*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.*

*• 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.* (115)
**– The forward price equals the future value at time T**

of the underlying asset.^{a}

*• Equation (115) 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*
(116)

in discrete time.

**– The analog to Eq. (116) 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* (117)

by Eq. (115) on p. 969.

*• Furthermore,*

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

*Δt* *→ −∂ ln P (t, T )*

*∂T*

= *−∂P (t, T )/∂T*
*P (t, T )* *.*

### Fundamental Relations (continued)

*• So*

*f (t, T ) ≡ lim*

*Δt→0**f (t, T, Δt) = −∂P (t, T )/∂T*

*P (t, T )* *, t ≤ T.*

(118)

*• Because Eq. (118) is equivalent to*
*P (t, T ) = e*^{−}

_{T}

*t* *f (t,s) ds**,* (119)
the spot rate curve is

*r(t, T ) =*

_{T}

*t* *f (t, s) ds*
*T − t* *.*

### Fundamental Relations (concluded)

*• The discrete analog to Eq. (119) 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).* (120)
**– Relation (120) in fact follows from the risk-neutral**

valuation principle.^{a}

aTheorem 17 on p. 503.

### Risk-Neutral Pricing (continued)

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

*• Rewrite Eq. (120) as*

*E*_{t}^{π}*[ P (t + 1, T ) ]*

*1 + r(t)* *= P (t, T ).*

**– 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 (continued)

*• 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))*

*.* (121)

### Risk-Neutral Pricing (concluded)

*• Equation (120) on p. 974 can also be expressed as*
*E*_{t}*[ P (t + 1, T ) ] = F (t, t + 1, T ).*

**– Verify that with, e.g., Eq. (115) on p. 969.**

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

aBut the forward rate is not an unbiased estimator of the expected future short rate (p. 925).

### 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.* (122)

*• 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}*.

*• 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}*.

*• For simplicity, assume t*_{i+1}*− t*_{i}*is a ﬁxed constant Δt*
*for all i, and the notional principal is one dollar.*

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

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

### Interest Rate Swaps (continued)

*• The amount to be paid out at time t*_{i+1}*is (f*_{i}*− c) Δt*
*for the floating-rate payer.*

*• Simple rates are adopted here.*

*• Hence f** _{i}* satisﬁes

*P (t*_{i}*, t** _{i+1}*) = 1

*1 + f*_{i}*Δt.*

### 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*

*[ P (t, t** _{i−1}*)

*− (1 + cΔt) × P (t, t*

*) ]*

_{i}= *P (t, t*_{0}) *− P (t, t** _{n}*)

*− cΔt*

*n*
*i=1*

*P (t, t*_{i}*).*

### Interest Rate Swaps (concluded)

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

*• In fact, it can be priced by simple present value*
calculations.

### Swap Rate

*• The swap rate, which gives the swap zero value, equals*
*S*_{n}*(t) ≡* *P (t, t*_{0}) *− P (t, t** _{n}*)

_{n}

*i=1* *P (t, t*_{i}*) Δt* *.* (123)

*• 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 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 s*_{1}
*and buy α units of a bond maturing at time s*_{2}.

### 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 ≡* *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*

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

### 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)* (124)
*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*
to preclude arbitrage opportunities.

### 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,*

(125)

*σ**p* =

*σ(r, t)* *∂P*

*∂r*

*/P,* (125* ^{}*)

*subject to P ( · , T, T ) = 1.*

### The Term Structure Equation (concluded)

*• Substitute μ*_{p}*and σ** _{p}* into Eq. (124) on p. 987 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.*

(126)

*• This is called the term structure equation.*

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

*T − t* *.*

*• Equation (126) applies to all interest rate derivatives,*
the diﬀerence being the terminal and the boundary
conditions.

### The Binomial Model

*• The analytical framework can be nicely illustrated with*
the binomial model.

