Foundations of Term Structure Modeling

Foundations of Term Structure Modeling

[Meriwether] scoring especially high marks in mathematics — an indispensable subject for a bond trader.

— Roger Lowenstein, When Genius Failed (2000)

[The] fixed-income traders I knew seemed smarter than the equity trader [· · · ] there’s no competitive edge to being smart in the equities business[.]

— Emanuel Derman, My Life as a Quant (2004) Bond market terminology was designed less to convey meaning than to bewilder outsiders.

— Michael Lewis, The Big Short (2011)

Terminology

• A period denotes a unit of elapsed time.

– Viewed at time t, the next time instant refers to time t + dt in the continuous-time model and time t + 1 in the discrete-time case.

• Bonds will be assumed to have a par value of one — unless stated otherwise.

• The time unit for continuous-time models will usually be measured by the year.

Standard Notations

The following notation will be used throughout.

t: a point in time.

r(t): the one-period riskless rate prevailing at time t for repayment one period later.^a

P (t, T ): the present value at time t of one dollar at time T .

aAlternatively, the instantaneous spot rate, or short rate, at time t.

Standard Notations (continued)

r(t, T ): the (T − t)-period interest rate prevailing at time t stated on a per-period basis and compounded once per period.^a

F (t, T, M ): the forward price at time t of a forward

contract that delivers at time T a zero-coupon bond maturing at time M ≥ T .

aIn other words, the (T − t)-period spot rate at time t.

Standard Notations (concluded)

f (t, T, L): the L-period forward rate at time T implied at time t stated on a per-period basis and compounded once per period.

f (t, T ): the one-period or instantaneous forward rate at time T as seen at time t stated on a per period basis and compounded once per period.

• It is f(t, T, 1) in the discrete-time model and f (t, T, dt) in the continuous-time model.

• Note that f(t, t) equals the short rate r(t).

Fundamental Relations

• The price of a zero-coupon bond equals

P (t, T ) =

⎧⎨

⎩

(1 + r(t, T ))^{−(T −t)}, in discrete time,

e−r(t,T )(T −t), in continuous time. (145)

• r(t, T ) as a function of T defines the spot rate curve at time t.

• By definition,

f (t, t) =

⎧⎨

⎩

r(t, t + 1), in discrete time, r(t, t), in continuous time.

Fundamental Relations (continued)

• Forward prices and zero-coupon bond prices are related:

F (t, T, M ) = P (t, M )

P (t, T ) , T ≤ M. (146) – The forward price equals the future value at time T

of the underlying asset.^a

• The above identity holds whether the model is discrete-time or continuous-time.

aSee Exercise 24.2.1 of the textbook for proof.

Fundamental Relations (continued)

• Forward rates and forward prices are related definitionally by

f(t, T, L) =

 1

F (t, T, T + L)

_1/L

− 1 =

 P (t, T ) P (t, T + L)

_1/L

− 1 (147)

in discrete time.

• The analog under simple compounding is f (t, T, L) = 1

L

 P (t, T )

P (t, T + L) − 1

 .

Fundamental Relations (continued)

• In continuous time,

f (t, T, L) = −ln F (t, T, T + L)

L = ln(P (t, T )/P (t, T + L))

L (148)

by Eq. (146) on p. 1095.

• Furthermore,

f (t, T, Δt) = ln(P (t, T )/P (t, T + Δt))

Δt → −∂ ln P (t, T )

∂T

−∂P (t, T )/∂T

Fundamental Relations (continued)

• So

f (t, T ) =^Δ −∂ ln P (t, T )

∂T = −∂P (t, T )/∂T

P (t, T ) , t ≤ T.

(149)

• Because the above identity is equivalent to P (t, T ) = e⁻

_T

t f (t,s) ds, (150) the spot rate curve is

r(t, T ) =

 _T

t f (t, s) ds T − t .

Fundamental Relations (concluded)

• The discrete analog to Eq. (150) is

P (t, T ) = 1

(1 + r(t))(1 + f (t, t + 1))· · · (1 + f(t, T − 1)).

• The short rate and the market discount function are related by

r(t) = − ∂P (t, T )

∂T



T =t

.

Risk-Neutral Pricing

• Assume the local expectations theory.

• The expected rate of return of any riskless bond over a single period equals the prevailing one-period spot rate.

