3 Binomial Pricing of Forward Rate Agreements, Swaps, Caplets, Floorlets, Swaptions, and Other Derivatives
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16.3 Binomial Pricing of Forward Rate Agreements
385
The pricing of options such as caps (or caplets) is not as straightforward as that for
FRAs. In particular, recall from Chapter 2, the pay-off of a caplet struck at fixed interest rate
RK , maturing at time T, on a floating reference rate R T of tenor applied to the period
T T + in the future. The floating rate is typically the three- or six-month LIBOR. The
pay-off of this caplet option is given by
T
Cpl R
T = R
T − RK
+
Z T rT T +
(16.19)
where ZT rT T + is the discount function over that period, since the cash flow occurs at
time T + . Here we define x + ≡ max x 0 as usual. In order to obtain the price at current
time t = 0 of this caplet one must take an expectation, or integral, of the pay-off with a
risk-neutral distribution in the reference rate Rt , where t = T , i.e., the expiry or maturity
time of the option on the (call-type) pay-off. The latter is, however, expressed in terms of a
rate applied to the period of the tenor (i.e., the reference forward rate) and not the short rate
used in the rate lattice calibration of the previous section. In particular, the short rate lattice
gives the conditional distribution of the short rate.
To price the caplet, one must relate the short rate to this reference forward rate. In
particular, the values of the short rate at the nodes j i , r j i must be related to the values of
the reference forward rates, denoted by R j i , at these nodes. This is achieved by using the
continuous-time relation for the forward rate, and this is where the conditional zero-coupon
prices Z r t T are useful. In particular, for continuous compounding,
R
t =
1
R
t =
1
log
Zr t t
Z r t t+
(16.20)
log
1
Z r t t+
(16.21)
and since Z r t t = 1,
Choosing t = Ti and T = Ti + n t, where it is assumed that the tenor is exactly n periods of
the lattice time step, for some integer n, = n t, we arrive at the discrete time value at the
jth node:
R
j i =
1
log
1
Z j Ti Ti + n
t
(16.22)
Here Z j Ti Ti + n t ≡ Z j Ti Ti+n is the zero-coupon value maturing at time Ti+n (n
time steps in the future), conditional on the short rate’s having value r j i at time Ti . Based
on equations (16.19) and (16.22), we can write all components of the payoff vector of the
caplet at each node j i , denoted by C j i , as
C
j i = R
j i − RK
+
Z j Ti Ti+n
(16.23)
where equation (16.22) is plugged in for R j i . Note that the preceding equations assume
continuous compounding, while a similar set of equations obtain for the case of discrete
compounding, where the log x function is simply replaced by x. The foregoing payoff vector
introduces an extra procedural step, requiring one to compute the quantities Z j Ti Ti+n ,
which involve a separate forward induction starting from the nodes r j i . In practice, these
are computed using the discrete-time version of equation (16.7):
j+n
Z j Ti Ti+n =
G j Ti k Ti+n
k=j
(16.24)
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where the n + 1 Arrow–Debreu values (conditional on beginning at a jth node at time Ti
and ending at node k = j j + 1
j + n at time Ti+n ) on the right-hand side of this equation
are computed by forward recursion using an adaptation of equation (16.12), rewritten here in
a slightly different form:
G j Ti k Ti+m =
1 −r
e
2
s=k k−1 j≤s≤j+m−1
s i+m−1
t
G j Ti s Ti+m−1
(16.25)
Here m = 1 2
n and the iteration is readily carried out from time T = Ti to final time
Ti+n , where one initially has G j Ti j Ti = 1 for any j value. It is instructive to write out
the Arrow–Debreu values explicitly for the first two time steps. For a single step (for m = 1)
the terminal time is Ti+1 , and we simply have
1
G j Ti k Ti+1 = e−r j i
2
t
(16.26)
where k = j j + 1 are the only two possible values for k. Not surprisingly, this is consistent with the relation in equation (16.13). Propagating out to the second step (m = 2),
