### Sample Term Structure

*• We shall construct interest rate trees consistent with the*
sample term structure in the following table.

*• Assume the short rate volatility is such that*
*v ≡ r*_{h}*/r*_{`}*= 1.5, independent of time.*

Period 1 2 3

Spot rate (%) 4 *4.2* *4.3*

One-period forward rate (%) 4 *4.4* *4.5*

Discount factor *0.96154* *0.92101* *0.88135*

### An Approximate Calibration Scheme

*• Start with the implied one-period forward rates and*
then equate the expected short rate with the forward
rate (see Exercise 5.6.6 in text).

*• For the first period, the forward rate is today’s*
one-period spot rate.

*• In general, let f*_{j}*denote the forward rate in period j.*

*• This forward rate can be derived from the market*
*discount function via f*_{j}*= (d(j)/d(j + 1)) − 1 (see*
Exercise 5.6.3 in text).

### An Approximate Calibration Scheme (continued)

*• Since the ith short rate, 1 ≤ i ≤ j, occurs with*
probability 2* ^{−(j−1)}* ¡

_{j−1}*i−1*

¢, this means

X*j*
*i=1*

2^{−(j−1)}

µ*j − 1*
*i − 1*

¶

*r*_{j}*v*_{j}^{i−1}*= f*_{j}*.*

*• Thus*

*r** _{j}* =

µ 2

*1 + v*_{j}

¶_{j−1}

*f*_{j}*.* (78)

*• The binomial interest rate tree is trivial to set up.*

### An Approximate Calibration Scheme (concluded)

*• The ensuing tree for the sample term structure appears*
in figure next page.

*• For example, the price of the zero-coupon bond paying*

$1 at the end of the third period is

1

4 *×* 1
*1.04* *×*

µ 1

*1.0352* *×*

µ 1

*1.0288* + 1
*1.0432*

¶

+ 1

*1.0528* *×*

µ 1

*1.0432* + 1
*1.0648*

¶¶

or 0.88155, which exceeds discount factor 0.88135.

*• The tree is thus not calibrated.*

*• Indeed, this bias is inherent (see text).*

4.0%

3.52%

2.88%

5.28%

4.32%

6.48%

Baseline rates

A C

B B

C

C

D

D

D

D

period 2 period 3 period 1

4.0% 4.4% 4.5%

Implied forward rates:

### Issues in Calibration

*• The model prices generated by the binomial interest rate*
tree should match the observed market prices.

*• Perhaps the most crucial aspect of model building.*

*• Treat the backward induction for the model price of the*
*m-period zero-coupon bond as computing some function*
*of the unknown baseline rate r*_{m}*called f (r** _{m}*).

*• A root-finding method is applied to solve f (r*_{m}*) = P for*
*r*_{m}*given the zero’s price P and r*_{1}*, r*_{2}*, . . . , r** _{m−1}*.

*• This procedure is carried out for m = 1, 2, . . . , n.*

*• It runs in cubic time, hopelessly slow.*

### Binomial Interest Rate Tree Calibration

*• Calibration can be accomplished in quadratic time by*
the use of forward induction (Jamshidian, 1991).

*• The scheme records how much $1 at a node contributes*
to the model price.

*• This number is called the state price.*

– It is the price of a state contingent claim that pays

$1 at that particular node (state) and 0 elsewhere.

*• The column of state prices will be established by moving*
*forward from time 1 to time n.*

### Binomial Interest Rate Tree Calibration (continued)

*• Suppose we are at time j and there are j + 1 nodes.*

*– The baseline rate for period j is r ≡ r** _{j}*.

*– The multiplicative ratio be v ≡ v*

*.*

_{j}*– P*_{1}*, P*_{2}*, . . . , P** _{j}* are the state prices a period prior,

*corresponding to rates r, rv, . . . , rv*

*.*

^{j−1}*• By definition,* P_{j}

*i=1* *P*_{i}*is the price of the (j − 1)-period*
zero-coupon bond.

