*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

*• We now survey equilibrium models.*

*• Recall that the spot rates satisfy*

*r(t, T ) =* *−ln P (t, T )*
*T* *− t*
by Eq. (129) on p. 1005.

*• Hence 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. (78) on p. 585.

aVasicek (1977). Vasicek co-founded KMV, which was sold to Moody’s for USD$210 million in 2002.

### 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)**,* (142)
where

*A(t, T ) =*

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

exp

(*B(t,T )−T +t)(β2μ−σ2/2)*

*β2* *−* ^{σ2B(t,T )2}_{4}_{β}

*if β = 0,*

exp

*σ2(T −t)3*
6

*if β = 0.*

and

*B(t, T ) =*

⎧⎨

⎩

*1−e*^{−β(T −t)}

*β* *if β* *= 0,*
*T* *− t* *if β = 0.*

### The Vasicek Model (concluded)

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

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

*• Even if β = 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* =^{Δ} 1
*σ** _{v}* ln

*P (t, s)*
*P (t, T ) X*

+ *σ** _{v}*
2

*,*

*σ*

_{v}*≡ v(t, T ) B(T, s),*

*v(t, T )*^{2} =^{Δ}

⎧⎨

⎩

*σ*^{2}[*1−e** ^{−2β(T −t)}*]

*2β* *,* *if β* *= 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 Δt* *= T /n and*^{Δ}

*p(r)* =^{Δ} 1

2 + *β(μ* *− r)* *√*
*Δt*

*2σ* *.*

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

*r(k + 1) = r(k) + σ√*

*Δt ξ(k), 0* *≤ k < n.*

aNelson & 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.*

### The Cox-Ingersoll-Ross Model

^{a}

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

*dr = β(μ* *− r) dt + σ√*

*r dW.* (143)

*• 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.*

aCox, Ingersoll, & Ross (1985).

### Binomial CIR

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

*• Divide it into n periods of duration Δt* *= T /n.*^{Δ}

*• 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^{Δ} *√*

*r/σ.*

*• By Ito’s lemma (p. 562),*

*dx = m(x) dt + dW,*
where

*m(x)* *= 2βμ/(σ*^{Δ} ^{2}*x)* *− (βx/2) − 1/(2x).*

*• This new process has a constant volatility.*

*• Thus its binomial tree combines.*

### 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)* =^{Δ} *x*^{2}*σ*^{2}
4
(see p. 1054).

*• But when x ≈ 0 (so r ≈ 0), the moments may not be*
matched well.^{a}

aNawalkha & Beliaeva (2007).

*x + 2**√*

Δ*t* *f(x + 2**√*

Δ*t)*

*x +* *√*

Δ*t* *f(x +* *√*

Δ*t)*

*x* *x* *f(x)* *f(x)*

*x −**√*

Δ*t* *f(x −**√*

Δ*t)*

*x − 2**√*

Δ*t* *f(x − 2**√*

Δ*t)*

### Binomial CIR (continued)

*• The probability of an up move at each node r is*
*p(r)* =^{Δ} *β(μ* *− r) Δt + r − r*^{−}

*r*^{+} *− r*^{−}*.*
**– r**^{+ Δ}*= f (x +* *√*

*Δt) denotes the result of an up move*
*from r.*

**– r**^{− Δ}*= f (x* *−* *√*

*Δt) 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.

### Binomial CIR (concluded)

*• It can be shown that*
*p(r) =*

*βμ* *−* *σ*^{2}
4

*Δt*

*r* *− B√*

*rΔt + C,*
*for some B* *≥ 0 and C > 0.*^{a}

*• If βμ − (σ*^{2}*/4)* *≥ 0, the up-move probability p(r)*
*decreases if and only if short rate r increases.*

*• Even if βμ − (σ*^{2}*/4) < 0, p(r) tends to decrease as r*
*increases and decrease as r declines.*

*• This phenomenon agrees with mean reversion.*

aThanks to a lively class discussion on May 28, 2014.

### 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 Δt = 0.2 (year) for the binomial*
approximation.

*• See p. 1058(a) for the resulting binomial short rate tree*
with the up-move probabilities in parentheses.

