Equilibrium Term Structure Models

Equilibrium Term Structure Models

8. What’s your problem? Any moron can understand bond pricing models.

— Top Ten Lies Finance Professors Tell Their Students

Introduction

• This chapter surveys equilibrium models.

• Since the spot rates satisfy

r(t, T ) = −ln P (t, T ) T − t ,

the discount function P (t, T ) suffices to establish the spot rate curve.

• All models to follow are short rate models.

• Unless stated otherwise, the processes are risk-neutral.

The Vasicek Model

• The short rate follows

dr = β(µ − r) dt + σ dW.

• The short rate is pulled to the long-term mean level µ at rate β.

• Superimposed on this “pull” is a normally distributed stochastic term σ dW .

• Since the process is an Ornstein-Uhlenbeck process, E[ r(T ) | r(t) = r ] = µ + (r − µ) e^{−β(T −t)} from Eq. (53) on p. 494.

aVasicek (1977).

The Vasicek Model (continued)

• The price of a zero-coupon bond paying one dollar at maturity can be shown to be

P (t, T ) = A(t, T ) e−B(t,T ) r(t), (97) where

A(t, T ) =













exp

·

(B(t,T )−T +t)(β2µ−σ2/2)

β2 − ^{σ2B(t,T )2}_4β

¸

if β 6= 0,

exp

·

σ2(T −t)3 6

¸

if β = 0.

and

B(t, T ) =

 ^{1−e−β(T −t)}

β if β 6= 0,

The Vasicek Model (concluded)

• If β = 0, then P goes to infinity as T → ∞.

• Sensibly, P goes to zero as T → ∞ if β 6= 0.

• Even if β 6= 0, P may exceed one for a finite T .

• The spot rate volatility structure is the curve (∂r(t, T )/∂r) σ = σB(t, T )/(T − t).

• When β > 0, the curve tends to decline with maturity.

• The speed of mean reversion, β, controls the shape of the curve.

• Indeed, higher β leads to greater attenuation of volatility with maturity.

2 4 6 8 10 Term 0.05

0.1 0.15 0.2

Yield

humped

inverted

normal

The Vasicek Model: Options on Zeros

• Consider a European call with strike price X expiring at time T on a zero-coupon bond with par value $1 and maturing at time s > T .

• Its price is given by

P (t, s) N (x) − XP (t, T ) N (x − σ_v).

aJamshidian (1989).

The Vasicek Model: Options on Zeros (concluded)

• Above

x ≡ 1

σ_v ln

µ P (t, s) P (t, T ) X

¶

+ σ_v 2 , σ_v ≡ v(t, T ) B(T, s),

v(t, T )² ≡





σ²[^{1−e−2β(T −t)}]

2β , if β 6= 0 σ²(T − t), if β = 0 .

• By the put-call parity, the price of a European put is XP (t, T ) N (−x + σ_v) − P (t, s) N (−x).

Binomial Vasicek

• Consider a binomial model for the short rate in the time interval [ 0, T ] divided into n identical pieces.

• Let ∆t ≡ T /n and

p(r) ≡ 1

2 + β(µ − r)√

∆t

2σ .

• The following binomial model converges to the Vasicek model,^a

r(k + 1) = r(k) + σ√

∆t ξ(k), 0 ≤ k < n.

aNelson and Ramaswamy (1990).

Binomial Vasicek (continued)

• Above, ξ(k) = ±1 with

Prob[ ξ(k) = 1 ] =









p(r(k)) if 0 ≤ p(r(k)) ≤ 1 0 if p(r(k)) < 0

1 if 1 < p(r(k))

.

• Observe that the probability of an up move, p, is a decreasing function of the interest rate r.

• This is consistent with mean reversion.

Binomial Vasicek (concluded)

• The rate is the same whether it is the result of an up move followed by a down move or a down move followed by an up move.

• The binomial tree combines.

• The key feature of the model that makes it happen is its constant volatility, σ.

• For a general process Y with nonconstant volatility, the resulting binomial tree may not combine.

The Cox-Ingersoll-Ross Model

• It is the following square-root short rate model:

dr = β(µ − r) dt + σ√

r dW. (98)

• The diffusion differs from the Vasicek model by a multiplicative factor √

r .

