices vs. Model Cap Prices

Cap Maturity

Pric

e

Model Cap PriceMarket Cap Price

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2Market Cap Volatilities vs. Model Implied Cap Volatilities

Cap Maturity

Vol

atili

ty

Model Implied Cap VolatilityMarket Cap Volatility

Figure 4.3: Market- versus Vasicek Model Cap Prices and Volatilities, 31March 2006


the yields produced by the Vasicek model with α, β and σ from Table 4.3 arenot in accordance with the market yields.

So the Vasicek model is not an appropriate model for fitting cap volatil-ities, something that aims at all affine models in alleviated form. Generallyspeaking, the basic problem with this approach is the fact that the cap pricecalculated by an affine model is not compatible with Black’s market formula,since the model forward rate dynamics are never of form (4.5). More specif-ically, for no choice of the model parameters does the distribution of theforward rate produced by an affine model coincide with the distribution ofthe “Black”-like forward rate following (4.5). For this reason the so-called“log-normal forward LIBOR models” were developed, where the dynamics ofF are specified by (4.5).

Nevertheless we investigate once again the calibration to cap volatilitiesby means of an another short rate model, namely the Hull and White model,since it enables an exact fit of the term structure of interest rates while beingcalibrated to cap volatilities at the same time. Nevertheless the above statedobjections are still true.

4.2.3 Calibrating the Hull-White Extended

Vasicek Model

In this section we consider the following extension of the Vasicek model whichwas analyzed by Hull and White

dr(t) = (α(t) − βr(t))dt+ σdW ∗t , (4.14)

where α(t) is time-dependent and chosen so as to exactly fit the current termstructure of interest rates. It can be shown that α must be

α(t) =∂fM(0, t)

∂t+ βfM(0, t) +

σ2

2β(1 − e−2βt), (4.15)

where fM denotes the market instantaneous forward rate. The bond priceand B(t, T ) have the same form as in the classical Vasicek model (3.10),however A(t, T ) is now given by

A(t, T ) = lnPM(0, T ))

PM(0, t)+B(t, T )fM(0, t) − σ2

4β(1 − e−2βt)B(t, T )2, (4.16)

where PM is the market bond price. It is therefore obvious that current (i.e.t = 0) market yields can always be exactly reproduced by the model.

We now try to find the other parameters β and σ by calibrating the modelto the market cap volatilities taking the same approach as in the previous


Param./ Date 10/05 11/05 12/05 01/06 02/06 03/06β 0.0221 0.0219 0.0214 0.0236 0.0201 0.0255σ 0.0072 0.0072 0.0071 0.0069 0.0066 0.0068

Table 4.4: Parameters for the Hull-White Model - Calibration to Cap Volatil-ities

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Market Cap Volatilities vs. H&W Model Implied Cap Volatilities

Cap Maturity

Vo

latil

ity

Model Implied Cap VolatilityMarket Cap Volatility

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60.005

0.01

0.015

0.02

0.025

0.03H&W Parameters over Time

Oktober 2005 − March 2006

Par

amet

ers

betasigma

Figure 4.4: Hull-White Model Implied Cap Volatilities, 31 March 2006 andStability of the Parameters over 6 month (October 2005 - March 2006)

section. The cap volatility curve implied by the Hull and White model isshown in figure 4.4. Notice that it generates a decreasing cap volatilitycurve instead of reproducing the humped market curve. Furthermore thecalibrated value of the mean-reversion parameter β shown in table 4.4 isextremely small, which is a common situation and can be traced back to thedominance of the increasing part of the market volatility curve. In spite ofthese disadvantages the parameters remain relatively stable as illustrated infigure 4.4.

It is worth mentioning that the Hull and White extension of the Vasicekmodel is - despite some disadvantages that we have already stated and thefact that it can produce negative interest rates - one of the historically mostimportant interest rate models. Nevertheless, the time-dependency of α is adisadvantage, since it creates the necessity to justify the right starting time.

4.3 Historical Estimation

Although it usually preferable to estimate parameter values from prices byimplied calibration methods as described in the last section, it is still nec-


essary to compare these results to historical parameter estimates. Herewe exemplify the two main methods, namely the General Method of Mo-ments(GMM) and the Maximum Likelihood Method(ML) again with the Va-sicek model.

4.3.1 Maximum Likelihood Method

The idea behind the Maximum Likelihood Method is to find parameter valuesfor which the actual outcome has the maximum probability. Suppose wehave observed a time series r(ti), i = 1, . . . , n whose transition density, i.e.the likelihood that r(ti) will move from state (ti, r(ti)) at time ti to state(ti+1, r(ti+1)) at time ti+1,

p(ti+1, r(ti); ti, r(ti)|θ) (4.17)

is known. The density depends on the parameter set θ. In the generalcase the transition density at time t is conditional on Fti , however when theprocess is Markov it is only conditional on values at time ti. As our interestrate models are based on Markov processes we consider only this case.The joint density of our observations is

p(r(t1), . . . , r(tn)|θ) = p0(r(t1)|θ)n∏

i=1

p(ti+1, r(ti); ti, r(ti)|θ), (4.18)

where p0 is some former density for r(t1). The likelihood function is thengiven by

L(θ) =n∏

i=1

p(ti+1, r(ti); ti, r(ti)|θ). (4.19)

Finally, θ = arg maxθ L(θ) is the maximum likelihood estimate of θ. Maxi-mizing L places the observed time series at the maximum of the joint densityfunction. Since lnL is monotonically increasing, it is often more convenientto maximize lnL instead of L.