(0.472049150276)0.04

0.05988854382 (0.44081188025)

0.03155572809 (0.489789553691)

0.02411145618 (0.50975924867)

0.0713328157297 (0.426604457655)

0.08377708764

0.01222291236 0.01766718427

(0.533083330907) (0.472049150276)0.04

0.0494442719102

(0.455865503068) 0.0494442719102 (0.455865503068)

0.03155572809 (0.489789553691)

0.05988854382

0.04

0.02411145618

=

0.992031914837 0.984128889634 0.976293244408 0.968526861261 0.960831229521

0.992031914837 0.984128889634

0.976293244408 0.992031914837 0.990159879565

0.980492588317 0.970995502019 0.961665706744

0.993708727831 0.987391576942 0.981054487259 0.974702907786

0.988093738447 0.976486896485 0.965170249273

0.990159879565 0.980492588317

0.995189317343 0.990276851751 0.985271123591

0.993708727831 0.987391576942 0.98583472203 0.972116454453

0.996472798388 0.992781347933

0.983384173756

0.988093738447

0.995189317343

### Numerical Examples (concluded)

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

*• Since the root has x = 2*

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

*Δt = 4.4472135955.*

*• Use the inverse transformation to obtain the short rate*
*x*^{2} *× (0.1)*^{2}

4 *≈ 0.0494442719102.*

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

*• Convergence is quite good (see p. 369 of the textbook).*

### A General Method for Constructing Binomial Models

^{a}

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

*• Need to make sure the binomial model’s drift and*
diﬀusion converge to the above process.

*• Set the probability of an up move to*
*α(y, t) Δt + y* *− y*_{d}

*y*_{u} *− y*_{d} *.*

*• Here y*_{u} *= y + σ(y, t)*^{Δ} *√*

*Δt and y*_{d} *= y*^{Δ} *− σ(y, t)√*
*Δt*
*represent the two rates that follow the current rate y.*

aNelson & Ramaswamy (1990).

### A General Method (continued)

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

*Δt .*

*• But the binomial tree may not combine as*
*σ(y, t)√*

*Δt* *− σ(y*_{u}*, t + Δt)√*
*Δt*

*= −σ(y, t)√*

*Δt + σ(y*_{d}*, t + Δt)√*
*Δt*
in general.

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

### A General Method (continued)

*• To achieve this, deﬁne the transformation*
*x(y, t)* =^{Δ}

_{y}

*σ(z, t)*^{−1}*dz.*

*• Then x follows*

*dx = m(y, t) dt + dW*
*for some m(y, t).*^{a}

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

aSee Exercise 25.2.13 of the textbook.

### A General Method (concluded)

*• The transformation is unique.*^{a}

*• The probability of an up move remains*

*α(y(x, t), t) Δt + 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)* =^{Δ} *y(x +* *√*

*Δt, t + Δt),*
*y*_{d}*(x, t)* =^{Δ} *y(x* *−* *√*

*Δt, t + Δt).*

aChiu (R98723059) (2012).

### Examples

*• The transformation is*
_{r}

*(σ√*

*z)*^{−1}*dz =* 2*√*
*r*
*σ*
for the CIR model.

*• The transformation is*
_{S}

*(σz)*^{−1}*dz =* *ln S*
*σ*
for the Black-Scholes model.

*• The familiar BOPM and CRR in fact discretize ln S not*
*S.*

### 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- or three-factor ones.

### Options on Coupon Bonds

^{a}

*• Assume the market discount function P is a*

*monotonically decreasing function of the short rate r.*

**– Such as a one-factor short rate model.**

*• 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.*

aJamshidian (1989).

### Options on Coupon Bonds (continued)

*• 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

_{n}

*i=1*

*c*_{i}*P (r(T ), T, t** _{i}*)

*− X, 0*

*.*

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

### Options on Coupon Bonds (continued)

*• This r** ^{∗}* can be obtained by solving

*X =*

*n*
*i=1*

*c*_{i}*P (r, T, t** _{i}*)

*numerically for r.*

*• Let*

*X*_{i}*= P (r*^{Δ} ^{∗}*, T, t*_{i}*),*

*the value at time T of a zero-coupon bond with par*
*value $1 and maturing at time t*_{i}*if r(T ) = r** ^{∗}*.