• The parameter β determines the speed of adjustment.

• The short rate can reach zero only if 2βµ < σ².

• See text for the bond pricing formula.

a

Binomial CIR

• We want to approximate the short rate process in the time interval [ 0, T ].

• Divide it into n periods of duration ∆t ≡ T /n.

• Assume µ, β ≥ 0.

• A direct discretization of the process is problematic because the resulting binomial tree will not combine.

Binomial CIR (continued)

• Instead, consider the transformed process x(r) ≡ 2√

r/σ.

• It follows

dx = m(x) dt + dW, where

m(x) ≡ 2βµ/(σ²x) − (βx/2) − 1/(2x).

• Since this new process has a constant volatility, its

Binomial CIR (continued)

• Construct the combining tree for r as follows.

• First, construct a tree for x.

• Then transform each node of the tree into one for r via the inverse transformation r = f (x) ≡ x²σ²/4 (p. 870).

x + 2√

∆t f (x + 2√

∆t)

% %

x +√

∆t f (x +√

∆t)

% & % &

x x f (x) f (x)

& % & %

x −√

∆t f (x −√

∆t)

& &

x − 2√

∆t f (x − 2√

∆t)

Binomial CIR (concluded)

• The probability of an up move at each node r is p(r) ≡ β(µ − r) ∆t + r − r⁻

r⁺ − r⁻ . (99)

– r⁺ ≡ f (x + √

∆t) denotes the result of an up move from r.

– r⁻ ≡ f (x − √

∆t) the result of a down move.

• Finally, set the probability p(r) to one as r goes to zero to make the probability stay between zero and one.

Numerical Examples

• Consider the process,

0.2 (0.04 − r) dt + 0.1√

r dW,

for the time interval [ 0, 1 ] given the initial rate r(0) = 0.04.

• We shall use ∆t = 0.2 (year) for the binomial approximation.

• See p. 873(a) for the resulting binomial short rate tree with the up-move probabilities in parentheses.

0.04 (0.472049150276)

0 . 0 5 9 8 8 8 5 4 3 8 2 (0.44081188025)

0.03155572809 (0.489789553691)

0.02411145618 (0.50975924867)

0.0713328157297 (0.426604457655)

0 . 0 8 3 7 7 7 0 8 7 6 4

0.01222291236 0.01766718427

(0.533083330907) 0.04

(0.472049150276) 0.0494442719102

(0.455865503068)

0.0494442719102 (0.455865503068)

0.03155572809 (0.489789553691)

0 . 0 5 9 8 8 8 5 4 3 8 2

0.04

0.02411145618

(a)

(b) 0.992031914837

0.984128889634 0 . 9 7 6 2 9 3 2 4 4 4 0 8 0.968526861261 0.960831229521

0.992031914837 0.984128889634 0 . 9 7 6 2 9 3 2 4 4 4 0 8

0.992031914837 0 . 9 9 0 1 5 9 8 7 9 5 6 5

0.980492588317 0.970995502019 0.961665706744

0 . 9 9 3 7 0 8 7 2 7 8 3 1 0.987391576942 0.981054487259 0 . 9 7 4 7 0 2 9 0 7 7 8 6

0 . 9 8 8 0 9 3 7 3 8 4 4 7 0 . 9 7 6 4 8 6 8 9 6 4 8 5 0.965170249273

0 . 9 9 0 1 5 9 8 7 9 5 6 5 0.980492588317

0.995189317343 0.990276851751 0.985271123591

0 . 9 9 3 7 0 8 7 2 7 8 3 1 0.987391576942 0 . 9 8 5 8 3 4 7 2 2 0 3 0.972116454453

0 . 9 9 6 4 7 2 7 9 8 3 8 8 0.992781347933

0 . 9 8 3 3 8 4 1 7 3 7 5 6

0 . 9 8 8 0 9 3 7 3 8 4 4 7

0.995189317343

0 . 9 9 7 5 5 8 4 0 3 0 8 6

Numerical Examples (continued)

• Consider the node which is the result of an up move from the root.