– For all t + 1 < T ,

E_t[ P (t + 1, T ) ]

P (t, T ) = 1 + r(t). (151) – Relation (151) in fact follows from the risk-neutral

valuation principle.^a

aRecall Theorem 17 on p. 566.

Risk-Neutral Pricing (continued)

• The local expectations theory is thus a consequence of the existence of a risk-neutral probability π.

• Equation (151) on p. 1100 can also be expressed as E_t[ P (t + 1, T ) ] = F (t, t + 1, T ).

– Verify that with, e.g., Eq. (146) on p. 1095.

• Hence the forward price for the next period is an unbiased estimator of the expected bond price.^a

– But the forward rate is not an unbiased estimator of the expected future short rate.^b

Risk-Neutral Pricing (continued)

• Rewrite Eq. (151) on p. 1100 as E_t^π[ P (t + 1, T ) ]

1 + r(t) = P (t, T ). (152) – It says the current market discount function equals

the expected market discount function one period from now discounted by the short rate.

Risk-Neutral Pricing (concluded)

• Apply the above equality iteratively to obtain

P (t, T )

= E_t^π

 P (t + 1, T ) 1 + r(t)



= E_t^π

 E_t+1^π [ P (t + 2, T ) ] (1 + r(t))(1 + r(t + 1))



= · · ·

= E_t^π

 1

(1 + r(t))(1 + r(t + 1))· · · (1 + r(T − 1))

 .

Continuous-Time Risk-Neutral Pricing

• In continuous time, the local expectations theory implies P (t, T ) = E_t

e⁻

_T

t r(s) ds

, t < T. (153)

• Note that e^{tT r(s) ds} is the bank account process, which denotes the rolled-over money market account.

Interest Rate Swaps

• Consider an interest rate swap made at time t (now) with payments to be exchanged at times t₁, t₂, . . . , t_n.

• For simplicity, assume t_i+1 − t_i is a fixed constant Δt for all i, and the notional principal is one dollar.

• The fixed rate is c per annum.

• The floating-rate payments are based on the future annual rates f₀, f₁, . . . , f_n−1 at times t₀, t₁, . . . , t_n−1.

• The payoff at time t_i+1 for the fixed-rate payer is

Interest Rate Swaps (continued)

- 6

?

6

t0 t1 t2 tn

(f0 − c) Δt

(f1 − c) Δt

· · ·

(fn−1 − c) Δt

Interest Rate Swaps (continued)

• Simple rates are adopted here.

• Hence f_i satisfies

P (t_i, t_i+1) = 1

1 + f_iΔt.

• If t < t₀, we have a forward interest rate swap.

• The ordinary swap corresponds to t = t₀.

Interest Rate Swaps (continued)

• The value of the swap at time t is thus n

i=1

E_t^π

e⁻

_ti

t r(s) ds(f_i−1 − c) Δt

=

n i=1

E_t^π 

e⁻

_ti

t r(s) ds

 1

P (t_i−1, t_i) − (1 + cΔt)



=

n i=1

E_t^π 

e⁻

_ti

t r(s) ds

 e

_ti

ti−1 r(s) ds − (1 + cΔt)



=

n i=1

[ P (t, t_i−1) − (1 + cΔt) × P (t, t_i) ]

= P (t, t ) − P (t, t ) − cΔt

n

P (t, t ).

Interest Rate Swaps (concluded)

• So a swap can be replicated as a portfolio of bonds.

• In fact, it can be priced by simple PV calculations.

Swap Rate

• The swap rate, which gives the swap zero value, equals S_n(t) =^Δ P (t, t₀) − P (t, t_n)

_n

i=1 P (t, t_i) Δt . (154)

• The swap rate is the fixed rate that equates the present values of the fixed payments and the floating payments.

• For an ordinary swap, P (t, t₀) = 1.

• The swap rate is called a forward swap rate if t₀ > t.

The Term Structure Equation

• Let us start with the zero-coupon bonds and the money market account.

• Let the zero-coupon bond price P (r, t, T ) follow dP

P = μ_p dt + σ_p dW.

• At time t, short one unit of a bond maturing at time s₁ and buy α units of a bond maturing at time s₂.

aVasicek (1977).

