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# Foundations of Term Structure Modeling

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### Foundations of Term Structure Modeling

(2)

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

— Roger Lowenstein, When Genius Failed (2000)

(3)

[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)

(4)

### 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.

(5)

### 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.

(6)

### 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.

(7)

### 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).

(8)

### 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.

(9)

### 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.

(10)

### 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

 .

(11)

### 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

(12)

### 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 .

(13)

### 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

.

(14)

### 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 ,

Et[ 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.

(15)

### 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 Et[ 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

(16)

### Risk-Neutral Pricing (continued)

• Rewrite Eq. (151) on p. 1100 as Etπ[ 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.

(17)

### Risk-Neutral Pricing (concluded)

• Apply the above equality iteratively to obtain

P (t, T )

= Etπ

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



= Etπ

 Et+1π [ P (t + 2, T ) ] (1 + r(t))(1 + r(t + 1))



= · · ·

= Etπ

 1

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

 .

(18)

### Continuous-Time Risk-Neutral Pricing

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

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.

(19)

### Interest Rate Swaps

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

• For simplicity, assume ti+1 − ti 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, . . . , fn−1 at times t0, t1, . . . , tn−1.

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

(20)

- 6

?

6

t0 t1 t2 tn

(f0 − c) Δt

(f1 − c) Δt

· · ·

(fn−1 − c) Δt

(21)

### Interest Rate Swaps (continued)

• Simple rates are adopted here.

• Hence fi satisﬁes

P (ti, ti+1) = 1

1 + fiΔt.

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

• The ordinary swap corresponds to t = t0.

(22)

### Interest Rate Swaps (continued)

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

i=1

Etπ

e

ti

t r(s) ds(fi−1 − c) Δt

=

n i=1

Etπ

e

ti

t r(s) ds

 1

P (ti−1, ti) − (1 + cΔt)



=

n i=1

Etπ

e

ti

t r(s) ds

 e

ti

ti−1 r(s) ds − (1 + cΔt)



=

n i=1

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

= P (t, t ) − P (t, t ) − cΔt

n

P (t, t ).

(23)

### 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.

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### Swap Rate

• The swap rate, which gives the swap zero value, equals Sn(t) =Δ P (t, t0) − P (t, tn)

n

i=1 P (t, ti) 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.

(25)

### The Term Structure Equation

a

• 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 = μp dt + σp 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).

(26)

### 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) σp(r, t, s1) P (r, t, s2) σp(r, t, s2).

(27)

### The Term Structure Equation (continued)

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

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

−P (r, t, s1) + αP (r, t, s2) = r.

• Simplify the above to obtain

σp(r, t, s1) μp(r, t, s2) − σp(r, t, s2) μp(r, t, s1)

σp(r, t, s1) − σp(r, t, s2) = r.

• This becomes

μp(r, t, s2) − r

= μp(r, t, s1) − r

(28)

### The Term Structure Equation (continued)

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

σp(r, t, s)

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

• As μp = r + λσp, 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

(29)

### 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,

μp =



∂P

∂T + μ(r, t) ∂P

∂r + σ(r, t)2 2

2P

∂r2

 /P,

(156)

σp = σ(r, t) ∂P

∂r /P, (156)

(30)

### The Term Structure Equation (concluded)

• Substitute μp and σp 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 .

(31)

### 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%

(32)

### 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.

(33)

### 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.

(34)

### 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.

(35)

### Numerical Examples (concluded)

• Solving the equation leads to p = 0.319.

• Interest rate contingent claims can be priced under this probability.

(36)

### 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.

(37)

### 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

(38)

### 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.

(39)

### 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.

(40)

### 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.

(41)

### 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

aBy Eq. (146) on p. 1095.

bUnlike the nonstochastic case on p. 509.

(42)

### Equilibrium Term Structure Models

(43)

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

(44)

### 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.

(45)

### 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)

(46)

### 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 )24β



, 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,

(47)

### 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

2 ,

independent of the current short rate.

(48)

### 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

(49)

2 4 6 8 10 Term 0.05

0.1 0.15 0.2

Yield

humped

inverted

normal

(50)

### 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).

(51)

### The Vasicek Model: Options on Zeros (concluded)

• Above

x =Δ 1 σv ln

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



+ σv 2 , σv ≡ v(t, T ) B(T, s),

v(t, T )2 =Δ

⎧⎨

σ2[1−e−2β(T −t)]

, 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).

(52)

### Binomial Vasicek

a

• Consider a binomial model for the short rate in the time interval [ 0, T ] divided into n identical pieces.

• Let Δt = T /n andΔ b p(r) =Δ 1

2 + β(μ − r) Δt

.

• 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.

(53)

### 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.

(54)

### 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.

(55)

### The Cox-Ingersoll-Ross Model

a

• 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.

(56)

### 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.

(57)

### Binomial CIR (continued)

• Instead, consider the transformed processa x(r) = 2Δ

r/σ.

• By Ito’s lemma (p. 609),

dx = m(x) dt + dW, where

m(x) = 2βμ/(σΔ 2x) − (βx/2) − 1/(2x).

• This new process has a constant volatility.

(58)

### 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).

(59)

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)

(60)

### 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.

(61)

### 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.

(62)

### 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.

(63)

(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

(64)

### 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

(65)

### 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(n1.5) 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).

(66)

### A Comparison

a

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

(67)

### A General Method for Constructing Binomial Models

a

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

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

• Set the probability of an up move to α(y, t) Δt + y − yd

yu − yd .