• Since the root has x = 2p

r(0)/σ = 4, this particular node’s x value equals 4 + √

∆t = 4.4472135955.

• Use the inverse transformation to obtain the short rate x² × (0.1)²/4 ≈ 0.0494442719102.

Numerical Examples (concluded)

• Once the short rates are in place, computing the probabilities is easy.

• Note that the up-move probability decreases as interest rates increase and decreases as interest rates decline.

• This phenomenon agrees with mean reversion.

• Convergence is quite good (see text).

A General Method for Constructing Binomial Models

• We are given a continuous-time process dy = α(y, t) dt + σ(y, t) dW .

• Make sure the binomial model’s drift and diffusion

converge to the above process by setting the probability of an up move to

α(y, t) ∆t + y − y_d y_u − y_d .

• Here y_u ≡ y + σ(y, t)√

∆t and y_d ≡ y − σ(y, t)√

∆t represent the two rates that follow the current rate y.

• The displacements are identical, at σ(y, t)√

∆t .

A General Method (continued)

• But the binomial tree may not combine:

σ(y, t)√

∆t − σ(y_u, t)√

∆t 6= −σ(y, t)√

∆t + σ(y_d, t)√

∆t in general.

• When σ(y, t) is a constant independent of y, equality holds and the tree combines.

• To achieve this, define the transformation x(y, t) ≡

Z _y

σ(z, t)⁻¹ dz.

• Then x follows dx = m(y, t) dt + dW for some m(y, t) (see text).

A General Method (continued)

• The key is that the diffusion term is now a constant, and the binomial tree for x combines.

• The probability of an up move remains

α(y(x, t), t) ∆t + y(x, t) − y_d(x, t) y_u(x, t) − y_d(x, t) ,

where y(x, t) is the inverse transformation of x(y, t) from x back to y.

• Note that y_u(x, t) ≡ y(x + √

∆t, t + ∆t) and y (x, t) ≡ y(x − √

∆t, t + ∆t) .

A General Method (concluded)

• The transformation is Z _r

(σ√

z)⁻¹ dz = 2√ r/σ for the CIR model.

• The transformation is Z _S

(σz)⁻¹ dz = (1/σ) ln S for the Black-Scholes model.

• The familiar binomial option pricing model in fact discretizes ln S not S.

Model Calibration

• In the time-series approach, the time series of short rates is used to estimate the parameters of the process.

• This approach may help in validating the proposed interest rate process.

• But it alone cannot be used to estimate the risk premium parameter λ.

• The model prices based on the estimated parameters may also deviate a lot from those in the market.

Model Calibration (concluded)

• The cross-sectional approach uses a cross section of bond prices observed at the same time.

• The parameters are to be such that the model prices closely match those in the market.

• After this procedure, the calibrated model can be used to price interest rate derivatives.

• Unlike the time-series approach, the cross-sectional

approach is unable to separate out the interest rate risk premium from the model parameters.

On One-Factor Short Rate Models

• By using only the short rate, they ignore other rates on the yield curve.

• Such models also restrict the volatility to be a function of interest rate levels only.

• The prices of all bonds move in the same direction at the same time (their magnitudes may differ).

• The returns on all bonds thus become highly correlated.

• In reality, there seems to be a certain amount of independence between short- and long-term rates.

On One-Factor Short Rate Models (continued)

• One-factor models therefore cannot accommodate

nondegenerate correlation structures across maturities.

• Derivatives whose values depend on the correlation structure will be mispriced.

• The calibrated models may not generate term structures as concave as the data suggest.

• The term structure empirically changes in slope and curvature as well as makes parallel moves.

• This is inconsistent with the restriction that all

segments of the term structure be perfectly correlated.

On One-Factor Short Rate Models (concluded)

• Multi-factor models lead to families of yield curves that can take a greater variety of shapes and can better

represent reality.

• But they are much harder to think about and work with.

• They also take much more computer time—the curse of dimensionality.

• These practical concerns limit the use of multifactor models to two-factor ones.

Options on Coupon Bonds

• The price of a European option on a coupon bond can be calculated from those on zero-coupon bonds.

• Consider a European call expiring at time T on a bond with par value $1.

• Let X denote the strike price.

