### The Extended Vasicek Model

^{a}

*• Hull and White proposed models that extend the*
Vasicek model and the CIR model.

*• They are called the extended Vasicek model and the*
extended CIR model.

*• The extended Vasicek model adds time dependence to*
the original Vasicek model,

*dr = (θ(t) − a(t) r) dt + σ(t) dW.*

*• Like the Ho-Lee model, this is a normal model.*

*• The inclusion of θ(t) allows for an exact ﬁt to the*
current spot rate curve.

### The Extended Vasicek Model (concluded)

*• Function σ(t) deﬁnes the short rate volatility, and a(t)*
determines the shape of the volatility structure.

*• Many European-style securities can be evaluated*
analytically.

*• Eﬃcient numerical procedures can be developed for*
American-style securities.

### The Hull-White Model

*• The Hull-White model is the following special case,*

*dr = (θ(t) − ar) dt + σ dW.* (166)

*• When the current term structure is matched,*^{a}
*θ(t) =* *∂f (0, t)*

*∂t* *+ af (0, t) +* *σ*^{2}
*2a*

1 *− e*^{−2at}*.*
**– Recall that f (0, t) deﬁnes the forward rate curve.**

aHull & White (1993).

### The Extended CIR Model

*• In the extended CIR model the short rate follows*
*dr = (θ(t) − a(t) r) dt + σ(t)√*

*r dW.*

*• The functions θ(t), a(t), and σ(t) are implied from*
market observables.

*• With constant parameters, there exist analytical*
solutions to a small set of interest rate-sensitive
securities.

### The Hull-White Model: Calibration

^{a}

*• We describe a trinomial forward induction scheme to*
*calibrate the Hull-White model given a and σ.*

*• As with the Ho-Lee model, the set of achievable short*
rates is evenly spaced.

*• Let r*0 be the annualized, continuously compounded
short rate at time zero.

*• Every short rate on the tree takes on a value*
*r*_{0} *+ jΔr*

*for some integer j.*

### The Hull-White Model: Calibration (continued)

*• Time increments on the tree are also equally spaced at*
*Δt apart.*

*• Hence nodes are located at times iΔt for i = 0, 1, 2, . . . .*

*• We shall refer to the node on the tree with*
*t** _{i}* =

^{Δ}

*iΔt,*

*r** _{j}* =

^{Δ}

*r*

_{0}

*+ jΔr,*(167)

*as the (i, j) node.*

*• The short rate at node (i, j), which equals r** _{j}*, is

*eﬀective for the time period [ t*

_{i}*, t*

*).*

_{i+1}### The Hull-White Model: Calibration (continued)

*• Use*

*μ*_{i,j}*= θ(t*^{Δ} * _{i}*)

*− ar*

*(168) to denote the drift rate*

_{j}^{a}of the short rate as seen from

*node (i, j).*

*• The three distinct possibilities for node (i, j) with three*
branches incident from it are displayed on p. 1215.

*• The middle branch may be an increase of Δr, no*
*change, or a decrease of Δr.*

a

### The Hull-White Model: Calibration (continued)

*(i, j)*

*(i + 1, j + 2)*

**(i + 1, j + 1)*

- _{(i + 1, j)}

*(i, j)*

**(i + 1, j + 1)*

- * _{(i + 1, j)}*
j

*(i + 1, j − 1)*

*(i, j)* - * _{(i + 1, j)}*
j

*(i + 1, j − 1)*

R*(i + 1, j − 2)*

### The Hull-White Model: Calibration (continued)

*• The upper and the lower branches bracket the middle*
branch.

*• Deﬁne*

*p1(i, j)* Δ

= the probability of following the upper branch from node (*i, j),*
*p2(i, j)* Δ

= the probability of following the middle branch from node (*i, j),*
*p3(i, j)* Δ

= the probability of following the lower branch from node (*i, j).*

*• The root of the tree is set to the current short rate r*_{0}.

*• Inductively, the drift μ*_{i,j}*at node (i, j) is a function of*
*(the still unknown) θ(t** _{i}*).

### The Hull-White Model: Calibration (continued)

*• Once θ(t*_{i}*) is available, μ** _{i,j}* can be derived via
Eq. (168) on p. 1214.

*• This in turn determines the branching scheme at every*
*node (i, j) for each j, as we will see shortly.*

*• The value of θ(t**i*) must thus be made consistent with
*the spot rate r(0, t** _{i+2}*).

^{a}

aNot *r(0, t** _{i+1}*)!

### The Hull-White Model: Calibration (continued)

*• The branches emanating from node (i, j) with their*
probabilities^{a} *must be chosen to be consistent with μ*_{i,j}*and σ.*

*• This is done by selecting the middle node to be as*

*closest to the current short rate r*_{j}*plus the drift μ*_{i,j}*Δt.*^{b}

aThat is, *p*1(*i, j), p*2(*i, j), and p*3(*i, j).*

bA precursor of Lyuu and C. Wu’s (R90723065) (2003, 2005) mean- tracking idea, which in turn is the precursor of the binomial-trinomial tree of Dai (B82506025, R86526008, D8852600) & Lyuu (2006, 2008, 2010).