*• Note that P (r, T, t**i*) *≥ X**i* *if and only if r* *≤ r** ^{∗}*.

### Options on Coupon Bonds (concluded)

*• As X =*

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

_{n}

*i=1*

*c*_{i}*P (r(T ), T, t** _{i}*)

*−*

_{n}

*i=1*

*c*_{i}*X*_{i}

*, 0*

=

*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.

*• Why can’t we do the same thing for Asian options?*^{a}

aContributed by Mr. Yang, Jui-Chung (D97723002) on May 20, 2009.

*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 & Lee (1986). Thomas Lee is a “billionaire founder” of Thomas
*H. Lee Partners LP, according to Bloomberg on May 26, 2012.*

### 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 & 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}^{)}

(144) (see p. 376 of the textbook).

### The Ho-Lee Model (continued)

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

*y*_{d}*(n)* =^{Δ} *−ln P*_{d}*(t + 1, t + n)*
*n* *− 1*

*y*_{u}*(n)* =^{Δ} *−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* Δ

=

*py*u*(n)*^{2} *+ (1 − p) y*d*(n)*^{2} *− [ py*u*(n) + (1 − p) y*d*(n) ]*^{2}

=

*p(1 − p) (y*u*(n) − y*d*(n))*

=

*p(1 − p) v*^{2} *+ · · · + v**n*

*n − 1* *.*

### The Ho-Lee Model (concluded)

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

*σ =*

*p(1* *− p) v*_{2}*.* (145)

*• 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 (122) on p. 946.

### The Ho-Lee Model: Volatility Term Structure

*• The volatility term structure is composed of*
*κ*_{2}*, κ*_{3}*, . . . .*

**– It is independent of**

*r*_{2}*, r*_{3}*, . . . .*

*• It is easy to compute the v** _{i}*s from the volatility
structure, and vice versa (review p. 1080).

*• 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. (144) on p. 1079 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* ]*,*
*P*_{u}*(t + 1, t + n) =* *P (t, t + n)*

*P (t, t + 1)*

2

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

aIn the limit, only the volatility matters; the ﬁrst formula is similar to multiple logistic regression.

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

*• The bond price tree combines.*^{a}

*• Suppose all v**i* *equal some constant v and δ* *= e*^{Δ} ^{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 = v/2 by Eq. (145) on p. 1081.*

*• Price derivatives by taking expectations under the*
risk-neutral probability.

aSee Exercise 26.2.3 of the textbook.

### Calibration

*• The Ho-Lee model can be calibrated in O(n*^{2}) time using
state prices.

*• But it can actually be calibrated in O(n) time.*

**– Derive the v*** _{i}*’s in linear time.

**– Derive the r***i*’s in linear time.^{a}

aSee Programming Assignment 26.2.6 of the textbook.

### 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^{Δ}

*P (t + 1, t + n)*
*P (t, t + n)*

*.*

*• Its two possible value are*
ln *P*_{d}*(t + 1, t + n)*

*P (t, t + n)* and ln *P*_{u}*(t + 1, t + n)*
*P (t, 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*^{a}
*(n* *− 1)(m − 1) σ*^{2}*.*

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

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

aSee Exercise 26.2.7 of the textbook.

### The Ho-Lee Model: Short Rate Process

*• The continuous-time limit of the Ho-Lee model is*^{a}
*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,

*dr = θ(t) dt + σ(t) dW.*

*• This corresponds to the discrete-time model in which v**i*

are not all identical.

aSee Exercise 26.2.10 of the textbook.

### 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. 942ﬀ.^{b}

*• 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, & Toy (BDT) (1990), but essentially ﬁnished in 1986 according to Mehrling (2005).

bRepeated on next page.

*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}

*r*_{4}*v*_{4}^{3}

### The Black-Derman-Toy Model (concluded)

*• Our earlier binomial interest rate tree, in contrast,*
*assumes v** _{i}* are given a priori.