### Binomial Interest Rate Tree Calibration (continued)

*• One dollar at time j has a known market value of*
*1/[ 1 + S(j) ]*^{j}*, where S(j) is the j-period spot rate.*

*• Alternatively, this dollar has a present value of*
*g(r) ≡* *P*_{1}

*(1 + r)* + *P*_{2}

*(1 + rv)* + *P*_{3}

*(1 + rv*^{2}) *+ · · · +* *P*_{j}

*(1 + rv** ^{j−1}*)

*.*

*• So we solve*

*g(r) =* 1

*[ 1 + S(j) ]** ^{j}* (79)

*for r.*

### Binomial Interest Rate Tree Calibration (continued)

*• Given a decreasing market discount function, a unique*
*positive solution for r is guaranteed.*

*• The state prices at time j can now be calculated (see*
figure (a) next page).

*• We call a tree with these state prices a binomial state*
price tree (see figure (b) next page).

*• The calibrated tree is depicted on p. 761.*

A C B

B

C

C

D

D D

D 4.00%

3.526%

2.895%

0.480769

0.460505

0.228308 A

C

B

C

C

D

D D D

B

period 2 period 3 period 1

4.0% 4.4% 4.5%

Implied forward rates:

0.480769

1

0.112832

(b)

0.333501

0.327842

0.107173 0.232197

(a) 1

r

rv

P rv

2

2 1

### a f

P r

P rv

1 2

2 1

### a f

2 1### a f

P r

1

2 1

### a f

P1

P2

4.00%

3.526%

2.895%

5.289%

4.343%

6.514%

A

C B

C

C

D

D D D

B

period 2 period 3 period 1

4.0% 4.4% 4.5%

Implied forward rates:

### Binomial Interest Rate Tree Calibration (concluded)

*• The Newton-Raphson method can be used to solve for*
*the r in Eq. (79) on p. 758 as g*^{0}*(r) is easy to evaluate.*

*• The monotonicity and the convexity of g(r) also*
facilitate root finding.

*• The total running time is O(Cn*^{2}*), where C is the*
maximum number of times the root-finding routine
*iterates, each consuming O(n) work.*

*• With a good initial guess, the Newton-Raphson method*
converges in only a few steps^{a}

aLyuu (1999).

### A Numerical Example

*• One dollar at the end of the second period should have a*
present value of 0.92101 by the sample term structure.

*• The baseline rate for the second period, r*_{2}, satisfies
*0.480769*

*1 + r*_{2} + *0.480769*

*1 + 1.5 × r*_{2} *= 0.92101.*

*• The result is r*_{2} *= 3.526%.*

*• This is used to derive the next column of state prices*
*shown in figure (b) on p. 760 as 0.232197, 0.460505, and*
*0.228308.*

*• Their sum gives the correct market discount factor*
*0.92101.*

### A Numerical Example (concluded)

*• The baseline rate for the third period, r*_{3}, satisfies
*0.232197*

*1 + r*_{3} + *0.460505*

*1 + 1.5 × r*_{3} + *0.228308*

*1 + (1.5)*^{2} *× r*_{3} *= 0.88135.*

*• The result is r*_{3} *= 2.895%.*

*• Now, redo the calculation on p. 753 using the new rates:*

1

4 *×* 1

*1.04* *× [* 1

*1.03526* *× (* 1

*1.02895* + 1

*1.04343*) + 1

*1.05289* *× (* 1

*1.04343* + 1

*1.06514**)],*

which equals 0.88135, an exact match.

*• The tree on p. 761 prices without bias the benchmark*
securities.

*• The term structure dynamics is shown on p. 765.*

*[ 0.971865 ]*

" *%*

*0.965941*
*0.932250*

#

*%* *&*

*0.96154*
*0.92101*
*0.88135*

*[ 0.958378 ]*

*&* *%*

"

*0.949767*
*0.900959*

#

*&*

*[ 0.938844 ]*

### Spread of Nonbenchmark Bonds

*• Model prices calculated by the calibrated tree as a rule*
do not match market prices of nonbenchmark bonds.

*• The incremental return over the benchmark bonds is*
called spread.

*• We look for the spread that, when added uniformly over*
the short rates in the tree, makes the model price equal
the market price.

*• We will apply the spread concept to option-free bonds*
here.

### Spread of Nonbenchmark Bonds (continued)

*• We illustrate the idea with an example.*

*• Start with the tree on p. 768.*

*• Consider a security with cash flow C*_{i}*at time i for*
*i = 1, 2, 3.*

*• Its model price is p(s), which is equal to*

1

*1.04 + s* *×*

"

*C1 +* 1

2 *×* 1

*1.03526 + s* *×*
Ã

*C2 +* 1
2

Ã *C3*

*1.02895 + s* + *C3*
*1.04343 + s*

!!