### The Hull-White Model: Calibration (continued)

*• Let k be the number among { j − 1, j, j + 1 } that*

*makes the short rate reached by the middle branch, r** _{k}*,
closest to

*r*_{j}*+ μ*_{i,j}*Δt.*

**– But note that μ**_{i,j}*is still not computed yet.*

*• Then the three nodes following node (i, j) are nodes*
*(i + 1, k + 1), (i + 1, k), (i + 1, k − 1).*

*• See p. 1220 for a possible geometry.*

*• The resulting tree combines.*

* - j

*(0, 0)*

* - j

*(1, 1)*

* - j

*(1, 0)*

*

*(1, −1)* -

* - j

* - j

* - j

* - j

- j R

* - j

* - j

* - j

* - j

* --

*Δt*

6

?^{Δr}

### The Hull-White Model: Calibration (continued)

*• The probabilities for moving along these branches are*
*functions of μ*_{i,j}*, σ, j, and k:*

*p*_{1}*(i, j) =* *σ*^{2}*Δt + η*^{2}

*2(Δr)*^{2} + *η*

*2Δr,* (169)

*p*_{2}*(i, j) = 1 −* *σ*^{2}*Δt + η*^{2}

*(Δr)*^{2} *,* (169* ^{}*)

*p*

_{3}

*(i, j) =*

*σ*

^{2}

*Δt + η*

^{2}

*2(Δr)*^{2} *−* *η*

*2Δr,* (169* ^{}*)
where

*η* *= μ*^{Δ} _{i,j}*Δt + (j − k) Δr.*

### The Hull-White Model: Calibration (continued)

*• As trinomial tree algorithms are but explicit methods in*
disguise,^{a} *certain relations must hold for Δr and Δt to*
guarantee stability.

*• It can be shown that their values must satisfy*
*σ√*

*3Δt*

2 *≤ Δr ≤ 2σ√*
*Δt*

for the probabilities to lie between zero and one.

**– For example, Δr can be set to σ**√

*3Δt .*^{b}

*• Now it only remains to determine θ(t** _{i}*).

a

### The Hull-White Model: Calibration (continued)

*• At this point at time t** _{i}*,

*r(0, t*_{1}*), r(0, t*_{2}*), . . . , r(0, t** _{i+1}*)
have already been matched.

*• Let Q(i, j) be the state price at node (i, j).*

*• By construction, the state prices Q(i, j) for all j are*
known by now.

*• We begin with state price Q(0, 0) = 1.*

### The Hull-White Model: Calibration (continued)

*• Let ˆr(i) refer to the short rate value at time t** _{i}*.

*• The value at time zero of a zero-coupon bond maturing*
*at time t** _{i+2}* is then

*e*^{−r(0,t}^{i+2}^{)(i+2) Δt}

=

*j*

*Q(i, j) e*^{−r}^{j}^{Δt}*E*^{π}

*e**−ˆr(i+1) Δt** ˆr(i) = r**j*

*.(170)*

*• The right-hand side represents the value of $1 at time*
*t*_{i+2}*as seen at node (i, j) at time*^{a} *t** _{i}* before being

*discounted by Q(i, j).*

### The Hull-White Model: Calibration (continued)

*• The expectation in Eq. (170) can be approximated by*^{a}
*E*^{π}

*e**−ˆr(i+1) Δt* * ˆr(i) = r**j*

*≈ e*^{−r}^{j}^{Δt}

1 *− μ*_{i,j}*(Δt)*^{2} + *σ*^{2}*(Δt)*^{3}
2

*. (171)*
**– This solves the chicken-egg problem!**

*• Substitute Eq. (171) into Eq. (170) and replace μ*_{i,j}*with θ(t** _{i}*)

*− ar*

*to obtain*

_{j}*θ(ti) ≈*

*j Q(i, j) e**−2rjΔt *1 +*arj(Δt)**2 + σ2(Δt)3/2*

*− e−r*(0*,ti+2)(i+2) Δt*
(Δ*t)2*

*j Q(i, j) e**−2rjΔt* *.*

aSee Exercise 26.4.2 of the textbook.

### The Hull-White Model: Calibration (continued)

*• For the Hull-White model, the expectation in Eq. (171)*
is actually known analytically by Eq. (29) on p. 180:

*E*^{π}

*e**−ˆr(i+1) Δt** ˆr(i) = r**j*

= *e*^{−r}^{j}^{Δt+(−θ(t}^{i}^{)+ar}^{j}^{+σ}^{2}^{Δt/2)(Δt)}^{2}*.*

*• Therefore, alternatively,*

*θ(t** _{i}*) =

*r(0, t*

_{i+2}*)(i + 2)*

*Δt* +*σ*^{2}*Δt*

2 +ln

*j* *Q(i, j) e*^{−2r}^{j}^{Δt+ar}^{j}^{(Δt)}^{2}

*(Δt)*^{2} *.*

*• With θ(t*_{i}*) in hand, we can compute μ** _{i,j}*.

^{a}

### The Hull-White Model: Calibration (concluded)

*• With μ** _{i,j}* available, we compute the probabilities.

^{a}

*• Finally the state prices at time t** _{i+1}*:

*Q(i + 1, j)*

=

*(i, j*^{∗}*) is connected to (i + 1, j) with probability p*_{j∗}

*p**j*^{∗}*e*^{−r}^{j∗}^{Δt}*Q(i, j*^{∗}*).*

*• There are at most 5 choices for j** ^{∗}* (why?).

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

*• The space requirement is O(n) (why?).*

aSee Eqs. (169) on p. 1221.

### Comments on the Hull-White Model

*• One can try diﬀerent values of a and σ for each option.*

*• Or have an a value common to all options but use a*
*diﬀerent σ value for each option.*

*• Either approach can match all the option prices exactly.*

*• But suppose the demand is for a single set of parameters*
*to apply to all option prices.*

*• Then the Hull-White model can be calibrated to all the*
*observed option prices by choosing a and σ that*

minimize the mean-squared pricing error.^{a}

### The Hull-White Model: Calibration with Irregular Trinomial Trees

*• The previous calibration algorithm is quite general.*

*• For example, it can be modiﬁed to apply to cases where*
*the diﬀusion term has the form σr** ^{b}*.

*• But it has at least two shortcomings.*

*• First, the resulting trinomial tree is irregular (p. 1220).*

**– So it is harder to program (for nonprogrammers).**

*• The second shortcoming is a consequence of the tree’s*
irregular shape.

### The Hull-White Model: Calibration with Irregular Trinomial Trees (concluded)

*• Recall that the algorithm ﬁgured out θ(t** _{i}*) that matches

*the spot rate r(0, t*

*) in order to determine the*

_{i+2}*branching schemes for the nodes at time t** _{i}*.