Maximum Likelihood Estimation for the Vasicek Model

For the Vasicek model with θ = (α, β, σ), the transition density function forthe short rate r(t) process is because of the gaussian distribution

p(ti+1, r(ti+1; ti, r(ti))|θ) =1√

2πvarti

exp(− 1

2v2(r(ti), r(ti+1),∆ti)

),

(4.20)


where ∆ti = ti+1 − ti,

varti =σ2

2β(1 − e−2β∆ti), (4.21)

v(r(ti), r(ti+1),∆ti) =r(ti+1) − (α

β+ (r(ti) − α

β)e−β∆ti)

√varti

. (4.22)

Hence the log-likelihood which we will maximize is

lnL = −n− 1

2ln 2π − 1

2

n∑

i=1

[ln

(σ2

2β(1 − e−2β∆ti)

)+ v(r(ti), r(ti+1),∆ti)

].

(4.23)Using now daily data6 of the one-month rate R(0, 1

12) between March 1991

and March 2006 as surrogate for the short-term rate we find the followingparameters, which are not consistent with the values that we obtained whencalibrating to current market data (compare with table 4.1).

α β σ0.002472 0.1546 0.0056

Table 4.5: Maximum Likelihood Parameters for the Vasicek Model

Alternatively not regarding R(0, 112

) as substitute for the short-term rate,the “real” transition density function is7

p(ti+1, Rti+1; ti, Rti)|θ) =

1√2πvarti

exp(− 1

2v2(Rti , Rti+1

,∆ti)), (4.24)

where

varti =(B(1/12)

1/12

)2

varti , (4.25)

v(Rti , Rti+1,∆ti) =

Rti+1−Rtie

−β∆ti −(− A(1/12)

1/12+ B(1/12)

1/12αβ

)(1 − e−β∆ti)

√varti

. (4.26)

and where A and B are the functions from (3.10). Although this approachis more precise it leads to the same results.

6i.e. ∆ti = 1

256, considering public holidays

7We now write R instead of r in order to emphasize that we regard the one-month spotrate and not the short-term rate, although using the same data.


One reason why the parameter values are not in accordance with the val-ues from our calibration to current yields, is the fact that we only regardR(0, 1

12) for the historical estimation, whereas for the calibration to current

market data all yields with different maturities are taken into account. Fur-thermore, we have already seen that the parameters obtained by impliedcalibration methods vary a lot over time, so it would be rather surprising ifthe historical estimates exactly match the current calibration.

4.3.2 General Method of Moments

Basically, the idea of this method is to compare certain functions of a sample,called moments, with their theoretical values. The values of the parametersare then chosen so that the values of the theoretical moments are close totheir sample values.

We suppose to have a process X(t) that depends on a parameter vector θ.Given a function f , one may in principle compute the theoretical expectationg(θ) of f(X),

g(θ) = E(f(X)|θ). (4.27)

Using a set of observations X(ti), i = 1, . . . , n,8 the sample moment f iscalculated by

f =1

n

n∑

i=1

f(X(ti)). (4.28)

The aim is to choose θ so that f = g(θ). In fact, when f is vector valued, θis given by

θ = arg minθJ(θ) = (f − g)′W (θ)(f − g), (4.29)

where W is a weighting matrix that can be chosen in an optimal way, whichwas investigated by Hansen [9].

Of course the estimated value θ of θ depends crucially on the functionf . Thus, this fact immediately arises two questions. Firstly, how should fbe chosen and secondly, how can g(θ) be calculated? In general, computingthe theoretical moments g(θ) might be a serious practical problem. So, thesolution is to generate f , so that by construction g(θ) has known values, forinstance g(θ) = 0. Hansen and Scheinkman [10] give construction possibilitiesin order to generate moments with this property.

8This set of observations could be the short-term rate but also other yields, for instance.


Finding Moments for the Vasicek Model

In order to obtain some moment conditions in the case of the Vasicek Model,one could use the Euler discretization of (3.4), i.e.

r(t+ 1) = α∆t+ (1 − β∆t)r(t) + σ√

∆tz(t+ 1), (4.30)

where z is a standard gaussian variable. We set ε(t+ 1) = σ√

∆tz(t+ 1). Ifthe model is correctly specified, then ε(t + 1) ∼ N(0, σ2∆t) is iid normallydistributed and serially uncorrelated, which gives us the following momentconditions:9

• ε(t+ 1) ∼ N(0, σ2∆t):

E[ε(t+ 1)] = 0, (4.31)

E[ε(t+ 1)2 − σ2∆t] = 0, (4.32)

E[ε(t+ 1)3] = 0, (4.33)

. . .