+

1

2 *×* 1

*1.05289 + s* *×*
Ã

*C2 +* 1
2

Ã *C3*

*1.04343 + s* + *C3*
*1.06514 + s*

!!#

*.*

*• Given a market price of P , the spread is the s that*
*solves P = p(s).*

4.00%+s

3.526%+s

2.895%+s

5.289%+s

4.343%+s

6.514%+s

A C

B

period 2 period 3 period 1

4.0% 4.4% 4.5%

Implied forward rates:

B

C

C

D

D D D

### Spread of Nonbenchmark Bonds (continued)

*• The model price p(s) is a monotonically decreasing,*
*convex function of s.*

*• We will employ the Newton-Raphson root-finding*
*method to solve p(s) − P = 0 for s.*

*• But a quick look at the equation above reveals that*
*evaluating p*^{0}*(s) directly is infeasible.*

*• Fortunately, the tree can be used to evaluate both p(s)*
*and p*^{0}*(s) during backward induction.*

### Spread of Nonbenchmark Bonds (continued)

*• Consider an arbitrary node A in the tree associated with*
*the short rate r.*

*• In the process of computing the model price p(s), a*
*price p*_{A}*(s) is computed at A.*

*• Prices computed at A’s two successor nodes B and C are*
*discounted by r + s to obtain p*_{A}*(s) as follows,*

*p*_{A}*(s) = c +* *p*_{B}*(s) + p*_{C}*(s)*
*2(1 + r + s)* *,*
*where c denotes the cash flow at A.*

### Spread of Nonbenchmark Bonds (continued)

*• To compute p*^{0}_{A}*(s) as well, node A calculates*
*p*^{0}_{A}*(s) =* *p*^{0}_{B}*(s) + p*^{0}_{C}*(s)*

*2(1 + r + s)* *−* *p*_{B}*(s) + p*_{C}*(s)*

*2(1 + r + s)*^{2} *.* (80)

*• This is easy if p*^{0}_{B}*(s) and p*^{0}_{C}*(s) are also computed at*
nodes B and C.

*• Apply the above procedure inductively to yield p(s) and*
*p*^{0}*(s) at the root (p. 772).*

*• This is called the differential tree method.*^{a}

aLyuu (1999).

1 1 c

### a

s### f

1 1 cv

### a

s### f

1 1c cv^{2} sh

1 1 a

### a

s### f

1 1 bv

### a

s### f

1 1 b

### a

s### f

^{1 1}

^{a}

^{b}

^{s}

^{f}

^{2}

1 1a c sf^{2}

1 1a cv sf^{2}

1 1

### c

cv^{2}s

### h

^{2}

1 1a bv sf^{2}

1 1a a sf^{2}

A C

B B

C

C

D

D D D

A C

B B

C

C

D

D D D

(a) (b)

A

C

(c)

p s_{B}a f

B

p s c p s p s r s

A

B C

a f a f

( ) ( ) 2 1

p s p s p s r s

p s p s r s

A

B C B C

a f a f a f

( ) ( ) ( ) ( )

2 1 2 1 ^{2}

p_{C}a fs
p_{C}a fs
p s_{B}a f

a

b

c

a

b

c

r

### Spread of Nonbenchmark Bonds (continued)

*• Let C represent the number of times the tree is*
*traversed, which takes O(n*^{2}) time.

*• The total running time is O(Cn*^{2}).

*• In practice C is a small constant.*

*• The memory requirement is O(n).*

### Spread of Nonbenchmark Bonds (continued)

Number of Running Number of Number of Running Number of
*partitions n* time (s) iterations partitions time (s) iterations

500 7.850 5 10500 3503.410 5

1500 71.650 5 11500 4169.570 5

2500 198.770 5 12500 4912.680 5

3500 387.460 5 13500 5714.440 5

4500 641.400 5 14500 6589.360 5

5500 951.800 5 15500 7548.760 5

6500 1327.900 5 16500 8502.950 5

7500 1761.110 5 17500 9523.900 5

8500 2269.750 5 18500 10617.370 5

9500 2834.170 5 . . . . . . . . . . . .

