### The Black-Derman-Toy Model

^{a}

*• This model is extensively used by practitioners.*

*• The BDT short rate process is the lognormal binomial*
interest rate process described on pp. 1002ﬀ.^{b}

*• The volatility structure*^{c} is given by the market.

*• From it, the short rate volatilities (thus v** _{i}*) are

*determined together with the baseline rates r*

*.*

_{i}aBlack, Derman, & Toy (BDT) (1990), but essentially ﬁnished in 1986 according to Mehrling (2005).

bRepeated on next page.

cRecall Eq. (136) on p. 1053.

*r*_{4}

*r*_{3}

*r*_{2} *r*_{4}*v*_{4}

*r*_{1} *r*_{3}*v*_{3}

*r*_{2}*v*_{2} *r*_{4}*v*_{4}^{2}

*r*_{3}*v*_{3}^{2}

*r*_{4}*v*_{4}^{3}

### The Black-Derman-Toy Model (concluded)

*• Our earlier binomial interest rate tree, in contrast,*
*assumes v** _{i}* are given a priori.

*• Lognormal models preclude negative short rates.*

### The BDT Model: Volatility Structure

*• The volatility structure deﬁnes the yield volatilities of*
zero-coupon bonds of various maturities.

*• Let the yield volatility of the i-period zero-coupon bond*
*be denoted by κ** _{i}*.

*• P*_{u} *is the price of the i-period zero-coupon bond one*
period from now if the short rate makes an up move.

*• P*_{d} *is the price of the i-period zero-coupon bond one*
period from now if the short rate makes a down move.

### The BDT Model: Volatility Structure (concluded)

*• Corresponding to these two prices are the following*
yields to maturity,

*y*_{u} =^{Δ} *P*_{u}^{−1/(i−1)}*− 1,*
*y*_{d} =^{Δ} *P*_{d}^{−1/(i−1)}*− 1.*

*• The yield volatility is deﬁned as*^{a}
*κ** _{i}* =

^{Δ}

*ln(y*

_{u}

*/y*

_{d})

2 *.*

aRyecall Eq. (136) on p. 1053.

### The BDT Model: Calibration

*• The inputs to the BDT model are riskless zero-coupon*
bond yields and their volatilities.

*• For economy of expression, all numbers are period based.*

*• Suppose inductively that we have calculated*
*(r*_{1}*, v*_{1}*), (r*_{2}*, v*_{2}*), . . . , (r*_{i−1}*, v*_{i−1}*).*

**– They deﬁne the binomial tree up to time i − 2 (thus***period i* *− 1).*^{a}

* – Thus the spot rates up to time i − 1 have been*
matched.

a*Recall that (r*_{i−1}*, v*_{i−1}*) deﬁnes i**−1 short rates at time i−2, which*
*are applicable to period i* *− 1.*

### The BDT Model: Calibration (continued)

*• We now proceed to calculate r*_{i}*and v** _{i}* to extend the

*tree to cover period i.*

*• Assume the price of the i-period zero can move to P*_{u}
*or P*_{d} one period from now.

*• Let y denote the current i-period spot rate, which is*
known.

*• In a risk-neutral economy,*
*P*_{u} *+ P*_{d}

*2(1 + r** _{i}*) = 1

*(1 + y)*^{i}*.* (155)

*• Obviously, P*_{u} *and P*_{d} *are functions of the unknown r*_{i}*and v** _{i}*.

### The BDT Model: Calibration (continued)

*• Viewed from now, the future (i − 1)-period spot rate at*
time 1 is uncertain.

*• Recall that y*_{u} *and y*_{d} represent the spot rates at the
up node and the down node, respectively.^{a}

*• With κ*^{2}* _{i}* denoting their variance, we have

*κ** _{i}* = 1
2 ln

*P*_{u}^{−1/(i−1)}*− 1*
*P*_{d}^{−1/(i−1)}*− 1*

*.* (156)

aRecall p. 1162.

### The BDT Model: Calibration (continued)

*• Solving Eqs. (155)–(156) for r*_{i}*and v** _{i}* with backward

*induction takes O(i*

^{2}) time.

