### 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).* (101)
**– Relation (101) in fact follows from the risk-neutral**

valuation principle.^{a}

aTheorem 16 on p. 457.

### Risk-Neutral Pricing (continued)

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

*• Rewrite Eq. (101) 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))*
]

*.* (102)

### Risk-Neutral Pricing (concluded)

*• Equation (101) on p. 889 can also be expressed as*
*E*_{t}*[ P (t + 1, T ) ] = F (t, t + 1, T ).*

**– Verify that with, e.g., Eq. (96) on p. 884.**

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

### 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.* (103)

*• 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 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 ∆t*
*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* .

### Interest Rate Swaps (continued)

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

*• Simple rates are adopted here.*

*• Hence f** ^{i}* satisﬁes

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

*1 + f*_{i}*∆t.*

### 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) ∆t*]

=

∑*n*
*i=1*

*E*_{t}* ^{π}*
[

*e*^{−}

∫_{ti}

*t* *r(s) ds*

( 1

*P (t*_{i}_{−1}*, t** _{i}*)

*− (1 + c∆t)*)]

=

∑*n*
*i=1*

*[ P (t, t*_{i}* _{−1}*)

*− (1 + c∆t) × P (t, t*

*) ]*

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

∑*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}*) ∆t* *.* (104)

*• 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})

### 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)* (105)
*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 in the text,*

*µ** _{p}* =
(

*−**∂P*

*∂T* *+ µ(r, t)* *∂P*

*∂r* + *σ(r, t)*^{2}
2

*∂*^{2}*P*

*∂r*^{2}
)

*/P,*

(106)

*σ** _{p}* =
(

*σ(r, t)* *∂P*

*∂r*
)

*/P,* (106* ^{′}*)

*subject to P (· , T, T ) = 1.*

### The Term Structure Equation (concluded)

*• Substitute µ*^{p}*and σ** _{p}* into Eq. (105) on p. 902 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.*

(107)

*• 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 (107) applies to all interest rate derivatives,*
the diﬀerence being the terminal and the boundary

### The Binomial Model

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

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

*P*

** P d*
1 *− q*

*q* *j Pu*

### The Binomial Model (continued)

*• Over the period, the bond’s expected rate of return is*
*bµ ≡* *qP u + (1* *− q) P d*

*P* *− 1 = qu + (1 − q) d − 1.*

(108)

*• The variance of that return rate is*

*bσ*^{2} *≡ q(1 − q)(u − d)*^{2}*.* (109)

### The Binomial Model (continued)

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

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

$1.

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

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

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

### The Binomial Model (concluded)

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

*q(1* *− q) =* *(1 + r)* *− d*

*u* *− d* *,* (110)
*which is independent of bond maturity and q.*

**– Recall the BOPM.**

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

*P* *− 1 = pu + (1 − p) d − 1 = r.*

*• The local expectations theory hence holds under the*

### Numerical Examples

*• Assume this spot rate curve:*

Year 1 2

Spot rate 4% 5%

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

4%

* 8%

j 2%

### Numerical Examples (continued)

*• No real-world probabilities are speciﬁed.*

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

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

*• They follow the binomial processes on p. 911.*

### Numerical Examples (continued)

90.703

** 92.593 (= 100/1.08)*

*j 98.039 (= 100/1.02)* 96.154

* 100 j 100

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

### Numerical Examples (continued)

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

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

*• Then*

(1 *− p) ×* *92.593*

*90.703* *+ p* *×* *98.039*

*90.703* *− 1 = 4%,*

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

### Numerical Examples (concluded)

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

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

### Numerical Examples: Fixed-Income Options

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

$95 strike price has the payoﬀs,
*C*

** 0.000*
*j 3.039*

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

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

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

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

*• Consequently,*

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

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

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

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

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

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

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

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

*• The above call option is worth*

*C =* (1 *− p) × 0 + p × 3.039*

*1.04* *≈ 0.93,*

the same as before.

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

### Numerical Examples: Futures and Forward Prices

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

maturity:

*F*

** 92 (= 100 − 8)*
*j 98 (= 100 − 2)*

*• As the futures price F is the expected future payoﬀ (see*
*text or p. 458), F = (1* *− p) × 92 + p × 98 = 93.914.*

*• The forward price for a one-year forward contract on a*
*one-year zero-coupon bond is 90.703/96.154 = 94.331%.*

*• The forward price exceeds the futures price.*^{a}

### Numerical Examples: Mortgage-Backed Securities

*• Consider a 5%-coupon, two-year mortgage-backed*

security without amortization, prepayments, and default risk.

*• Its cash ﬂow and price process are illustrated on p. 920.*

*• Its fair price is*

*M =* (1 *− p) × 102.222 + p × 107.941*

*1.04* *= 100.045.*

*• Identical results could have been obtained via arbitrage*
considerations.

105

*↗*
5

*↗* *↘* *102.222 (= 5 + (105/1.08))*

105 *↗*

0 *M*

105 *↘*

*↘* *↗* *107.941 (= 5 + (105/1.02))*

5

*↘*

105

The left diagram depicts the cash ﬂow; the right diagram

### Numerical Examples: MBSs (continued)

*• Suppose that the security can be prepaid at par.*

*• It will be prepaid only when its price is higher than par.*

*• Prepayment will hence occur only in the “down” state*
when the security is worth 102.941 (excluding coupon).