75MHz Sun SPARCstation 20.

### Spread of Nonbenchmark Bonds (concluded)

*• Consider a three-year, 5% bond with a market price of*
100.569.

*• Assume the bond pays annual interest.*

*• The spread can be shown to be 50 basis points over the*
tree (p. 776).

*• Note that the idea of spread does not assume parallel*
shifts in the term structure.

*• It also differs from the yield spread and static spread of*
the nonbenchmark bond over an otherwise identical

benchmark bond.

4.50%

100.569 A

C B

5 5 105

Cash flows:

B

C

C

D

D D 4.026% D

3.395%

5.789%

4.843%

7.014%

105

105

105

105 106.552

105.150

103.118 106.754

103.436

### More Applications of the Differential Tree: Calibrating Black-Derman-Toy (in seconds)

Number Running Number Running Number Running

of years time of years time of years time

3000 398.880 39000 8562.640 75000 26182.080 6000 1697.680 42000 9579.780 78000 28138.140 9000 2539.040 45000 10785.850 81000 30230.260 12000 2803.890 48000 11905.290 84000 32317.050 15000 3149.330 51000 13199.470 87000 34487.320 18000 3549.100 54000 14411.790 90000 36795.430 21000 3990.050 57000 15932.370 120000 63767.690 24000 4470.320 60000 17360.670 150000 98339.710 27000 5211.830 63000 19037.910 180000 140484.180 30000 5944.330 66000 20751.100 210000 190557.420 33000 6639.480 69000 22435.050 240000 249138.210 36000 7611.630 72000 24292.740 270000 313480.390

75MHz Sun SPARCstation 20, one period per year.

### More Applications of the Differential Tree: Calculating Implied Volatility (in seconds)

American call American put

Number of Running Number of Number of Running Number of partitions time iterations partitions time iterations

100 0.008210 2 100 0.013845 3

200 0.033310 2 200 0.036335 3

300 0.072940 2 300 0.120455 3

400 0.129180 2 400 0.214100 3

500 0.201850 2 500 0.333950 3

600 0.290480 2 600 0.323260 2

700 0.394090 2 700 0.435720 2

800 0.522040 2 800 0.569605 2

Intel 166MHz Pentium, running on Microsoft Windows 95.

### Fixed-Income Options

*• Consider a two-year 99 European call on the three-year,*
5% Treasury.

*• Assume the Treasury pays annual interest.*

*• From p. 780 the three-year Treasury’s price minus the $5*
interest could be $102.046, $100.630, or $98.579 two

years from now.

*• Since these prices do not include the accrued interest,*
we should compare the strike price against them.

*• The call is therefore in the money in the first two*

scenarios, with values of $3.046 and $1.630, and out of the money in the third scenario.

A

C B

B

C

C

D

D D D

105

105

105

105 4.00%

101.955 1.458

3.526%

102.716 2.258

2.895%

102.046 3.046

5.289%

99.350 0.774

4.343%

100.630 1.630

6.514%

98.579 0.000

(a)

A

C B

B

C

C

D

D D D

105

105

105

105 4.00%

101.955 0.096

3.526%

102.716 0.000

2.895%

102.046 0.000

5.289%

99.350 0.200

4.343%

100.630 0.000

6.514%

98.579 0.421

(b)

### Fixed-Income Options (continued)

*• The option value is calculated to be $1.458 on p. 780(a).*

*• European interest rate puts can be valued similarly.*

*• Consider a two-year 99 European put on the same*
security.

*• At expiration, the put is in the money only if the*

Treasury is worth $98.579 without the accrued interest.

*• The option value is computed to be $0.096 on p. 780(b).*

### Fixed-Income Options (concluded)

*• The present value of the strike price is*
*PV(X) = 99 × 0.92101 = 91.18.*

*• The Treasury is worth B = 101.955.*

*• The present value of the interest payments during the*
life of the options is

*PV(I) = 5 × 0.96154 + 5 × 0.92101 = 9.41275.*

*• The call and the put are worth C = 1.458 and*
*P = 0.096, respectively.*

*• Hence the put-call parity is preserved:*

*C = P + B − PV(I) − PV(X).*

### Delta or Hedge Ratio

*• How much does the option price change in response to*
changes in the price of the underlying bond?