*• Suppose the bond price P can move with probability q*
*to P u and probability 1 − q to P d, where u > d:*

*P*

** P d*
1 *− q*

*q* *j Pu*

### The Binomial Model (continued)

*• Over the period, the bond’s expected rate of return is*

*μ ≡* *qP u + (1 − q) P d*

*P* *− 1 = qu + (1 − q) d − 1.*

(127)

*• The variance of that return rate is*

*σ*^{2} *≡ q(1 − q)(u − d)*^{2}*.* (128)

### The Binomial Model (continued)

*• In particular, the bond whose maturity is one period*

*away will move from a price of 1/(1 + r) to its par value*

$1.

*• This is the money market account modeled by the short*
*rate r.*

*• The market price of risk is deﬁned as λ ≡ (μ − r)/σ.*

*• As in the continuous-time case, it can be shown that λ*
is independent of the maturity of the bond (see text).

### The Binomial Model (concluded)

*• Now change the probability from q to*
*p ≡ q − λ*

*q(1 − q) =* *(1 + r) − d*

*u − d* *,* (129)
*which is independent of bond maturity and q.*

**– Recall the BOPM.**

*• The bond’s expected rate of return becomes*
*pP u + (1 − p) P d*

*P* *− 1 = pu + (1 − p) d − 1 = r.*

*• The local expectations theory hence holds under the*
*new probability measure p.*

### Numerical Examples

*• Assume this spot rate curve:*

Year 1 2

Spot rate 4% 5%

*• Assume the one-year rate (short rate) can move up to*
8% or down to 2% after a year:

4%

* 8%

j 2%

### Numerical Examples (continued)

*• No real-world probabilities are speciﬁed.*

*• The prices of one- and two-year zero-coupon bonds are,*
respectively,

*100/1.04 = 96.154,*
*100/(1.05)*^{2} = *90.703.*

*• They follow the binomial processes on p. 996.*

### Numerical Examples (continued)

90.703

** 92.593 (= 100/1.08)*

*j 98.039 (= 100/1.02)* 96.154

* 100 j 100

The price process of the two-year zero-coupon bond is on the left; that of the one-year zero-coupon bond is on the right.

### Numerical Examples (continued)

*• The pricing of derivatives can be simpliﬁed by assuming*
investors are risk-neutral.

*• Suppose all securities have the same expected one-period*
rate of return, the riskless rate.

*• Then*

(1 *− p) ×* *92.593*

*90.703* *+ p ×* *98.039*

*90.703* *− 1 = 4%,*

*where p denotes the risk-neutral probability of a down*
move in rates.

### Numerical Examples (concluded)

*• Solving the equation leads to p = 0.319.*

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

### Numerical Examples: Fixed-Income Options

*• A one-year European call on the two-year zero with a*

$95 strike price has the payoﬀs,
*C*

** 0.000*
*j 3.039*

*• To solve for the option value C, we replicate the call by*
*a portfolio of x one-year and y two-year zeros.*

### Numerical Examples: Fixed-Income Options (continued)

*• This leads to the simultaneous equations,*
*x × 100 + y × 92.593 = 0.000,*
*x × 100 + y × 98.039 = 3.039.*

*• They give x = −0.5167 and y = 0.5580.*

*• Consequently,*

*C = x × 96.154 + y × 90.703 ≈ 0.93*
to prevent arbitrage.

### Numerical Examples: Fixed-Income Options (continued)

*• This price is derived without assuming any version of an*
expectations theory.

*• Instead, the arbitrage-free price is derived by replication.*

*• The price of an interest rate contingent claim does not*
depend directly on the real-world probabilities.

*• The dependence holds only indirectly via the current*
bond prices.

### Numerical Examples: Fixed-Income Options (concluded)

*• An equivalent method is to utilize risk-neutral pricing.*

*• The above call option is worth*

*C =* (1 *− p) × 0 + p × 3.039*

*1.04* *≈ 0.93,*

the same as before.

*• This is not surprising, as arbitrage freedom and the*
existence of a risk-neutral economy are equivalent.