equation (16.25) gives
G j Ti k Ti+2 =
1 −r
e
2
s=k k−1 j≤s≤j+1
t 1 −r s i+1
ji
2
e
t
(16.27)
where possible values for k are j j + 1 j + 2. Summing up the terms explicitly, these three
Arrow–Debreu prices are
1
G j Ti j Ti+2 = e−r
4
1
G j Ti j + 1 Ti+2 = e−r
4
1
G j Ti j + 2 Ti+2 = e−r
4
ji
t −r j i+1
ji
t
ji
t −r j+1 i+1
e
e−r
t
j i+1
t
e
+ e−r
t
j+1 i+1
t
(16.28)
Specializing equation (16.14) we therefore finally have the binomial lattice pricing formula
for a caplet valued at current time T0 = 0 and maturing at time Ti of tenor = n t:
i
Cpl0 RK Ti =
G 0 0 j Ti C
j i
(16.29)
j=0
To summarize then, the application of this pricing formula contains two components. The first
part is the computation of the G 0 0 j Ti , which are already computed from the calibration
step, as discussed in the previous section. The second part consists of computing the payoff
components C j i . These are obtained by first computing the conditional Arrow–Debreu
prices G j Ti k Ti+n by forward induction using equation (16.25). These quantities are
then summed up to give the Z j Ti Ti+n , as in equation (16.24). In turn, the latter are
plugged into equation (16.22), giving the forward rates R j i , and hence C j i , using
equation (16.23).
Figure 16.4 depicts, schematically, this procedure for pricing a caplet. For implementation
considerations, note that the inputs within the ir sheet (for pricing a caplet) are the expiry
time Ti , which for simplicity is assumed chosen as an integer number of time steps from
current time T0 , and the tenor of the caplet is chosen as an integer number of time steps
16.3 Binomial Pricing of Forward Rate Agreements
387
r ( k = j + n, i + n )
•••
C (τ)
( j, i )
r ( j, i )
r ( k = j + 1, i + n )
r ( k = j, i + n )
r (0, 0)
T0
Ti
Ti + n
τ
FIGURE 16.4 Schematic representation of the separate components used for the pricing of a caplet
option of tenor , expiring at time Ti . The initial leg starting from the current time node r 0 0 gives the
Arrow–Debreu prices G 0 0 j Ti at each jth node r j i at time Ti . The payoff vector of the caplet
with jth component C j i (for the jth node at time Ti ) is obtained by summing all the Arrow–Debreu
j + n) that are conditional on starting at the node r j i at time Ti
prices G j Ti k Ti+n (k = j
and ending at nodes r k i + n at time Ti+n for the period of the caplet.
past Ti , where the time step is the previously computed t. Note that, if needed, this apparent
restriction can be readily lifted by using a different time-step value for the lattice past-maturity
time Ti . The spreadsheet contains inputs for the number of time steps to reach the caplet
(or floorlet) option expiry time from today, i.e., an integer M with M t = Ti and another
integer for the number of steps defining the tenor.
The entire analysis for pricing a floorlet of the same maturity, struck at rate RK , follows
almost identically as in the case of the caplet, except the pay-off is now that of a put,
RK − R + , instead of that of a call, R − RK + . Within this project one should allow for a
computation of both types of options as well as the pricing of swaptions.
Next, we consider the pricing of European swaptions. Such options, as discussed in
Chapter 2, come in two flavors: The payer swaption has pay-off given by equation (2.41),
while the receiver swaption has the put type of pay-off. Let us consider a payer swaption,
struck at rate rK , on an underlying swap to start at time T = Tns in the future and having a
lifetime of n periods of fixed tenor :
n
PSOT =
rTs − rK
ZT T + p
+
(16.30)
p=1
Here rts denotes the equilibrium swap rate at time t. Hence, the first reset time of the swap is
assumed as T = Tns , with first payment time at T + , the latter being the second reset time
with second payments occurring at T + 2 , etc. Note that, as in the case of caplets, within
the ir application spreadsheet the user enters both the option expiry time T and the tenor .