The Term Structure Equation (continued)

• The net wealth change follows

−dP (r, t, s1) + α dP (r, t, s2)

= (−P (r, t, s1) μp(r, t, s1) + αP (r, t, s2) μp(r, t, s2)) dt + (−P (r, t, s1) σp(r, t, s1) + αP (r, t, s2) σp(r, t, s2)) dW.

• Pick

α =^Δ P (r, t, s₁) σ_p(r, t, s₁) P (r, t, s₂) σ_p(r, t, s₂).

The Term Structure Equation (continued)

• Then the net wealth has no volatility and must earn the riskless return:

−P (r, t, s₁) μ_p(r, t, s₁) + αP (r, t, s₂) μ_p(r, t, s₂)

−P (r, t, s₁) + αP (r, t, s₂) = r.

• Simplify the above to obtain

σ_p(r, t, s₁) μ_p(r, t, s₂) − σ_p(r, t, s₂) μ_p(r, t, s₁)

σ_p(r, t, s₁) − σ_p(r, t, s₂) = r.

• This becomes

μ_p(r, t, s₂) − r

= μ_p(r, t, s₁) − r

The Term Structure Equation (continued)

• Since the above equality holds for any s₁ and s₂, μ_p(r, t, s) − r

σ_p(r, t, s)

= λ(r, t)Δ (155) for some λ independent of the bond maturity s.

• As μ_p = r + λσ_p, all assets are expected to appreciate at a rate equal to the sum of the short rate and a constant times the asset’s volatility.

• The term λ(r, t) is called the market price of risk.

• The market price of risk must be the same for all bonds

The Term Structure Equation (continued)

• Assume a Markovian short rate model, dr = μ(r, t) dt + σ(r, t) dW.

• Then the bond price process is also Markovian.

• By Eq. (14.15) on p. 202 of the textbook,

μ_p =



−∂P

∂T + μ(r, t) ∂P

∂r + σ(r, t)² 2

∂²P

∂r²

 /P,

(156)

σ_p = σ(r, t) ∂P

∂r /P, (156^)

The Term Structure Equation (concluded)

• Substitute μ_p and σ_p into Eq. (155) on p. 1114 to obtain

− ∂P

∂T + [ μ(r, t)− λ(r, t) σ(r, t) ] ∂P

∂r + 1

2 σ(r, t)² ∂²P

∂r² = rP.

(157)

• This is called the term structure equation.

• It applies to all interest rate derivatives: The differences are the terminal and boundary conditions.

• Once P is available, the spot rate curve emerges via r(t, T ) = −ln P (t, T )

T − t .

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 given.

• 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. 1119.

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 (= 98.039 − 95)

• 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

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,^a F = (1 − p) × 92 + p × 98 = 93.914.

aSee Exercise 13.2.11 of the textbook or p. 567.

Numerical Examples: Futures and Forward Prices (concluded)

• The forward price for a one-year forward contract on a one-year zero-coupon bond is^a

90.703/96.154 = 94.331%.

• The forward price exceeds the futures price.^b

aBy Eq. (146) on p. 1095.

bUnlike the nonstochastic case on p. 509.

Equilibrium Term Structure Models

The nature of modern trade is to give to those who have much and take from those who have little.

— Walter Bagehot (1867), The English Constitution 8. What’s your problem? Any moron can understand bond pricing models.

— Top Ten Lies Finance Professors

Introduction

• We now survey equilibrium models.

• Recall that the spot rates satisfy

r(t, T ) = −ln P (t, T ) T − t by Eq. (145) on p. 1094.

• 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)}

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), (158) 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,

The Vasicek Model (continued)

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

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

• But even if β = 0, P may exceed one for a finite T .

• The long rate r(t, ∞) is the constant μ − σ²

2β² ,

independent of the current short rate.

The Vasicek Model (concluded)

• The spot rate volatility structure is the curve σ ∂r(t, T )

∂r = σB(t, T ) T − t .

• As it depends only on T − t not on t by itself, the same curve is maintained for any future time t.

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

– The long rate’s volatility is zero unless β = 0.

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

• Higher β leads to greater attenuation of volatility with

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

−x + σ − 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 b} p(r) =^Δ 1

2 + β(μ − r) √ Δt

2σ .