• Here yu = y + σ(y, t)Δ

Δt and yd = yΔ − σ(y, t)√ Δt

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### 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(yu, t + Δt)√ Δt

= −σ(y, t)√

Δt + σ(yd, t + Δt)√ Δt in general.

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

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### 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.

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### 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) − 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) =Δ y(x +

Δt, t + Δt), yd(x, t) =Δ y(x

Δt, t + Δt).

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### 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 .

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### 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.

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### 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

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### 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

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### Options on Coupon Bonds

a

• Assume the market discount function P is a

monotonically decreasing function of the short rate r.

– Such as a one-factor short rate model.

• The price of a European option on a coupon bond can be calculated from those on zero-coupon bonds.

• Consider a European call expiring at time T on a bond with par value \$1.

• Let X denote the strike price.

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### Options on Coupon Bonds (continued)

• The bond has cash ﬂows c1, c2, . . . , cn at times t1, t2, . . . , tn, where ti > T for all i.

• 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.

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### Options on Coupon Bonds (continued)

• This r can be obtained by solving X =

n i=1

ciP (r, T, ti) numerically for r.

• Let

Xi = P (rΔ , 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.

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### Options on Coupon Bonds (concluded)

• As X = 

i ciXi, the option’s payoﬀ equals max

 n

i=1

ciP (r(T ), T, ti)



 n

i=1

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.

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### No-Arbitrage Term Structure Models

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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)

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### 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.

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### 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.

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### 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.

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### 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.

aT. Ho & S. B. Lee (1986).

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 r3

 

r2

  

r1 r3 + v3

  

r2 + v2

 

r3 + 2v3

(86)

### 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,a

Pu(t + 1, t + n) = Pd(t + 1, t + n) e−(v2+···+vn).

(160)

(87)

### The Ho-Lee Model (continued)

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

yd(n) =Δ −ln Pd(t + 1, t + n) n − 1

yu(n) =Δ −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 Δ

= 

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

= 

p(1 − p) (yu(n) − yd(n))

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### 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) (ru − rd),

where ru and rd are the two successor rates.a

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

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### 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 vis from the volatility structure, and vice versa.a

• The r

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### The Ho-Lee Model: Bond Price Process

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

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. (160) on p. 1172 and assume p = 1/2 to obtainb

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

2 × exp[ v2 + · · · + vn ] 1 + exp[ v2 + · · · + vn ], Pu(t + 1, t + n) = P (t, t + n)

P (t, t + 1)

2

1 + exp[ v2 + · · · + vn ].

aRecall Eq. (152) on p. 1102.

b

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

• The bond price tree combines.a

• Suppose all vi equal some constant v and δ = eΔ v > 0.

• Then

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

n−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

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### 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 vi’s in linear time.

– Derive the ri’s in linear time.

aSee Programming Assignment 26.2.6 of the textbook.

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### The Ho-Lee Model: Yields and Their Covariances

• The one-period rate of return of an n-period zero-coupon bond isa

r(t, t + n) = lnΔ

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

 .

• Its two possible value are ln Pd(t + 1, t + n)

P (t, t + n) and ln Pu(t + 1, t + n) P (t, t + n) .

• Thus the variance of return isb

Var[ r(t, t + n) ] = p(1 − p) [ (n − 1) v ]2 = (n − 1)2σ2.

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### The Ho-Lee Model: Yields and Their Covariances (concluded)

• The covariance between r(t, t + n) and r(t, t + m) isa (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.

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

• The continuous-time limit of the Ho-Lee model isa

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 vi

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### 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.

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### 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.

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### 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.

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### 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. 1029ﬀ.b

• The volatility structurec is given by the market.

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

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

b

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r4

 r3

 

r2 r4v4

  

r1 r3v3

  

r2v2 r4v42

 

r3v32



r v3

(101)

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

• Our earlier binomial interest rate tree, in contrast, assumes vi are given a priori.

• Lognormal models preclude negative short rates.

(102)

### 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.

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

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

yu =Δ Pu−1/(i−1) − 1, yd =Δ Pd−1/(i−1) − 1.

• The yield volatility is deﬁned asa κi =Δ ln(yu/yd)

2 .

(104)

### 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 time i − 2 (thus period i − 1).a

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

aRecall that (ri−1, vi−1) deﬁnes i−1 short rates at time i−2, which

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### The BDT Model: Calibration (continued)

• We now proceed to calculate ri and vi to extend the tree to cover period i.

• 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 + ri) = 1

(1 + y)i. (163)

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### 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.a

• With κ2i denoting their variance, we have

κi = 1 2 ln



Pu−1/(i−1) − 1 Pd−1/(i−1) − 1



. (164)

aRecall p. 1189.

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### The BDT Model: Calibration (continued)

• Solving Eqs. (163)–(164) for ri and vi 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.

zero-coupon bond prices, forward rates, or the short rate.. • Bond price and forward rate models are usually non-Markovian

• 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

• 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

zero-coupon bond prices, forward rates, or the short rate.. • Bond price and forward rate models are usually non-Markovian

zero-coupon bond prices, forward rates, or the short rate. • Bond price and forward rate models are usually non-Markovian

• The XYZ.com bonds are equivalent to a default-free zero-coupon bond with \$X par value plus n written European puts on Merck at a strike price of \$30. – By the

• The binomial interest rate tree can be used to calculate the yield volatility of zero-coupon bonds.. • Consider an n-period

• The XYZ.com bonds are equivalent to a default-free zero-coupon bond with \$X par value plus n written European puts on Merck at a strike price of \$30. – By the