### The Black-Karasinski Model

^{a}

*• The BK model stipulates that the short rate follows*
*d ln r = κ(t)(θ(t) − ln r) dt + σ(t) dW.*

*• This explicitly mean-reverting model depends on time*
*through κ( · ), θ( · ), and σ( · ).*

*• The BK model hence has one more degree of freedom*
than the BDT model.

*• The speed of mean reversion κ(t) and the short rate*
*volatility σ(t) are independent.*

aBlack and Karasinski (1991).

### The Black-Karasinski Model: Discrete Time

*• The discrete-time version of the BK model has the same*
representation as the BDT model.

*• To maintain a combining binomial tree, however,*
requires some manipulations.

*• The next plot illustrates the ideas in which*
*t*_{2} *≡ t*1 *+ Δt*_{1}*,*

*t*3 *≡ t*2 *+ Δt*2*.*

*ln r*d*(t*2)

*ln r(t*1) *ln r*du*(t*3*) = ln r*ud*(t*3)

*ln r*u*(t*2)

### The Black-Karasinski Model: Discrete Time (continued)

*• Note that*

ln*r*_{d}(*t*2) = ln*r(t*1) +*κ(t*1)(*θ(t*1) *− ln r(t*1)) Δ*t*1 *− σ(t*1)

Δ*t*1 *,*
ln*r*u(*t*2) = ln*r(t*1) +*κ(t*1)(*θ(t*1) *− ln r(t*1)) Δ*t*1 + *σ(t*1)

Δ*t*1 *.*

*• To make sure an up move followed by a down move*
coincides with a down move followed by an up move,

*ln r*d*(t*2*) + κ(t*2*)(θ(t*2*) − ln r*d*(t*2*)) Δt*2 *+ σ(t*2)*√*

*Δt*2 *,*

*= ln r*u*(t*2*) + κ(t*2*)(θ(t*2*) − ln r*u*(t*2*)) Δt*2 *− σ(t*2)*√*

*Δt*2 *.*

### The Black-Karasinski Model: Discrete Time (continued)

*• They imply*

*κ(t*_{2}) = 1 *− (σ(t*2*)/σ(t*_{1}))

*Δt*_{2}*/Δt*_{1}

*Δt*_{2} *.*

(151)

*• So from Δt*_{0}*, we can calculate the Δt*_{1} that satisﬁes the
combining condition and then iterate.

* – t*0

*→ Δt*0

*→ t*1

*→ Δt*1

*→ t*2

*→ Δt*2

*→ · · · → T*(roughly).

^{a}

aAs *κ(t), θ(t), σ(t) are independent of r, the Δt**i*s will not depend on
*r.*

### The Black-Karasinski Model: Discrete Time (concluded)

*• Unequal durations Δt**i* are often necessary to ensure a
combining tree.^{a}

aAmin (1991); Chen (R98922127) (2011); Lok (D99922028) and Lyuu (2016).

### Problems with Lognormal Models in General

*• Lognormal models such as BDT and BK share the*
*problem that E*^{π}*[ M (t) ] = ∞ for any ﬁnite t if they*
model the continuously compounded rate.^{a}

*• So periodically compounded rates should be modeled.*^{b}

*• Another issue is computational.*

*• Lognormal models usually do not admit of analytical*
solutions to even basic ﬁxed-income securities.

*• As a result, to price short-dated derivatives on long-term*
bonds, the tree has to be built over the life of the

underlying asset instead of the life of the derivative.

aHogan and Weintraub (1993).

bSandmann and Sondermann (1993).

### Problems with Lognormal Models in General (concluded)

*• This problem can be somewhat mitigated by adopting*
diﬀerent time steps.^{a}

**– Use a ﬁne time step up to the maturity of the**
short-dated derivative.

**– Use a coarse time step beyond the maturity.**

*• A down side of this procedure is that it has to be*
tailor-made for each derivative.

*• Finally, empirically, interest rates do not follow the*
lognormal distribution.

aHull and White (1993).

### 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.

aHull and White (1990).

### 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.*

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

*∂t* *+ af (0, t) +* *σ*^{2}
*2a*

1 *− e*^{−2at}*.*

aHull and 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.*

aHull and White (1993).

### 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}*≡ iΔt,*

*r*_{j}*≡ r*_{0} *+ jΔr,*
*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**i*) *− ar**j* (152)
to denote the drift rate^{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. 1109.

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

aOr, the annualized expected change.

### 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)* *≡* the probability of following the upper branch from node (*i, j)*
*p2(i, j)* *≡* the probability of following the middle branch from node (*i, j)*
*p3(i, j)* *≡* 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}*).