*• This relation is called delta (or hedge ratio) defined as*
*O*_{h} *− O*_{`}

*P*_{h} *− P*_{`}*.*

*• In the above P*_{h} *and P** _{`}* denote the bond prices if the
short rate moves up and down, respectively.

*• Similarly, O*_{h} *and O** _{`}* denote the option values if the
short rate moves up and down, respectively.

### Delta or Hedge Ratio (concluded)

*• Since delta measures the sensitivity of the option value*
to changes in the underlying bond price, it shows how to
hedge one with the other.

*• Take the call and put on p. 780 as examples.*

*• Their deltas are*

*0.774 − 2.258*

*99.350 − 102.716* *= 0.441,*
*0.200 − 0.000*

*99.350 − 102.716* *= −0.059,*

respectively.

### Volatility Term Structures

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

*• Consider an n-period zero-coupon bond.*

*• First find its yield to maturity y*_{h} *(y** _{`}*, respectively) at
the end of the initial period if the rate rises (declines,
respectively).

*• The yield volatility for our model is defined as*
*(1/2) ln(y*_{h}*/y** _{`}*).

### Volatility Term Structures (continued)

*• For example, based on the tree on p. 761, the two-year*
zero’s yield at the end of the first period is 5.289% if the
rate rises and 3.526% if the rate declines.

*• Its yield volatility is therefore*
1

2 ln

µ*0.05289*
*0.03526*

¶

*= 20.273%.*

### Volatility Term Structures (continued)

*• Consider the three-year zero-coupon bond.*

*• If the rate rises, the price of the zero one year from now*
will be

1

2 *×* 1

*1.05289* *×*

µ 1

*1.04343* + 1
*1.06514*

¶

*= 0.90096.*

*• Thus its yield is*

q 1

*0.90096* *− 1 = 0.053531.*

*• If the rate declines, the price of the zero one year from*
now will be

1

2 *×* 1

*1.03526* *×*

µ 1

*1.02895* + 1
*1.04343*

¶

*= 0.93225.*

### Volatility Term Structures (continued)

*• Thus its yield is*

q 1

*0.93225* *− 1 = 0.0357.*

*• The yield volatility is hence*
1

2 ln

µ*0.053531*
*0.0357*

¶

*= 20.256%,*
slightly less than the one-year yield volatility.

*• This is consistent with the reality that longer-term*
bonds typically have lower yield volatilities than
shorter-term bonds.

*• The procedure can be repeated for longer-term zeros to*
obtain their yield volatilities.

0 100 200 300 400 500 Time period

0.1 0.101 0.102 0.103 0.104

Spot rate volatility

Short rate volatility given flat %10 volatility term structure.

### Volatility Term Structures (continued)

*• We started with v** _{i}* and then derived the volatility term
structure.

*• In practice, the steps are reversed.*

*• The volatility term structure is supplied by the user*
along with the term structure.

*• The v** _{i}*—hence the short rate volatilities via Eq. (76) on

*p. 741—and the r*

*are then simultaneously determined.*

_{i}*• The result is the Black-Derman-Toy model.*

### Volatility Term Structures (concluded)

*• Suppose the user supplies the volatility term structure*
*which results in (v*_{1}*, v*_{2}*, v*_{3}*, . . . ) for the tree.*

*• The volatility term structure one period from now will*
*be determined by (v*_{2}*, v*_{3}*, v*_{4}*, . . . ) not (v*_{1}*, v*_{2}*, v*_{3}*, . . . ).*

*• The volatility term structure supplied by the user is*
hence not maintained through time.

*• This issue will be addressed by other types of (complex)*
models.

*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)*

### 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 (the instantaneous spot rate,
*or short rate, at time t).*

*P (t, T ): the present value at time t of one dollar 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—in other words, the (T − t)-period spot rate at*
*time t.*

*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 .*

### 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.* (81)
*– The forward price equals the future value at time T*

of the underlying asset (see text for proof).

*• Equation (81) holds whether the model is discrete-time*
or continuous-time.

### Fundamental Relations (continued)

*• Forward rates and forward prices are related*
definitionally by

*f (t, T, L) =*

µ 1

*F (t, T, T + L)*

¶_{1/L}

*− 1 =*

µ *P (t, T )*

*P (t, T + L)*

¶_{1/L}

*− 1*
(82)

in discrete time.