### Numerical Examples: Futures and Forward Prices

*• A one-year futures contract on the one-year rate has a*
payoﬀ of 100 *− r, where r is the one-year rate at*

maturity:

*F*

** 92 (= 100 − 8)*
*j 98 (= 100 − 2)*

*• As the futures price F is the expected future payoﬀ,*^{a}
*F = (1 − p) × 92 + p × 98 = 93.914.*

aSee Exercise 13.2.11 of the textbook or p. 504.

### 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. (115) on p. 969.

bRecall p. 448.

*Equilibrium Term Structure Models*

8. What’s your problem? Any moron can understand bond pricing models.

*— Top Ten Lies Finance Professors*
*Tell Their Students*

### Introduction

*• This chapter surveys equilibrium models.*

*• Since the spot rates satisfy*

*r(t, T ) = −ln P (t, T )*
*T − t* *,*

*the discount function P (t, T ) suﬃces to establish the*
spot rate curve.

*• All models to follow are short rate models.*

*• Unless stated otherwise, the processes are risk-neutral.*

### The Vasicek Model

^{a}

*• The short rate follows*

*dr = β(μ − r) dt + σ dW.*

*• The short rate is pulled to the long-term mean level μ*
*at rate β.*

*• Superimposed on this “pull” is a normally distributed*
*stochastic term σ dW .*

*• Since the process is an Ornstein-Uhlenbeck process,*
*E[ r(T ) | r(t) = r ] = μ + (r − μ) e** ^{−β(T −t)}*
from Eq. (64) on p. 562.

aVasicek (1977).

### The Vasicek Model (continued)

*• The price of a zero-coupon bond paying one dollar at*
maturity can be shown to be

*P (t, T ) = A(t, T ) e**−B(t,T ) r(t)**,* (130)
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 − t* *if β = 0.*

### The Vasicek Model (concluded)

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

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

*• Even if β = 0, P may exceed one for a ﬁnite T .*

*• The spot rate volatility structure is the curve*
*(∂r(t, T )/∂r) σ = σB(t, T )/(T − t).*

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

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

*• Indeed, higher β leads to greater attenuation of*
volatility with maturity.

2 4 6 8 10 Term 0.05

0.1 0.15 0.2

Yield

humped

inverted

normal

### The Vasicek Model: Options on Zeros

^{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 ≡* 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} *≡*

⎧⎨

⎩

*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*
*XP (t, T ) N (−x + σ** _{v}*)

*− 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*

*p(r) ≡* 1

2 + *β(μ − r)√*
*Δt*

*2σ* *.*

*• The following binomial model converges to the Vasicek*
model,^{a}

*r(k + 1) = r(k) + σ√*

*Δt ξ(k), 0 ≤ k < n.*

aNelson and Ramaswamy (1990).

### Binomial Vasicek (continued)

*• Above, ξ(k) = ±1 with*

*Prob[ ξ(k) = 1 ] =*

⎧⎪

⎪⎨

⎪⎪

⎩

*p(r(k)) if 0 ≤ p(r(k)) ≤ 1*
0 *if p(r(k)) < 0*

1 *if 1 < p(r(k))*

*.*

*• Observe that the probability of an up move, p, is a*
*decreasing function of the interest rate r.*

*• This is consistent with mean reversion.*

### Binomial Vasicek (concluded)

*• The rate is the same whether it is the result of an up*
move followed by a down move or a down move followed
by an up move.

*• The binomial tree combines.*

*• The key feature of the model that makes it happen is its*
*constant volatility, σ.*

### The Cox-Ingersoll-Ross Model

^{a}

*• It is the following square-root short rate model:*

*dr = β(μ − r) dt + σ√*

*r dW.* (131)

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

*r .*

*• The parameter β determines the speed of adjustment.*

*• The short rate can reach zero only if 2βμ < σ*^{2}.

*• See text for the bond pricing formula.*

aCox, Ingersoll, and Ross (1985).