In addition, the swaption contract is defined by entering the number of periods n, with each
time period assumed constant and given by . In particular, given a maturity T, we choose
a number of time steps ns up to maturity with ns t = T , thereby defining a fixed time step
t = T/ns . The contract is assumed to be specified as having tenor = ms t. The number
of time steps within the swap is then Ns = ms n, giving a swap lifetime of Ns t; i.e., the
swap ends at calendar time given by the ns + Ns th time slice: T + n = Tns +Ns . Figure 16.5
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ns Δt
Ns Δt
ms Δt
T0
T+τ
T
T + nτ
FIGURE 16.5 Time spacing for a swaption expiring at time T. The underlying swap has n equal
periods of tenor .
shows a schematic of the time spacing for the swaption. Now recall from Chapter 2 that the
equilibrum swap rate at time T can be written as
rTs =
ZT T − ZT T + n
n
p=1 ZT T + p
=
1 − ZT T + n
n
p=1 ZT T + p
(16.31)
The pay-off then takes the form
PSOT = A − B
(16.32)
+
Here A is a floating-rate bond
A = 1 − ZT T + n
(16.33)
and
n
B = rK
ZT T + p
(16.34)
p=1
is an annuity or fixed-coupon bond originating at time T with fixed payments of amount rK
at n periods of time . The formula in equation (16.32) is directly suitable for implementation.
Based on equations (16.33) and (16.34), the components of the payoff vector of the payer
swaption at each jth node r j i = ns , denoted by P n j i , are given by
n
P
n
j i = ns = 1 − Z j Tns Tns +Ns − rK
Z j Tns Tns +pms
p=1
(16.35)
+
This pay-off therefore requires the evaluation of the zero-coupons Z j Tns Tns +pms conditional on the starting node r j ns at time slice Tns and maturing at times Tns +pms , p = 1
n.
These are computed in the same manner as described for the caplet case. Namely, equation (16.24) gives
Z j Tns Tns +pms =
j+pms
G j Tns k Tns +pms
(16.36)
k=j
where conditional Arrow–Debreu prices now need to be calculated at every time slice
n. The procedure for doing so is the same as in the caplet case,
ns + pms , i.e., for p = 1
where forward recursion equation (16.25) is used repeatedly. This time the recursion is carried
out for a total of Ns = ms n steps, and at each interval number p of ms steps we extract a
pms + 1 -dimensional vector of Arrow–Debreu prices G j Tns k Tns +pms with components
16.4 Trinomial Lattice Calibration and Pricing in the Hull–White Model
389
k = j j +1
j + pms . After having obtained the conditional zero coupons, the current
price of the payer swaption is given by discounting the payoff vector:
n
PSO0
rK T =
ns
G0 0 j T P
n
j ns
(16.37)
j=0
Lastly, note that the pricing of reciever swaptions follows in identical manner, except that
the pay-off is simply replaced by the put type of expression B − A + .
16.4 Trinomial Lattice Calibration and Pricing
in the Hull–White Model
The implementation of trinomial lattices for interest rate trees shares some similarities with the
case of stock price trees covered in the previous project on trinomial lattices for pricing equity
options. There are, however, some important differences, stemming from the fact that the
short rate is itself stochastic and, hence, discounting is inherently very different, as we have
seen in the binomial lattice implementation. Before proceeding to implement a specific shortrate lattice, it is useful to note that there are various possible acceptable tree implementations.
Namely, one could adapt the tree methodologies used in the previous trinomial lattice project,
which deals with stock price processes, over into the case of a short-rate process. This requires
appropriate modifications to account for the mean-reversion effect as well as calibration to discount bond prices across all time steps. The latter would require that the transition probabilites
(p+ , p0 , and p− ) also depend on the nodal positions. One can, moreover, also incorporate
a similar drift parameter (i.e., the parameter), which would now also depend on the ith
time slice Ti . Such a viable lattice makes use of only normal branching. Here we shall deviate slightly and follow Hull and White’s two-stage tree-building procedure [HW93, HW94,
Hul00]. As shown later, this procedure has the added advantage of separating out the reversion
term from the drift component. As well, the sampling of the short-rate nodes in the lattice is
done in a more efficient manner by incorporating three types of possible branching modes.