**– That leads to a cubic-time algorithm.**

*• We next employ forward induction to derive a*
quadratic-time calibration algorithm.^{a}

*• Forward induction ﬁgures out, by moving forward in*
time, how much $1 at a node contributes to the price.^{b}

*• This number is called the state price and is the price of*
the claim that pays $1 at that node and zero elsewhere.

aW. J. Chen (R84526007) & Lyuu (1997); Lyuu (1999).

bReview p. 1030(a).

### The BDT Model: Calibration (continued)

*• Let the unknown baseline rate for period i be r*_{i}*= r.*

*• Let the unknown multiplicative ratio be v*_{i}*= v.*

*• Let the state prices at time i − 1 be*
*P*_{1}*, P*_{2}*, . . . , P*_{i}*.*

*• They correspond to rates*

*r, rv, . . . , rv*^{i−1}*for period i, respectively.*

*• One dollar at time i has a present value of*
*f (r, v)* =^{Δ} *P*_{1}

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

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

*1 + rv*^{2} + *· · · +* *P*_{i}

*1 + rv*^{i−1}*.*

### The BDT Model: Calibration (continued)

*• By Eq. (156) on p. 1165, the yield volatility is*

*g(r, v)* ^{Δ}= 1
2 ln

⎛

⎜⎝

_{P}

u*,1*

*1+rv* + ^{P}^{u}^{,2}

*1+rv*^{2} + *· · · +* _{1+rv}^{P}^{u}^{,i−1}_{i−1}_{−1/(i−1)}

_{P}*− 1*

d*,1*

*1+r* + ^{P}^{d}^{,2}

*1+rv* + *· · · +* _{1+rv}^{P}^{d}^{,i−1}_{i−2}_{−1/(i−1)}

*− 1*

⎞

⎟*⎠ .*

*• Above, P*_{u,1}*, P*_{u,2}*, . . . denote the state prices at time*
*i* *− 1 of the subtree rooted at the up node.*^{a}

*• And P*_{d,1}*, P*_{d,2}*, . . . denote the state prices at time i* *− 1*
*of the subtree rooted at the down node.*^{b}

a*Like r*_{2}*v*_{2} on p. 1159.

b*Like r*_{2} on p. 1159.

### The BDT Model: Calibration (concluded)

*• Note that every node maintains three state prices:*

*P*_{i}*, P*_{u,i}*, P** _{d,i}*.

*• Now solve*

*f (r, v)* = 1

*(1 + y)*^{i}*,*
*g(r, v)* = *κ*_{i}*,*

*for r = r*_{i}*and v = v** _{i}*.

*• This O(n*^{2})-time algorithm appears on p. 382 of the
textbook.

### Calibrating the BDT Model with the Diﬀerential Tree (in seconds)

^{a}

Number Running Number Running Number Running

of years time of years time of years time

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

75MHz Sun SPARCstation 20, one period per year.

aLyuu (1999).

### The BDT Model: Continuous-Time Limit

*• The continuous-time limit of the BDT model is*^{a}
*d ln r =*

*θ(t) +* *σ*^{}*(t)*

*σ(t)* *ln r*

*dt + σ(t) dW.*

*• The short rate volatility σ(t) should be a declining*

function of time for the model to display mean reversion.

**– That makes σ**^{}*(t) < 0.*

*• In particular, constant σ(t) will not attain mean*
reversion.

aJamshidian (1991).

### 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 & 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} =^{Δ} *t*_{1} *+ Δt*_{1}*,*

*t*_{3} =^{Δ} *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*)) Δt*_{1} *− σ(t*1)

*Δt*_{1} *,*
*ln r*_{u}*(t*_{2}) = *ln r(t*_{1}*) + κ(t*_{1}*)(θ(t*_{1}) *− ln r(t*1*)) Δt*_{1} *+ σ(t*_{1})

*Δt*_{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} *.*

(157)

*• So from Δt*_{1}*, we can calculate the Δt*_{2} that satisﬁes the
combining condition and then iterate.