• The bond has cash flows c₁, c₂, . . . , c_n at times t₁, t₂, . . . , t_n, where t_i > T for all i.

• The payoff for the option is max

à _n X

i=1

c_iP (r(T ), T, t_i) − X, 0

! .

aJamshidian (1989).

Options on Coupon Bonds (continued)

• At time T , there is a unique value r^∗ for r(T ) that renders the coupon bond’s price equal the strike price X.

• This r^∗ can be obtained by solving X = P_n

i=1 c_iP (r, T, t_i) numerically for r.

• The solution is also unique for one-factor models whose bond price is a monotonically decreasing function of r.

• Let X_i ≡ P (r^∗, T, t_i), the value at time T of a

zero-coupon bond with par value $1 and maturing at

Options on Coupon Bonds (concluded)

• Note that P (r(T ), T, t_i) ≥ X_i if and only if r(T ) ≤ r^∗.

• As X = P

i c_iX_i, the option’s payoff equals Xn

i=1

c_i × max(P (r(T ), T, t_i) − X_i, 0).

• Thus the call is a package of n options on the underlying zero-coupon bond.

No-Arbitrage Term Structure Models

How much of the structure of our theories really tells us about things in nature, and how much do we contribute ourselves?

— Arthur Eddington (1882–1944)

Motivations

• Recall the difficulties facing equilibrium models mentioned earlier.

– They usually require the estimation of the market price of risk.

– They cannot fit the market term structure.

– But consistency with the market is often mandatory in practice.

No-Arbitrage Models

• No-arbitrage models utilize the full information of the term structure.

• They accept the observed term structure as consistent with an unobserved and unspecified equilibrium.

• From there, arbitrage-free movements of interest rates or bond prices over time are modeled.

• By definition, the market price of risk must be reflected in the current term structure; hence the resulting

interest rate process is risk-neutral.

aHo and Lee (1986).

No-Arbitrage Models (concluded)

• No-arbitrage models can specify the dynamics of

zero-coupon bond prices, forward rates, or the short rate.

• Bond price and forward rate models are usually non-Markovian (path dependent).

• In contrast, short rate models are generally constructed to be explicitly Markovian (path independent).

• Markovian models are easier to handle computationally.

The Ho-Lee Model

• The short rates at any given time are evenly spaced.

• Let p denote the risk-neutral probability that the short rate makes an up move.

• We shall adopt continuous compounding.

aHo and Lee (1986).

% r₃

% &

r₂

% & %

r₁ r₃ + v₃

& % &

r₂ + v₂

& %

r₃ + 2v₃

The Ho-Lee Model (continued)

• The Ho-Lee model starts with zero-coupon bond prices P (t, t + 1), P (t, t + 2), . . . at time t identified with the root of the tree.

• Let the discount factors in the next period be

P_d(t + 1, t + 2), P_d(t + 1, t + 3), . . . if short rate moves down P_u(t + 1, t + 2), P_u(t + 1, t + 3), . . . if short rate moves up

• By backward induction, it is not hard to see that for n ≥ 2,

P_u(t + 1, t + n) = P_d(t + 1, t + n) e^{−(v2+···+vn)}

(100) (see text).

The Ho-Lee Model (continued)

• It is also not hard to check that the n-period zero-coupon bond has yields

y_d(n) ≡ −ln P_d(t + 1, t + n) n − 1

y_u(n) ≡ −ln P_u(t + 1, t + n)

n − 1 = y_d(n) + v₂ + · · · + v_n n − 1

• The volatility of the yield to maturity for this bond is therefore

κ_n ≡ p

py_u(n)² + (1 − p) y_d(n)² − [ py_u(n) + (1 − p) y_d(n) ]²

= p

p(1 − p) (y_u(n) − y_d(n)) p v + · · · + v

The Ho-Lee Model (concluded)

• In particular, the short rate volatility is determined by taking n = 2:

σ = p

p(1 − p) v₂. (101)

• The variance of the short rate therefore equals p(1 − p)(r_u − r_d)², where r_u and r_d are the two successor rates.^a

aContrast this with the lognormal model .