• ε(t+ 1) is serially uncorrelated:

E[ε(t+ 1)ε(t)] = 0, (4.34)

E[ε(t+ 1)ε(t− 1)] = 0, (4.35)

. . .

Pros and Cons of GMM

A big advantage of GMM is that its use generally does not require a knowl-edge of the distribution of ε(t), just its moments f . The ignorance of thetransition density function in cases where it is available is disadvantageous,since GMM only uses information about the moments f and does not makeuse of other possible information. Beside that, there is also the already men-tioned problem of the choice of which moments to use.

9For instance, in the case of equation (4.31), f(r) = ε(t + 1) = r(t + 1) − (α∆t + (1 −β∆t)r(t)).

Chapter 5

Affine Yield-Factor Models

A yield factor model as analyzed by Duffie and Kan [8] is characterized bythe fact that the ith component of the state variable X(t) in an affine model(see section 2.1) can be viewed as the yield at time t on a zero-coupon bondof maturity τi, for fixed maturities τ1, . . . , τn. This is very convenient featuresince the abstract state variables now get a reasonable economic interpreta-tion.

5.1 General Affine Yield-Factor Model

In order to have an affine factor model with P (t, t + τ) = exp(A(τ) +B(τ)′X(t)), where Xi is the yield of maturity τi,

Xi(t) =− lnP (t, t+ τi)

τi, i ∈ 1, . . . , n, (5.1)

not only the initial conditions A(0) = 0 and B(0) = 0 (2.15) and the parame-ter restrictions from theorem 2.3 must hold, but also the following constraintsresulting from (5.1)

A(τi) = Bj(τi) = 0, j 6= i, Bi(τi) = −τi (5.2)

must be fulfilled for all i.There are two possible ways to construct an affine yield-factor model.

One is to suppose from the beginning that the state variables are yieldsand to ensure that the coefficients for the process (2.10) are chosen so that(5.2) is satisfied. The other, indirect approach is to allow X to be anygeneral state process for an arbitrary affine model and to attempt a changeof variables from the original state vector X(t) to a new yield state vectorR(t) = (R(t, t+ τ1), . . . , R(t, t+ τn))′.

58

CHAPTER 5. AFFINE YIELD-FACTOR MODELS 59

Proposition 5.1. If the model is affine in a general process X(t), meaningthat it is of form (2.10) and that P (t, t+ τ) can be expressed by exp(A(τ) +B(τ)′X(t)), then in can be reformulated in a way which is affine in R(t), i.e.

P (t, t+ τ) = exp(A(τ) + B(τ)′R(t)).

Proof. From (2.2) we know that R(t, t+ τi) is given by

R(t, t+ τi) = −A(τi)

τi− B(τi)

′

τiX(t), for i = 1, . . . , n.

In vector notation this is

R(t) = (R(t, t+ τ1), . . . , R(t, t+ τn))′ = k +KX(t)

for a constant vector k and a constant matrix K (given τ1, . . . , τn). Assumingthat K is invertible this implies that

X(t) = K−1((R(t) − k).

Hence

P (t, t+ τ) = exp(A(τ) +B(τ)′K−1(R(t) − k))

= exp(A(τ) + B(τ)′R(t)),

where A(τ) = A(τ) −B(τ)′K−1k and B(τ)′ = B(τ)′K−1.

So, provided that the matrix K is non-singular, the change of variablesfrom a general state vector X(t) given by (2.10) to R(t) is possible. In thiscase, we can write

dR(t) = (α+ βR(t))dt+ SD(R(t))dW ∗t , (5.3)

where

α = Kα−KβK−1k, (5.4)

β = KβK−1, (5.5)

S = KS, (5.6)

D(R(t)) =

√γ′1R(t) + δ1 0

. . .

0

√γ′nR(t) + δn

, (5.7)

γ′i = γ′iK−1, (5.8)

δi = δi − γ′iK−1k. (5.9)


Remarks. The volatility function (SD(·)) is rather difficult to calibrate toobserved volatilities from current, since the elements of the matrix K are ofthe form (−Bj(τi)/τi), which depend via the solution of the Ricatti equations(2.13) and (2.14) on the original parameters α, β, γ, δ and S.So, from a practical point of view it is advisable to start with an affine factormodel for which the state vector X(t) is already treated as a vector of yieldsfor fixed maturities τ1, . . . , τn. Then the parameters S and D could be chosendirectly from calibration under consideration that the boundary conditionsA(0) = 0 and B(0) = 0 as well as (5.2) must be fulfilled. Furthermore,theorem 2.3, which guarantees a solution to the differential equation (2.10),must also be respected .Although there are no theoretical results describing how certain coefficientscan be fixed in advance, while others can be adjusted afterward in order toachieve consistency with all the above mentioned conditions, in practice itis always possible to fix S and D at first and then adjust the drift, i.e. theparameters α and β. In the next sections this method is explained in detailfor a two-factor version of the model.