16.4.1 The First Stage: The Lattice with Zero Drift
As discussed in Chapter 2, the Hull–White (HW) model is defined by the stochastic short-rate
process, which can be written in the form
drt =
t − a t rt dt +
t dWt
(16.38)
where t is a time-dependent drift term. Throughout, we shall further restrict the mean
reversion a t = a and volatility t = to be time-independent parameters. For present
purposes this offers a reasonably good model that can be used to calibrate to zero-coupon
bonds and subsequently to price interest rate options. Extensions that allow for the reversion
speed and/or volatility functions to take on a time dependence (either numerically or analytically) can also be readily achieved. This would allow for exact calibration of the lattice model
to a larger basket of instruments besides zero-coupon bonds. We leave this as an optional
implementation exercise for the interested reader. The first step is to construct a tree for the
related process with zero drift (and nonzero reversion) defined by
drt∗ = −art∗ dt +
dWt
(16.39)
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Fixing rt∗ within a time step, we compute the mean and variance of the random variable
∗
∗
rt+
t − rt as given by the expectations
∗
∗
∗
E rt+
t − rt = −art
∗
∗
E rt+
t − rt
where only terms up to order
r ∗ j i , where
t
2
2
t
= a2 rt∗
(16.40)
t 2+
2
2
t
(16.41)
are included. The r ∗ lattice has nodes defined by r j i =
r ∗ j i = r0∗ + j
r
(16.42)
i − 1 i for any time slice Ti = i
with r0∗ = 0 and j = −i −i + 1
Using equation (16.42) within equations (16.40) and (16.41) gives
∗
∗
∗
E rt+
t − rt rt = j
E
∗
rt+
t
− rt∗ 2
rt∗
=j
r = −aj
r =a j
2 2
r
r
t (see Figure 16.6).
t
(16.43)
t +
2
2
2
t
(16.44)
At this point one finds explicit formulas for the transition probabilities p+ , p0 , and p− for,
respectively, the higher, middle, and lower branches emanating from a given node r j i . The
three possible branching modes considered are depicted in Figure 16.7. Note the difference
in convention with respect to the indexing of the nodes that was used in the binomial lattice.
As in the previous project on trinomial lattice models, an up (down) move changes the jth
index in r j i by +1 −1 , while only for a middle move j remains unchanged. For the
case of normal branching we compute the expectations
∗
∗
∗
E rt+
t − rt rt = j
r = p+ j + 1
= p+ − p−
r + p0 j
r + p− j − 1
(16.45)
r ( j + 1, i + 1)
r (1, 2)
r (0, 2)
r (0, 0)
r
r
r (2, 2)
r (1, 1)
r −j
r (0, 1)
•••
r ( j, i )
r ( j,i + 1)
•••
r ( j – 1,i + 1)
r (–1, 2)
r (–1, 1)
r (–2, 2)
T0
T1
T2
Ti
Ti + 1
FIGURE 16.6 Schematic of the (driftless) symmetric trinomial r ∗ -lattice for the short-rate process with
symmetric (normal) branching from all nodes.
16.4 Trinomial Lattice Calibration and Pricing in the Hull–White Model
r ( j + 1, i + 1)
r ( j + 2, i + 1)
r ( j, i )
p+
r ( j, i + 1)
r ( j + 1, i + 1)
p0
r ( j, i + 1)
r ( j, i )
391
r ( j –1, i + 1)
r ( j, i + 1)
p–
r ( j –1, i + 1)
r ( j, i )
r ( j – 2, i + 1)
normal
downward
upward
FIGURE 16.7 The three possible branching modes.