The Ho-Lee Model: Volatility Term Structure

• The volatility term structure is composed of κ₂, κ₃, . . . . – It is independent of the r_i.

• It is easy to compute the v_is from the volatility structure, and vice versa.

• The r_is can be computed by forward induction.

• The volatility structure is supplied by the market.

The Ho-Lee Model: Bond Price Process

• In a risk-neutral economy, the initial discount factors satisfy

P (t, t+n) = (pP_u(t+1, t+n)+(1−p) P_d(t+1, t+n)) P (t, t+1).

• Combine the above with Eq. (100) on p. 895 and assume p = 1/2 to obtain^a

P_d(t + 1, t + n) = P (t, t + n) P (t, t + 1)

2 × exp[ v₂ + · · · + v_n ] 1 + exp[ v₂ + · · · + v_n ],

(102)

P_u(t + 1, t + n) = P (t, t + n) P (t, t + 1)

2

1 + exp[ v₂ + · · · + v_n ].

(102⁰)

aIn the limit, only the volatility matters.

The Ho-Lee Model: Bond Price Process (concluded)

• The bond price tree combines.

• Suppose all v_i equal some constant v and δ ≡ e^v > 0.

• Then

P_d(t + 1, t + n) = P (t, t + n) P (t, t + 1)

2δⁿ⁻¹ 1 + δⁿ⁻¹, P_u(t + 1, t + n) = P (t, t + n)

P (t, t + 1)

2

1 + δⁿ⁻¹.

• Short rate volatility σ equals v/2 by Eq. (101) on p. 897.

• Price derivatives by taking expectations under the

The Ho-Lee Model: Yields and Their Covariances

• The one-period rate of return of an n-period zero-coupon bond is

r(t, t + n) ≡ ln

µP (t + 1, t + n) P (t, t + n)

¶ .

• Its value is either ln ^Pd_{P (t,t+n)}^(t+1,t+n) or ln ^Pu_{P (t,t+n)}^(t+1,t+n).

• Thus the variance of return is

Var[ r(t, t + n) ] = p(1 − p)((n − 1) v)² = (n − 1)²σ².

The Ho-Lee Model: Yields and Their Covariances (concluded)

• The covariance between r(t, t + n) and r(t, t + m) is (n − 1)(m − 1) σ² (see text).

• As a result, the correlation between any two one-period rates of return is unity.

• Strong correlation between rates is inherent in all one-factor Markovian models.

The Ho-Lee Model: Short Rate Process

• The continuous-time limit of the Ho-Lee model is dr = θ(t) dt + σ dW.

• This is Vasicek’s model with the mean-reverting drift replaced by a deterministic, time-dependent drift.

• A nonflat term structure of volatilities can be achieved if the short rate volatility is also made time varying, i.e., dr = θ(t) dt + σ(t) dW .

• This corresponds to the discrete-time model in which v_i are not all identical.

The Ho-Lee Model: Some Problems

• Future (nominal) interest rates may be negative.

• The short rate volatility is independent of the rate level.

Problems with No-Arbitrage Models in General

• Interest rate movements should reflect shifts in the model’s state variables (factors) not its parameters.

• Model parameters, such as the drift θ(t) in the

continuous-time Ho-Lee model, should be stable over time.

• But in practice, no-arbitrage models capture yield curve shifts through the recalibration of parameters.

– A new model is thus born everyday.

Problems with No-Arbitrage Models in General (concluded)

• This in effect says the model estimated at some time does not describe the term structure of interest rates and their volatilities at other times.

• Consequently, a model’s intertemporal behavior is

suspect, and using it for hedging and risk management may be unreliable.

The Black-Derman-Toy Model

• This model is extensively used by practitioners.

• The BDT short rate process is the lognormal binomial interest rate process described on pp. 747ff (repeated on next page).

• The volatility structure is given by the market.

• From it, the short rate volatilities (thus v_i) are determined together with r_i.

aBlack, Derman, and Toy (BDT) (1990).

r₄

% r₃

% &

r₂ r₄v₄

% & %

r₁ r₃v₃

& % &

r₂v₂ r₄v₄²

& %

r₃v₃²

&

The Black-Derman-Toy Model (concluded)

• Our earlier binomial interest rate tree, in contrast, assumes v_i are given a priori.