5.2 A Two-Factor Affine Model of the

“Long”- and the Short-Term Rate

We will now consider a two-factor affine yield model, in which one of thefactors is the short-term rate itself. For the second factor we take the yieldwith maturity τ2 = 10 years1, which could be interpreted as a long-term rate.Furthermore we simplify the matrix D(X(t)), so that all diagonal elementsare of the same form γ′X(t) + δ. In this special case (2.13) and (2.14) canbe written

∂A(τ)

∂τ= B(τ)′α+

δ

2q(τ), (5.10)

∂B1(τ)′

∂τ= B(τ)′β1 +

γ1

2q(τ) − 1, (5.11)

∂B2(τ)′

∂τ= B(τ)′β2 +

γ2

2q(τ), (5.12)

where βi is the ith column of the matrix β, γi the ith component of the vectorγ and

q(τ) = B(τ)′SS ′B(τ) =2∑

i=1

2∑

j=1

Bi(τ)Bj(τ)sis′j

1One could also choose any other maturity.


with si being the ith row of the matrix S.

5.2.1 Deterministic Volatility

In the case of deterministic volatility, meaning that the volatility is indepen-dent of X(t), thus defined by γ = 0, the equations (5.11) and (5.12) form alinear system and have the standard solution

Bi(τ) =2∑

j=1

ψij exp(λjτ) + φi, i ∈ 1, 2, (5.13)

where ψij and φi are constants and λj are the roots of the characteristicequation,

det

(β11 − λ β12

β21 β22 − λ

)= 0. (5.14)

Thus, we have

λ1,2 =1

2(β11 + β22 ±

√∆), (5.15)

where ∆ = β211 + β2

22 + 4β12β21 − 2β11β22. Considering the initial conditionB(0) = 0, the solution for B is then

B1(τ) =1

4√

∆(β12β21 − β11β22)[(β11 − β22 +

√∆)(β11 + β22 −

√∆)eλ1τ

− (β11 − β22 −√

∆)(β11 + β22 +√

∆)eλ2τ − 4β22

√∆,

B2(τ) =β12

2√

∆(β12β21 − β11β22)[(β11 + β22 −

√∆)eλ1τ

− (β11 + β22 +√

∆)eλ2τ + 2√

∆. (5.16)

The constraints (5.2)for B, in our case for τ2 = 10 can then be writtenexplicitly by setting

B1(10) = 0 and B2(10) = −10. (5.17)

As the first factor is the short-term rate itself, (5.2) only implies these twoconditions, since B(0) = 0 must hold anyway. By putting B into (5.10), Acan be obtained by integration.

5.2.2 Calibrating the Deterministic Volatility Model

to the Current Term Structure

As already indicated the calibration of a general affine yield model causessome difficulty since all the mentioned constraints must be obeyed. In this


special case things simplify significantly. Firstly, we have explicit solutions forthe Ricatti equations (5.10)- (5.12) and secondly, the constrains of theorem2.3 are automatically fulfilled because γ = 0 and δ1 = δ2. Hence we onlyhave to consider (5.17).

We choose the same approach as for the calibration of the Vasicek modelto the current term structure (see section 4.2.1), i.e. minimizing the sum ofsquare deviations between model and current market yields. However, it isnot as straightforward as in the case of the Vasicek model since we have toconsider which and how many parameters - depending on the other parametervalues that are ”free” for calibration - have to be adjusted in order to satisfy(5.17). To overcome this problem we apply the following approach:For fixed β11 we determine the parameters β12, β21 and β22 by solving thetwo non-linear equations (5.17). Having then fixed β, α1, S and δ we defineα2 by setting A(10) = 0, which is the necessary condition of (5.2) for A. Theexact optimization algorithm that we used is the following:

I. Generate start values:

• For j = 1, 2, . . . , jmax

- Generate βj11 randomly.

• For k = 1, 2, . . . , kmax

- Solve equations (5.17) for determining βjk12 , β

jk21 , β

jk22 .

2

• For i = 1, 2, . . . , imax

- Generate αi1, (s1s

′1)

i, (s1s′2)

i, (s2s′2)

i, δi randomly as startvalues in order to calculate some steps of the followingoptimization. 3

- Minimize

Resj,k,i =∑

Tn

(RM(0, Tn) −R(0, Tn))2 (5.18)

with respect to α1, s1s′1, s1s

′2, s2s

′2 and δ leaving βjk un-

changed.4 The model yields are computed by means ofthe explicit solution (5.16) of B and A, adjusting α2 sothat A(10) = 0 is fulfilled.

2As the solution of (5.17) for β12, β21 and β22 is not unique the k- loop is necessary tofind the most appropriate parameters.

3We only generate s1s′

1, s1s

′

2, s2s

′

2where si is the ith row of S as the values of S only

appear in these combinations.4We use again the Matlab function “lsqnonlin” for this least-square problem.


- Save the parameters if Resj,k,i < Resj,k,i−1.

II. Final optimization:

• Do a last optimization of (5.18) with respect to α1, s1s′1, s1s

′2, s2s

′2 and

δ using the values with minimal residual found by the above loops asstart parameters.

The reason why we do this kind of optimization is the fact that β issubject to non-linear constraints which can usually not be implemented instandard optimization routines for the non-linear least-squares problem. Infact, β12, β21, β22 and α2 are adjusted to match the constraints, whereas allthe other parameters are chosen to fit the market yields at best.