*• The price therefore follows the process,*
*M*

** 102.222*

j 105

*• The security is worth*

*M =* (1 *− p) × 102.222 + p × 105*

*1.04* *= 99.142.*

### Numerical Examples: MBSs (continued)

*• The cash ﬂow of the principal-only (PO) strip comes*
from the mortgage’s principal cash ﬂow.

*• The cash ﬂow of the interest-only (IO) strip comes from*
the interest cash ﬂow (p. 923(a)).

*• Their prices hence follow the processes on p. 923(b).*

*• The fair prices are*

PO = (1 *− p) × 92.593 + p × 100*

*1.04* *= 91.304,*

IO = (1 *− p) × 9.630 + p × 5*

*1.04* *= 7.839.*

PO: 100 IO: 5

*↗* *↗*

0 5

*↗* *↘* *↗* *↘*

100 5

0 0

0 0

*↘* *↗* *↘* *↗*

100 5

*↘* *↘*

0 0

(a)

*92.593* *9.630*

*↗* *↗*

po io

*↘* *↘*

100 5

(b)

### Numerical Examples: MBSs (continued)

*• Suppose the mortgage is split into half ﬂoater and half*
inverse ﬂoater.

*• Let the ﬂoater (FLT) receive the one-year rate.*

*• Then the inverse ﬂoater (INV) must have a coupon rate*
of

(10% *− one-year rate)*
to make the overall coupon rate 5%.

*• Their cash ﬂows as percentages of par and values are*
shown on p. 925.

FLT: 108 INV: 102

*↗* *↗*

4 6

*↗* *↘* *↗* *↘*

108 102

0 0

0 0

*↘* *↗* *↘* *↗*

104 106

*↘* *↘*

0 0

(a)

104 *100.444*

*↗* *↗*

flt inv

*↘* *↘*

104 106

(b)

### Numerical Examples: MBSs (concluded)

*• On p. 925, the ﬂoater’s price in the up node, 104, is*
*derived from 4 + (108/1.08).*

*• The inverse ﬂoater’s price 100.444 is derived from*
*6 + (102/1.08).*

*• The current prices are*

FLT = 1

2 *×* 104

*1.04* *= 50,*

INV = 1

2 *×* (1 *− p) × 100.444 + p × 106*

*1.04* *= 49.142.*

*Equilibrium Term Structure Models*

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

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

### Introduction

*• This chapter surveys equilibrium models.*

*• Since the spot rates satisfy*

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

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

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

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

### The Vasicek Model

^{a}

*• The short rate follows*

*dr = β(µ* *− r) dt + σ dW.*

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

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

*• Since the process is an Ornstein-Uhlenbeck process,*
*E[ r(T )| r(t) = r ] = µ + (r − µ) e** ^{−β(T −t)}*
from Eq. (55) on p. 517.

### 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)**,* (111)
where

*A(t, T ) =*

exp [

*(B(t,T )**−T +t)(β2µ−σ2/2)*

*β2* *−* *σ2 B(t,T )2*

*4β*

]

*if β* *̸= 0,*

exp [

*σ2 (T**−t)3*
6

]

*if β = 0.*

and

*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*

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 ∆t ≡ T/n and*

*p(r)* *≡* 1

2 + *β(µ* *− r)√*

*∆t*

*2σ* *.*

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

*r(k + 1) = r(k) + σ√*

*∆t ξ(k), 0* *≤ k < n.*

a

### Binomial Vasicek (continued)

*• Above, ξ(k) = ±1 with*

*Prob[ ξ(k) = 1 ] =*

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

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

*.*

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

*• This is consistent with mean reversion.*

### Binomial Vasicek (concluded)

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

*• The binomial tree combines.*

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

*• For a general process Y with nonconstant volatility, the*
resulting binomial tree may not combine, as we will see
next.

### The Cox-Ingersoll-Ross Model

^{a}

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

*dr = β(µ* *− r) dt + σ√*

*r dW.* (112)

*• 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 ∆t ≡ T/n.*

*• Assume µ, β ≥ 0.*

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

### Binomial CIR (continued)

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

*r/σ.*

*• It follows*

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

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

*• Since this new process has a constant volatility, 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 (p. 943).*

*x + 2**√*

*∆t* *f (x + 2**√*

*∆t)*

*↗* *↗*

*x +**√*

*∆t* *f (x +**√*

*∆t)*

*↗* *↘* *↗* *↘*

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

*↘* *↗* *↘* *↗*

*x**−**√*

*∆t* *f (x**−**√*

*∆t)*

*↘* *↘*

*x**− 2**√*

*∆t* *f (x**− 2**√*

*∆t)*

### Binomial CIR (concluded)

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

*r*^{+} *− r*^{−}*.* (113)

**– r**^{+} *≡ f(x +* *√*

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

**– r**^{−}*≡ f(x −* *√*

*∆t) the result of a down move.*

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

### Numerical Examples

*• Consider the process,*

*0.2 (0.04* *− r) dt + 0.1√*

*r dW,*

*for the time interval [ 0, 1 ] given the initial rate*
*r(0) = 0.04.*

*• We shall use ∆t = 0.2 (year) for the binomial*
approximation.

