*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 .*

a*Alternatively, 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 .*

a*In 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*^{}^{t}^{T}* ^{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 t*_{1}*, t*_{2}*, . . . , 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 f*_{0}*, f*_{1}*, . . . , f*_{n−1}*at times t*_{0}*, t*_{1}*, . . . , t** _{n−1}*.

*• The payoﬀ at time t*_{i+1}*for the fixed-rate payer is*

### Interest Rate Swaps (continued)

- 6

?

6

*t*0 *t*1 *t*2 *t**n*

*(f*0 *− c) Δt*

*(f*1 *− c) Δt*

*· · ·*

*(f**n−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*_{i}*Δt.*

*• If t < t*_{0}, we have a forward interest rate swap.

*• The ordinary swap corresponds to t = t*_{0}.

### 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, t*_{0}) *− 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, t*_{0}) = 1.

*• The swap rate is called a forward swap rate if t*_{0} *> t.*

### 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* *= μ*_{p}*dt + σ*_{p}*dW.*

*• At time t, short one unit of a bond maturing at time s*_{1}
*and buy α units of a bond maturing at time s*_{2}.

aVasicek (1977).

### The Term Structure Equation (continued)

*• The net wealth change follows*

*−dP (r, t, s*1*) + α dP (r, t, s*2)

= *(−P (r, t, s*1*) μ**p**(r, t, s*1*) + αP (r, t, s*2*) μ**p**(r, t, s*2*)) dt*
*+ (−P (r, t, s*1*) σ**p**(r, t, s*1*) + αP (r, t, s*2*) σ**p**(r, t, s*2*)) dW.*

*• Pick*

*α* =^{Δ} *P (r, t, s*_{1}*) σ*_{p}*(r, t, s*_{1})
*P (r, t, s*_{2}*) σ*_{p}*(r, t, s*_{2})*.*

### The Term Structure Equation (continued)

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

*−P (r, t, s*_{1}*) μ*_{p}*(r, t, s*_{1}*) + αP (r, t, s*_{2}*) μ*_{p}*(r, t, s*_{2})

*−P (r, t, s*_{1}*) + αP (r, t, s*_{2}) *= r.*

*• Simplify the above to obtain*

*σ*_{p}*(r, t, s*_{1}*) μ*_{p}*(r, t, s*_{2}) *− σ*_{p}*(r, t, s*_{2}*) μ*_{p}*(r, t, s*_{1})

*σ*_{p}*(r, t, s*_{1}) *− σ*_{p}*(r, t, s*_{2}) *= r.*

*• This becomes*

*μ*_{p}*(r, t, s*_{2}) *− r*

= *μ*_{p}*(r, t, s*_{1}) *− r*

### The Term Structure Equation (continued)

*• Since the above equality holds for any s*_{1} *and s*_{2},
*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

*∂*^{2}*P*

*∂r*^{2}

*/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} *∂*^{2}*P*

*∂r*^{2} *= 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

^{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)}

### 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}_{4}_{β}

*,* *if β = 0,*

exp

*σ2(T −t)3*
6

*,* *if β = 0,*

and

*B(t, T ) =*

⎧⎨

⎩

*1−e*^{−β(T −t)}

*β* *,* *if β* *= 0,*

*− t,*

### 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* *−* *σ*^{2}

*2β*^{2} *,*

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

^{a}

*• Consider a European call with strike price X expiring*
*at time T on a zero-coupon bond with par value $1 and*
*maturing at time s > T .*

*• Its price is given by*

*P (t, s) N (x)* *− XP (t, T ) N(x − σ*_{v}*).*

aJamshidian (1989).

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

*• Above*

*x* =^{Δ} 1
*σ** _{v}* ln

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

+ *σ** _{v}*
2

*,*

*σ*

_{v}*≡ v(t, T ) B(T, s),*

*v(t, T )*^{2} =^{Δ}

⎧⎨

⎩

*σ*^{2}[*1−e** ^{−2β(T −t)}*]

*2β* *,* *if β* *= 0*
*σ*^{2}*(T* *− t),* *if β = 0*

*.*

*• By the put-call parity, the price of a European put is*

*−x + σ* *− P (t, s) N(−x).*

### 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 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

^{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 < σ*^{2}*.*

**– 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} *x*^{2}*σ*^{2}
4 *.*

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

aNawalkha & Beliaeva (2007).

*x + 2**√*

Δ*t* *f(x + 2**√*

Δ*t)*

*x +* *√*

Δ*t* *f(x +* *√*

Δ*t)*

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

*x −**√*

Δ*t* *f(x −**√*

Δ*t)*

*x − 2**√*

Δ*t* *f(x − 2**√*

Δ*t)*

### Binomial CIR (continued)

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

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

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

**– r**^{− Δ}*= f (x* *−* *√*

*Δt) the result of a down move.*

*• Finally, set the probability p(r) to one as r goes to zero*
to make the probability stay between zero and one.