– A related model of Salomon Brothers takes v_i to be constants.

• Lognormal models preclude negative short rates.

The BDT Model: Volatility Structure

• The volatility structure defines the yield volatilities of zero-coupon bonds of various maturities.

• Let the yield volatility of the i-period zero-coupon bond be denoted by κ_i.

• P_u is the price of the i-period zero-coupon bond one period from now if the short rate makes an up move.

The BDT Model: Volatility Structure (concluded)

• P_d is the price of the i-period zero-coupon bond one period from now if the short rate makes a down move.

• Corresponding to these two prices are the following yields to maturity,

y_u ≡ P_u^−1/(i−1) − 1, y_d ≡ P_d^−1/(i−1) − 1.

• The yield volatility is defined as κ_i ≡ (1/2) ln(y_u/y_d).

The BDT Model: Calibration

• The inputs to the BDT model are riskless zero-coupon bond yields and their volatilities.

• For economy of expression, all numbers are period based.

• Suppose inductively that we have calculated r₁, v₁, r₂, v₂, . . . , r_i−1, v_i−1.

– They define the binomial tree up to period i − 1.

• We now proceed to calculate r_i and v_i to extend the tree to period i.

The BDT Model: Calibration (continued)

• Assume the price of the i-period zero can move to P_u or P_d one period from now.

• Let y denote the current i-period spot rate, which is known.

• In a risk-neutral economy, P_u + P_d

2(1 + r₁) = 1

(1 + y)ⁱ. (103)

• Obviously, P_u and P_d are functions of the unknown r_i and v_i.

The BDT Model: Calibration (continued)

• Viewed from now, the future (i − 1)-period spot rate at time one is uncertain.

• Let y_u and y_d represent the spot rates at the up node and the down node, respectively, with κ² denoting the variance, or

κ_i = 1 2 ln

ÃP_u^−1/(i−1) − 1 P_d^−1/(i−1) − 1

!

. (104)

The BDT Model: Calibration (continued)

• We will employ forward induction to derive a quadratic-time calibration algorithm.^a

• Recall that forward induction inductively figures out, by moving forward in time, how much $1 at a node

contributes to the price (review p. 772(a)).

• This number is called the state price and is the price of the claim that pays $1 at that node and zero elsewhere.

aChen and Lyuu (1997); Lyuu (1999).

The BDT Model: Calibration (continued)

• Let the unknown baseline rate for period i be r_i = r.

• Let the unknown multiplicative ratio be v_i = v.

• Let the state prices at time i − 1 be P₁, P₂, . . . , P_i, corresponding to rates r, rv, . . . , rvⁱ⁻¹, respectively.

• One dollar at time i has a present value of f (r, v) ≡ P₁

1 + r + P₂

1 + rv + P₃

1 + rv² + · · · + P_i

1 + rvⁱ⁻¹.

The BDT Model: Calibration (continued)

• The yield volatility is

g(r, v) ≡ 1 2 ln





³ P_u,1

1+rv + _1+rv^Pu,2₂ + · · · + _1+rv^Pu,i−1_i−1

´_−1/(i−1)

− 1

³P_d,1

1+r + _1+rv^Pd,2 + · · · + _1+rv^Pd,i−1_i−2

´_−1/(i−1)

− 1



 .

• Above, P_u,1, P_u,2, . . . denote the state prices at time

i − 1 of the subtree rooted at the up node (like r₂v₂ on p. 908).

• And P_d,1, P_d,2, . . . denote the state prices at time i − 1 of the subtree rooted at the down node (like r₂ on

p. 908).

The BDT Model: Calibration (concluded)

• Now solve

f (r, v) = 1

(1 + y)ⁱ and g(r, v) = κ_i for r = r_i and v = v_i.

• This O(n²)-time algorithm appears in the text.

The BDT Model: Continuous-Time Limit

• The continuous-time limit of the BDT model is d ln r = (θ(t) + σ⁰(t)

σ(t) ln r) dt + σ(t) dW.

• The short rate volatility clearly should be a declining function of time for the model to display mean reversion.

• In particular, constant volatility will not attain mean reversion.