### Set Things in Motion

*• The abstract process is now in place.*

*• We need the annualized rates of return of the riskless*
bonds that make up the benchmark yield curve and
their volatilities.

*• In the U.S., for example, the on-the-run yield curve*

obtained by the most recently issued Treasury securities may be used as the benchmark curve.

### Set Things in Motion (concluded)

*• The term structure of (yield) volatilities*^{a} can be
estimated from:

**– Historical data (historical volatility).**

**– Or interest rate option prices such as cap prices**
(implied volatility).

*• The binomial tree should be found that is consistent*
with both term structures.

*• Here we focus on the term structure of interest rates.*

aOr simply the volatility (term) structure.

### Model Term Structures

*• The model price is computed by backward induction.*

*• Refer back to the ﬁgure on p. 928.*

*• Given that the values at nodes B and C are P*B *and P*_{C},
respectively, the value at node A is then

*P*_{B} *+ P*_{C}

*2(1 + r)* *+ cash flow at node A.*

*• We compute the values column by column without*

explicitly expanding the binomial interest rate tree (see next page).

*• This takes O(n*^{2}*) time and O(n) space.*

HL

A C

B

Cash flows:

B

C

C

D

D

D D

+

+ 2 2 H

1 2

2 1

## = B

+ 2 2

HL

2 3

2 1

## = B

H

HL^{2}

+ 2 2

HL

3 4

2 1

## ?

2## D

2_{1}

2_{2}

2_{3}

2_{4}

### Term Structure Dynamics

*• An n-period zero-coupon bond’s price can be computed*
*by assigning $1 to every node at period n and then*

applying backward induction.

*• Repeating this step for n = 1, 2, . . . , one obtains the*
market discount function implied by the tree.

*• The tree therefore determines a term structure.*

*• It also contains a term structure dynamics.*

**– Taking any node in the tree as the current state**
induces a binomial interest rate tree and, again, a
term structure.

### Sample Term Structure

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

**– This was called calibration (the reverse of pricing).**

*• 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.*

*• 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 r*_{j}*v*_{j}* ^{i−1}*, 1

*≤ i ≤ j, occurs with*probability 2

^{−(j−1)}

_{j−1}*i−1*

, this means

*j*
*i=1*

2^{−(j−1)}

*j* *− 1*
*i* *− 1*

*r*_{j}*v*_{j}^{i−1}*= f*_{j}*.*

*• Thus*

*r** _{j}* =

2

*1 + v*_{j}

_{j−1}

*f*_{j}*.* (123)

*• This binomial interest rate tree is trivial to set up, in*
*O(n) time.*

### An Approximate Calibration Scheme (continued)

*• 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* *×*

1

*1.0352* *×*

1

*1.0288* + 1
*1.0432*

+ 1

*1.0528* *×*

1

*1.0432* + 1
*1.0648*

or 0.88155, which exceeds discount factor 0.88135.

*• The tree is thus not calibrated.*

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:

### An Approximate Calibration Scheme (concluded)

*• Indeed, this bias is inherent: The tree overprices the*
bonds.^{a}

*• Suppose we replace the baseline rates r*_{j}*by r*_{j}*v** _{j}*.

*• Then the resulting tree underprices the bonds.*^{b}

*• The true baseline rates are thus bounded between r*_{j}*and r*_{j}*v** _{j}*.

aSee Exercise 23.2.4 in text.

bLyuu and Wang (F95922018) (2009, 2011).

### 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*
*f (r*_{m}*) of the unknown baseline rate r*_{m}*for period 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 O(n*^{3}) time.

### Binomial Interest Rate Tree Calibration

*• Calibration can be accomplished in O(n*^{2}) time by the
use of forward induction.^{a}

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

*• This number is called the state price, the Arrow-Debreu*
price, or Green’s function.

**– 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 0 to time n.*

aJamshidian (1991).