*• But without those branches, the tree was not speciﬁed,*
and backward induction on the tree was not possible.

*• To avoid this chicken-egg dilemma, the algorithm turned*
to the continuous-time model to evaluate Eq. (170) on
*p. 1224 that helps derive θ(t** _{i}*).

*• The resulting θ(t** _{i}*) hence might not yield a tree that

### The Hull-White Model: Calibration with Regular Trinomial Trees

^{a}

*• The next, simpler algorithm exploits the fact that the*
*Hull-White model has a constant diﬀusion term σ.*

*• The resulting trinomial tree will be regular.*

*• All the θ(t** _{i}*) terms can be chosen by backward
induction to match the spot rates exactly.

*• The tree is constructed in two phases.*

aHull & White (1994).

### The Hull-White Model: Calibration with Regular Trinomial Trees (continued)

*• In the ﬁrst phase, a tree is built for the θ(t) = 0 case,*
which is an Ornstein-Uhlenbeck process:

*dr = −ar dt + σ dW,* *r(0) = 0.*

**– The tree is dagger-shaped (see p. 1233).**

**– The number of nodes above the r**_{0}*-line is j*_{max}, and
*that below the line is j*_{min}.

**– They will be picked so that the probabilities (169) on**
p. 1221 are positive for all nodes.

* - j

*(0, 0)*
*r*0

* - j

*(1, 1)*

* - j

*(1, 0)*

* -

*(1, −1)* j

* - j

* - j

* - j

* - j

* -

- j R

* - j

* - j

* - j

* - j

* -

- j R

* - j

* - j

* - j

* - j

* -

- j R

* - j

* - j

* - j

* - j

* - -

*Δt*

6?^{Δr}

*The short rate at node (0, 0) equals r*_{0} *= 0; here j*_{max} = 3
*and j*_{min} = 2.

### The Hull-White Model: Calibration with Regular Trinomial Trees (concluded)

*• The tree’s branches and probabilities are now in place.*

*• Phase two ﬁts the term structure.*

**– Backward induction is applied to calculate the β*** _{i}* to

*add to the short rates on the tree at time t*

*so that*

_{i}*the spot rate r(0, t*

*) is matched.*

_{i+1}^{a}

aContrast this with the previous algorithm, where it was *r(0, t**i+2*)
that was matched!

### The Hull-White Model: Calibration

*• Assume that a > 0.*

*• Set Δr = σ√*

*3Δt .*^{a}

*• Node (i, j) is a top node if j = j*_{max} and a bottom node
*if j = −j*_{min}.

*• Because the root has a short rate of r*_{0} = 0, phase one
*sets r*_{j}*= jΔr.*^{b}

*• Hence the probabilities in Eqs. (169) on p. 1221 use*
*η* =^{Δ} *−ajΔrΔt + (j − k) Δr.*

*• Recall that k tracks the middle branch.*

aRecall p. 1222.

bCompare it with formula (167) on p. 1213.

### The Hull-White Model: Calibration (continued)

*• The probabilities become*

*p1(i, j)*

= 1

6

+ *a2j2(Δt)2 − 2ajΔt(j − k) + (j − k)2 − ajΔt + (j − k)*

2 *,* (172)

*p2(i, j)*

= 2

3 *−*

*a2 j2(Δt)2 − 2ajΔt(j − k) + (j − k)2*

*,* (173)

*p3(i, j)*

= 1 6

+ *a2j2(Δt)2 − 2ajΔt(j − k) + (j − k)2 + ajΔt − (j − k)*

2 *.* (174)

*• p*_{1}*: up move; p*_{2}*: ﬂat move; p*_{3}: down move.

### The Hull-White Model: Calibration (continued)

*• The dagger shape dictates this:*

**– Let k = j − 1 if node (i, j) is a top node.**

**– Let k = j + 1 if node (i, j) is a bottom node.**

**– Let k = j for the rest of the nodes.**

*• Note that the probabilities are identical for nodes (i, j)*
*with the same j.*

*• Note also the symmetry,*

*p*_{1}*(i, j) = p*_{3}*(i, −j).*

### The Hull-White Model: Calibration (continued)

*• The inequalities*

3 *−* *√*
6

3 *< jaΔt <*

2

3 (175)

ensure that all the branching probabilities are positive in
*the upper half of the tree, that is, j > 0 (verify this).*

*• Similarly, the inequalities*

*−*
2

3 *< jaΔt < −*3 *−* *√*
6
3

ensure that the probabilities are positive in the lower

### The Hull-White Model: Calibration (continued)

*• To further make the tree symmetric across the r*_{0}-line,
*we let j*_{min} *= j*_{max}.

*• As*

3 *−* *√*
6

3 *≈ 0.184,*
a good choice is

*j*_{max} =

*0.184*
*aΔt*

*= O(n).*

### The Hull-White Model: Calibration (continued)

*• Phase two computes the β** _{i}*s to ﬁt the spot rates.

*• We begin with state price Q(0, 0) = 1.*

*• Inductively, suppose that spot rates*

*r(0, t*_{1}*), r(0, t*_{2}*), . . . , r(0, t** _{i}*)
have already been matched.

*• By construction, the state prices Q(i, j) for all j are*
known by now.

### The Hull-White Model: Calibration (continued)

*• The value of a zero-coupon bond maturing at time t** _{i+1}*
equals

*e*^{−r(0,t}^{i+1}* ^{)(i+1) Δt}* =

*j*

*Q(i, j) e*^{−(β}^{i}^{+r}^{j}* ^{)Δt}*
by risk-neutral valuation.

*• Hence*

*β** _{i}* =

*r(0, t*

_{i+1}*)(i + 1) Δt + ln*

*j* *Q(i, j) e*^{−r}^{j}^{Δt}

*Δt* *.*

(176)

### The Hull-White Model: Calibration (concluded)

*• The short rate at node (i, j) now equals β*_{i}*+ r** _{j}*.