### Binomial CIR

*• We want to approximate the short rate process in the*
*time interval [ 0, T ].*

*• Divide it into n periods of duration Δt ≡ T/n.*

*• Assume μ, β ≥ 0.*

*• A direct discretization of the process is problematic*
*because the resulting binomial tree will not combine.*

### Binomial CIR (continued)

*• Instead, consider the transformed process*
*x(r) ≡ 2√*

*r/σ.*

*• It follows*

*dx = m(x) dt + dW,*
where

*m(x) ≡ 2βμ/(σ*^{2}*x) − (βx/2) − 1/(2x).*

*• This new process has a constant volatility, and its*
associated binomial tree combines.

### Binomial CIR (continued)

*• Construct the combining tree for r as follows.*

*• First, construct a tree for x.*

*• Then transform each node of the tree into one for r via*
the inverse transformation

*r = f (x) ≡* *x*^{2}*σ*^{2}
4
(see p. 1021).

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

aNawalkha and 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 (concluded)

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

*r*^{+} *− r*^{−}*.* (132)

**– r**^{+} *≡ f(x +* *√*

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

**– r**^{−}*≡ 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.

### 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. 1024(a) for the resulting binomial short rate tree*
with the up-move probabilities in parentheses.

(0.472049150276)0.04

0.05988854382 (0.44081188025)

0.03155572809 (0.489789553691)

0.02411145618 (0.50975924867)

0.0713328157297 (0.426604457655)

0.08377708764

0.01222291236 0.01766718427

(0.533083330907) (0.472049150276)0.04

0.0494442719102

(0.455865503068) 0.0494442719102 (0.455865503068)

0.03155572809 (0.489789553691)

0.05988854382

0.04

0.02411145618

=

>

0.992031914837 0.984128889634 0.976293244408 0.968526861261 0.960831229521

0.992031914837 0.984128889634

0.976293244408 0.992031914837 0.990159879565

0.980492588317 0.970995502019 0.961665706744

0.993708727831 0.987391576942 0.981054487259 0.974702907786

0.988093738447 0.976486896485 0.965170249273

0.990159879565 0.980492588317

0.995189317343 0.990276851751 0.985271123591

0.993708727831 0.987391576942 0.98583472203 0.972116454453

0.996472798388 0.992781347933

0.983384173756

0.988093738447

0.995189317343

### Numerical Examples (continued)

*• Consider the node which is the result of an up move*
from the root.

*• Since the root has x = 2*

*r(0)/σ = 4, this particular*
*node’s x value equals 4 +* *√*

*Δt = 4.4472135955.*

*• Use the inverse transformation to obtain the short rate*
*x*^{2} *× (0.1)*^{2}

4 *≈ 0.0494442719102.*

### Numerical Examples (concluded)

*• Once the short rates are in place, computing the*
probabilities is easy.

*• Note that the up-move probability decreases as interest*
rates increase and decreases as interest rates decline.

**– I suspect that**

*p(r) = A*

*Δt*

*r* *+ B − C√*

*rΔt*
*for some A, B, C > 0.*^{a}

*• This phenomenon agrees with mean reversion.*

*• Convergence is quite good (see text).*

aThanks to a lively class discussion on May 28, 2014.

### 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)√*

*Δt and y*_{d} *≡ y − σ(y, t)√*
*Δt*
*represent the two rates that follow the current rate y.*

aNelson and Ramaswamy (1990).

### A General Method (continued)

*• The displacements are identical, at σ(y, t)√*

*Δt .*

*• But the binomial tree may not combine as*
*σ(y, t)√*

*Δt − σ(y*_{u}*, t + Δt)√*
*Δt*

*= −σ(y, t)√*

*Δt + σ(y*_{d}*, t + Δt)√*
*Δt*
in general.

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

### A General Method (continued)

*• To achieve this, deﬁne the transformation*
*x(y, t) ≡*

_{y}

*σ(z, t)*^{−1}*dz.*

*• Then x follows*

*dx = m(y, t) dt + dW*
*for some m(y, t).*^{a}

*• The diﬀusion term is now a constant, and the binomial*
*tree for x combines.*

aSee Exercise 25.2.13 of the textbook.

