The Binomial Model
• The analytical framework can be nicely illustrated with the binomial model.
• Suppose the bond price P can move with probability q to P u and probability 1 − q to P d, where u > d:
P
* P d 1 − q
q j Pu
The Binomial Model (continued)
• Over the period, the bond’s expected rate of return is bµ ≡ qP u + (1 − q) P d
P − 1 = qu + (1 − q) d − 1.
(112)
• The variance of that return rate is
bσ2 ≡ q(1 − q)(u − d)2. (113)
The Binomial Model (continued)
• In particular, the bond whose maturity is one period
away will move from a price of 1/(1 + r) to its par value
$1.
• This is the money market account modeled by the short rate r.
• The market price of risk is defined as λ ≡ (bµ − r)/bσ.
• As in the continuous-time case, it can be shown that λ is independent of the maturity of the bond (see text).
The Binomial Model (concluded)
• Now change the probability from q to p ≡ q − λ√
q(1 − q) = (1 + r) − d
u − d , (114) which is independent of bond maturity and q.
– Recall the BOPM.
• The bond’s expected rate of return becomes pP u + (1 − p) P d
P − 1 = pu + (1 − p) d − 1 = r.
• The local expectations theory hence holds under the
Numerical Examples
• Assume this spot rate curve:
Year 1 2
Spot rate 4% 5%
• Assume the one-year rate (short rate) can move up to 8% or down to 2% after a year:
4%
* 8%
j 2%
Numerical Examples (continued)
• No real-world probabilities are specified.
• The prices of one- and two-year zero-coupon bonds are, respectively,
100/1.04 = 96.154, 100/(1.05)2 = 90.703.
• They follow the binomial processes on p. 925.
Numerical Examples (continued)
90.703
* 92.593 (= 100/1.08)
j 98.039 (= 100/1.02) 96.154
* 100 j 100
The price process of the two-year zero-coupon bond is on the left; that of the one-year zero-coupon bond is on the right.
Numerical Examples (continued)
• The pricing of derivatives can be simplified by assuming investors are risk-neutral.
• Suppose all securities have the same expected one-period rate of return, the riskless rate.
• Then
(1 − p) × 92.593
90.703 + p × 98.039
90.703 − 1 = 4%,
where p denotes the risk-neutral probability of a down move in rates.
Numerical Examples (concluded)
• Solving the equation leads to p = 0.319.
• Interest rate contingent claims can be priced under this probability.
Numerical Examples: Fixed-Income Options
• A one-year European call on the two-year zero with a
$95 strike price has the payoffs, C
* 0.000 j 3.039
• To solve for the option value C, we replicate the call by a portfolio of x one-year and y two-year zeros.
Numerical Examples: Fixed-Income Options (continued)
• This leads to the simultaneous equations, x × 100 + y × 92.593 = 0.000, x × 100 + y × 98.039 = 3.039.
• They give x = −0.5167 and y = 0.5580.
• Consequently,
C = x × 96.154 + y × 90.703 ≈ 0.93 to prevent arbitrage.
Numerical Examples: Fixed-Income Options (continued)
• This price is derived without assuming any version of an expectations theory.
• Instead, the arbitrage-free price is derived by replication.
• The price of an interest rate contingent claim does not depend directly on the real-world probabilities.
• The dependence holds only indirectly via the current bond prices.
Numerical Examples: Fixed-Income Options (concluded)
• An equivalent method is to utilize risk-neutral pricing.
• The above call option is worth
C = (1 − p) × 0 + p × 3.039
1.04 ≈ 0.93,
the same as before.
• This is not surprising, as arbitrage freedom and the existence of a risk-neutral economy are equivalent.
Numerical Examples: Futures and Forward Prices
• A one-year futures contract on the one-year rate has a payoff of 100 − r, where r is the one-year rate at
maturity:
F
* 92 (= 100 − 8) j 98 (= 100 − 2)
• As the futures price F is the expected future payoff (see text or p. 464),
F = (1 − p) × 92 + p × 98 = 93.914.
Numerical Examples: Futures and Forward Prices (concluded)
• The forward price for a one-year forward contract on a one-year zero-coupon bond isa
90.703/96.154 = 94.331%.
• The forward price exceeds the futures price.b
aSee Eq. (100) on p. 898.
bRecall p. 410.
Numerical Examples: Mortgage-Backed Securities
• Consider a 5%-coupon, two-year mortgage-backed
security without amortization, prepayments, and default risk.
• Its cash flow and price process are illustrated on p. 935.
• Its fair price is
M = (1 − p) × 102.222 + p × 107.941
1.04 = 100.045.