*• The state prices at time t** _{i+1}*,

*Q(i + 1, j)*

for *− min(i + 1, j*_{max}) *≤ j ≤ min(i + 1, j*_{max}), can now be
calculated as before.^{a}

*• The total running time is O(nj*_{max}).

*• The space requirement is O(n).*

aRecall p. 1227.

### A Numerical Example

*• Assume a = 0.1, σ = 0.01, and Δt = 1 (year).*

*• Immediately, Δr = 1.73205% and j*_{max} = 2.

*• The plot on p. 1244 illustrates the 3-period trinomial*
tree after phase one.

*• For example, the branching probabilities for node E are*
*calculated by Eqs. (172)–(174) on p. 1236 with j = 2*
*and k = 1.*

* - j

A

* - j

B *

- j

C *

- j

D

- j R

E *

- j

F *

- j

G *

- j

H

* -

I

Node A, C, G B, F E D, H I

*r (%)* 0.00000 *1.73205* *3.46410* *−1.73205* *−3.46410*
*p*1 0.16667 0.12167 0.88667 0.22167 0.08667
*p*2 0.66667 0.65667 0.02667 0.65667 0.02667
*p*3 0.16667 0.22167 0.08667 0.12167 0.88667

### A Numerical Example (continued)

*• Suppose that phase two is to ﬁt the spot rate curve*
*0.08 − 0.05 × e*^{−0.18×t}*.*

*• The annualized continuously compounded spot rates are*
*r(0, 1) = 3.82365%, r(0, 2) = 4.51162%, r(0, 3) = 5.08626%.*

*• Start with state price Q(0, 0) = 1 at node A.*

### A Numerical Example (continued)

*• Now, by Eq. (176) on p. 1241,*

*β*_{0} *= r(0, 1) + ln Q(0, 0) e*^{−r}^{0} *= r(0, 1) = 3.82365%.*

*• Hence the short rate at node A equals*
*β*_{0} *+ r*_{0} *= 3.82365%.*

*• The state prices at year one are calculated as*
*Q(1, 1) = p*_{1}*(0, 0) e*^{−(β}^{0}^{+r}^{0}^{)} *= 0.160414,*
*Q(1, 0) = p*_{2}*(0, 0) e*^{−(β}^{0}^{+r}^{0}^{)} *= 0.641657,*

*−(β* *+r* )

### A Numerical Example (continued)

*• The 2-year rate spot rate r(0, 2) is matched by picking*

*β*1 = *r(0, 2)×2+ln*

*Q(1, 1) e** ^{−Δr}* +

*Q(1, 0) + Q(1, −1) e*

^{Δr}= 5*.20459%.*

*• Hence the short rates at nodes B, C, and D equal*
*β*_{1} *+ r*_{j}*,*

*where j = 1, 0, −1, respectively.*

*• They are found to be 6.93664%, 5.20459%, and*
*3.47254%.*

### A Numerical Example (continued)

*• The state prices at year two are calculated as*

*Q(2, 2) = p*1(1*, 1) e*^{−(β}^{1}^{+r}^{1}^{)}*Q(1, 1) = 0.018209,*

*Q(2, 1) = p*2(1*, 1) e*^{−(β}^{1}^{+r}^{1}^{)}*Q(1, 1) + p*1(1*, 0) e*^{−(β}^{1}^{+r}^{0}^{)}*Q(1, 0)*

= 0*.199799,*

*Q(2, 0) = p*3(1*, 1) e*^{−(β}^{1}^{+r}^{1}^{)}*Q(1, 1) + p*2(1*, 0) e*^{−(β}^{1}^{+r}^{0}^{)}*Q(1, 0)*
+*p*1(1*, −1) e*^{−(β}^{1}^{+r}^{−1}^{)}*Q(1, −1) = 0.473597,*

*Q(2, −1) = p*3(1*, 0) e*^{−(β}^{1}^{+r}^{0}^{)}*Q(1, 0) + p*2(1*, −1) e*^{−(β}^{1}^{+r}^{−1}^{)}*Q(1, −1)*

= 0*.203263,*

*Q(2, −2) = p*3(1*, −1) e*^{−(β}^{1}^{+r}^{−1}^{)}*Q(1, −1) = 0.018851.*

### A Numerical Example (concluded)

*• The 3-year rate spot rate r(0, 3) is matched by picking*

*β*2 *= r(0, 3) × 3 + ln*

*Q(2, 2) e*^{−2×Δr}*+ Q(2, 1) e*^{−Δr}*+ Q(2, 0)*
*+Q(2, −1) e*^{Δr}*+ Q(2, −2) e*^{2×Δr}

*= 6.25359%.*

*• Hence the short rates at nodes E, F, G, H, and I equal*
*β*_{2} *+ r*_{j}*, where j = 2, 1, 0, −1, −2, respectively.*

*• They are found to be 9.71769%, 7.98564%, 6.25359%,*
*4.52154%, and 2.78949%.*

*• The ﬁgure on p. 1250 plots β*_{i}*for i = 0, 1, . . . , 29.*

<HDU +L/

-L+/

### The (Whole) Yield Curve Approach

*• We have seen several Markovian short rate models.*

*• The Markovian approach is computationally eﬃcient.*

*• But it is diﬃcult to model the behavior of yields and*
bond prices of diﬀerent maturities.

*• The alternative yield curve approach regards the whole*
term structure as the state of a process and directly
speciﬁes how it evolves.

### The Heath-Jarrow-Morton (HJM) Model

^{a}

*• This inﬂuential model is a forward rate model.*

*• The HJM model speciﬁes the initial forward rate curve*
and the forward rate volatility structure.

**– The volatility structure describes the volatility of**
each forward rate for a given maturity date.

*• Like the Black-Scholes option pricing model, neither risk*
preference assumptions nor the drifts of forward rates
are needed.

aHeath, Jarrow, & Morton (1992).

