Term Structure Dynamics of Interest Rates by Exponential-Affine Models Marco Papi [email protected]...

Term Structure Dynamics of Interest Rates by

Exponential-Affine Models

Marco [email protected]

Istituto per le Applicazioni Istituto per le Applicazioni del Calcolo IAC - CNR del Calcolo IAC - CNR

Università di Varese Università di Varese Dipartimento di EconomiaDipartimento di Economia

Master in Calcolo Scientifico

Dipartimento di Matematica

Università degli Studi “La Sapienza” Roma

23 Maggio 2005

mailto:[email protected]

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).


The evolution of the term structure in the time frame January 1999-December 2002

• 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.


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 your positions?

• What is a good (parsimonious?) way to

describe the (partly observed) existing yield curve?


Definition: • Bonds: T-bond = zero coupon bond, paying 1€ at the date of maturity T.


.1),(

,bond-Ta for t at time price),(

TTB

TtB

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.)

Definition: • Yield to Maturity: the continuously compounded rate

of return that causes the bond price to rise to one at time T:


tT

TtBTtR

eTtB TtRtT

),(log),(

yieldfor the solving or,

1),( ),()(

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.

• 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,


)(

is investment euro one with0 at time

dinitializeaccount market money theof valueThe

),(lim)(

0)(

tdssr

tT

et

TtRtr

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,


)!,()( that Note . ),(

as rates forward of termsin price bond thedefine

can one tly, Equivalenable.differenti are prices bond that assuming

),(log),(

have We time.of period ousinstantane anfor

at time starting loana for at time contracts one that rate theisIt

),,( rate forward ousinstantane theisinterest particular of

),(log),(log),,(

),(

12

2121

ttftreTtB

T

TtBTtf

T

t

TTtf

TT

TtBTtBTTtf

T

tdsstf

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.


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

t

t dssrEtT

TtR ))((1

),(

There exist four continuous-time interpretations of the expectation hypothesis.

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.


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.


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.



The three most significant components computed from monthly yield changes, Jan 1999-Dec 2002


Simulation of the term structure evolution based on the PCA

Arbitrage-free versus Equilibrium models:

• Arbitrage-free models start with assumptions about the 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.


• All our models will be set up in a given probability space 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:


),,,( P

• 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:



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


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-premium .


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 risk-adjusted drift .


• 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

• 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.

)0,),(max()( KTTBTV CC


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

From one world to another • We start with the original risk-free interest rate

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


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 price of a traded asset.

• 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.


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.


Affine Models• Given n state variables X(t), spot rates take the following

form

• Taking the limit as τ → 0, we obtain an expression for the short rate r(t): r(t) = f + <g, X(t)> .


nib

ga

f

ba

tTtXba

ttR

ii

i

n

i

i

,.....,1 ),0( ),0(

have we,0)0( and 0)0( assuming , where

where),()()(

),( 1

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:

• 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


)()( )())(()(

. being ,),(

1

*

)()()(1

tXtHdWtHdtvtuXtdX

tTeTtB

i

n

i iiijijt

tXba ii

n

i

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

• There are fairly easy numerical solution methods available for this type of differential equations.


.0)0( ,)(,)( 2

1)(,)(

.0)0( ,)(,)()diag( 2

1)(,)(

T

T

bbbbugb

abbbvfa

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.


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

• is normally distributed under Q, with a mean

m and a variance v, and


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-premium λ.

• The explicit solution is

• Interest rates can become negative, which is incompatible with no-arbitrage.


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.


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.


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



Simulation of the term structure evolution based on the Vasicek model

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.


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.


Some Specific Examples (CIR)• Bond prices solve

• The solution is

with


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.



Simulation of the term structure evolution based on the CIR model

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.


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.


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.


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


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.


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.


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)


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.


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.


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.


Term Structure Dynamics of Interest Rates by Exponential-Affine Models Marco Papi [email protected]...

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