Since we want to find parameters that remain relatively stable over timeor at best are the same for each point in time, we minimize (5.18) for differentdates simultaneously. In fact we try to minimize

∑

ti

∑

Tn

(RM(ti, Tn) −R(ti, Tn))2, (5.19)

where ti corresponds to the end-of-month dates from October 2005 to March2006. Tn are again the different maturities 1/12, 2/12, . . . , 11/12, 1, 2, . . . , 30.For these data we found the following parameters.

β11 β12 β21 β22 δ-0.8671 0.8561 -0.1467 0.1452 0.0146α1 α2 s1s

′1 s1s

′2 s2s

′2

0.0046 0.0010 0.8835 0.1011 0.0133

Table 5.1: Parameters for the Two-Factor Deterministic Volatility Modelbetween October 2005 and March 2006

Table 5.2 summarizes the residuals obtained by using the above parame-ters for every month from October to March.

Oct. Nov. Dec. Jan. Feb. Mar. Sum1.88e-05 1.84e-05 0.21e-05 0.20e-05 0.59e-05 1.61e-05 6.34e-05

Table 5.2: Residuals for the Two-Factor Deterministic Volatility Model, Oc-tober 2005 - March 2006

The results of December, January and February are already satisfactory,whereas the other ones could be improved by slightly adapting the parameters


which we investigate later on. Figure 5.1 also confirms this findings. Whenapplying the above parameters to data from April we also achieve a promisingresult since the residual is only 1.069e-05, although we did not use data fromApril in our initial optimization. The graphical result of April is also shownin Figure 5.1.

As already indicated we now adjust our parameters for each month inorder to achieve better results, especially for October, November, March andApril. The values for β remain unchanged, whereas α1, s1s

′1, s1s

′2, s2s

′2 and

δ are slightly modified to fit the data in a better way. Indeed, we alwaysuse the optimal parameters of the previous month as start values for thenext optimization rather than generating new initial values for each monthas described in the above algorithm. This procedure allows to keep the valuesrelatively constant over time. Table 5.3 contains the so obtained values forα1, s1s

′1, s1s

′2, s2s

′2 and δ as well as the corresponding residuals for each month,

which have improved in all cases.

Month α1 α2 s1s′1 s1s

′2 s2s

′2 δ Res

Oct. 0.0032 0.0011 0.8887 0.0958 0.0136 0.0147 0.53e-05Nov. 0.0032 0.0011 0.8874 0.0963 0.0136 0.0146 0.38e-05Dec. 0.0037 0.0009 0.8701 0.0976 0.0129 0.0132 0.20e-05Jan. 0.004 0.0009 0.8714 0.098 0.0127 0.013 0.12e-05Feb. 0.0039 0.0009 0.8727 0.0991 0.0131 0.0128 0.33e-05Mar. 0.0049 0.0008 0.8708 0.1009 0.0125 0.0133 0.21e-05Apr. 0.0045 0.0008 0,8693 0,1019 0,0132 0,013 0.57e-05

Table 5.3: Monthly Parameters for the Two-Factor Deterministic VolatilityModel, October 2005 - April 2006

The behavior of the parameters α1, α2, s2s′2, δ over time is then graph-

ically illustrated by figure 5.2, which points out that the variation of theparameters is very small. Considering the results of March as an example,figure 5.3 shows that the market- and the model yield curves are now nearlyidentical.

5.2.3 Stochastic Volatility

If γ 6= 0, we have to assure the non-negativity of v(X(t)) := γ′X(t) + δ(see (2.12)) by restricting the coefficients. For this reason we consider the“hyperplane”

H = (x1, x2) : γ1x1 + γ2x2 + δ = 0, (5.20)


0 5 10 15 20 25 300.02

0.022

0.024

0.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04Market Yields vs. Model Yields Oct.

Maturity

Inte

rest

rate

model spotsmarket spots

0 5 10 15 20 25 300.024

0.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04Market Yields vs. Model Yields Nov.

Maturity

Inte

rest

rate


0 5 10 15 20 25 300.024

0.026

0.028

0.03

0.032

0.034

0.036

0.038Market Yields vs. Model Yields Dec.

Maturity

Inte

rest

rate


0 5 10 15 20 25 300.024

0.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04

0.042Market Yields vs. Model Yields Jan.

Maturity

Inte

rest

rate


0 5 10 15 20 25 300.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04Market Yields vs. Model Yields Feb.