### The HJM Model (continued)

*• Within a ﬁnite-time horizon [ 0, U ], we take as given the*
*time-zero forward rate curve f (0, T ) for T ∈ [ 0, U ].*

*• Since this curve is used as the boundary value at t = 0,*
*perfect ﬁt to the observed term structure is automatic.*

*• The forward rates are driven by k stochastic factors.*

### The HJM Model (continued)

*• Speciﬁcally the forward rate movements are governed by*
the stochastic process,

*df (t, T ) = μ(t, T ) dt +*

*k*
*i=1*

*σ*_{i}*(t, T ) dW*_{i}*,*

(177)
*where μ and σ** _{i}* may depend on the past history of the

*independent Wiener processes W*

_{1}

*, W*

_{2}

*, . . . , W*

*.*

_{k}*• One-factor models seem to perform better than*
*multifactor models empirically, at least for pricing*
short-dated options.^{a}

### The HJM Model (continued)

*• But two-factor models perform better in hedging caps*
and ﬂoors.^{a}

*• Kamakura (2019) has a 10-factor*^{b} (14-factor^{c}) HJM
model for the U.S. Treasuries (German bonds,

respectively).

*• A unique equivalent martingale measure π can be*
established under which the prices of interest rate

derivatives do not depend on the market prices of risk.

aGupta & Subrahmanyam (2001, 2005).

bSee http://www.kamakuraco.com/KamakuraReleasesNewStochasticVolatilityModel

cSee http://www.kamakuraco.com/KamakuraReleases14FactorHeathJarrowandMorto.

### The HJM Model (continued)

**Theorem 23 (1) For all 0 < t ≤ T ,**

*μ(t, T ) =*

*k*
*i=1*

*σ*_{i}*(t, T )*

_{T}

*t*

*σ*_{i}*(t, u) du* (178)
*holds under π almost surely. (2) The bond price dynamics*
*under π is given by*

*dP (t, T )*

*P (t, T )* *= r(t) dt −*

*k*
*i=1*

*σ*_{p,i}*(t, T ) dW*_{i}*,* (179)

*where σ*_{p,i}*(t, T ) ≡* _{T}

*t* *σ*_{i}*(t, u) du.*

### The HJM Model (concluded)

*• Hence choosing the volatility functions σ*_{i}*(t, T ) of the*
*forward rate dynamics under π uniquely determines the*
*drift parameters under π and the prices of all claims.*

### The Use of the HJM Model

*• Take the one-factor model,*

*df (t, T ) = μ(t, T ) dt + σ(t, T ) dW*_{t}*.*

*• To use the HJM model, we ﬁrst pick σ(t, T ).*

*• This is the modeling part.*

*• The drift parameters are then determined by Eq. (178)*
on p. 1256.

*• Now fetch today’s forward rate curve { f(0, T ), T ≥ 0 }*
and integrate it to obtain the forward rates,

*f (t, T ) = f (0, T ) +*

_{t}

*μ(s, T ) ds +*

_{t}

*σ(s, T ) dW* *.*

### The Use of the HJM Model (concluded)

*• Compute the future bond prices by*
*P (t, T ) = e*^{−}

_{T}

*t* *f (t,s) ds*

if necessary.

*• European-style derivatives can be priced by simulating*
many paths and taking average.

### Short Rate under the HJM Model

*• From Eq. (26.19) of the textbook, the short rate follows*
the following SDE,

*dr(t) =* *∂f (0, t)*

*∂t* *dt*
+

_{t}

0

*σ*_{p}*(s, t)* *∂σ(s, t)*

*∂t* *+ σ(s, t)*^{2}

*ds*

*dt*
+

_{t}

0

*∂σ(s, t)*

*∂t* *dW*_{s}

*dt + σ(t, t) dW*_{t}*. (180)*

*• Since the second and the third terms on the right-hand*
*side depend on the history of σ*_{p} *and/or dW , they can*

### Short Rate under the HJM Model (concluded)

*• If σ*_{p}*(t, T ) = σ(T − t) for a constant σ, the short rate*
*process r becomes Markovian.*

*• Then Eq. (180) on p. 1260 is reduced to*
*dr =*

*∂f (0, t)*

*∂t* *+ σ*^{2}*t*

*dt + σ dW.*

*• This is the continuous-time Ho-Lee model (162) on p.*

1182.^{a}

*• See Carverhill (1994) and Jeﬀrey (1995) for conditions*
for the short rate to be Markovian.

aSee p. 392 of the textbook.

### The Alternative HJM Model

*• Alternatively, we can start with the bond process under*
*π:*

*dP (t, T )*

*P (t, T )* *= r(t) dt +*

*k*
*i=1*

*σ*_{p,i}*(t, T ) dW*_{i}*.* (181)

*• Then*^{a}

*df (t, T ) =*

*k*
*i=1*

*σ*_{p,i}*(t, T )* *∂σ*_{p,i}*(t, T )*

*∂T* *dt*

*−*

*k*
*i=1*

*∂σ*_{p,i}*(t, T )*

*∂T* *dW*_{i}*.*

### Gaussian HJM Models

^{a}

*• A nonstochastic volatility depends on only t and T .*

*• When the forward rate volatilities σ*_{i}*(t, T ) are*
nonstochastic, we have a Gaussian HJM model.

*• For Gaussian HJM models, the bond price volatilities*
*σ*_{p,i}*(t, T ) must also be nonstochastic.*

*• The forward rates have a normal distribution, whereas*
the bond prices have a lognormal distribution.

aMusiela & Rutkowski (1997).

### Gaussian HJM Models (concluded)

*• σ(t, T ) = σ: The Ho-Lee model (162) on p. 1182 obtains.*

*• σ(t, T ) = σe** ^{−a(T −t)}*: The Hull-White model (166) on p.

1210 obtains.

*• σ(t, T ) = σ*_{0} *+ σ*_{1}*(T − t): The linear absolute model.*^{a}

*• σ(t, T ) = σ [ γ(T − t) + 1 ] e**−(λ/2)(T −t)*: The
Mercurio-Moraleda (2000) model.

aGupta & Subrahmanyam (2001, 2005).

