Term Structure Dynamics of Interest Rates by ExponentialAffine 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_at_iac.cnr.it
Istituto per le Applicazioni del Calcolo IAC -
CNR
Università di Varese Dipartimento di Economia
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Term Structure Dynamics of Interest Rates
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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).
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Term Structure Dynamics of Interest Rates
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The evolution of the term structure in the time
frame January 1999-December 2002
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Term Structure Dynamics of Interest Rates
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Term Structure Dynamics of Interest Rates
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• 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.

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Term Structure Dynamics of Interest Rates
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• 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?

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Term Structure Dynamics of Interest Rates
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• Definition
• Bonds T-bond zero coupon bond, paying 1 at
  the
• date of maturity T.

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.)
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Term Structure Dynamics of Interest Rates
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• Definition
• Yield to Maturity the continuously compounded
  rate of return that causes the bond price to rise
  to one at time 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.
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Term Structure Dynamics of Interest Rates
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• 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,

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Term Structure Dynamics of Interest Rates
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• 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,

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Term Structure Dynamics of Interest Rates
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• 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.

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Term Structure Dynamics of Interest Rates
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• Theories of term structure
• The expectation hypothesis the term structure is
  driven by the investors 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,

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

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Term Structure Dynamics of Interest Rates
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• 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.

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Term Structure Dynamics of Interest Rates
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• 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.

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Term Structure Dynamics of Interest Rates
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The three most significant components computed
from monthly yield changes, Jan 1999-Dec 2002
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Term Structure Dynamics of Interest Rates
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Simulation of the term structure evolution based
on the PCA
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Term Structure Dynamics of Interest Rates
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• 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.

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Term Structure Dynamics of Interest Rates
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• 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

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Term Structure Dynamics of Interest Rates
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• 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 Itos lemma,
• Dividing both sides by V(t) yields the rate of
  return

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Term Structure Dynamics of Interest Rates
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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
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Term Structure Dynamics of Interest Rates
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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
?(t,r(t)) is called the market risk-premium .
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Term Structure Dynamics of Interest Rates
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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- sr?r is called
the risk-adjusted drift .
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Term Structure Dynamics of Interest Rates
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• 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 lt T, satisfies F-K equation
• For a swap of fixed rate d against a floating
  rate r with maturity date T, we have
• with the boundary condition V(0) 0.

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Term Structure Dynamics of Interest Rates
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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. Therefo
re, we can write

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Term Structure Dynamics of Interest Rates
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• 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) ?(t) dt
• Under technical conditions, using Girsanovs
  Theorem , there exists a probability measure Q
  s.t. W(t) is a Q-Brownian motion, where
• dQ ?(T,?) dP
• where for any t T

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Term Structure Dynamics of Interest Rates
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• 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 ? of observable parameters enters in
  a pde collectively with ?. We need to use market
  traded assets to find the combination (?, ?) that
  fits prices.

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Term Structure Dynamics of Interest Rates
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• 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.

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Term Structure Dynamics of Interest Rates
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• Affine Models
• Given n state variables X(t), spot rates take the
  following form
• Taking the limit as t ? 0, we obtain an
  expression for the short rate r(t) r(t) f
  ltg, X(t)gt .

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Term Structure Dynamics of Interest Rates
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• 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

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Term Structure Dynamics of Interest Rates
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• 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.

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Term Structure Dynamics of Interest Rates
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• Types of Affine Models
• Commonly used affine models can be conveniently
  separated into three main types
• Gaussian affine models all state variables have
  constant volatilitiesVasicek (77), Hull-White
  (90), Babbs (93).
• CIR affine models all state variables have
  square-root volatilitiesCIR (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.

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Term Striature Dynamics of Interest Rates
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• 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 (sr)2 are affine in
  r(t), then the model is affine.

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Term Striature Dynamics of Interest Rates
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• One Factor Gaussian Model
• In a Gaussian model, dr(t) µ1(t)µ2(t)dts2(t)
  dW(t),
• r(t) is normally distributed, and
• where
• is normally distributed under
  Q, with a mean m and a variance v, and

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Term Striature Dynamics of Interest Rates
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• Some Specific Examples
• Vasicek (1977)
• when r goes over ?, the expected variation of r
  becomes negative and r tends to come back to its
  average long term level ? at an adjustment speed
  k. Vasicek postulates a constant risk-premium ?.
• The explicit solution is
• Interest rates can become negative, which is
  incompatible with no-arbitrage.

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Term Striature Dynamics of Interest Rates
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• 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.

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Term Striature Dynamics of Interest Rates
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• Some Specific Examples (Vasicek)
• The term structure can be positively shaped when
• r(t) lt R(t,8)-0.5(sr/k)2, negatively shaped
  for r(t) gt R(t,8)-0.5(sr/k)2.
• Given the set of information at time s t,
  R(t,T) is normally distributed.

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Term Striature Dynamics of Interest Rates
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• 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

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Term Structure Dynamics of Interest Rates
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Simulation of the term structure evolution based
on the Vasicek model
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Term Striature Dynamics of Interest Rates
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• 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) gt 0, k 0, ? 0, and k? 0.5(s)2 , the
  SDE admits a unique solution, that is strictly
  positive, for all t gt 0.

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Term Striature Dynamics of Interest Rates
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• 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 2q2 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.

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Term Striature Dynamics of Interest Rates
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• Some Specific Examples (CIR)
• Bond prices solve
• The solution is
• with

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Term Striature Dynamics of Interest Rates
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• 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,8), 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.

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Term Structure Dynamics of Interest Rates
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Simulation of the term structure evolution based
on the CIR model
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Term Striature Dynamics of Interest Rates
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• 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
  s constant and ? as time-varying.

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Term Striature Dynamics of Interest Rates
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• Other Models
• Black, Derman, and Toys (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.

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Term Striature Dynamics of Interest Rates
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• 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 p(t), following indipendent
  processes
• Applying Itos formula, we obtain the pde solved
  by the price of a T-bond
• It is possible to express r as a function of p
  and q.
• Richard obtained a complicated, but analytical,
  solution for the Zcb price.

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Term Striature Dynamics of Interest Rates
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• 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

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Term Striature Dynamics of Interest Rates
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• 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.

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Term Striature Dynamics of Interest Rates
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• 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 (?). Who
  determines ?? The market.
• We must get price information from the market in
  order to estimate parameters.

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Term Striature Dynamics of Interest Rates
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• Inverting the Yield Curve
• Q-dynamics with a parameter vector a
• Theoretical term structure
• Observed term structure
• Want A model such that theoretical prices fit
  the observed prices of today, i.e. choose
  parameter vector a such that
• Number of equations 8 (one for each T).
• Number of unknowns dim(a)

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Term Striature Dynamics of Interest Rates
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• GMM Estimation
• Suppose we have a set of observations of r, whose
  evolution depends upon a set of parameters a of
  dimension k
• It will possible to find functions fi(r(t) a) ,
  i1,.,m, m k, s.t.
• 
  Efi(r(t) a) 0.
• The GMM estimates a of a are those values of a
  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 a to
  be
• arg mina
  lt M f, f gt
• M being is a positive definite matrix.
• This is the GMM estimate contingent upon M and f.
  

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• 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(ti1,ri1ti,ria)
• The process is assumed to be Markov.
• The Likelihood function is L(a) ?i
  p(ti1,ri1ti,ria)
• An estimate of a 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.

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• 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 ? with this
  structure is that the functional form for g(.) is
  often unknown or intractable.