*– f (t, T, L) =* _{L}^{1} (_{P (t,T +L)}^{P (t,T )}*− 1) is the analog to*
Eq. (82) under simple compounding.

### 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* (83)

by Eq. (81) on p. 800.

*• Furthermore,*

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

*∆t* *→ −∂ ln P (t, T )*

*∂T*

*= −∂P (t, T )/∂T*
*P (t, T )* *.*

### Fundamental Relations (continued)

*• So*

*f (t, T ) ≡ lim*

*∆t→0**f (t, T, ∆t) = −∂P (t, T )/∂T*

*P (t, T )* *, t ≤ T.*

(84)

*• Because Eq. (84) is equivalent to*

*P (t, T ) = e*^{−}^{R}^{t}^{T}^{f (t,s) ds}*,* (85)
the spot rate curve is

*r(t, T ) =* 1
*T − t*

Z _{T}

*t*

*f (t, s) ds.*

### Fundamental Relations (concluded)

*• The discrete analog to Eq. (85) 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).* (86)
– Relation (86) in fact follows from the risk-neutral

valuation principle.^{a}

aTheorem 14 on p. 429.

### Risk-Neutral Pricing (continued)

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

*• Rewrite Eq. (86) as*

*E*_{t}^{π}*[ P (t + 1, T ) ]*

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

– It says the current spot rate curve equals the expected spot rate curve 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))*

¸

*.* (87)

### Risk-Neutral Pricing (concluded)

*• Equation (86) on p. 805 can also be expressed as*
*E*_{t}*[ P (t + 1, T ) ] = F (t, t + 1, T ).*

*• 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}

h

*e*^{−}^{R}^{t}^{T}* ^{r(s) ds}*
i

*, t < T.* (88)

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

*• When the local expectations theory holds, riskless*
arbitrage opportunities are impossible.

### 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*_{0}.

### 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*
X*n*

*i=1*

*E*_{t}* ^{π}*
h

*e*^{−}^{R}^{t}^{ti}^{r(s) ds}*(f*_{i−1}*− c) ∆t*
i

=

X*n*
*i=1*

*E*_{t}^{π}

·

*e*^{−}^{R}^{t}^{ti}^{r(s) ds}

µ 1

*P (t*_{i−1}*, t** _{i}*)

*− (1 + c∆t)*

¶¸

=

X*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*
X*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}*)

P_{n}

*i=1* *P (t, t*_{i}*) ∆t* *.* (89)

*• 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 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*
b

*µ ≡* *qP u + (1 − q) P d*

*P* *− 1 = qu + (1 − q) d − 1.*

(90)

*• The variance of that return rate is*
b

*σ*^{2} *≡ q(1 − q)(u − d)*^{2}*.* (91)

*• The bond whose maturity is only 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.

### The Binomial Model (continued)

*• 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*p

*q(1 − q) =* *(1 + r) − d*

*u − d* *,* (92)

*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. 821.*

### 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 an up*
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), F = (1 − p) × 92 + p × 98 = 93.914.*

*• On the other hand, the forward price for a one-year*

forward contract on a one-year zero-coupon bond equals
*90.703/96.154 = 94.331%.*

*• The forward price exceeds the futures price.*

### 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. 830.*

*• 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 illustrates the price process.

### 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. 833(a)).

*• Their prices hence follow the processes on p. 833(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)

*The price 9.630 is derived from 5 + (5/1.08).*

### 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. 835.

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. 835, 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. (52) on p. 485.

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)**,* (93)
where

*A(t, T ) =*

exp

·

*(B(t,T )−T +t)(β2µ−σ2/2)*

*β2* *−* ^{σ2B(t,T )2}_{4β}

¸

*if β 6= 0,*

exp

·

*σ2(T −t)3*
6

¸

*if β = 0.*

and

*B(t, T ) =*

*1−e*^{−β(T −t)}

*β* *if β 6= 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 6= 0.*

*• Even if β 6= 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.

*• Iindeed, 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

µ *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 6= 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.*

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.*

*• For a general process Y with nonconstant volatility, the*
resulting binomial tree may not combine.

### The Cox-Ingersoll-Ross Model

^{a}

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

*dr = β(µ − r) dt + σ√*

*r dW.* (94)

*• 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. 853).*