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3 Binomial Pricing of Forward Rate Agreements, Swaps, Caplets, Floorlets, Swaptions, and Other Derivatives

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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C H A P T E R 16



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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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C H A P T E R 16



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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)



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