Maturity

Inte

rest

rate


0 5 10 15 20 25 300.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04

0.042

0.044Market Yields vs. Model Yields March

Maturity

Inte

rest

rate

model yieldsmarket yields

0 5 10 15 20 25 300.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04

0.042

0.044

0.046Market Yields vs. Model Yields April

Maturity

Inte

rest

rate


Figure 5.1: Market- versus Model Yields, Two-Factor Deterministic VolatilityModel, October 2005 - April 2006


1 2 3 4 5 6 70

0.005

0.01

0.015Parameters over time

October 2005 − April 2006

Para

met

ers

alpha1alpha2s2s2’ delta

Figure 5.2: Parameters over Time, Two-Factor Deterministic VolatilityModel, October 2005 - April 2006

where v = 0. Without loss of generality we take γ2 = 1, if γ1 6= 0, so thatx2 = −(γ1x1 + δ) on H. There, the drift function of v(X(t)) is consequently

γ′(α+ βx) = γ1[α1 + β11x1 + β12x2] + γ2[α2 + β21x1 + β22x2]

= γ1[α1 + β11x1 − β12(γ1x1 + δ)]

+[α2 + β21x1 − β22(γ1x1 + δ)],

= k1 + k2x1 (5.21)

where

k1 = γ1(α1 − β12δ) + α2 − β22δ. (5.22)

k2 = γ1β11 − γ21β12 + β21 − γ1β22. (5.23)

Thus, in order to guarantee v(X(t)) to be non-negative, we will have toassume that it has a sufficiently strong positive drift on H. Therefore themodel has to satisfy another condition

k1 > 0, and k2 = 0 (5.24)

besides the general requirements (5.10)-(5.12) and (5.2).


0 5 10 15 20 25 300.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04

0.042


Maturity

Inte

rest

rate

model yieldsmarket yields

Figure 5.3: Market- versus Model Yields, Two-Factor Deterministic VolatilityModel, March 2006

5.2.4 Calibrating the Stochastic Volatility Model

to the Current Term Structure

As we do not have explicit solutions for the differential equations (5.10)in this case, we need numerical methods for solving them. In effect, wehave to solve a two-point boundary value problem (BVP) for the differentialequations (5.10), where the boundary conditions are

A(0) = 0 A(10) = 0,

B1(0) = 0 B1(10) = 0,

B1(10) = 0 B2(10) = −10.

Having fixed the parameters β12, γ1, S5, α1, δ, setting γ2 = 1 and deriving β21

from k2 = 0 (5.24), it is then possible to find the 3 parameters β11, β22 and α2

by solving the above BVP. More precisely, we determine all our parametersin the following way:

I. Generate start values:

5We fix again s1s′

1, s2s

′

2, s1s

′

2.


α1 α2 β11 β12 β21 β22

-0.0101 0.0045 -0.6592 1.0675 -0.1609 0.1733

s1s′1 s2s

′2 s1s

′2 γ1 δ

0.2945 0.0586 0.0204 -0.4263 0.0246

Table 5.4: Parameters for the Two-Factor Stochastic Volatility Model, Oc-tober 2005 - March 2006

• For j = 1, 2, . . . , jmax

1. Generate βj11, β

j22 and αj

2 randomly as initial guess for the BVPalgorithm.

2. Generate βj12, γ

j1, S

j, αj1, δ

j randomly as start values in order tocalculate some steps of the following minimization. Set γ2 = 1and calculate β21 by means of (5.23).

3. MinimizeResj =

∑

Tn

(RM(0, Tn) −R(0, Tn))2 (5.25)

with respect to βj12, γ

j1, S

j, αj1, δ

j. In each optimization step themodel yields are calculated by solving the above BVP problemincluding the determination of the parameters β11, β22 and α2.Having then found all the parameters, the solution of (5.10) iscalculated by a Runge-Kutta method for all maturities (until 30),since the solution of the BVP problem is only obtained in theinterval [0, 10].

4. Save the parameters if (Resj < Resj−1).

II. Final optimization:

• Do a last optimization as described in item 3, using the parameterswith minimal residual found by the above loop as start parameters.

Finally one has to check if the parameters are chosen so as to satisfy theconditions of of theorem 2.3. For solving our BVP problem we use the Matlabfunction “bvp4c” which is a finite difference code that implements the threestage Lobatto Illa formula.

Analogously to the deterministic case we minimize (5.19) for the samedata as before, which allows to obtain constant parameters for different pointsin time. With this method we found the parameters listed in table 5.4, whichalso satisfy the conditions of theorem 2.3. The sum of the residuals obtained


by using these parameters for every month from October to March is smallerthan in the deterministic volatility case, which can be read off table 5.5.Similar to the deterministic volatility case these results can be improved by

Oct. Nov. Dec. Jan. Feb. Mar. Sum0.46e-05 0.75e-05 1.52e-05 0.38e-05 0.43e-05 1.18e-05 4.73e-5

Table 5.5: Residuals for the Two-Factor Stochastic Volatility Model, October2005 - March 2006

optimizing (5.25) separately for each month using the parameters from table5.4 as initial values. Table 5.6 presents the parameters for each month whichfulfill the conditions of theorem 2.3 as well.