*• 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.

*• P*_{d} *is the price of the i-period zero-coupon bond one*
period from now if the short rate makes a down move.

### The BDT Model: Volatility Structure (concluded)

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

*y*_{u} =^{Δ} *P*_{u}^{−1/(i−1)}*− 1,*
*y*_{d} =^{Δ} *P*_{d}^{−1/(i−1)}*− 1.*

*• The yield volatility is deﬁned as*
*κ** _{i}* =

^{Δ}

*ln(y*

_{u}

*/y*

_{d})

2 (recall Eq. (128) on p. 992).

### 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}*.* (146)

*• 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 1 is uncertain.

*• Recall that y*_{u} *and y*_{d} represent the spot rates at the
up node and the down node, respectively (p. 1096).

*• With κ*^{2}* _{i}* denoting their variance, we have

*κ** _{i}* = 1
2 ln

*P*_{u}^{−1/(i−1)}*− 1*
*P*_{d}^{−1/(i−1)}*− 1*

*.* (147)

*• Solving Eqs. (146)–(147) for r and v with backward*
*induction takes O(i*^{2}) time.

**– That leads to a cubic-time algorithm.**

### The BDT Model: Calibration (continued)

*• We next 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.^{b}

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

aW. J. Chen (R84526007) & Lyuu (1997); Lyuu (1999).

bReview p. 969(a).

### 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}*.*

*• They correspond to rates*

*r, rv, . . . , rv*^{i−1}*for period i, respectively.*

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

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

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

*1 + rv*^{2} + *· · · +* *P*_{i}

*1 + rv*^{i−1}*.*

### The BDT Model: Calibration (continued)

*• By Eq. (147) on p. 1099, the yield volatility is*

*g(r, v)* ^{Δ}= 1
2 ln

⎛

⎜⎝

_{P}

u*,1*

*1+rv* + ^{P}^{u}^{,2}

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

*− 1*

_{P}

d*,1*

*1+r* + ^{P}^{d}^{,2}

*1+rv* + *· · · +* _{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. 1093).

*• 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. 1093).

### The BDT Model: Calibration (concluded)

*• Note that every node maintains three state prices:*

*P*_{i}*, P*_{u,i}*, P** _{d,i}*.

*• Now solve*

*f (r, v)* = 1

*(1 + y)*^{i}*,*
*g(r, v)* = *κ*_{i}*,*

*for r = r*_{i}*and v = v** _{i}*.

*• This O(n*^{2})-time algorithm appears on p. 382 of the
textbook.

### Calibrating the BDT Model with the Diﬀerential Tree (in seconds)

^{a}

Number Running Number Running Number Running

of years time of years time of years time

3000 398.880 39000 8562.640 75000 26182.080 6000 1697.680 42000 9579.780 78000 28138.140 9000 2539.040 45000 10785.850 81000 30230.260 12000 2803.890 48000 11905.290 84000 32317.050 15000 3149.330 51000 13199.470 87000 34487.320 18000 3549.100 54000 14411.790 90000 36795.430 21000 3990.050 57000 15932.370 120000 63767.690 24000 4470.320 60000 17360.670 150000 98339.710 27000 5211.830 63000 19037.910 180000 140484.180 30000 5944.330 66000 20751.100 210000 190557.420 33000 6639.480 69000 22435.050 240000 249138.210 36000 7611.630 72000 24292.740 270000 313480.390

75MHz Sun SPARCstation 20, one period per year.

aLyuu (1999).

### The BDT Model: Continuous-Time Limit

*• The continuous-time limit of the BDT model is*^{a}
*d ln r =*

*θ(t) +* *σ*^{}*(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.

**– That makes σ**^{}*(t) < 0.*

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

aJamshidian (1991).

### The Black-Karasinski Model

^{a}

*• The BK model stipulates that the short rate follows*
*d ln r = κ(t)(θ(t)* *− ln r) dt + σ(t) dW.*

*• This explicitly mean-reverting model depends on time*
*through κ(· ), θ( · ), and σ( · ).*

*• The BK model hence has one more degree of freedom*
than the BDT model.

