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Stochastic Differential Equations and Interest Rate Models
Trinomial Tree Implementation of the Hull-White Model
12 February 2008
The model is
drt = (θ(t)− λrt)dt + σdwt,so that
rt = e−λtr0 +
t0
e−λ(t−s)θ(s)ds +
t0
e−λ(t−s)σdws
≡ a(t) + e−λtX t, (1)
where X t is the zero-mean gaussian
martingale
X t =
t0
eλsσdws.
The tree models X t. The final maturity time is
T years, and there are M time
steps per year,
so the discrete time increment is δ =
1/M with a total of T M steps.
The spatial step is h. If
we take
h = σ√
3δ (2)
and up and down probabilities equal to 1/6, then the trinomial
step matches the first 5 moments
of the normal distribution N (0, σ2). Here
X t does not have stationary increments, so we let
σ2
be an ‘average’ variance, defined by
σ2 = σ2
2λT
(e2λT
−1) =
1
T
E [X 2T ]
≡σ2b2.
Thus
h = σ
3b2δ.
Now
E (X t+δ − X t)2 = σ2 t+δt
e2λsds
= σ2e2λt
2λ (e2λδ − 1)
= σ2t δ,
so
σ2t = σ2e2λtb1,
where
b1 = 1
2λδ (e2λδ − 1).
To match the incremental variance at time t =
kδ , and recalling that h is given by (2),
the up
and down probability pk must satisfy 2 pkh2
= σ2t δ , from which we find
pk = 1
6
b1b2
e2λδk.
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Calibration to the Yield Curve for given σ, λ
Let us denote the discrete-time process on the tree by
(Y k), so that Y k ∈ { jh : −k ≤
j ≤ k}.We want to ensure that the tree correctly
calculates the zero-coupon bond values
Bk = p(0, kδ ), 1
≤k
≤T M, (3)
which are today’s market data. Index the nodes at time k
as (k, j) where −k ≤ j ≤ +k.
In(1) we take a(t) to be piecewise constant: a(t) =
ak, t ∈ [kδ, (k + 1)δ [. We thus have
T M constraints to determine the T M
values a0, . . . , aTM −1. This is done by a forwards
induction
procedure involving the so-called Arrow-Debreu
prices q k,j . The price q k,j is
the value at time 0
of a contract that delivers $1 if Y k
= jh and zero otherwise. The calibration
condition is thus
Bk =k
j=−k
q k,j , k = 1, . . . , T M .
(Of course, B0
= q 0,0
= 1.) At node (k, j) the short rate is, from (1),
ak
+ e−λδk jh. We interpret
this as the period-δ rate set at time kδ ,
so the discount factor for the period [kδ, (k + 1)δ ]
when
Y k = jh is1
1 + δ (ak + e−λδk jh) ≡
1
1 + αk + jβ k
We recursively compute the coefficients αk and the
Arrow-Debreu (AD) prices, and then we
throw away the latter. They are not needed subsequently. Note
that the β k are known.
At time k = 0, the discount factor is 1/(1
+ α0), so
B1 = 1
1 + α0, α0 =
1
B1− 1.
The AD prices at time k = 1 are
q 1,1 = q 1,−1 = p01
+ α0
, q 1,0 = (1 − 2 p0)
1 + α0.
Suppose we have determined α0, . . . , αk−1 and
all the AD prices q k,j , at some time index
k.
Then the next ZCB price is
Bk+1 =k
j=−k
q k,j1 + αk + jβ k
. (4)
The right-hand side of (4) is a monotone decreasing function
of αk, so we can determine the
unique value of αk satisfying (4), by a binary
search procedure.If we consider a node (k+1, j) with −k+1 ≤
j ≤ k−1, then Y k+1 = j only
if Y k = j−1, j , j+1.
The AD price for (k + 1, j) is therefore
q k+1,j = pkq k,j−1
1 + αk + ( j − 1)β k+
(1 − 2 pk)q k,j1
+ αk + jβ k
+ pkq k,j+1
1 + αk + ( j + 1)β k.
There are similar expressions for the ‘boundary’ cases j
= −k − 1,−k,k,k + 1. The order of business is
thus to use the above expressions to compute α0 →
q 1,· → α1 → q 2,· → . . . →
αTM −1.This completes calibration to market zero-coupon bond
prices.
Calibration of the volatility parameters σ, λ must
use volatility-sensitive market data, i.e.
cap or swaption prices.
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