### Binomial Interest Rate Tree Calibration (continued)

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

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

**– The multiplicative ratio is v ≡ v***.*

_{j}* – P*1

*, P*

_{2}

*, . . . , P*

_{j}*are the known state prices at earlier*

*time j*

*− 1, corresponding to rates r, rv, . . . , rv*

*for*

^{j−1}*period j.*

*• By deﬁnition,* _{j}

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

*• We want to ﬁnd r based on P*_{1}*, P*_{2}*, . . . , P** _{j}* and the price

*of the j-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}*)
(see next plot).

*• So we solve*

*g(r) =* 1

*[ 1 + S(j) ]** ^{j}* (124)

*for r.*

1

1

*P*

_{i}*rv*

^{i}^{}

^{1}

### 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*
panel (a) next page).

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

*• The calibrated tree is depicted on p. 954.*

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

H

HL

2 HL

2

2 1

### = B

2 H

2 HL

1 2

2 1

### = B = B

2 1 2H

1

2 1

### = B

21

2_{2}

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. (124) on p. 950 as g*^{}*(r) is easy to evaluate.*

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

*• The total running time is O(n*^{2}), as each root-ﬁnding
*routine consumes O(j) time.*

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

a*Such as the r**j* = (_{1+v}^{2}

*j* )^{j−1}*f**j* on p. 943.

bLyuu (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 panel (b) on p. 953 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. 944 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. 954 prices without bias the benchmark*
securities.

### 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.

*• If we add the spread uniformly over the short rates in*
the tree, the model price will equal the market price.

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

### Spread of Nonbenchmark Bonds (continued)

*• We illustrate the idea with an example.*

*• Start with the tree on p. 960.*

*• 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* *×*

*C2 +* 1
2

*C3*

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

+

1

2 *×* 1

*1.05289 + s* *×*

*C2 +* 1
2

*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%+I

3.526%+I

2.895%+I

5.289%+I

4.343%+I

6.514%+I

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 for p(s) reveals that*
*evaluating p*^{}*(s) directly is infeasible.*

*• Fortunately, the tree can be used to evaluate both p(s)*
*and p*^{}*(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*^{}_{A}*(s) as well, node A calculates*
*p*^{}_{A}*(s) =* *p*^{}_{B}*(s) + p*^{}_{C}*(s)*

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

(125)

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

*• When A is a terminal node, simply use the payoﬀ*
*function for p*_{A}*(s).*^{a}

aContributed by Mr. Chou, Ming-Hsin (R02723073) on May 28, 2014.

1 1 ? I

### = B

1 1 ?L I

### = B

1 1? ?L^{2} ID

1 1 = I

### = B

1 1 >L I

### = B

1 1 > I

### = B

^{1 1}

^{=}

^{> I}

^{B}

^{2}

1 1= ? IB^{2}

1 1= ?L IB^{2}

1 1

### ?

?L^{2}I

### D

^{2}

1 1= >L IB^{2}

1 1= = IB^{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)

F I*= B

B F I ? F I F I

)= B *^{2 1}=^{( )} H I+^{( )}B

F I F I F I H I

F I F I

)= B = B = B*^{2 1}^{( )} +^{( )} ^{2 1}*^{( )}H I+^{( )}^{2}

F I+= B

F I+= B

F I*= B

=

>

?

=

>

?

H

### Spread of Nonbenchmark Bonds (continued)

*• Apply the above procedure inductively to yield p(s) and*
*p*^{}*(s) at the root (p. 964).*

*• This is called the diﬀerential tree method.*^{a}
**– Similar ideas can be found in automatic**

diﬀerentiation (AD)^{b} and backpropagation^{c} in
artiﬁcial neural networks.

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

*• The memory requirement is O(n).*

aLyuu (1999).

bRall (1981).

c

### 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. 968).

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

*• It also diﬀers from the yield spread (p. 121) and static*
spread (p. 122) 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: Calculating Implied Volatility (in seconds)

^{a}

**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.

aLyuu (1999).

### Fixed-Income Options

*• Consider a 2-year 99 European call on the 3-year, 5%*

Treasury.