### A General Method (concluded)

*• The transformation is unique.*^{a}

*• The probability of an up move remains*

*α(y(x, t), t) Δt + y(x, t) − y*_{d}*(x, t)*
*y*_{u}*(x, t) − y*_{d}*(x, t)* *,*

*where y(x, t) is the inverse transformation of x(y, t)*
*from x back to y.*

*• Note that*

*y*_{u}*(x, t) ≡ y(x +* *√*

*Δt, t + Δt),*
*y*_{d}*(x, t) ≡ y(x −* *√*

*Δt, t + Δt).*

aChiu (R98723059) (2012).

### Examples

*• The transformation is*

_{r}

*(σ√*

*z)*^{−1}*dz =* 2*√*
*r*
*σ*
for the CIR model.

*• The transformation is*

_{S}

*(σz)*^{−1}*dz =* *ln S*
*σ*
for the Black-Scholes model.

*• The familiar binomial option pricing model in fact*
*discretizes ln S not S.*

### On One-Factor Short Rate Models

*• By using only the short rate, they ignore other rates on*
the yield curve.

*• Such models also restrict the volatility to be a function*
*of interest rate levels only.*

*• The prices of all bonds move in the same direction at*
the same time (their magnitudes may diﬀer).

*• The returns on all bonds thus become highly correlated.*

*• In reality, there seems to be a certain amount of*
independence between short- and long-term rates.

### On One-Factor Short Rate Models (continued)

*• One-factor models therefore cannot accommodate*

nondegenerate correlation structures across maturities.

*• Derivatives whose values depend on the correlation*
structure will be mispriced.

*• The calibrated models may not generate term structures*
as concave as the data suggest.

*• The term structure empirically changes in slope and*
curvature as well as makes parallel moves.

*• This is inconsistent with the restriction that all*

segments of the term structure be perfectly correlated.

### On One-Factor Short Rate Models (concluded)

*• Multi-factor models lead to families of yield curves that*
can take a greater variety of shapes and can better

represent reality.

*• But they are much harder to think about and work with.*

*• They also take much more computer time—the curse of*
dimensionality.

*• These practical concerns limit the use of multifactor*
models to two- or three-factor ones.

### Options on Coupon Bonds

^{a}

*• Assume a one-factor short rate model.*

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

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

*• Let X denote the strike price.*

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

*t*

_{1}

*, t*

_{2}

*, . . . , t*

_{n}*, where t*

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

aJamshidian (1989).

### Options on Coupon Bonds (continued)

*• 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.*

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

*X =*

*n*
*i=1*

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

*numerically for r.*

### Options on Coupon Bonds (continued)

*• The solution is unique for one-factor models whose bond*
*price is a monotonically decreasing function of r.*

*• Let*

*X*_{i}*≡ P (r*^{∗}*, T, t*_{i}*),*

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

*• Note that P (r, T, t** _{i}*)

*≥ X*

*if and only if*

_{i}*≤ 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}*)

*−*

*i*

*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)

### Motivations

*• Recall the diﬃculties facing equilibrium models*
mentioned earlier.

**– They usually require the estimation of the market**
price of risk.

**– They cannot ﬁt the market term structure.**

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

### No-Arbitrage Models

^{a}

*• No-arbitrage models utilize the full information of the*
term structure.

*• They accept the observed term structure as consistent*
with an unobserved and unspeciﬁed equilibrium.

*• From there, arbitrage-free movements of interest rates or*
bond prices over time are modeled.

*• By deﬁnition, the market price of risk must be reﬂected*
in the current term structure; hence the resulting

interest rate process is risk-neutral.

aHo and Lee (1986). Thomas Lee is a “billionaire founder” of Thomas
*H. Lee Partners LP, according to Bloomberg on May 26, 2012.*

### No-Arbitrage Models (concluded)

*• No-arbitrage models can specify the dynamics of*

zero-coupon bond prices, forward rates, or the short rate.