**– Both describe the expected change from node (i, j).**

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

*• Once θ(t*_{i}*) is available, μ** _{i,j}* can be derived via
Eq. (152) on p. 1108.

*• 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 letting the middle node be as closest to*
*the current short rate r*_{j}*plus the drift μ*_{i,j}*Δt.*^{b}

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

bA predecessor to Lyuu and Wu’s (R90723065) (2003, 2005) mean- tracking idea, which is the precursor of the binomial-trinomial tree of Dai (B82506025, R86526008, D8852600) and 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. 1114 for a possible geometry.*

*• The resulting tree combines because of the constant*
*jump sizes to reach k from j.*

* - 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* (153)

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

*(Δr)*^{2} (153* ^{}*)

*p*3*(i, j) =* *σ*^{2}*Δt + η*^{2}

*2(Δr)*^{2} *−* *η*

*2Δr* (153* ^{}*)
where

*η ≡ μ*_{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}*).

aRecall p. 761.

bHull and White (1988).

### 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*

*.(154)*

*• The right-hand side represents the value of $1 obtained*
*by holding a zero-coupon bond until time t** _{i+1}* and then
reinvesting the proceeds at that time at the prevailing
short rate ˆ

*r(i + 1), which is stochastic.*

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

*• The expectation in Eq. (154) can be approximated by*
*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

*. (155)*
**– This solves the chicken-egg problem!**

*• Substitute Eq. (155) into Eq. (154) and replace μ**i,j*

*with θ(t**i*) *− ar**j* to obtain

*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* *.*

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

*• For the Hull-White model, the expectation in Eq. (155)*
is actually known analytically by Eq. (25) on p. 162:

*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} *.*

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

*• With θ(t**i**) in hand, we can compute μ**i,j*,^{a} the

probabilities,^{b} *and 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 Eq. (152) on p. 1108.

bSee Eqs. (153) on p. 1115.

### 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*
*that replicate 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}

aHull and White (1995).

### 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. 1114).*

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

*• The second shortcoming is again 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. (154) on
*p. 1118 that helps derive θ(t** _{i}*).

*• The resulting θ(t**i*) hence might not yield a tree that
matches the spot rates exactly.

### 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 and 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 (preview p. 1127).**

**– 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 (153) on**
p. 1115 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** _{i}* so that

*the spot rate r(0, t*

*) is matched.*

_{i+1}^{a}

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

### The Hull-White Model: Calibration

*• Set Δr = σ√*

*3Δt and assume that a > 0.*

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

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

*• Hence the probabilities in Eqs. (153) on p. 1115 use*
*η ≡ −ajΔrΔt + (j − k) Δr.*

*• Recall that k denotes the middle branch.*

### 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 *,* (156)

*p2(i, j)*

= 2

3 *−*

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

*,* (157)

*p3(i, j)*

= 1 6

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

2 *.* (158)

*• 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.*

*• Furthermore, p*_{1}*(i, j) = p*_{3}*(i, −j).*

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

*• The inequalities*

3 *−* *√*
6

3 *< jaΔt <*

2

3 (159)

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
*half of the tree, that is, j < 0.*

### 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) .*

### 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 as of time t**i*.

*• 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* *.*

(160)

### 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),* *− min(i + 1, j*max) *≤ j ≤ min(i + 1, j*max*),*
can now be calculated as before.

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

*• The space requirement is O(n).*

### A Numerical Example

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

*• Immediately, Δr = 0.0173205 and j*max = 2.

*• The plot on p. 1138 illustrates the 3-period trinomial*
tree after phase one.

*• For example, the branching probabilities for node E are*
*calculated by Eqs. (156)–(158) on p. 1130 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. (160) on p. 1135,*

*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,*
*Q(1, −1) = p*_{3}*(0, 0) e*^{−(β}^{0}^{+r}^{0}^{)} *= 0.160414.*

### 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. 1144 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, and Morton (1992).

*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

### 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.

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 mortgage sector is one of the largest in the debt*
market (see p. 3 of the textbook).^{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.

aThe outstanding balance was US$8.1 trillion as of 2012 vs. the US Treasury’s US$10.9 trillion according to SIFMA.

### 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.

aFirst issued by Ginnie Mae in 1970.

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 with the help of First Boston, which was acquired by Credit Suisse in 1990.

### 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. 1162.

*• 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*
starts principal paydown at the end of month 4.

### 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. 46 times (0.95)** ^{i}*.

*• This done for all the months, the interest payment at*
any month is the remaining principal of the previous
month times 1%.

### 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 1165 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. 1162.

*• 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.*

*• 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. 1161 and p. 1163:*

**– 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.

### 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.