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

r(k + 1) = r(k) + σ√

Δt ξ(k), 0 ≤ k < n.

aNelson & Ramaswamy (1990).

bThe same form as Eq. (42) on p. 296 for the BOPM.

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. (159)

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

r .

• The parameter β determines the speed of adjustment.

• If r(0) > 0, then the short rate can reach zero only if 2βμ < σ².

– This is called the Feller (1951) condition.

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^a x(r) = 2^Δ √

r/σ.

• By Ito’s lemma (p. 609),

dx = m(x) dt + dW, where

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

• This new process has a constant volatility.

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 (see next page)

r = f (x) =^Δ x²σ² 4 .

• 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.

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. 1149(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

0.988093738447 0.976486896485 0.965170249273

0.990159879565 0.980492588317

0.993708727831 0.987391576942 0.98583472203 0.972116454453

0.983384173756

0.988093738447

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.^a

Trinomial CIR

• The binomial CIR tree does not have the degree of freedom to match the mean and variance exactly.

• It actually fails to match them at very low x.

• A trinomial tree for the CIR model with O(n^1.5) nodes that matches the mean and variance exactly is recently obtained using the ideas on pp. 803ff and others.^a

aZ. Lu (D00922011) & Lyuu (2018); H. Huang (R03922103) (2019).

A Comparison

r(0) = 0.01, μ = 0.05, σ = 0.2, β = 1.2, T = 5, principal is 10,000.

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

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.

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).

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 dS = μS dt + σS dW .

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)

• Multifactor 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.^a

aKamakura (2019) has a 10-factor

HJM model for the U.S. Treasuries (see

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.

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^∗.

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) How can I apply this model if I don’t understand it?

— Edward I. Altman (2019)

Motivations

• Recall the difficulties facing equilibrium models mentioned earlier.

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

– They cannot fit the market term structure.

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

aRecall p. 1114.

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.

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.

aT. Ho & S. B. 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,^a

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

(160)

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 + · · · + v_n 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))

The Ho-Lee Model (concluded)

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

σ = 

p(1 − p) v₂. (161)

• The volatility of the short rate therefore equals

p(1 − p) (r_u − r_d),

where r_u and r_d are the two successor rates.^a

aContrast this with the lognormal model (138) on p. 1033.

The Ho-Lee Model: Volatility Term Structure

• The volatility term structure is composed of κ₂, κ₃, . . . .

– The volatility structure is supplied by the market.

– For the Ho-Lee model, it is independent of r₂, r₃, . . . .

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

• The r

The Ho-Lee Model: Bond Price Process

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

P (t, t+n) = [ pP_u(t+1, t+n)+(1−p) Pd(t+1, t+n) ] P (t, t+1).

• Combine the above with Eq. (160) on p. 1172 and assume p = 1/2 to obtain^b

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 ].

aRecall Eq. (152) on p. 1102.

b

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. (161) on p. 1174.

• Price derivatives by taking expectations under the

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.^a – Derive the v_i’s in linear time.

– Derive the r_i’s in linear time.

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^a

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^b

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. (162)

• 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

The Ho-Lee Model: Some Problems

• Future (nominal) interest rates may be negative.

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

• It has all the problems associated with a one-factor model.^a

aRecall pp. 1158ff. See T. Ho & S. B. Lee (2004) for a multifactor Ho-Lee model.

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 every day.

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. 1029ff.^b

• The volatility structure^c is given by the market.

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

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

b

r₄

 r₃

 

r₂ r₄v₄

  

r₁ r₃v₃

  

r₂v₂ r₄v₄²

 

r₃v₃²



r v³

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^a κ_i =^Δ ln(y_u/y_d)

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), . . . , (r_i−1, v_i−1).

– They define the binomial tree up to time i − 2 (thus period i − 1).^a

– Thus the spot rates up to time i − 1 have been matched.

aRecall that (r_i−1, v_i−1) defines i−1 short rates at time i−2, which

The BDT Model: Calibration (continued)

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

• 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_i) = 1

(1 + y)ⁱ. (163)

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.^a

• 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



. (164)

aRecall p. 1189.

The BDT Model: Calibration (continued)

• Solving Eqs. (163)–(164) for r_i and v_i with backward induction takes O(i²) time.

– That leads to a cubic-time algorithm.

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

• Forward induction 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.