• Identical results could have been obtained via arbitrage
105
↗ 5
↗ ↘ 102.222 (= 5 + (105/1.08))
105 ↗
0 M
105 ↘
↘ ↗ 107.941 (= 5 + (105/1.02))
5
↘
105
The left diagram depicts the cash flow; the right diagram illustrates the price process.
Numerical Examples: MBSs (continued)
• Suppose that the security can be prepaid at par.
• It will be prepaid only when its price is higher than par.
• Prepayment will hence occur only in the “down” state when the security is worth 102.941 (excluding coupon).
• The price therefore follows the process, M
* 102.222
j 105
• The security is worth
(1 − p) × 102.222 + p × 105
Numerical Examples: MBSs (continued)
• The cash flow of the principal-only (PO) strip comes from the mortgage’s principal cash flow.
• The cash flow of the interest-only (IO) strip comes from the interest cash flow (p. 938(a)).
• Their prices hence follow the processes on p. 938(b).
• The fair prices are
PO = (1 − p) × 92.593 + p × 100
1.04 = 91.304,
IO = (1 − p) × 9.630 + p × 5
1.04 = 7.839.
PO: 100 IO: 5
↗ ↗
0 5
↗ ↘ ↗ ↘
100 5
0 0
0 0
↘ ↗ ↘ ↗
100 5
↘ ↘
0 0
(a)
92.593 9.630
↗ ↗
po io
↘ ↘
100 5
(b)
Numerical Examples: MBSs (continued)
• Suppose the mortgage is split into half floater and half inverse floater.
• Let the floater (FLT) receive the one-year rate.
• Then the inverse floater (INV) must have a coupon rate of
(10% − one-year rate) to make the overall coupon rate 5%.
• Their cash flows as percentages of par and values are shown on p. 940.
FLT: 108 INV: 102
↗ ↗
4 6
↗ ↘ ↗ ↘
108 102
0 0
0 0
↘ ↗ ↘ ↗
104 106
↘ ↘
0 0
(a)
104 100.444
↗ ↗
flt inv
↘ ↘
104 106
Numerical Examples: MBSs (concluded)
• On p. 940, the floater’s price in the up node, 104, is derived from 4 + (108/1.08).
• The inverse floater’s price 100.444 is derived from 6 + (102/1.08).
• The current prices are
FLT = 1
2 × 104
1.04 = 50,
INV = 1
2 × (1 − p) × 100.444 + p × 106
1.04 = 49.142.
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. (58) on p. 523.
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), (115) where
A(t, T ) =
exp [
(B(t,T )−T +t)(β2µ−σ2/2)
β2 − σ2 B(t,T )2
4β
]
if β ̸= 0,
exp [
σ2 (T−t)3 6
]
if β = 0.
and
1−e−β(T −t)
β 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 − σv).
aJamshidian (1989).
The Vasicek Model: Options on Zeros (concluded)
• Above
x ≡ 1
σv ln
( P (t, s) P (t, T ) X
)
+ σv 2 , σv ≡ v(t, T ) B(T, s),
v(t, T )2 ≡
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 ∆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, as we will see next.
The Cox-Ingersoll-Ross Model
a• It is the following square-root short rate model:
dr = β(µ − r) dt + σ√
r dW. (116)
• 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.
Binomial CIR
• We want to approximate the short rate process in the time interval [ 0, T ].
• Divide it into n periods of duration ∆t ≡ T/n.
• Assume µ, β ≥ 0.
• A direct discretization of the process is problematic because the resulting binomial tree will not combine.
Binomial CIR (continued)
• Instead, consider the transformed process x(r) ≡ 2√
r/σ.
• It follows
dx = m(x) dt + dW, where
m(x) ≡ 2βµ/(σ2x) − (βx/2) − 1/(2x).
• Since this new process has a constant volatility, its
Binomial CIR (continued)
• Construct the combining tree for r as follows.
• First, construct a tree for x.
• Then transform each node of the tree into one for r via the inverse transformation r = f (x) ≡ x2σ2/4 (p. 958).
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− . (117)
– 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. 961(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)
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
Numerical Examples (continued)
• 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 + √
∆t = 4.4472135955.
• Use the inverse transformation to obtain the short rate x2 × (0.1)2/4 ≈ 0.0494442719102.
Numerical Examples (concluded)
• Once the short rates are in place, computing the probabilities is easy.
• Note that the up-move probability decreases as interest rates increase and decreases as interest rates decline.
• This phenomenon agrees with mean reversion.
• Convergence is quite good (see text).
A General Method for Constructing Binomial Models
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 − yd yu − yd .
• Here yu ≡ y + σ(y, t)√
∆t and yd ≡ y − σ(y, t)√
∆t represent the two rates that follow the current rate y.
