*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

^{a}

*• 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

^{a}

*• 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} *≡*

*σ*^{2}[^{1−e}* ^{−2β(T −t)}*]

*2β* *, if β 6= 0*
*σ*^{2}*(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

^{a}

*• 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 < σ*^{2}.

*• 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/(σ*^{2}*x) − (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) ≡ x*^{2}*σ*^{2}*/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 = 2*p

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

*∆t = 4.4472135955.*

*• Use the inverse transformation to obtain the short rate*
*x*^{2} *× (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

^{a}

*• 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

^{a}

*• 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 c*_{1}*, c*_{2}*, . . . , c** _{n}* at times

*t*

_{1}

*, t*

_{2}

*, . . . , t*

_{n}*, where t*

_{i}*> T for all i.*

*• The payoff for the option is*
max

Ã * _{n}*
X

*i=1*

*c*_{i}*P (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*_{i}*P (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*_{i}*X** _{i}*, the option’s payoff equals
X

*n*

*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

^{a}

*• 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

^{a}

*• 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).

*%*
*r*_{3}

*%* *&*

*r*_{2}

*%* *&* *%*

*r*_{1} *r*_{3} *+ v*_{3}

*&* *%* *&*

*r*_{2} *+ v*_{2}

*&* *%*

*r*_{3} *+ 2v*_{3}

### 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*^{−(v}^{2}^{+···+v}^{n}^{)}

(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) +* *v*_{2} *+ · · · + 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) v*_{2}*.* (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 κ*_{2}*, κ*_{3}*, . . . .*
*– It is independent of the r** _{i}*.

*• It is easy to compute the v** _{i}*s from the volatility
structure, and vice versa.

*• The r** _{i}*s 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[ v*_{2} *+ · · · + v** _{n}* ]

*1 + exp[ v*

_{2}

*+ · · · + v*

*]*

_{n}*,*

(102)

*P*_{u}*(t + 1, t + n) =* *P (t, t + n)*
*P (t, t + 1)*

2

*1 + exp[ v*_{2} *+ · · · + v** _{n}* ]

*.*

(102* ^{0}*)

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)*

*2δ*^{n−1}*1 + δ*^{n−1}*,*
*P*_{u}*(t + 1, t + n)* = *P (t, t + n)*

*P (t, t + 1)*

2

*1 + δ*^{n−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*

µ*P (t + 1, t + n)*
*P (t, t + n)*

¶
*.*

*• Its value is either ln* ^{P}^{d}_{P (t,t+n)}* ^{(t+1,t+n)}* or ln

^{P}^{u}

_{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)*^{2}*σ*^{2}*.*

### 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) σ*^{2} (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

^{a}

*• 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).

*r*_{4}

*%*
*r*_{3}

*%* *&*

*r*_{2} *r*_{4}*v*_{4}

*%* *&* *%*

*r*_{1} *r*_{3}*v*_{3}

*&* *%* *&*

*r*_{2}*v*_{2} *r*_{4}*v*_{4}^{2}

*&* *%*

*r*_{3}*v*_{3}^{2}

*&*

### 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*
*r*_{1}*, v*_{1}*, r*_{2}*, v*_{2}*, . . . , 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 + r*_{1}) = 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 κ*^{2} denoting the
variance, or

*κ** _{i}* = 1
2 ln

Ã*P*_{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 P*_{1}*, P*_{2}*, . . . , P** _{i}*,

*corresponding to rates r, rv, . . . , rv*

*, respectively.*

^{i−1}*• One dollar at time i has a present value of*
*f (r, v) ≡* *P*_{1}

*1 + r* + *P*_{2}

*1 + rv* + *P*_{3}

*1 + rv*^{2} *+ · · · +* *P*_{i}

*1 + rv*^{i−1}*.*

### The BDT Model: Calibration (continued)

*• The yield volatility is*

*g(r, v) ≡* 1
2 ln

³ *P*_{u,1}

*1+rv* + _{1+rv}^{P}^{u,2}_{2} *+ · · · +* _{1+rv}^{P}^{u,i−1}_{i−1}

´_{−1/(i−1)}

*− 1*

³*P*_{d,1}

*1+r* + _{1+rv}^{P}^{d,2}*+ · · · +* _{1+rv}^{P}^{d,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 r*_{2}*v*_{2} 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 r*_{2} 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(n*^{2})-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) +* *σ*^{0}*(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.