# Term Structure Dynamics of Interest Rates by Exponential-Affine Models

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```					           Term Structure Dynamics of
Interest Rates by
Exponential-Affine Models
Master in Calcolo Scientifico
Dipartimento di Matematica
Università degli Studi “La Sapienza” Roma
23 Maggio 2005

Marco Papi
m.papi@iac.cnr.it
Università di Varese                               Istituto per le Applicazioni
Dipartimento di Economia                           del Calcolo IAC - CNR
Term Structure Dynamics of Interest Rates                                                      Marco Papi

The evolution of the interest rates with maturity from three months up to thirty years in the
time frame June 1999-December 2002. The thick line is a linear interpolation of the ECB
offcial rate (represented by cross points).
Term Structure Dynamics of Interest Rates                                       Marco Papi

The evolution of the term structure in the time frame January 1999-December 2002
Term Structure Dynamics of Interest Rates   Marco Papi
Term Structure Dynamics of Interest Rates               Marco Papi

• Understanding and modelling the term structure of
interest rates represents one of the most challenging
topics of financial research.
• There are many benefits from a better understanding
of the term structure: pricing and hedging interest
rate dependent assets or managing the risk of interest
rates contingent portfolios.
• Bond prices and interest rates derivatives are
dependent on interest rates, which exhibit a complex
stochastic behavior and are not directly tradable,
which means that the dynamic replication strategy is
more complex.
• Thus each of existing models has its own advantages
and drawbacks.
Term Structure Dynamics of Interest Rates              Marco Papi

Main problems:

• How do you build a model to explain the yield
curve.
• How do you build a model in order to price
derivatives?
• How do you build a model to help you to hedge
• What is a good (parsimonious?) way to
describe the (partly observed) existing yield curve?
Term Structure Dynamics of Interest Rates                        Marco Papi

Definition:
• Bonds: T-bond = zero coupon bond, paying 1€ at the
date of maturity T.
B (t , T )  price at time t for a T - bond,
B (T , T )  1.
Main Objectives:
1. Investigate the term structure, i.e. how prices of bonds
with different dates of maturity are related to each other.

2. Compute arbitrage free prices of interest rate derivatives
(bond options, swaps, caps, floors etc.)
Term Structure Dynamics of Interest Rates                  Marco Papi

Definition:
• Yield to Maturity: the continuously compounded rate
of return that causes the bond price to rise to one at
time T:
( T t ) R ( t ,T )
B (t , T )e                         1
or, solving for the yield
log B(t , T )
R(t , T )  
T t
For a fixed time t, the shape of R(t,T) as T increases
determines the term structure of interest rates. In our
framework, the yield curve is the same as the term structure
of interest rates, as we only work with ZCBs.
Term Structure Dynamics of Interest Rates                  Marco Papi

• Finance traditionally views bonds as contingent claims
and interest rates as underlying assets.

Definitions:
• Instantaneous risk-free interest rate (short term rate):
the yield on the currently maturing bond,

r (t )  lim R(t , T )
T t

The value of the money market account initialize d
at time 0 with one euro investment is
t

 (t )  e 0 r ( s ) ds
Term Structure Dynamics of Interest Rates                                  Marco Papi

Definitions:
• Forward rate: the rate that can be agreed upon at time t for a
risk-free loan starting at time T1 and finishing at time T2,
log B(t , T1 )  log B (t , T2 )
f (t , T1 , T2 ) 
T2  T1
of particular interest is the instantaneous forwardrate f (t , T , T )
It is the rate that one contractsat time t for a loan starting at time
T for an instantaneous period of time. We have
 log B (t , T )
f (t , T )  
T
assuming that bond prices are differentiable. Equivalen tly, one can
define the bond price in terms of forwardrates as
T

B (t , T )  e t
 f ( t , s ) ds
. Note that r (t )  f (t , t )!
Term Structure Dynamics of Interest Rates             Marco Papi

No-Arbitrage Restrictions:

• A bond price will never exceed its terminal value plus
the outstanding coupon payments.
• A zero-coupon price cannot exceed the price of any
zero-coupon with a shorter maturity.
• The value of a zero-coupon bond must be equal to a
value of a replicating portfolio composed of zero-
coupon bonds.
• Interest rates should not be negative.
Term Structure Dynamics of Interest Rates                   Marco Papi

Theories of term structure :
• The expectation hypothesis: the term structure is
driven by the investor‟s expectations on future spot
rates. The rate of return on a T-bond should be equal to
the average of the expected short-term rate from t to T,

T
1
R(t , T )        Et (r( s))ds
T t t

There exist four continuous-time interpretations of
the expectation hypothesis.
Term Structure Dynamics of Interest Rates             Marco Papi

Continuous versus discrete models:

• Should we model the term structure in discrete or a
continuous framework?