Month α1 α2 s1s′1 s2s

′2 s1s

′2 γ1 δ

Oct. -0.0115 0.0036 0.2759 0.0638 0.0208 -0.5352 0.0161Nov. -0.0085 0.0046 0.3068 0.0588 0.0189 -0.4189 0.0275Dez. -0.0116 0.0035 0.2711 0.0606 0.0212 -0.4421 0.0147Jan. -0.0082 0.0043 0.3072 0.0589 0.0189 -0.4554 0.0262Feb. -0.0082 0.0039 0.3039 0.0594 0.0191 -0.4806 0.0224Mar. -0.0117 0.0045 0.2924 0.0579 0.0213 -0.4434 0.0224Apr. -0.0112 0.0045 0.2942 0.0580 0.0210 -0.4466 0.0234

Month β11 β12 β21 β22 ResOct. -0.7097 1.0617 -0.1726 0.1811 0.4203e-06Nov. -0.6561 1.0667 -0.1580 0.168 0.8981e-06Dec. -0.6652 1.0669 -0.1640 0.1774 1.8474e-06Jan. -0.6699 1.0645 -0.1614 0.1693 0.8083e-06Feb. -0.6792 1.0594 -0.1642 0.1715 1.7042e-06Mar. -0.6654 1.0661 -0.1641 0.1774 0.9317e-06Apr. -0.6666 1.0657 -0.1640 0.1765 1.1840e-06

Table 5.6: Monthly parameters for the Two-Factor Stochastic Volatilitymodel, October 2005 - April 2006

From the following figures one can gather that the parameters remainstable while the result are really satisfactory since they could be improvedwith respect to the results presented in table (5.5), where we used in contrastto the current approach exactly the same parameters for each month. Figure5.5 shows the comparison between model and market yields from March andOctober as example. It is obvious that the fit is very good for both monthalthough the parameters do not change much.


1 2 3 4 5 6−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06


October 2005−March 2006

Par

amet

ers

alpha1alpha2s1s2’s2s2’ delta

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3


October 2005 − March 2006

Par

amet

ers

s1gamma1

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


October 2005 − March 2006

Par

amet

ers

beta11beta12beta21beta22

Figure 5.4: Parameters over Time, Two-Factor Stochastic Volatility Model,October 2005 - March 2006

0 5 10 15 20 25 300.02

0.022

0.024

0.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04Market Yields vs. Model Yields Oct.

Maturity

Inte

rest

rat

e


0 5 10 15 20 25 300.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04

0.042


Maturity

Inte

rest

rat

e


Figure 5.5: Market- versus Model Yields, Two-Factor Stochastic VolatilityModel, October 2005, March 2006


Using the parameters from table 5.4 as initial values for the calibrationto data from April6 we also found satisfying results with parameters that donot change significantly which can also be seen in table 5.6.

5.2.5 Conclusion

On account of our calibration results we can conclude that both models, thedeterministic and the stochastic volatility model are appropriate means tofit the term structure of interest rates. Both models enable a good fit tomarket yields from different dates with unchanging parameters. This can beseen as advantage over models with time dependent parameters, as the Hull-White model described in section 4.2.3, where one parameter is a function ofthe market instantaneous forward rate. The fitting results of the stochasticvolatility model are slightly better than those of the deterministic volatilitymodel, whereas the numeric calibration procedure is more stable in the lat-ter case, since there are explicit solutions to the Ricatti equations. In thestochastic volatility case however, we have to solve them in each optimiza-tion step numerically respecting the boundary constraints, which makes thecalibration very time-intensive and less stable.

6April data was not taken into consideration when optimizing (5.19).

Appendix A

Numerical Methods for

Calibration

We explain here some methods that we used when calibrating our models tomarket data. For most of theses procedures Matlab already provides goodalgorithms which we incorporated into our own programs. In the followingsections we dwell on the methods that form a basis of our nonlinear least-squares data fitting problems.

A.1 Trust-Region Methods for Nonlinear

Minimization

Our calibration issues described in chapter 4 are principally a matter ofnonlinear least-squares problems. However, before referring to this specialkind of minimization problem, we consider the general case, i.e. the generalunconstrained minimization problem

minx∈Rn

f(x), (A.1)

where f : Rn → R is a twice continuously differentiable function. The basic

idea is to approximate f with a simpler function ψ, which reasonably reflectsthe behavior of f in a neighborhood around the current point xk, the so-calledtrust region. In the standard trust region method [13] for problem (A.1), ψis the quadratic approximation for f defined by the first two terms of theTaylor approximation to f at xk. The trust-region subproblem is typicallyformulated by

mins∈Rn

ψk(s) = g′ks+

1

2s′Hks : ||Dks|| ≤ ∆k

, (A.2)

72

APPENDIX A. NUMERICAL METHODS FOR CALIBRATION 73

where gk = ∇f(xk), the gradient of f at the current point xk, and Hk is theHessian matrix ∇2f(xk). Dk is a non-singular scaling matrix and ∆k is apositive scalar representing the trust region size. || · || denotes the 2-norm.

In Newton’s method with a trust region strategy, each iterate xk hastherefore a neighborhood regulated by ∆k and Dk so that

f(xk + s) ≈ f(xk) + ψk(s), ||Dks|| ≤ ∆k. (A.3)

In other words ψk is a model of the reduction in f within a neighborhood ofthe current point xk. So, by finding sk that minimizes ψk we also minimizef(xk + s). If the step is satisfactory, in the sense that f(xk + sk) < f(xk),xk+1 = xk + sk, otherwise the xk remains unchanged and the trust region isshrunk by adapting ∆k.