### Local-Volatility HJM Models

^{a}

*• If the forward rate volatilities σ*_{i}*(t, T, f (t, T )) depend on*
*t, T , and f (t, T ) only, we have a local-volatility HJM*
model.

*• The same term may also apply to HJM models whose*
*bond price volatilities σ*_{p,i}*(t, T, P (t, T )) depend on t, T ,*
*and P (t, T ) only.*

aBrigo & Mercurio (2006).

### Local-Volatility HJM Models (continued)

*• The (nearly) proportional volatility model:*^{a}

*σ(t, T, f (t, T )) = σ*_{0} *min(κ, f (t, T )),* *σ*_{0}*, κ > 0.*

*• The proportional volatility model:*^{b}

*σ(t, T, f (t, T )) = σ*_{0}*f (t, T ).* (182)

*• The linear proportional model:*^{c}

*σ(t, T, f (t, T )) = [ σ*_{0} *+ σ*_{1}*(T − t) ] f (t, T ).*

aHeath, Jarrow, & Morton (1992); Jarrow (1996). The large positive
constant *κ prevents explosion in ﬁnite time.*

### Local-Volatility HJM Models (continued)

*• Exponentially dampened volatility proportional to the*
short rate:^{a}

*σ(t, T ) = σf (t, t) e*^{−a(T −t)}*.*

*• The Ritchken-Sankarasubramanian (1995) model:*^{b}
*σ(t, T ) = σ(t, t) e*^{−}

_{T}

*t* *κ(x) dx**.*
**– For example,**^{c}

*σ(t, t) = σr(t)*^{γ}*.*

aGrant & Vora (1999).

bThe short rate volatility *σ(t, t) may depend on the short rate r(t).*

cRitchken & Sankarasubramanian (1995); Li, Ritchken, & Sankara- subramanian (1995).

### Local-Volatility HJM Models (concluded)

*• A model attributed to Ian Cooper (1993):*^{a}
*σ*_{p}*(t, T, P (t, T )) = ψ(t) ln P (t, T )*
in Eq. (181) on p. 1262:

aRebonato (1996). It is equivalent to the proportional volatility model
(182) when *ψ(t) is a constant.*

### Trees for HJM Models

*• Obtain today’s forward rate curve:*

*f (0, 0), f (0, Δt), f (0, 2Δt), f (0, 3Δt), . . . , f (0, T ).*

*• For binomial trees, generate the two forward rate curves*
*at time Δt:*

*f*_{u}*(Δt, Δt), f*_{u}*(Δt, 2Δt), f*_{u}*(Δt, 3Δt), . . . , f*_{u}*(Δt, T ),*
*f*_{d}*(Δt, Δt), f*_{d}*(Δt, 2Δt), f*_{d}*(Δt, 3Δt), . . . , f*_{d}*(Δt, T ).*

*by Eq. (177) on p. 1254 with μ(t, T ) from Eq. (178) on*
p. 1256.

### Trees for HJM Models (continued)

*• Iterate until the maturity t ≤ T of the derivative.*

*• A straightforward implementation of the HJM model*
results in noncombining trees.

**– For a binomial tree with n time steps, O(2*** ^{n}*) nodes

*for one-factor HJM models; O(3*

^{n}*) or O(4*

*) for*

^{n}two-factor models.^{a}

aClewlow & Strickland (1998); Hull (1999); Nawalkha, Beliaeva, &

Soto (2007).

H(0) B(0.1) B(0,2) B(0,3)

H(1) B(1,2) B(1,3)

H(1) B(1,2) B(1,3)

H(2) B(2,3)

H(2) B(2,3)

H(2) B(2,3)

H(2) B(2,3)

H(3) H(3) H(3) H(3) H(3) H(3) H(3) H(3)

### Trees for HJM Models (continued)

*• Jarrow (1996): “a large number of time steps is not*
always essential for obtaining good approximations.”

*• Rebonato (1996): “it is diﬃcult to see how a ﬁve-year*
cap with quarterly resets (let alone an option thereon)
could be priced using [10 or 12 time steps].”

*• Some trees are not analyzed.*^{a}

aBrace (1996); G¸atarek & Kolakowski (2003); Ferris (2012).

### Trees for HJM Models (concluded)

*• Nawalkha & J. Zhang (2004) has a combining tree for*
the proportional volatility model with a positive lower
bound.

**– It is described in Nawalkha, Beliaeva, & Soto (2007)**
but not published.

*• For Gaussian HJM models, O(n*^{2}) nodes may suﬃce.^{a}

aLok (D99922028), Lu (D00922011), & Lyuu (2020); Lyuu (2019).

*Introduction to Mortgage-Backed Securities*

Anyone stupid enough to promise to be responsible for a stranger’s debts deserves to have his own property held to guarantee payment.

— Proverbs 27:13 I’m not putting my money in real estate.

I prefer bonds.

— Margaret Mitchell (1900–1949),
*Gone with the Wind (1936)*

### Mortgages

*• A mortgage is a loan secured by the collateral of real*
estate property.

*• Suppose the borrower (the mortgagor) defaults, that is,*
fails to make the contractual payments.

*• The lender (the mortgagee) can foreclose the loan by*
seizing the property.

### Mortgage-Backed Securities

*• A mortgage-backed security (MBS) is a bond backed by*
an undivided interest in a pool of mortgages.^{a}

*• MBSs traditionally enjoy high returns, wide ranges of*
products, high credit quality, and liquidity.

*• The mortgage market has witnessed tremendous*
innovations in product design.

*• The collapse of MBSs also triggered the ﬁnancial crisis*
of 2008.

aThey can be traced to 1880s (Levy, 2012).

### Mortgage-Backed Securities (concluded)

*• The complexity of the products and the prepayment*

option require advanced models and software techniques.

