The Vasicek Model

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

• 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}_4β



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

• 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

⎧⎨

⎩

σ²[1−e^{−2β(T −t)}]

2β , if β = 0 σ²(T − t), if β = 0

.

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

• 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 < σ².

• 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 x²σ² 4 (see p. 1054).

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

aNawalkha & Beliaeva (2007).

x + 2√

Δt f(x + 2√

Δt)

 

x + √

Δt f(x + √

Δt)

   

x x f(x) f(x)

   

x −√

Δt f(x −√

Δt)

 

x − 2√

Δt f(x − 2√

Δt)

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 − σ² 4

 Δt

r − B√

rΔt + C, for some B ≥ 0 and C > 0.^a

• If βμ − (σ²/4) ≥ 0, the up-move probability p(r) decreases if and only if short rate r increases.

• Even if βμ − (σ²/4) < 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 x² × (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

• 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

• 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 c₁, c₂, . . . , c_n at times t₁, t₂, . . . , t_n, where t_i > T for all i.

• The payoff for the option is max

 _n

i=1

c_iP (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_iP (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, ti) ≥ Xi if and only if r ≤ r^∗.

Options on Coupon Bonds (concluded)

• As X = 

i c_iX_i, the option’s payoff equals max

 _n

i=1

c_iP (r(T ), T, t_i)



−

 _n

i=1

c_iX_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

• 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

• 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).

 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)}

(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) + v₂ + · · · + vn

n − 1

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

κn Δ

= 

pyu(n)² + (1 − p) yd(n)² − [ pyu(n) + (1 − p) yd(n) ]²

= 

p(1 − p) (yu(n) − yd(n))

= 

p(1 − p) v² + · · · + vn

n − 1 .

The Ho-Lee Model (concluded)

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

σ =

p(1 − p) v₂. (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 κ₂, κ₃, . . . .

– It is independent of

r₂, r₃, . . . .

• It is easy to compute the v_is from the volatility structure, and vice versa (review p. 1080).

• 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. (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[ v2 + · · · + vn ] 1 + exp[ v₂ + · · · + vn ], P_u(t + 1, t + n) = P (t, t + n)

P (t, t + 1)

2

1 + exp[ v₂ + · · · + vn ].

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 vi equal some constant v and δ = e^{Δ v} > 0.

• Then

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

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

P (t, t + 1)

2

1 + δⁿ⁻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(n²) time using state prices.

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

– Derive the v_i’s in linear time.

– Derive the ri’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)²σ².

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) σ².

• 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 vi

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

• 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 vi) are determined together with r_i.

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

bRepeated on next page.

r₄

 r₃

 

r₂ r₄v₄

  

r₁ r₃v₃

  

r₂v₂ r₄v₄²

 

r₃v₃²



r₄v₄³

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 (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. (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 κ²_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(i²) 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 P₁, P₂, . . . , P_i.

• They correspond to rates

r, rv, . . . , rvⁱ⁻¹ for period i, respectively.

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

1 + r + P₂

1 + rv + P₃

1 + rv² + · · · + P_i

1 + rvⁱ⁻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 + ^Pu,2

1+rv² + · · · + _1+rv^Pu,i−1_i−1_−1/(i−1)

− 1

_P

d,1

1+r + ^Pd,2

1+rv + · · · + _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. 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 r₂ 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(n²)-time algorithm appears on p. 382 of the textbook.

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

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

• 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 t₂ =^Δ t₁ + Δt₁,

t₃ =^Δ t₂ + Δt₂.

 ln rd(t2)

 

ln r(t1) ln rdu(t3) = ln rud(t3)

 

ln ru(t2)



The Black-Karasinski Model: Discrete Time (continued)

• Note that

ln r_d(t₂) = ln r(t₁) + κ(t₁)(θ(t₁) − ln r(t1)) Δt₁ − σ(t1)

Δt₁ , ln r_u(t₂) = ln r(t₁) + κ(t₁)(θ(t₁) − ln r(t1)) Δt₁ + σ(t₁)

Δt₁ .

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

ln rd(t2) + κ(t2)(θ(t2) − ln rd(t2)) Δt2 + σ(t2)√

Δt2 ,

= ln ru(t2) + κ(t2)(θ(t2) − ln ru(t2)) Δt2 − σ(t2)√

Δt2 .

The Black-Karasinski Model: Discrete Time (continued)

• They imply

κ(t₂) = 1 − (σ(t2)/σ(t₁))

Δt₂/Δt₁

Δt₂ .

(148)

• So from Δt₁, we can calculate the Δt₂ that satisfies the combining condition and then iterate.

– t₀ → Δt₁ → t₁ → Δt₂ → t₂ → Δt₃ → · · · → T (roughly).^a

aAs κ(t), θ(t), σ(t) are independent of r, the Δt_is will not depend on r.

The Black-Karasinski Model: Discrete Time (concluded)

• Unequal durations Δti 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

• 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) + σ² 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

• 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 r0 be the annualized, continuously compounded short rate at time zero.

• Every short rate on the tree takes on a value r₀ + 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 r₀ + 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) − arj (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 r₀.

• Inductively, the drift μi,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(ti) must thus be made consistent with the spot rate r(0, t_i+2).^a

aNot 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,jΔt.^b

ap₁(i, j), p₂(i, j), and p₃(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,jΔt.

– 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:

p₁(i, j) = σ²Δt + η²

2(Δr)² + η

2Δr (150)

p₂(i, j) = 1 − σ²Δt + η²

(Δr)² (150^)

p₃(i, j) = σ²Δt + η²

2(Δr)² − η

2Δr (150^) where

η = μ^Δ _i,jΔt + (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, t₁), r(0, t₂), . . . , 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,ti+2)(i+2) Δt}

=

j

Q(i, j) e^−rjΔt E^π



e−ˆr(i+1) Δt ˆr(i) = rj



.(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 ˆr(i + 1), which is stochastic.