*• Bond price and forward rate models are usually*
non-Markovian (path dependent).

*• In contrast, short rate models are generally constructed*
to be explicitly Markovian (path independent).

*• Markovian models are easier to handle computationally.*

### The Ho-Lee Model

^{a}

*• The short rates at any given time are evenly spaced.*

*• Let p denote the risk-neutral probability that the short*
rate makes an up move.

*• We shall adopt continuous compounding.*

aHo and Lee (1986).

*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,*

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

(133) (see p. 376 of the textbook).

### The Ho-Lee Model (continued)

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

*y*_{d}*(n) ≡ −ln P*_{d}*(t + 1, t + n)*
*n − 1*

*y*_{u}*(n) ≡ −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* *≡*

*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))*

=

*p(1 − p) v*^{2} *+ · · · + v**n*

*n − 1* *.*

### The Ho-Lee Model (concluded)

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

*σ =*

*p(1 − p) v*_{2}*.* (134)

*• The variance of the short rate therefore equals*
*p(1 − p)(r*_{u} *− r*_{d})^{2}*,*

*where r*_{u} *and r*_{d} are the two successor rates.^{a}

aContrast this with the lognormal model (108) on p. 909.

### The Ho-Lee Model: Volatility Term Structure

*• The volatility term structure is composed of*
*κ*_{2}*, κ*_{3}*, . . . .*

**– It is independent of**

*r*_{2}*, r*_{3}*, . . . .*

*• It is easy to compute the v** _{i}*s from the volatility
structure, and vice versa (review p. 1047).

*• The r** _{i}*s can be computed by forward induction.

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

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

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

*P (t, t+n) = (pP*_{u}*(t+1, t+n)+(1−p) P*_{d}*(t+1, t+n)) P (t, t+1).*

*• Combine the above with Eq. (133) on p. 1046 and*
*assume p = 1/2 to obtain*^{a}

*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* ]*,*

(135)

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

2

*1 + exp[ v*2 + *· · · + v**n* ]*.*

(135* ^{}*)

aIn the limit, only the volatility matters.

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

*• The bond price tree combines.*

*• Suppose all v*_{i}*equal some constant v and δ ≡ e*^{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. (134) on p. 1048.*

*• Price derivatives by taking expectations under the*
risk-neutral probability.

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

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

*r(t, t + n) ≡ ln*

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

*.*

*• Its value is either ln* ^{P}^{d}_{P (t,t+n)}* ^{(t+1,t+n)}* or ln

^{P}^{u}

_{P (t,t+n)}*.*

^{(t+1,t+n)}*• Thus the variance of return is*

*Var[ r(t, t + n) ] = p(1 − p)((n − 1) v)*^{2} *= (n − 1)*^{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 unity.

*• 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*
*dr = θ(t) dt + σ dW.*

*• This is Vasicek’s model with the mean-reverting drift*
replaced by a deterministic, time-dependent drift.

*• A nonﬂat term structure of volatilities can be achieved if*
the short rate volatility is also made time varying,

*dr = θ(t) dt + σ(t) dW.*

*• This corresponds to the discrete-time model in which v** _{i}*
are not all identical.

### The Ho-Lee Model: Some Problems

*• Future (nominal) interest rates may be negative.*

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

### Problems with No-Arbitrage Models in General

*• Interest rate movements should reﬂect shifts in the*
model’s state variables (factors) not its parameters.

*• Model parameters, such as the drift θ(t) in the*

continuous-time Ho-Lee model, should be stable over time.

*• But in practice, no-arbitrage models capture yield curve*
shifts through the recalibration of parameters.

**– A new model is thus born everyday.**

### Problems with No-Arbitrage Models in General (concluded)

*• This in eﬀect says the model estimated at some time*
does not describe the term structure of interest rates
and their volatilities at other times.

*• Consequently, a model’s intertemporal behavior is*

suspect, and using it for hedging and risk management may be unreliable.