**– In fact, the mortgage market probably could not**
have operated eﬃciently without them.^{a}

*• They also consume lots of computing power.*

*• Our focus will be on residential mortgages.*

*• But the underlying principles are applicable to other*
types of assets.

aMerton (1994).

### Types of MBSs

*• An MBS is issued with pools of mortgage loans as the*
collateral.

*• The cash ﬂows of the mortgages making up the pool*
naturally reﬂect upon those of the MBS.

*• There are three basic types of MBSs:*

1. Mortgage pass-through security (MPTS).

2. Collateralized mortgage obligation (CMO).

3. Stripped mortgage-backed security (SMBS).

### Problems Investing in Mortgages

*• The MBS sector is one of the largest in the debt*
market.^{a}

*• Individual mortgages are unattractive for many*
investors.

*• Often at hundreds of thousands of U.S. dollars or more,*
they demand too much investment.

*• Most investors lack the resources and knowledge to*
assess the credit risk involved.

aSee p. 3 of the textbook. In the U.S., the outstanding balance was US$9.3 trillion as of 2017 vs. the US Treasury’s US$14.5 trillion and corporate debt’s US$9.0 trillion (SIFMA, 2018). The residential property

### Problems Investing in Mortgages (concluded)

*• Recall that a traditional mortgage is ﬁxed rate, level*
payment, and fully amortized.

*• So the percentage of principal and interest (P&I) varying*
from month to month, creating accounting headaches.

*• Prepayment levels ﬂuctuate with a host of factors.*

*• That makes the size and the timing of the cash ﬂows*
unpredictable.

### Mortgage Pass-Throughs

^{a}

*• The simplest kind of MBS.*

*• Payments from the underlying mortgages are passed*

from the mortgage holders through the servicing agency, after a fee is subtracted.

*• They are distributed to the security holder on a pro rata*
basis.

**– The holder of a $25,000 certiﬁcate from a $1 million**
pool is entitled to 21/2% (or 1/40th) of the cash ﬂow.

*• Because of higher marketability, a pass-through is easier*
to sell than its individual loans.

a

Rule for distribution of cash flows: pro rata Loan 2

Loan 10 Loan 1

Pass-through: $1 million par pooled mortgage loans

### Collateralized Mortgage Obligations (CMOs)

*• A pass-through exposes the investor to the total*
prepayment risk.

*• Such risk is undesirable from an asset/liability*
perspective.

*• To deal with prepayment uncertainty, CMOs were*
created.^{a}

*• Mortgage pass-throughs have a single maturity and are*
backed by individual mortgages.

aIn June 1983 by Freddie Mac (Federal Home Loan Mortgage Cor- poration), bailed out by the U.S. government in 2008, with the help of

### Collateralized Mortgage Obligations (CMOs) (continued)

*• CMOs are multiple-maturity, multiclass debt*

instruments collateralized by pass-throughs, stripped mortgage-backed securities, and whole loans.

*• The total prepayment risk is now divided among classes*
of bonds called classes or tranches.^{a}

*• The principal, scheduled and prepaid, is allocated on a*
*prioritized basis so as to redistribute the prepayment*
risk among the tranches in an unequal way.

a*Tranche is a French word for “slice.”*

### Collateralized Mortgage Obligations (CMOs) (concluded)

*• CMOs were the ﬁrst successful attempt to alter*

mortgage cash ﬂows in a security form that attracts a wide range of investors

**– The outstanding balance of agency CMOs was**
US$1.1 trillion as of the ﬁrst quarter of 2015.^{a}

aSIFMA (2015).

### Sequential Tranche Paydown

*• In the sequential tranche paydown structure, Class A*
receives principal paydown and prepayments before

Class B, which in turn does it before Class C, and so on.

*• Each tranche thus has a diﬀerent eﬀective maturity.*

*• Each tranche may even have a diﬀerent coupon rate.*

### An Example

*• Consider a two-tranche sequential-pay CMO backed by*

$1,000,000 of mortgages with a 12% coupon and 6 months to maturity.

*• The cash ﬂow pattern for each tranche with zero*

prepayment and zero servicing fee is shown on p. 1289.

*• The calculation can be carried out ﬁrst for the Total*
columns, which make up the amortization schedule.

*• Then the cash ﬂow is allocated.*

*• Tranche A is retired after 4 months, and tranche B*

### CMO Cash Flows without Prepayments

Interest Principal Remaining principal

Month A B Total A B Total A B Tota

500,000 500,000 1,000,0

1 5,000 5,000 10,000 162,548 0 162,548 337,452 500,000 837,4

2 3,375 5,000 8,375 164,173 0 164,173 173,279 500,000 673,2

3 1,733 5,000 6,733 165,815 0 165,815 7,464 500,000 507,4

4 75 5,000 5,075 7,464 160,009 167,473 0 339,991 339,9

5 0 3,400 3,400 0 169,148 169,148 0 170,843 170,8

6 0 1,708 1,708 0 170,843 170,843 0 0

Total 10,183 25,108 35,291 500,000 500,000 1,000,000

*The total monthly payment is $172,548. Month-i numbers*
*reﬂect the ith monthly payment.*

### Another Example

*• When prepayments are present, the calculation is only*
slightly more complex.

*• Suppose the single monthly mortality (SMM) per month*
is 5%.

*• This means the prepayment amount is 5% of the*
*remaining principal.*

*• The remaining principal at month i after prepayment*
then equals the scheduled remaining principal as

*computed by Eq. (7) on p. 56 times (0.95)** ^{i}*.

*• This done for all the months, the interest payment at*
any month is the remaining principal of the previous

### Another Example (continued)

*• The prepayment amount equals the remaining principal*
*times 0.05/0.95.*

**– The division by 0.95 yields the remaining principal**
*before prepayment.*

*• Page 1292 tabulates the cash ﬂows of the same*
two-tranche CMO under 5% SMM.