Thus, the essential step is the solution of the trust-region subproblem.The Matlab method follows the Branch, Coleman and Li [2] approach, whichrestricts the trust region subproblem to a two-dimensional subspace S =〈s1, s2〉. This is necessary, because accurate solutions of (A.2) require toomuch time in the case of large-scale problems. So the first direction of oursubspace s1 is the gradient whereas the second direction s2 is determinedwith the aid of a preconditioned conjugate gradient process (PCG). PCG isa popular way to solve general large symmetric positive definite systems oflinear equations

Hs = −g. (A.4)

In our context we obtain this equation by differentiating ψ with respect to sand setting the derivative 0. So the solution s would be the arg min of ψ.1

The Hessian matrix can assumed to be symmetric, however it is guaranteedto be positive definite only in the neighborhood of a strong minimizer. ThePCG algorithm exists when a direction of negative curvature, i.e. s′Hs ≤ 0,is encountered. So the output direction s2 is either a direction of negativecurvature or an approximate solution of the Newton system Hs2 = −g.Solving (A.2) becomes much easier and faster than in the unrestricted casesince in the subspace the problem is only two-dimensional.

Summarizing the mentioned ideas the whole minimization algorithm con-sists basically in 4 steps:

• Formulation of the 2-dimensional trust region subproblem using PCGto determine the subspace directions.

• Resolution of the now two-dimensional problem (A.2) to determine sk.

• If f(xk + sk) < f(xk) then xk+1 = xk + sk.

1provided that H is positive definite.

APPENDIX A. NUMERICAL METHODS FOR CALIBRATION 74

• Adjustment of ∆k

These four steps are repeated until convergence. For some special cases of fthe procedure can be simplified, for instance for our nonlinear least-squares,as we see in section (A.1.2).

A.1.1 Box Constraints

The box constrained problem is of the form

minl≤x≤u

f(x), (A.5)

where l is a vector of lower bounds and u is vector of upper bounds, which canalso be equal to −∞ and ∞. Two techniques are used to generate a sequenceof strictly feasible points. Firstly, instead of the unconstrained Newton step,i.e. solving (A.4) to define the two dimensional subspace S, a scaled modifiedNewton step is introduced and secondly reflections are used to increase thestepsize (see Coleman and Li [5] for details).

A.1.2 Nonlinear Least-Squares

An important special case for f(x) which is relevant for our purposes is thenonlinear least-square problem

f(x) =1

2

n∑

i=1

f 2i (x) =

1

2||F (x)||2 (A.6)

F (x) is a vector-valued function whose ith-component is fi(x)2. The structure

of this function is exploited to enhance efficiency. Instead of trying to find s2

of the two-dimensional subspace by solving (A.4), the normal equations, i.e.

J ′Js2 = J ′F, (A.7)

where J is the Jacobian of F (x) are solved for determining s2. These equa-

tions are derived by differentiating ψ = ||Js + F ||2 with respect to s andsetting the derivative 0. In this particular case s′g + s′Hs can be approxi-mated by ψ which allows to avoid the calculation of the second derivatives.

2In our calibration problems F is always the difference between the market- and themodel yields.

Bibliography

[1] Tomas Bjork. Arbitrage Theory in Continuous Time. Oxford UniversityPress, 1998.

[2] Mary A. Branch, Thomas Coleman, and Yuying Li. A subspace, inte-rior and conjugate gradient method for large-scale bound- constrainedminimization problems. SIAM Journal of Scientific Computing, 21.

[3] Damiano Brigo and Fabio Mercurio. Interest Rate Models - Theory andPractice. Springer Finance, 2001.

[4] Andrew J.G. Cairns. Interest Rate Models, An Introduction. PrincetonUniversity Press, 2004.

[5] Thomas Coleman and Yuying Li. An interior trust region approach fornonlinear minimization subject to bounds. SIAM Journal of Optimiza-tion, 6:418–445, 1996.

[6] Qiang Dai and Kenneth J. Singelton. Specification analysis of affineterm structure models. The Journal of Finance, LV, No.5, 2000.

[7] Duffie, Ma, and Yong. Black’s consol rate conjecture. Annals of AppliedProbability 5(2), 1995.

[8] Darrell Duffie and Rui Kan. A yield-factor model of interest rates.Graduate School of Business, Stanford University, 1996.

[9] L.P. Hansen. Large sample properties of generalized method of momentsestimators. Econometrica, 1982.

[10] L.P. Hansen and J.A. Scheinkman. Back to the future: Generating mo-ment implications for continuous-time markov processes. Econometrica,1995.

[11] Jessica James and Nick Webber. Interest Rate Modeling. John Wileyand Sons,LTD, 2001.

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[12] J.Tice and N.J.Webber. A non-linear model of the term structure ofinterest rates. Mathematical Finance, 7, 1997.

[13] Jorge J. More and D.C. Sorensen. Computing a trust region step. SIAMJournal on Scientific and Statistical Computing, 3:553–572, 1983.

[14] Marek Musiela and Marek Rutkowski. Martingale Methods in FinancialModeling, volume 26. Springer, 1998.

[15] Bernt Øksendal. Stochastic Differential Equations. Springer, 2000.

[16] P.Balduzzi, S.R.Das, and S.Foresi. A central tendency: A second factorin bond yields. 1996.

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