and
∗
∗
E rt+
t − rt
2
rt∗ = j
r = p+
r 2 + p0 0 2 + p−
= p+ + p−
r
2
r
2
(16.46)
where we have used probability conservation p+ + p0 + p− = 1. It has been observed in the
past [HW94] that numerical efficiency is maximized by fixing the spacing to
r=
√
t
3
(16.47)
Using this value for the spacing and equating expectations in equations (16.45) and (16.43)
and the expectation in equation (16.46) with that in equation (16.44) gives a linear system of
two equations in p+ and p− with unique solution
p± j =
1 1
+ aj
6 2
t∓1
t aj
(16.48)
Probability conservation gives
p0 j =
2
− aj
3
t
2
(16.49)
Note that the argument j is used to explicitly denote the dependence of the transition probabilities on the nodal j-position value.
A similar analysis gives the probabilities for downward branching:
7 1
+ aj
6 2
1 1
d
p−
j = + aj
6 2
1
p0d j = − − aj
3
d
p+
j =
t aj
t−3
(16.50)
t aj
t−1
(16.51)
t aj
t−2
(16.52)
The superscipt d is used to denote the transition probabilities for downward branching.
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Lastly, for upward branching we have
1 1
+ aj
6 2
7 1
u
p−
j = + aj
6 2
1
p0u j = − − aj
3
u
j =
p+
t aj
t+1
(16.53)
t aj
t+3
(16.54)
t aj
t+2
(16.55)
The superscipt u denotes the transition probabilities for upward branching.
Note that the foregoing expressions are either concave or convex quadratic functions of j.
One can readily derive conditions on j for the transition probabilities to be strictly positive.
Namely, for normal branching
√
√
2/3
2/3
(16.56)
−
a t
a t
for upward branching
√
√
−1 − 2/3
−1 + 2/3
a t
a t
(16.57)
and for downward branching
√
√
1 − 2/3
1 + 2/3
a t
a t
(16.58)
Throughout we assume
a > 0. Let us define a maximum value jmax as the smallest integer
√
greater than 1 − 2/3 / a t ≈ 0 1835/ a t , for the index j at any time slice, and a
minimum value as jmin = −jmax . This leads to the branching methodology for each node j i ,
whereby normal branching is used for jmin < j < jmax , downward branching is used for extreme
positive value j = jmax , and upward branching is used for extreme negative value j = jmin .
16.4.2 The Second Stage: Lattice Calibration with Drift and Reversion
The purpose of the first stage is to build the component of the r-tree (i.e., the r ∗ -tree) that
encapsulates the mean-reversion and volatility aspect of the short-rate process. In the final
tree implementation, considered in this section, one needs to incorporate a drift component.
Namely, at each time slice the nodes will be drifted by an amount determined by the market
prices of the zero-coupon bonds. The drift component is incorporated by considering the
difference t = rt − rt∗ . This satisfies an ordinary differential equation where
d
t
=
t −a
dt
(16.59)
eas s ds
(16.60)
t
with solution
t
= e−at
0+
t
0
Here 0 = r0 = r 0 0 since r0∗ = 0. Equation (16.59) provides an apparently trivial analytical
link between the actual r-tree and the driftless r ∗ -tree since t can be obtained exactly
from the initial-term structure [i.e., as function of the yield y0 t ]. Indeed, the right-hand
16.4 Trinomial Lattice Calibration and Pricing in the Hull–White Model
393
side of equation (16.60) can be computed explicitly by applying the formulas derived in
Chapter 2. Namely, one can use equation (2.110) or (2.109) [note that there the drift function
t is called a t and the function b is called a here] into the integral of equation (16.60) to
obtain t . We will not adopt this methodology here since it leads to inaccurate results and,
moreover, bypasses the importance of the pricing algorithm for the drifted trinomial lattice,
which we now present.