*• Assume the Treasury pays annual interest.*

*• From p. 971 the 3-year Treasury’s price minus the $5*
interest at year 2 could be $102.046, $100.630, or

$98.579 two years from now.

**– The accrued interest is not included as it belongs to**
the original bondholder.

*• Now compare the strike price against the bond prices.*

*• The call is in the money in the ﬁrst two scenarios out of*
the money in the third.

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. 971(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 when the*
Treasury is worth $98.579 without the accrued interest.

*• The option value is computed to be $0.096 on p. 971(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)

*• Delta measures the sensitivity of the option value to*
changes in the underlying bond price.

*• So it shows how to hedge one with the other.*

*• Take the call and put on p. 971 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,*

### 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 short rate rises

(declines, respectively).

*• The yield volatility for our model is deﬁned as*
1

2 ln

*y*_{h}
*y*_{}

*.* (126)

### Volatility Term Structures (continued)

*• For example, based on the tree on p. 954, 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

*0.05289*
*0.03526*

*= 20.273%.*

### Volatility Term Structures (continued)

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

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

1

2 *×* 1

*1.05289* *×*

1

*1.04343* + 1
*1.06514*

*= 0.90096.*

*• Thus its yield is*

*0.90096*1 *− 1 = 0.053531.*

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

1

2 *×* 1

*1.03526* *×*

1

*1.02895* + 1
*1.04343*

*= 0.93225.*

### Volatility Term Structures (continued)

*• Thus its yield is*

*0.93225*1 *− 1 = 0.0357.*

*• The yield volatility is hence*
1

2 ln

*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.^{a}

*• 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 (concluded)

*• 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. (121) on
*p. 932—and the r** _{i}* are then simultaneously determined.

*• The result is the Black-Derman-Toy model of Goldman*
Sachs.^{a}

aBlack, Derman, and Toy (1990).

*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.* (127)
**– The forward price equals the future value at time T**

of the underlying asset.^{a}

*• Equation (127) 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*
(128)

in discrete time.

**– The analog to Eq. (128) 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* (129)

by Eq. (127) on p. 990.

*• 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.*

(130)

*• Because Eq. (130) is equivalent to*
*P (t, T ) = e*^{−}

_{T}

*t* *f (t,s) ds**,* (131)
the spot rate curve is

*r(t, T ) =*
_{T}

*t* *f (t, s) ds*
*T* *− t* *.*

### Fundamental Relations (concluded)

*• The discrete analog to Eq. (131) 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).* (132)
**– Relation (132) in fact follows from the risk-neutral**

valuation principle.^{a}

aTheorem 17 on p. 514.

### Risk-Neutral Pricing (continued)

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

*• Rewrite Eq. (132) 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))*

*.* (133)

### Risk-Neutral Pricing (concluded)

*• Equation (132) on p. 995 can also be expressed as*
*E*_{t}*[ P (t + 1, T ) ] = F (t, t + 1, T ).*

**– Verify that with, e.g., Eq. (127) on p. 990.**

*• 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. 946).

### 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.* (134)

*• 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* *.* (135)

*• 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

^{a}

*• Let us start with the zero-coupon bonds and the money*
market account.

*• Let the zero-coupon bond price P (r, t, T ) follow*
*dP*

*P* *= μ*_{p}*dt + σ*_{p}*dW.*

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

aVasicek (1977).

### The Term Structure Equation (continued)

*• The net wealth change follows*

*−dP (r, t, s*1*) + α dP (r, t, s*2)

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

*• Pick*

*α* *≡* *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)* (136)
*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,*

(137)

*σ**p* =

*σ(r, t)* *∂P*

*∂r*

*/P,* (137* ^{}*)

*subject to P (· , T, T ) = 1.*

### The Term Structure Equation (concluded)

*• Substitute μ*_{p}*and σ** _{p}* into Eq. (136) on p. 1008 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.*

(138)

*• This is called the term structure equation.*

*• It applies to all interest rate derivatives: The diﬀerences*
are the terminal and boundary conditions.

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

*T* *− t* *.*