*• The speed of mean reversion κ(t) and the short rate*
*volatility σ(t) are independent.*

aBlack & Karasinski (1991).

### The Black-Karasinski Model: Discrete Time

*• The discrete-time version of the BK model has the same*
representation as the BDT model.

*• To maintain a combining binomial tree, however,*
requires some manipulations.

*• The next plot illustrates the ideas in which*
*t*_{2} =^{Δ} *t*_{1} *+ Δt*_{1}*,*

*t*_{3} =^{Δ} *t*_{2} *+ Δt*_{2}*.*

*ln r*d*(t*2)

*ln r(t*1) *ln r*du*(t*3*) = ln r*ud*(t*3)

*ln r*u*(t*2)

### The Black-Karasinski Model: Discrete Time (continued)

*• Note that*

*ln r*_{d}*(t*_{2}) = *ln r(t*_{1}*) + κ(t*_{1}*)(θ(t*_{1}) *− ln r(t*1*)) Δt*_{1} *− σ(t*1)

*Δt*_{1} *,*
*ln r*_{u}*(t*_{2}) = *ln r(t*_{1}*) + κ(t*_{1}*)(θ(t*_{1}) *− ln r(t*1*)) Δt*_{1} *+ σ(t*_{1})

*Δt*_{1} *.*

*• To make sure an up move followed by a down move*
coincides with a down move followed by an up move,

*ln r*d*(t*2*) + κ(t*2*)(θ(t*2*) − ln r*d*(t*2*)) Δt*2 *+ σ(t*2)*√*

*Δt*2 *,*

*= ln r*u*(t*2*) + κ(t*2*)(θ(t*2*) − ln r*u*(t*2*)) Δt*2 *− σ(t*2)*√*

*Δt*2 *.*

### The Black-Karasinski Model: Discrete Time (continued)

*• They imply*

*κ(t*_{2}) = 1 *− (σ(t*2*)/σ(t*_{1}))

*Δt*_{2}*/Δt*_{1}

*Δt*_{2} *.*

(148)

*• So from Δt*_{1}*, we can calculate the Δt*_{2} that satisﬁes the
combining condition and then iterate.

**– t**_{0} *→ Δt*_{1} *→ t*_{1} *→ Δt*_{2} *→ t*_{2} *→ Δt*_{3} *→ · · · → T*
(roughly).^{a}

a*As κ(t), θ(t), σ(t) are independent of r, the Δt** _{i}*s will not depend on

*r.*

### The Black-Karasinski Model: Discrete Time (concluded)

*• Unequal durations Δt**i* are often necessary to ensure a
combining tree.^{a}

aAmin (1991); C. Chen (R98922127) (2011); Lok (D99922028) & Lyuu (2016, 2017).

### Problems with Lognormal Models in General

*• Lognormal models such as BDT and BK share the*
*problem that E*^{π}*[ M (t) ] =* *∞ for any ﬁnite t if they*
model the continuously compounded rate.^{a}

*• So periodically compounded rates should be modeled.*^{b}

*• Another issue is computational.*

*• Lognormal models usually do not admit of analytical*
solutions to even basic ﬁxed-income securities.

*• As a result, to price short-dated derivatives on long-term*
bonds, the tree has to be built over the life of the

underlying asset instead of the life of the derivative.

aHogan & Weintraub (1993).

bSandmann & Sondermann (1993).

### Problems with Lognormal Models in General (concluded)

*• This problem can be somewhat mitigated by adopting*
diﬀerent time steps.^{a}

**– Use a ﬁne time step up to the maturity of the**
short-dated derivative.

**– Use a coarse time step beyond the maturity.**

*• A down side of this procedure is that it has to be*
tailor-made for each derivative.

*• Finally, empirically, interest rates do not follow the*
lognormal distribution.

aHull & White (1993).

### The Extended Vasicek Model

^{a}

*• Hull and White proposed models that extend the*
Vasicek model and the CIR model.

*• They are called the extended Vasicek model and the*
extended CIR model.