• The power of continuous-time stochastic calculus allows
more elegant derivations and proofs, and provides an
adequate framework to produce more precise theoretical
solutions and refined empirical hypothesis, unfortunately
at the cost of a considerably higher degree of
mathematical sophistication.
Term Structure Dynamics of Interest Rates                Marco Papi

Bond prices, interest rate Vs yield curve models:
• Early models attempted to model the bond price
dynamics. Their results did not allow for a better
understanding of the term structure.

• Many interest rate models describe the evolution of a
given interest rate (often the short term i.r.). This
translates the valuation problem into a partial differential
equation that can be solved analytically or numerically.

• An alternative is to specify the stochastic dynamics of
the entire term structure, either by using all yields or
all forward rates. The model complexity increases
significantly.
Term Structure Dynamics of Interest Rates              Marco Papi

Single Vs multi-factor models:
• Factor models assume that the structrure of interest rates
is driven by a set of variables or factors. Empirical
studies used Principal Component Analysis to
decompose the motion of the interest rate term
structure into 3 i.i.d factors:
• Shift: it is parallel movement of all rates. It usually
accounts for up 80-90 percent of the total variance.
• Twist: it describes a situation in which long rates and
short term rates move in opposite directions. It usually
accounts for an additional 5-10 percent of the total
variance.
• Butterfly: the intermediate rate moves in the opposite
direction of the short and long term rate. 1-2 percent of
influence.
Term Structure Dynamics of Interest Rates                                       Marco Papi

The three most significant components computed from monthly
yield changes, Jan 1999-Dec 2002
Term Structure Dynamics of Interest Rates                                    Marco Papi

Simulation of the term structure evolution based on the PCA
Term Structure Dynamics of Interest Rates               Marco Papi

Arbitrage-free versus Equilibrium models:

stochastic behavior of one or many interest rates and a
specific market price of risk and derive the price of all
contingent claims.

• Equilibrium models start from a description of the
economy, including the utility function of a
representative investor and derive the term structure of
interest rates and the risk premium endogenously,
assuming that the market is at equilibrium.
Term Structure Dynamics of Interest Rates                    Marco Papi

•     All our models will be set up in a given probability
space (, , P ), and a filtration Ft generated by a st.
Brownian motion W(t).
Single factor models:
•     All the information can be summarized by one single
specific factor, the short term rate r(t). For a Zcb
maturing at time T (T ≥ t), we have B(t,T) = B(t,T,r(t)).
•     Short term rate dynamics:
Term Structure Dynamics of Interest Rates                  Marco Papi

•  Let us denote by V(t) the value at time t of an interest
rate claim with maturity T.
• From the one factor model assumption, we can write
V(t) = V(t,T,r(t))
By Ito‟s lemma,

Dividing both sides by V(t) yields the rate of return:
Term Structure Dynamics of Interest Rates             Marco Papi

Now let us consider two distinct interest rate contingent
claims V1 and V2 with maturities T1 and T2 and let us form
a portfolio of value

The prices satisfy the equations

The variations of the portfolio value are given by
Term Structure Dynamics of Interest Rates                Marco Papi

We can select x1 and x2 to cancel out the instantaneous risk
of the position, i.e to reduce the volatility of P(t) to zero.
This gives the following system of equations:

Actually, in order to avoid arbitrage opportunities, the
return must be equal to r(t). The system has a non trivial
solution iff

This common value λ(t,r(t)) is called the market risk-
Term Structure Dynamics of Interest Rates              Marco Papi

This allows us to express the instantaneous return on the
bond as

We obtain the second order pde (called the Feynman-Kac
equation) that must satisfy any interest rate contingent claim
in a no-arbitrage one factor model:

with one boundary condition. The term µr- σrλr is called the
Term Structure Dynamics of Interest Rates                     Marco Papi