**– t**_{0} *→ Δt*_{1} *→ t*_{1} *→ Δt*_{2} *→ t*_{2} *→ Δt*_{3} *→ · · · → T*
(roughly).^{a}

a*As κ(t), θ(t), σ(t) are independent of r, the Δt** _{i}* will not depend on

*r either.*

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

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

^{a}

aAmin (1991); C. I. Chen (R98922127) (2011); Lok (D99922028) &

Lyuu (2016, 2017).

### 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 & Weintraub (1993).

bSandmann & Sondermann (1993).

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

*• This problem can be somewhat mitigated by adopting*
variable-duration 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 & 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 & 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.* (158)

*• 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.*

aHull & 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}* =

^{Δ}

*iΔt,*

*r** _{j}* =

^{Δ}

*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*^{Δ} * _{i}*)

*− ar*

*(159) 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. 1187.

*• 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)* 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}*).

**– It describes 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. (159) on p. 1186.

*• 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}

a*Not 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}

a*That 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 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. 1192 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,* (160)

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

*(Δr)*^{2} *,* (160* ^{}*)

*p*

_{3}

*(i, j) =*

*σ*

^{2}

*Δt + η*

^{2}

*2(Δr)*^{2} *−* *η*

*2Δr,* (160* ^{}*)
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}*).

aRecall p. 826.

bHull & 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*

*.(161)*

*• 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).*

aThus ˆ*r(i + 1) is stochastic.*

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

*• The expectation in Eq. (161) 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

*.* (162)
**– This solves the chicken-egg problem!**

*• Substitute Eq. (162) into Eq. (161) 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. (162)*
is actually known analytically by Eq. (30) on p. 179:

*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}

aSee Eq. (159) on p. 1186.

### 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. (160) on p. 1193.

### 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 & 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. 1192).*

**– 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. (161) on
*p. 1196 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 & 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. 1205).**

**– 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 (160) on**
p. 1193 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}

a*Contrast this with the previous algorithm, where it was r(0, t** _{i+2}*)
that was 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. (160) on p. 1193 use*
*η* =^{Δ} *−ajΔrΔt + (j − k) Δr.*

*• Recall that k tracks 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 *,* (163)

*p2(i, j)*

= 2

3 *−*

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

*,* (164)

*p3(i, j)*

= 1 6

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

2 *.* (165)

*• 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 (166)

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.

*• 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* *.*

(167)

### 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.^{a}

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

*• The space requirement is O(n).*

aRecall p. 1199.

### 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. 1216 illustrates the 3-period trinomial*
tree after phase one.

*• For example, the branching probabilities for node E are*
*calculated by Eqs. (163)–(165) on p. 1208 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. (167) on p. 1213,*

*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. 1222 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}*,*

(168)
*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}

aAmin & Morton (1994).

### 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 22 (1) For all 0 < t ≤ T ,***μ(t, T ) =*

*k*
*i=1*

*σ*_{i}*(t, T )*

_{T}

*t*

*σ*_{i}*(t, u) du* (169)
*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}*,* (170)

*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. (169)*
on p. 1228.

*• 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}

0

*μ(s, T ) ds +*

_{t}

0

*σ(s, T ) dW*_{s}*.*

### 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}*. (171)*

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

### 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. (171) on p. 1232 is reduced to*
*dr =*

*∂f (0, t)*

*∂t* *+ σ*^{2}*t*

*dt + σ dW.*

*• This is the continuous-time Ho-Lee model (154) on p.*

1154.^{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}*.* (172)

*• 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}*.*

aCarverhill (1995); Musiela & Rutkowski (1997); Hull (1999).

### 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 (154) on p. 1154 obtains.*

*• σ(t, T ) = σe** ^{−a(T −t)}*: The Hull-White model (158) on p.

1182 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 ).* (173)

*• 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.*

bGupta & Subrahmanyam (2001, 2005).

cGupta & Subrahmanyam (2001, 2005).

### 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).

b*The 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. (172) on p. 1234:

aRebonato (1996). It is equivalent to the proportional volatility model
*(173) 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. (168) on p. 1226 with μ(t, T ) from Eq. (169) on*
p. 1228.

### 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)