*• The extended Vasicek model adds time dependence to*
the original Vasicek model,

*dr = (θ(t)* *− a(t) r) dt + σ(t) dW.*

*• Like the Ho-Lee model, this is a normal model.*

*• The inclusion of θ(t) allows for an exact ﬁt to the*
current spot rate curve.

aHull & White (1990).

### The Extended Vasicek Model (concluded)

*• Function σ(t) deﬁnes the short rate volatility, and a(t)*
determines the shape of the volatility structure.

*• Many European-style securities can be evaluated*
analytically.

*• Eﬃcient numerical procedures can be developed for*
American-style securities.

### The Hull-White Model

*• The Hull-White model is the following special case,*
*dr = (θ(t)* *− ar) dt + σ dW.*

*• When the current term structure is matched,*^{a}
*θ(t) =* *∂f (0, t)*

*∂t* *+ af (0, t) +* *σ*^{2}
*2a*

1 *− e*^{−2at}*.*

aHull & White (1993).

### The Extended CIR Model

*• In the extended CIR model the short rate follows*
*dr = (θ(t)* *− a(t) r) dt + σ(t)√*

*r dW.*

*• The functions θ(t), a(t), and σ(t) are implied from*
market observables.

*• With constant parameters, there exist analytical*
solutions to a small set of interest rate-sensitive
securities.

### The Hull-White Model: Calibration

^{a}

*• We describe a trinomial forward induction scheme to*
*calibrate the Hull-White model given a and σ.*

*• As with the Ho-Lee model, the set of achievable short*
rates is evenly spaced.

*• Let r*0 be the annualized, continuously compounded
short rate at time zero.

*• Every short rate on the tree takes on a value*
*r*_{0} *+ jΔr*

*for some integer j.*

aHull & White (1993).

### The Hull-White Model: Calibration (continued)

*• Time increments on the tree are also equally spaced at*
*Δt apart.*

*• Hence nodes are located at times iΔt for i = 0, 1, 2, . . . .*

*• We shall refer to the node on the tree with*
*t** _{i}* =

^{Δ}

*iΔt,*

*r** _{j}* =

^{Δ}

*r*

_{0}

*+ jΔr,*

*as the (i, j) node.*

*• The short rate at node (i, j), which equals r** _{j}*, is

*eﬀective for the time period [ t*

_{i}*, t*

*).*

_{i+1}### The Hull-White Model: Calibration (continued)

*• Use*

*μ*_{i,j}*= θ(t*^{Δ} * _{i}*)

*− ar*

*j*(149) to denote the drift rate

^{a}of the short rate as seen from

*node (i, j).*

*• The three distinct possibilities for node (i, j) with three*
branches incident from it are displayed on p. 1121.

*• The middle branch may be an increase of Δr, no*
*change, or a decrease of Δr.*

aOr, the annualized expected change.

### The Hull-White Model: Calibration (continued)

*(i, j)*

*(i + 1, j + 2)*

**(i + 1, j + 1)*

-_{(i + 1, j)}

*(i, j)*

**(i + 1, j + 1)*

-* _{(i + 1, j)}*
j

*(i + 1, j − 1)*

*(i, j)* -* _{(i + 1, j)}*
j

*(i + 1, j − 1)*

R*(i + 1, j − 2)*

### The Hull-White Model: Calibration (continued)

*• The upper and the lower branches bracket the middle*
branch.

*• Deﬁne*

*p1(i, j)* Δ

= the probability of following the upper branch from node (*i, j),*
*p2(i, j)* Δ

= the probability of following the middle branch from node (*i, j),*
*p3(i, j)* Δ

= the probability of following the lower branch from node (*i, j).*

*• The root of the tree is set to the current short rate r*_{0}.

*• Inductively, the drift μ**i,j* *at node (i, j) is a function of*
*(the still unknown) θ(t** _{i}*).

**– It describes the expected change from node (i, j).**

### The Hull-White Model: Calibration (continued)

*• Once θ(t*_{i}*) is available, μ** _{i,j}* can be derived via
Eq. (149) on p. 1120.

*• This in turn determines the branching scheme at every*
*node (i, j) for each j, as we will see shortly.*

*• The value of θ(t**i*) must thus be made consistent with
*the spot rate r(0, t** _{i+2}*).