### Binomial CIR (concluded)

*• It can be shown that*
*p(r) =*

*βμ* *−* *σ*^{2}
4

*Δt*

*r* *− B√*

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

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

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

*• This phenomenon agrees with mean reversion.*

### 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*
*x*^{2} *× (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

^{a}

*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

^{a}

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

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

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

*y*_{u} *− y*_{d} *.*

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

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

### 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

^{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.*

### Options on Coupon Bonds (continued)

*• The bond has cash ﬂows c*_{1}*, c*_{2}*, . . . , c** _{n}* at times

*t*

_{1}

*, t*

_{2}

*, . . . , t*

_{n}*, where t*

_{i}*> T for all i.*

*• The payoﬀ for the option is*
max

_{n}

*i=1*

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

*− X, 0*

*.*

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

### Options on Coupon Bonds (continued)

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

*X =*

*n*
*i=1*

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

*numerically for r.*

*• Let*

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

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

### Options on Coupon Bonds (concluded)

*• As X =*

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

_{n}

*i=1*

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

*−*

_{n}

*i=1*

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

*, 0*

=

*n*
*i=1*

*c*_{i}*× max(P (r(T ), T, t** _{i}*)

*− X*

_{i}*, 0).*

*• Thus the call is a package of n options on the*
underlying zero-coupon bond.

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

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

*No-Arbitrage Term Structure Models*

How much of the structure of our theories really tells us about things in nature, and how much do we contribute ourselves?

— Arthur Eddington (1882–1944) 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

^{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).

*r*3

*r*2

*r*1 *r*3 *+ v*3

*r*2 *+ v*2

*r*3 *+ 2v*3

### The Ho-Lee Model (continued)

*• The Ho-Lee model starts with zero-coupon bond prices*
*P (t, t + 1), P (t, t + 2), . . . at time t identiﬁed with the*
root of the tree.

*• Let the discount factors in the next period be*

*P*_{d}*(t + 1, t + 2), P*_{d}*(t + 1, t + 3), . . . ,* *if short rate moves down,*
*P*_{u}*(t + 1, t + 2), P*_{u}*(t + 1, t + 3), . . . ,* *if short rate moves up.*

*• By backward induction, it is not hard to see that for*
*n* *≥ 2,*^{a}

*P*_{u}*(t + 1, t + n) = P*_{d}*(t + 1, t + n) e*^{−(v}^{2}^{+···+v}^{n}^{)}*.*

(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) +* *v*_{2} + *· · · + v*_{n}*n* *− 1*

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

*κ**n* Δ

=

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

=

*p(1 − p) (y*u*(n) − y*d*(n))*

### The Ho-Lee Model (concluded)

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

*σ =*

*p(1* *− p) v*_{2}*.* (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*
*κ*_{2}*, κ*_{3}*, . . . .*

**– The volatility structure is supplied by the market.**

**– For the Ho-Lee model, it is independent of**
*r*_{2}*, r*_{3}*, . . . .*

*• It is easy to compute the v** _{i}*s 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) P*d*(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[ v*2 + *· · · + v**n* ]
*1 + exp[ v*_{2} + *· · · + v**n* ]*,*
*P*_{u}*(t + 1, t + n) =* *P (t, t + n)*

*P (t, t + 1)*

2

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

aRecall Eq. (152) on p. 1102.

b

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

*• The bond price tree combines.*^{a}

*• Suppose all v**i* *equal some constant v and δ* *= e*^{Δ} ^{v}*> 0.*

*• Then*

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

*2δ*^{n−1}*1 + δ*^{n−1}*,*
*P*u*(t + 1, t + n)* = *P (t, t + n)*

*P (t, t + 1)*

2

*1 + δ*^{n−1}*.*

*• Short rate volatility σ = v/2 by Eq. (161) on p. 1174.*

*• Price derivatives by taking expectations under the*

### Calibration

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

*• But it can actually be calibrated in O(n) time.*^{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)*^{2}*σ*^{2}*.*

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

*• The covariance between r(t, t + n) and r(t, t + m) is*^{a}
*(n* *− 1)(m − 1) σ*^{2}*.*

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

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

aSee Exercise 26.2.7 of the textbook.

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

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

*dr = θ(t) dt + σ dW.* (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

^{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 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

*r*_{4}

*r*_{3}

*r*_{2} *r*_{4}*v*_{4}

*r*_{1} *r*_{3}*v*_{3}

*r*_{2}*v*_{2} *r*_{4}*v*_{4}^{2}

*r*_{3}*v*_{3}^{2}

*r* *v*^{3}

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

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

*• Lognormal models preclude negative short rates.*

### The BDT Model: Volatility Structure

*• The volatility structure deﬁnes the yield volatilities of*
zero-coupon bonds of various maturities.

*• Let the yield volatility of the i-period zero-coupon bond*
*be denoted by κ** _{i}*.

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

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

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

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

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

*• The yield volatility is deﬁned as*^{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*
*(r*_{1}*, v*_{1}*), (r*_{2}*, v*_{2}*), . . . , (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.

a*Recall 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 κ*^{2}* _{i}* denoting their variance, we have

*κ** _{i}* = 1
2 ln

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

*.* (164)

aRecall p. 1189.

### The BDT Model: Calibration (continued)

*• Solving Eqs. (163)–(164) for r*_{i}*and v** _{i}* with backward

*induction takes O(i*

^{2}) 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.