• The displacements are identical, at σ(y, t)√
A General Method (continued)
• But the binomial tree may not combine as σ(y, t)√
∆t − σ(yu, t + ∆t)√
∆t
̸= −σ(y, t)√
∆t + σ(yd, t + ∆t)√
∆t 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) ≡
∫ y
σ(z, t)−1 dz.
• Then x follows
dx = m(y, t) dt + dW for some m(y, t) (see text).
• The key is that the diffusion term is now a constant, and the binomial tree for x combines.
• The transformation that turns a 1-dim stochastic process
A General Method (concluded)
• The probability of an up move remains
α(y(x, t), t) ∆t + y(x, t) − yd(x, t) yu(x, t) − yd(x, t) ,
where y(x, t) is the inverse transformation of x(y, t) from x back to y.
• Note that yu(x, t) ≡ y(x + √
∆t, t + ∆t) and yd(x, t) ≡ y(x − √
∆t, t + ∆t) .
Examples
• The transformation is
∫ r
(σ√
z)−1 dz = 2√ r/σ for the CIR model.
• The transformation is
∫ S
(σz)−1 dz = (1/σ) ln S for the Black-Scholes model.
• The familiar binomial option pricing model in fact
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
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• Assume 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.
• The bond has cash flows c1, c2, . . . , cn at times t1, t2, . . . , tn, where ti > T for all i.
Options on Coupon Bonds (continued)
• The payoff for the option is max
( n
∑
i=1
ciP (r(T ), T, ti) − 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.
• This r∗ can be obtained by solving X =
∑n i=1
ciP (r, T, ti) numerically for r.
Options on Coupon Bonds (continued)
• The solution is unique for one-factor models whose bond price is a monotonically decreasing function of r.
• Let
Xi ≡ P (r∗, T, ti),
the value at time T of a zero-coupon bond with par value $1 and maturing at time ti if r(T ) = r∗.
• Note that P (r(T ), T, ti) ≥ Xi if and only if r(T ) ≤ r∗.
Options on Coupon Bonds (concluded)
• As X = ∑
i ciXi, the option’s payoff equals max
( n
∑
i=1
ciP (r(T ), T, ti) − ∑
i
ciXi, 0 )
=
∑n i=1
ci × max(P (r(T ), T, ti) − Xi, 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 and 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 and Lee (1986).
↗ r3
↗ ↘
r2
↗ ↘ ↗
r1 r3 + v3
↘ ↗ ↘
r2 + v2
↘ ↗
r3 + 2v3
The Ho-Lee Model (continued)
• The Ho-Lee model starts with zero-coupon bond prices P (t, t + 1), P (t, t + 2), . . . at time t identified with the root of the tree.
• Let the discount factors in the next period be
Pd(t + 1, t + 2), Pd(t + 1, t + 3), . . . if short rate moves down Pu(t + 1, t + 2), Pu(t + 1, t + 3), . . . if short rate moves up
• By backward induction, it is not hard to see that for n ≥ 2,
Pu(t + 1, t + n) = Pd(t + 1, t + n) e−(v2+···+vn)
(118) (see text).
The Ho-Lee Model (continued)
• It is also not hard to check that the n-period zero-coupon bond has yields
yd(n) ≡ −ln Pd(t + 1, t + n) n − 1
yu(n) ≡ −ln Pu(t + 1, t + n)
n − 1 = yd(n) + v2 + · · · + vn n − 1
• The volatility of the yield to maturity for this bond is therefore
κn ≡ √
pyu(n)2 + (1 − p) yd(n)2 − [ pyu(n) + (1 − p) yd(n) ]2
= √
p(1 − p) (yu(n) − yd(n))
√ · · · + v
The Ho-Lee Model (concluded)
• In particular, the short rate volatility is determined by taking n = 2:
σ = √
p(1 − p) v2. (119)
• The variance of the short rate therefore equals p(1 − p)(ru − rd)2, where ru and rd are the two successor rates.a
aContrast this with the lognormal model.
The Ho-Lee Model: Volatility Term Structure
• The volatility term structure is composed of κ2, κ3, . . . . – It is independent of the ri.
• It is easy to compute the vis from the volatility structure, and vice versa.
• The ris 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) = (pPu(t+1, t+n)+(1−p) Pd(t+1, t+n)) P (t, t+1).
• Combine the above with Eq. (118) on p. 983 and assume p = 1/2 to obtaina
Pd(t + 1, t + n) = P (t, t + n) P (t, t + 1)
2 × exp[ v2 + · · · + vn ] 1 + exp[ v2 + · · · + vn ],
(120)
Pu(t + 1, t + n) = P (t, t + n) P (t, t + 1)
2
1 + exp[ v2 + · · · + vn ].
(120′)
aIn the limit, only the volatility matters.
The Ho-Lee Model: Bond Price Process (concluded)
• The bond price tree combines.