• A Zcb B(t,T) satisfies F-K equation with B(T,T) = 1.
• A plain vanilla call option on B(t,T) with maturity TC < T,
satisfies F-K equation
V (TC )  max(B(TC , T )  K ,0)
• For a swap of fixed rate δ against a floating rate r with
maturity date T, we have

with the boundary condition V(0) = 0.
Term Structure Dynamics of Interest Rates            Marco Papi

Theorem 1 . The solution to F-K equation for B(t,T) under
the terminal condition B(T,T) = 1 is given by

Theorem 2 [Harrison Kreps (1979)]. Under some
regularity conditions, a market is arbitrage-free if there
exists an equivalent market measure Q, such that the
discounted price process of any security is a Q-martingale.
Therefore, we can write
Term Structure Dynamics of Interest Rates              Marco Papi

From one world to another
dynamics, then we define a new process
dW*(t) = dW(t) – λ(t) dt
• Under technical conditions, using Girsanov’s
Theorem , there exists a probability measure Q s.t.
W*(t) is a Q-Brownian motion, where
dQ = ρ(T,λ) dP
where for any t ≤ T
Term Structure Dynamics of Interest Rates                Marco Papi

Model specification: P or Q ?
• The specification of the dynamics of the rate r(t)
under P causes problems as the equivalent
probability measure Q may not be unique.
• As in the B-S framework, we have one source of
randomness and one state process, but r(t) is not the
• The market is clearly incomplete and Q in not
necessarily unique.
• There are consequences on the parameter estimation:
the set θ of observable parameters enters in a pde
collectively with λ. We need to use market traded
assets to find the combination (θ, λ) that fits prices.
Term Structure Dynamics of Interest Rates               Marco Papi

Affine Models
• Their popularity is due both to their tractability and
flexibility.
• In some cases explicit solutions to the F-K can be
found, and it is relatively easy to price other
instruments with this models.
• They have sufficient free parameters so that they can
fit term structures quite well.
• Affine models were first investigated as a category
by Brown and Schaefer (1994).
• Duffie and Kan (1994,1996) developed a general
theory.
• Dai and Singleton (1998) provided a classification.
Term Structure Dynamics of Interest Rates                                     Marco Papi

Affine Models
• Given n state variables X(t), spot rates take the following
form
a ( )      n
bi ( )
R (t , t   )                              X i (t ), where   T  t
      i 1 
• Taking the limit as τ → 0, we obtain an expression for the
short rate r(t):  r(t) = f + <g, X(t)> .

where assuming a (0)  0 and b(0)  0, we have
,
a                         bi
f     (0),             gi       (0), i  1,.....,n
                         
Term Structure Dynamics of Interest Rates                                     Marco Papi

Affine Models
• If the model is affine, then price of a T-bond can be
written in the form of an exponential-affine function
of X:                       n
a ( )   bi ( ) X i ( t )
B (t , T )  e i 1                        , being   T  t.
• Once the processes for the vector of state variables have
been specified (under Q, say) is sufficient to establish
prices in the model.
• Duffie and Kan (1996) show that X(t) must be of the form

H ij (t )   ij i  i 1 i X i (t )
n
dX (t )  (uX (t )  v )dt   H (t )dWt    *
Term Structure Dynamics of Interest Rates                         Marco Papi

Affine Models
• To find bond prices we need to solve for a(.) and b(.) the F-K
equation. It is not diffult to see that

a                            1
( )   f   v, b( )     diag( ) T b( ), b( ) , a (0)  0.
                            2

b                            1
( )   g   u, b( )      T b( ), b( ) , b(0)  0.
                            2

• There are fairly easy numerical solution methods available for this
type of differential equations.
Term Structure Dynamics of Interest Rates             Marco Papi

Types of Affine Models
• Commonly used affine models can be conveniently
separated into three main types:
• Gaussian affine models: all state variables have constant
volatilities[Vasicek („77), Hull-White (‟90), Babbs (‟93)].
• CIR affine models: all state variables have square-root
volatilities[CIR (‟85), Longstaff (‟90), El-Karoui (‟92)].
• A three-factor affine family. [Sorensen (‟94), Chen (‟96)].