*• See p. 946(a) for the resulting binomial short rate tree*
with the up-move probabilities in parentheses.

0.04 (0.472049150276)

0 . 0 5 9 8 8 8 5 4 3 8 2 (0.44081188025)

0.03155572809 (0.489789553691)

0.02411145618 (0.50975924867)

0.0713328157297 (0.426604457655)

0 . 0 8 3 7 7 7 0 8 7 6 4

0.01222291236 0.01766718427

(0.533083330907) 0.04

(0.472049150276) 0.0494442719102

(0.455865503068)

0.0494442719102 (0.455865503068)

0.03155572809 (0.489789553691)

0 . 0 5 9 8 8 8 5 4 3 8 2

0.04

0.02411145618

(a)

0.992031914837 0.984128889634 0 . 9 7 6 2 9 3 2 4 4 4 0 8 0.968526861261 0.960831229521

0.992031914837 0.984128889634 0 . 9 7 6 2 9 3 2 4 4 4 0 8

0.992031914837 0 . 9 9 0 1 5 9 8 7 9 5 6 5

0.980492588317 0.970995502019 0.961665706744

0 . 9 9 3 7 0 8 7 2 7 8 3 1 0.987391576942 0.981054487259 0 . 9 7 4 7 0 2 9 0 7 7 8 6

0 . 9 8 8 0 9 3 7 3 8 4 4 7 0 . 9 7 6 4 8 6 8 9 6 4 8 5 0.965170249273

0 . 9 9 0 1 5 9 8 7 9 5 6 5 0.980492588317

0.995189317343 0.990276851751

0 . 9 9 3 7 0 8 7 2 7 8 3 1 0.987391576942 0 . 9 8 5 8 3 4 7 2 2 0 3 0.972116454453

0 . 9 8 3 3 8 4 1 7 3 7 5 6

0 . 9 8 8 0 9 3 7 3 8 4 4 7

0.995189317343

### Numerical Examples (continued)

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

*• Since the root has x = 2*√

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

*∆t = 4.4472135955.*

*• Use the inverse transformation to obtain the short rate*
*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.

*• This phenomenon agrees with mean reversion.*

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

### A General Method for Constructing Binomial Models

^{a}

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

*• Make sure the binomial model’s drift and diﬀusion*

converge to the above process by setting the probability of an up move to

*α(y, t) ∆t + y* *− y*^{d}
*y*_{u} *− y*^{d} *.*

*• Here y*^{u} *≡ y + σ(y, t)√*

*∆t and y*_{d} *≡ y − σ(y, t)√*

*∆t*
*represent the two rates that follow the current rate y.*

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

*∆t .*

### A General Method (continued)

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

*∆t* *− σ(y*^{u}*, t + ∆t)√*

*∆t*

*̸= −σ(y, t)√*

*∆t + σ(y*_{d}*, t + ∆t)√*

*∆t*
in general.

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

### A General Method (continued)

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

∫ *y*

*σ(z, t)*^{−1}*dz.*

*• Then x follows dx = m(y, t) dt + dW for some m(y, t)*
(see text).

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

*• The transformation that turns a 1-dim stochastic process*
into one with a constant diﬀusion term is unique.^{a}

aChiu (R98723059) (2012).

### A General Method (concluded)

*• The probability of an up move remains*

*α(y(x, t), t) ∆t + 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 +* *√*

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

*∆t, t + ∆t) .*

### Examples

*• The transformation is*

∫ *r*

*(σ√*

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

*• The transformation is*

∫ *S*

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

*• The familiar binomial option pricing model in fact*
*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-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}*)

### 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, t** ^{i}*)

*≥ X*

^{i}*if and only 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}*)

*−*∑

*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}

*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

### 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}^{)}

(114)

### 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}

*− 1* *.*

### The Ho-Lee Model (concluded)

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

*σ =* √

*p(1* *− p) v*^{2}*.* (115)

*• 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.

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

*• The volatility term structure is composed of κ*^{2}*, κ*_{3}*, . . . .*
**– It is independent of the r*** _{i}*.

*• It is easy to compute the v** ^{i}*s from the volatility
structure, and vice versa.

*• 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. (114) on p. 968 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}*,*

(116)

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

2

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

*.*

(116* ^{′}*)

### 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 equals v/2 by Eq. (115) on*
p. 970.

*• Price derivatives by taking expectations under the*

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

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

*r(t, t + n)* *≡ ln*

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

)
*.*

*• Its value is either ln* ^{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*
*(n* *− 1)(m − 1) σ*^{2} (see text).

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

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

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

*• The continuous-time limit of the Ho-Lee model is*
*dr = θ(t) dt + σ dW.*

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

*• A nonﬂat term structure of volatilities can be achieved if*
the short rate volatility is also made time varying, i.e.,
*dr = θ(t) dt + σ(t) dW .*

*• This corresponds to the discrete-time model in which 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.