To apply the trinomial lattice pricing methodology we simply extend the equations of
Section 16.2 into the trinomial lattice case. In general, we must distinguish between the
different possible branching. Let us first assume normal branching. In this case the Green’s
function propagation (i.e., the Arrow–Debreu forward recursion) equation (16.12) is modified
to read
G 0 0 j Ti =
G 0 0 k Ti−1 G k Ti−1 j Ti
(16.61)
k=j±1 j k ≤i−1
where the Arrow–Debreu prices for a single time step are nonzero for k ≤ i − 1 and k =
j j ± 1 and given by
G k Ti−1
⎧
p+ k e−r k i−1
⎪
⎪
⎪
⎪
⎪
⎨
j Ti = p0 k e−r k i−1
⎪
⎪
⎪
⎪
⎪
⎩
p− k e−r k i−1
t
k = j −1
t
k=j
t
k = j +1
(16.62)
In contrast to the binomial case, the forward time propagation of Arrow–Debreu prices is
now obtained by summing contributions up to three (as opposed to two) possible two-legged
paths. Note that the probabilities for up/down and middle moves in equations (16.61) and
(16.62) are the ones corresponding to normal branching. For terminal node values of j close
to either jmin or jmax , equations (16.61) and (16.62) need to be slightly modified. Namely,
for any given value of j, equation (16.61) must be modified to the more general case
G 0 0 k Ti−1 p j k e−r k i−1
G 0 0 j Ti =
t
(16.63)
k k ≤i−1
This formula takes into account all (generally mixed) branching types. The quantities p j k
denote the nodal transition probabilities for all possible nonzero contributions from intermediate nodes at positions k for time Ti−1 . The sum of the corresponding probability values to
be used in equation (16.63) now depend on the terminal j value. Assuming jmax > 2, there
are possibly seven distinct cases to consider after jmax time steps.
d
1. j = jmax gives two terms (one down branch and one normal branch) with p j j = p+
j,
p j j − 1 = p+ j − 1 .
2. j = jmax − 1 gives three terms (one down branch and two normal branches) with
p j j + 1 = p0d j + 1 , p j j = p0 j , p j j − 1 = p+ j − 1 .
3. jmin + 2 < j < jmax − 2 gives three terms (three normal branches) with p j j + 1 =
p− j + 1 , p j j = p0 j , p j j − 1 = p+ j − 1 .
4. j = jmin , gives two terms (one up branch and one normal branch) with p j k = j + 1 =
u
p− j + 1 , p j j = p−
j.
5. j = jmin + 1 gives three terms (one up branch and two normal branches) with p j k =
j + 1 = p− j + 1 , p j j = p0 j , p j j − 1 = p0u j − 1 .
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C H A P T E R 16
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6. j = jmax −2 gives four terms (one down branch and three normal branches) with p j k =
d
j + 2 p j j + 1 = p − j + 1 p j j = p 0 j p j j − 1 = p+ j − 1 .
j + 2 = p−
7. j = jmin + 2 gives four terms (one up branch and three normal branches) with p j k =
u
j −2 .
j + 1 = p − j + 1 p j j = p 0 j p j j − 1 = p+ j − 1 p j j − 2 = p +
The forward propagation of Arrow–Debreu prices therefore involves a sum of two, three
or four terms in cases where the terminal node is close to jmax or jmin . Most values of j,
however, involve normal branching, with the use of a three-term sum.
The pricing of zero-coupon bonds is essentially similar to the binomial lattice case, in
the sense that one iterates out to any given time slice Ti to obtain the Arrow–Debreu prices
G 0 0 j Ti . The analogue of equation (16.14) takes the form
i
P0 r0 Ti =
G 0 0 j Ti
r j Ti Ti
(16.64)
j=−i
Specializing to the case of zero-coupon bonds, the equation analogous to equation (16.15)
for pricing zero-coupon bonds is
i
Z0 Ti =
G 0 0 j Ti
(16.65)
j=−i
Inserting equation (16.63) into equation (16.65) gives
i
G 0 0 k Ti−1 p j k e−r
Z0 T i =
k i−1
t
(16.66)
j=−i k ≤i−1
Hence, in general, one finds that the trinomial lattice calibration for a short-rate model can
be achieved using a numerical root-finding procedure in equation (16.66) analogous to the
binomial lattice methodology. The HW model, however, offers extra flexibility since one can
actually solve the calibration problem analytically in the case of continuous compounding.