• In addition, an affine model may be “extended”, that is
some of its parameters may be allowed to be
deterministic functions of time.
Term Striature Dynamics of Interest Rates               Marco Papi

One Factor Affine Models

• The term structure of interest rates is an affine function
of the short rate r(t):

• Proposition. If under Q, µr (σr)2 are affine in r(t), then
the model is affine.
Term Striature Dynamics of Interest Rates            Marco Papi

One Factor Gaussian Model
• In a Gaussian model, dr(t) = [µ1(t)+µ2(t)]dt+σ2(t)dW(t),
r(t) is normally distributed, and

where

m and a variance v, and
Term Striature Dynamics of Interest Rates                  Marco Papi

Some Specific Examples
• Vasicek (1977)
• when r goes over θ, the expected variation of r becomes
negative and r tends to come back to its average long term level
θ at an adjustment speed k. Vasicek postulates a constant risk-
• The explicit solution is

• Interest rates can become negative, which is incompatible with
no-arbitrage.
Term Striature Dynamics of Interest Rates                  Marco Papi

Some Specific Examples (Vasicek)

• Under the original measure P, the bond price dynamics is given
by

• This implies that both prices are lognormally distributed. Note
that the volatility increases with T.
Term Striature Dynamics of Interest Rates                    Marco Papi

Some Specific Examples (Vasicek)

• The term structure can be positively shaped when
r(t) < R(t,∞)-0.5(σr/k)2, negatively shaped for r(t) > R(t,∞)-
0.5(σr/k)2.

• Given the set of information at time s ≤ t, R(t,T) is normally
distributed.
Term Striature Dynamics of Interest Rates                   Marco Papi

Some Specific Examples (Vasicek)
• Option prices [Jamshidian (’89)]: The option pricing formula
has similarities with the Black & Scholes formula, since bond
prices are lognormally distributed in the model:

with
Term Structure Dynamics of Interest Rates                                            Marco Papi

Simulation of the term structure evolution based on the Vasicek model
Term Striature Dynamics of Interest Rates                    Marco Papi

Some Specific Examples
• CIR (1985). Cox, Ingersoll and Ross devoloped an equilibrium
model in which interest rates are determined by the supply and
demand of individuals having a logarithmic utility function.

• The risk premium at equilibrium is

• The short term process is known as the square-root process and
has a variance proportional to the short rate rather than constant.
• If r(0) > 0, k ≥ 0, θ ≥ 0, and kθ ≥ 0.5(σ)2 , the SDE admits a
unique solution, that is strictly positive, for all t > 0.
Term Striature Dynamics of Interest Rates                       Marco Papi

Some Specific Examples (CIR)
• The unique solution is

• Given the information at time s, then the short term rate r(t) is
distributed as a non central chi-squared [Feller, 1951]:

with 2q+2 degrees of freedom and non central parameter 2u.
• The distribution can be written explicitly as

Iq is the modified Bessel function of the first type and order q.
Term Striature Dynamics of Interest Rates   Marco Papi

Some Specific Examples (CIR)
• Bond prices solve

• The solution is

with
Term Striature Dynamics of Interest Rates                  Marco Papi

Some Specific Examples (CIR)
• Under the original measure P, bond price dynamics is given by

• Term structure. The rate R(t,T) depends linearly on r(t) and
R(t,∞), where

• The value of r(t) determines the level of the term structure at
time t, but not its shape.

• Cox, Ingersoll and Ross provide formulas for the price of
European call and put options.
Term Structure Dynamics of Interest Rates                                            Marco Papi

Simulation of the term structure evolution based on the CIR model
Term Striature Dynamics of Interest Rates                   Marco Papi

Time-varying processes: Hull and White
• Hull and White (1993) introduced a class of models which is
consistent with a whole class of existing models.