^{a}

a*Not r(0, t** _{i+1}*)!

### The Hull-White Model: Calibration (continued)

*• The branches emanating from node (i, j) with their*
probabilities^{a} *must be chosen to be consistent with μ*_{i,j}*and σ.*

*• This is done by letting the middle node be as closest to*
*the current short rate r*_{j}*plus the drift μ*_{i,j}*Δt.*^{b}

a*p*_{1}*(i, j), p*_{2}*(i, j), and p*_{3}*(i, j).*

bA precursor of Lyuu and C. N. Wu’s (R90723065) (2003, 2005) mean- tracking idea, which is the precursor of the binomial-trinomial tree of Dai (B82506025, R86526008, D8852600) & Lyuu (2006, 2008, 2010).

### The Hull-White Model: Calibration (continued)

*• Let k be the number among { j − 1, j, j + 1 } that*

*makes the short rate reached by the middle branch, r** _{k}*,
closest to

*r*_{j}*+ μ*_{i,j}*Δt.*

**– But note that μ**_{i,j}*is still not computed yet.*

*• Then the three nodes following node (i, j) are nodes*
*(i + 1, k + 1), (i + 1, k), (i + 1, k* *− 1).*

*• See p. 1126 for a possible geometry.*

*• The resulting tree combines because of the constant*
*jump sizes to reach k from j.*

* - j

*(0, 0)*

* - j

*(1, 1)*

* - j

*(1, 0)*

*

*(1, −1)* -

* - j

* - j

* - j

* - j

- j R

* - j

* - j

* - j

* - j

* --

*Δt*

6

?^{Δr}

### The Hull-White Model: Calibration (continued)

*• The probabilities for moving along these branches are*
*functions of μ*_{i,j}*, σ, j, and k:*

*p*_{1}*(i, j) =* *σ*^{2}*Δt + η*^{2}

*2(Δr)*^{2} + *η*

*2Δr* (150)

*p*_{2}*(i, j) = 1* *−* *σ*^{2}*Δt + η*^{2}

*(Δr)*^{2} (150* ^{}*)

*p*_{3}*(i, j) =* *σ*^{2}*Δt + η*^{2}

*2(Δr)*^{2} *−* *η*

*2Δr* (150* ^{}*)
where

*η* *= μ*^{Δ} _{i,j}*Δt + (j* *− k) Δr.*

### The Hull-White Model: Calibration (continued)

*• As trinomial tree algorithms are but explicit methods in*
disguise,^{a} *certain relations must hold for Δr and Δt to*
guarantee stability.

*• It can be shown that their values must satisfy*
*σ√*

*3Δt*

2 *≤ Δr ≤ 2σ√*
*Δt*

for the probabilities to lie between zero and one.

**– For example, Δr can be set to σ**√

*3Δt .*^{b}

*• Now it only remains to determine θ(t** _{i}*).

aRecall p. 777.

bHull & White (1988).

### The Hull-White Model: Calibration (continued)

*• At this point at time t** _{i}*,

*r(0, t*_{1}*), r(0, t*_{2}*), . . . , r(0, t** _{i+1}*)
have already been matched.

*• Let Q(i, j) be the state price at node (i, j).*

*• By construction, the state prices Q(i, j) for all j are*
known by now.

*• We begin with state price Q(0, 0) = 1.*

### The Hull-White Model: Calibration (continued)

*• Let ˆr(i) refer to the short rate value at time t** _{i}*.

*• The value at time zero of a zero-coupon bond maturing*
*at time t** _{i+2}* is then

*e*^{−r(0,t}^{i+2}^{)(i+2) Δt}

=

*j*

*Q(i, j) e*^{−r}^{j}^{Δt}*E*^{π}

*e**−ˆr(i+1) Δt** ˆr(i) = r**j*

*.(151)*

*• The right-hand side represents the value of $1 obtained*
*by holding a zero-coupon bond until time t** _{i+1}* and then
reinvesting the proceeds at that time at the prevailing
short rate ˆ

*r(i + 1), which is stochastic.*