• Suppose all vi equal some constant v and δ ≡ ev > 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 equals v/2 by Eq. (119) on p. 985.
• 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
(P (t + 1, t + n) P (t, t + n)
) .
• Its value is either ln PdP (t,t+n)(t+1,t+n) or ln PuP (t,t+n)(t+1,t+n).
• Thus the variance of return is
Var[ r(t, t + n) ] = p(1 − p)((n − 1) v)2 = (n − 1)2σ2.
The Ho-Lee Model: Yields and Their Covariances (concluded)
• The covariance between r(t, t + n) and r(t, t + m) is (n − 1)(m − 1) σ2 (see text).
• As a result, the correlation between any two one-period rates of return is unity.
• Strong correlation between rates is inherent in all one-factor Markovian models.
The Ho-Lee Model: Short Rate Process
• The continuous-time limit of the Ho-Lee model is dr = θ(t) dt + σ dW.
• This is Vasicek’s model with the mean-reverting drift replaced by a deterministic, time-dependent drift.
• A nonflat term structure of volatilities can be achieved if the short rate volatility is also made time varying, i.e., dr = θ(t) dt + σ(t) dW .
• This corresponds to the discrete-time model in which vi 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. 834ff (repeated on next page).
• The volatility structure is given by the market.
• From it, the short rate volatilities (thus vi) are determined together with ri.
aBlack, Derman, and Toy (BDT) (1990), but essentially finished in 1986 according to Mehrling (2005).
r4
↗ r3
↗ ↘
r2 r4v4
↗ ↘ ↗
r1 r3v3
↘ ↗ ↘
r2v2 r4v42
↘ ↗
r3v32
↘
The Black-Derman-Toy Model (concluded)
• Our earlier binomial interest rate tree, in contrast, assumes vi are given a priori.
– A related model of Salomon Brothers takes vi to be a given constant.a
• Lognormal models preclude negative short rates.
aTuckman (2002).
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 κi.
• Pu is the price of the i-period zero-coupon bond one period from now if the short rate makes an up move.
• Pd 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,
yu ≡ Pu−1/(i−1) − 1, yd ≡ Pd−1/(i−1) − 1.
• The yield volatility is defined as κi ≡ ln(yu/yd)
2 .
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), . . . , (ri−1, vi−1).
– They define the binomial tree up to period i − 1.
• We now proceed to calculate ri and vi to extend the tree to period i.
The BDT Model: Calibration (continued)
• Assume the price of the i-period zero can move to Pu or Pd one period from now.
• Let y denote the current i-period spot rate, which is known.
• In a risk-neutral economy, Pu + Pd
2(1 + r1) = 1
(1 + y)i. (121)
• Obviously, Pu and Pd are functions of the unknown ri and vi.
The BDT Model: Calibration (continued)
• Viewed from now, the future (i − 1)-period spot rate at time 1 is uncertain.
• Recall that yu and yd represent the spot rates at the up node and the down node, respectively (p. 999).
• With κ2 denoting their variance, we have κi = 1
2 ln (
Pu−1/(i−1) − 1 Pd−1/(i−1) − 1
)
. (122)
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. 860(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 (R84526007) and Lyuu (1997); Lyuu (1999).
The BDT Model: Calibration (continued)
• Let the unknown baseline rate for period i be ri = r.
• Let the unknown multiplicative ratio be vi = v.
• Let the state prices at time i − 1 be P1, P2, . . . , Pi, corresponding to rates r, rv, . . . , rvi−1 for period i, respectively.
• One dollar at time i has a present value of f (r, v) ≡ P1
1 + r + P2
1 + rv + P3
1 + rv2 + · · · + Pi
1 + rvi−1.
The BDT Model: Calibration (continued)
• The yield volatility is
g(r, v) ≡ 1 2 ln
( P
u,1
1+rv + Pu,2
1+rv2 + · · · + 1+rvPu,i−1i−1
)−1/(i−1) (P − 1
d,1
1+r + 1+rvPd,2 + · · · + 1+rvPd,i−1i−2
)−1/(i−1)
− 1
.
• Above, Pu,1, Pu,2, . . . denote the state prices at time
i − 1 of the subtree rooted at the up node (like r2v2 on p. 996).
• And Pd,1, Pd,2, . . . denote the state prices at time i − 1 of the subtree rooted at the down node (like r2 on
p. 996).
The BDT Model: Calibration (concluded)
• Note that every node maintains 3 state prices.
• Now solve
f (r, v) = 1
(1 + y)i, g(r, v) = κi,
for r = ri and v = vi.
• This O(n2)-time algorithm appears in the text.
The BDT Model: Continuous-Time Limit
• The continuous-time limit of the BDT model is d ln r =
(
θ(t) + 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.