• with an exogeneously specified risk premium
• The time-varying coefficients can be used to calibrate exactly
the current market prices.
• The price to be paid for this exact calibration is that bond and
bond options prices are no longer analytically obtainable.
• Using all the degrees of freedom in a model to fit the volatility
constitutes an over-parametrization of the model.
• In practice, the model is implemented with k and σ constant and
θ as time-varying.
Term Striature Dynamics of Interest Rates                   Marco Papi

Other Models
• Black, Derman, and Toy‟s (1987, 1990)

• The model is very popular among practitioners for various
reasons. Unfortunately, the model lack analytical properties, and
its implications and implicit assumptions are unknown.
• Dothan (1978), Brennan and Schwartz (1980)

• but there is no known distribution for r(t), and contingent claim
prices must be computed using numerical procedures.
• In particular Brennan and Schwartz used the model to price
convertible bonds.
Term Striature Dynamics of Interest Rates                    Marco Papi

Multi-Factor Models
• Richard (1978), Cox, Ingersoll, Ross (1985) considered
multiple factors: the real short term rate q(t) and the expected
instantaneous inflation rate π(t), following indipendent
processes:

• Applying Ito‟s formula, we obtain the pde solved by the price of
a T-bond:

• It is possible to express r as a function of π and q.
• Richard obtained a complicated, but analytical, solution for the
Zcb price.
Term Striature Dynamics of Interest Rates                   Marco Papi

Multi-Factor Models
• Longstaff and Schwartz (1992) developed a two factor model:

• In their framework, there is only one good available in the
economy

• The factors can be related to observable quantities
Term Striature Dynamics of Interest Rates                    Marco Papi

Multi-Factor Models
• Chen (1996) proposed a three-factor model of the term
structure:

• r depends on its stochastic mean and stochastic volatility.

• Closed form solutions for T-bonds and some interest rate
derivatives are obtained in very specific cases.
Term Striature Dynamics of Interest Rates               Marco Papi

Estimation

• Suppose that we have chosen a specific model, e.g. H-W . How
do we estimate the parameters?
• Naive answer: Use standard methods from statistical theory.
• The parameters are Q-parameters.
• Our observations are not under Q, but under P. Standard
statistical techniques can not be used.
• We need to know the market price of risk (λ). Who determines
λ? The market.
• We must get price information from the market in order to
estimate parameters.
Term Striature Dynamics of Interest Rates                Marco Papi

Inverting the Yield Curve
• Q-dynamics with a parameter vector α :

• Theoretical term structure

• Observed term structure:

• Want: A model such that theoretical prices fit the observed
prices of today, i.e. choose parameter vector α such that

• Number of equations = ∞ (one for each T).
• Number of unknowns = dim(α)
Term Striature Dynamics of Interest Rates                      Marco Papi

GMM Estimation
• Suppose we have a set of observations of r, whose evolution
depends upon a set of parameters α of dimension k
• It will possible to find functions fi(r(t)| α) , i=1,….,m, m = k, s.t.
E[fi(r(t)| α)] = 0.
• The GMM estimates α* of α are those values of α that set the
sample estimates as close to zero as possible.
• In the „classic‟ method of moments the number of parameters
equals the number of functions.
• We can relax the assumption m = k, defining α* to be
arg minα < M f, f >
M being is a positive definite matrix.
• This is the GMM estimate contingent upon M and f.
Term Striature Dynamics of Interest Rates                Marco Papi

ML Estimation
• Miximum-Likelihood methods find parameters values for
which the actual outcome has the maximum probability.
• They choose parameter values so that the actual outcome lies in
the mode of the density function over sample paths.
• This can be calculated using the transition density function
p(ti+1,ri+1;ti,ri|α)
• The process is assumed to be Markov.
• The Likelihood function is L(α) = Πi p(ti+1,ri+1;ti,ri|α)
• An estimate of α if found by maximizing L, this places the
observed time series at the maximum of the joint density
function.
• It may be more convenient to maximize log L instead L.
Term Striature Dynamics of Interest Rates                       Marco Papi

ML Estimation
• The data are a panel of bond yields. They are equally spaced in
the time series, at intervals t = 1, . . . , T.
• The random vector y(t) represents a length-m vector of bond
yields. Denote the history of yields through t as Y(t) = (y(1) , . . .
, y(t) ). Yields are functions of a length-n state vector X(t) and
(perhaps) a latent noise vector W(t):

• We are interested in the resulting probability distribution of
yields:

• The primary difficulty in estimating ρ with this structure is that
the functional form for g(.) is often unknown or intractable.

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