### Another Example (continued)

Interest Principal Remaining principal

Month A B Total A B Total A B Total

500,000 500,000 1,000,00

1 5,000 5,000 10,000 204,421 0 204,421 295,579 500,000 795,57

2 2,956 5,000 7,956 187,946 0 187,946 107,633 500,000 607,63

3 1,076 5,000 6,076 107,633 64,915 172,548 0 435,085 435,08

4 0 4,351 4,351 0 158,163 158,163 0 276,922 276,92

5 0 2,769 2,769 0 144,730 144,730 0 132,192 132,19

6 0 1,322 1,322 0 132,192 132,192 0 0

Total 9,032 23,442 32,474 500,000 500,000 1,000,000

*Month-i numbers reﬂect the ith monthly payment.*

### Another Example (continued)

*• For instance, the total principal payment at month one,*

$204,421, can be veriﬁed as follows.

*• The scheduled remaining principal is $837,452 from*
p. 1289.

*• The remaining principal is hence*

837452 *× 0.95 = 795579.*

*• That makes the total principal payment*
1000000 *− 795579 = 204421.*

### Another Example (concluded)

*• As tranche A’s remaining principal is $500,000, all*
204,421 dollars go to tranche A.

*• Incidentally, the prepayment is*

837452 *× 5% = 41873.*

**– Aalternatively, 795579** *× 0.05/0.95 = 41873.*

*• Tranche A is retired after 3 months, and tranche B*
starts principal paydown at the end of month 3.

### Stripped Mortgage-Backed Securities (SMBSs)

^{a}

*• The principal and interest are divided between the PO*
strip and the IO strip.

*• In the scenarios on p. 1288 and p. 1290:*

**– The IO strip receives all the interest payments under**
the Interest/Total column.

**– The PO strip receives all the principal payments**
under the Principal/Total column.

aThey were created in February 1987 when Fannie Mae issued its Trust 1 stripped MBS. Fannie Mae was bailed out by the U.S. govern- ment in 2008.

### Stripped Mortgage-Backed Securities (SMBSs) (concluded)

*• These new instruments allow investors to better exploit*
anticipated changes in interest rates.^{a}

*• The collateral for an SMBS is a pass-through.*

*• CMOs and SMBSs are usually called derivative MBSs.*

aSee p. 357 of the textbook.

### Prepayments

*• The prepayment option sets MBSs apart from other*
ﬁxed-income securities.

*• The exercise of options on most securities is expected to*
be “rational.”

*• This kind of “rationality” is weakened when it comes to*
the homeowner’s decision to prepay.

*• For example, even when the prevailing mortgage rate*
exceeds the mortgage’s loan rate, some loans are

prepaid.

### Prepayment Risk

*• Prepayment risk is the uncertainty in the amount and*
timing of the principal prepayments in the pool of

mortgages that collateralize the security.

*• This risk can be divided into contraction risk and*
extension risk.^{a}

*• Contraction risk is the risk of having to reinvest the*

prepayments at a rate lower than the coupon rate when interest rates decline.

aSimilar to mortality risk and longevity risk in life insurance.

### Prepayment Risk (continued)

*• Extension risk is due to the slowdown of prepayments*
when interest rates climb, making the investor earn the
security’s lower coupon rate rather than the market’s
higher rate.

*• Prepayments can be in whole or in part.*

**– The former is called liquidation.**

**– The latter is called curtailment.**

### Prepayment Risk (concluded)

*• The holder of a pass-through security is exposed to the*
total prepayment risk associated with the underlying
pool of mortgage loans.

*• CMOs are designed to alter the distribution of that risk*
among investors.

*• They contributed to the subprime mortgage crisis of*
2008.

### Other Risks

*• Investors in mortgages are exposed to at least three*
other risks.

**– Interest rate risk is inherent in any ﬁxed-income**
security.

**– Credit risk is the risk of loss from default.**

*∗ For privately insured mortgage, the risk is related*
to the credit rating of the company that insures
the mortgage.

**– Liquidity risk is the risk of loss if the investment**
must be sold quickly.

### Prepayment: Causes

Prepayments have at least ﬁve components.

**Home sale (“housing turnover”). The sale of a home**
generally leads to the prepayment of mortgage because
of the full payment of the remaining principal.

**Reﬁnancing. Mortgagors can reﬁnance their home**

mortgage at a lower mortgage rate. This is the most volatile component of prepayment and constitutes the bulk of it when prepayments are extremely high.

**Default. Caused by foreclosure and subsequent liquidation**
of a mortgage. Relatively minor in most cases.^{a}

a

### Prepayment: Causes (concluded)

**Curtailment. As the extra payment above the scheduled**
payment, curtailment applies to the principal and

shortens the maturity of ﬁxed-rate loans. Its contribution to prepayments is minor.

**Full payoﬀ (liquidation). There is evidence that many**
mortgagors pay oﬀ their mortgage completely when it is
very seasoned and the remaining balance is small. Full
payoﬀ can also be due to natural disasters.

### Prepayment: Characteristics

*• Prepayments usually increase as the mortgage ages —*
ﬁrst at an increasing rate and then at a decreasing rate.

*• They are higher in the spring and summer and lower in*
the fall and winter.

*• They vary by the geographic locations of the underlying*
properties.

*• They increase when interest rates drop but with a time*
lag.

### Prepayment: Characteristics (continued)

*• If prepayments were higher for some time because of*
high reﬁnancing rates, they tend to slow down.

**– Perhaps, homeowners who do not prepay when rates**
have been low for a prolonged time tend never to
prepay.

*• Plot on p. 1306 illustrates the typical price/yield curves*
of the Treasury and pass-through.

0.05 0.1 0.15 0.2 0.25 0.3Interest rate 50

100 150 200

Price

The cusp

Treasury MBS

Price compression occurs as yields fall through a threshold.

### Prepayment: Characteristics (concluded)

*• As yields fall and the pass-through’s price moves above*
a certain price, it ﬂattens and then follows a downward
slope.

*• This phenomenon is called the price compression of*
premium-priced MBSs.

*• It demonstrates the negative convexity of such securities.*