The calibration of the lattice nodes for the HW model proceeds as follows.
The preceding formulas are specialized to the case where the actual drifted lattice is
represented by
r j i =
i +j
−i ≤ j ≤ i
r
(16.67)
with
0 = r 0 0 as the initial node and the spacing given by equation (16.47). The
coefficients i represent the central node r 0 i along each time slice Ti and will therefore
account for the drift of the lattice. Plugging this into equation (16.66) and taking logarithms
we obtain the simple analytical form for the coefficients:
log
i =
i+1
j=−i−1
k k ≤i G
0 0 k Ti p j k e−k
t
r t
− log Z0 Ti+1
(16.68)
Note that we have shifted the time slice index i to i + 1. This gives the central node at each
time slice Ti , and hence from equation (16.67) all nodes r j i for time Ti are obtained,
based on the market price of the zero-coupon bond maturing at time Ti+1 and knowledge of
the Arrow–Debreu prices out to time Ti . These Arrow–Debreu prices are in turn given by
forward induction using equation (16.63) by using the already-known values for the node
positions at time slice i − 1.
16.4 Trinomial Lattice Calibration and Pricing in the Hull–White Model
395
One begins the calibration procedure with G 0 0 0 0 = 1, and the initial node r 0 0 =
0 is given in terms of the interpolated zero-coupon price at the first maturity time T1 = t
using equation (16.16). Based on this and the zero-coupon prices at further maturities, one
obtains the rest of the lattice nodes using equations (16.68) and (16.63). For instance, after
the first step we have normal branching with G 0 0 0 T1 = p0 e− 0 t , G 0 0 ±1 T1 =
p± e− 0 t . Assuming normal branching, at the second time step we obtain 1 :
log
1 =
=
1
−j
j=−1 e
r
t
G 0 0 j T1
− log Z0 T2
t
log p− e
r
t
+ p0 + p+ e −
r
t
− log Z0 T2
−
t
0
(16.69)
This procedure is continued for the rest of the time steps, hence giving the calibrated lattice
for as many time steps as needed.
For the calibration of short-rate models that do not admit a simple analytical solution,
such as the Black–Karasinski model covered in Section 16.5, one can readily proceed to find
the central nodes numerically via a root-finding routine similar to what was described earlier
for the binomial lattice.
16.4.3 Pricing Options
Once the calibrated lattice is built, the procedure for pricing options (e.g., caplets, floorlets,
swaptions) follows similar steps as described for the binomial lattices given in Section 16.3.
The conditional zero-coupon bonds are now obtained using
j+n
Z j Ti Ti+n =
G j Ti k Ti+n
(16.70)
k=j−n
where the 2n + 1 Arrow–Debreu values (conditional on beginning at a jth node at time Ti
and ending at node k = j − n
j + n at time Ti+n ) are computed by a general extension
of equation (16.63), i.e., using the forward recursion relation
p k s e−r s i+m−1 t G j Ti s Ti+m−1
G j Ti k Ti+m =
(16.71)
s s ≤i+m−1
Just as in equation (16.63), this forward propagation formula takes into account all possible
mixed branchings. Note that the starting node is denoted by index j, while the terminal node
now has index k. The nodal transition probabilities p k s are again given as described just
following equation (16.63). For instance, when jmin + 2 < k < jmax − 2, normal branching is
used with three possible nonzero values for p k s : p k s = k ± 1 = p∓ k ± 1 , p k s =
k = p0 k .
Based on knowledge of the conditional zero-coupon prices, all option-pricing formulas
are indentical in form to those for the binomial lattice, except for the obvious modification in
having to compute and sum up more terms due to n extra terminal nodes for every n steps.
Hence, for example, the caplet price is obtained by modifying equation (16.29) slightly:
i
Cpl0 RK Ti =
G 0 0 j Ti C
j=−i
j i
(16.72)