Abstract
The LIBOR Market Model: A Recombining Binomial Tree Methodology
.
We propose an implementation of the LIBOR Market Model, adapting the recombining
node methodology of Ho, Stapleton and Subrahmanyam (1995). Initial tests suggest that
the method provides a fast and accurate approach for the valuation of path dependent
interest rate derivatives such as Bermudanstyle swaptions. The lattice based approach
illustrated here provides an efficient alternative to MonteCarlo simulation implementation
of the LMM.
1
Introduction and Motivation
The LIBOR Market Model (LMM) is the most common implementation in practice of the
general Heath, Jarrow and Morton (1990) forward rate approach to the valuation of interest
rate derivatives. First proposed by Miltersen, Sandmann and Sondermann (1997) (MSS)
and Brace, Gatarek and Musiella (1997) (BGM), the model assumes that the London Inter
bank Offer Rate LIBOR has a conditional probability distribution which is lognormal. This
paper addresses two problems that arise with the LMM. First, the proof of the continuous
time LMM is somewhat obscure. Hull (2002), for example simply states the drift of the
forward rates, under the riskneutral measure, without proof. However, without a knowl
edge of why the drift is as stated in the model, it is hard to use the model with confidence.
We show here that the drift of the forward rate can be derived very simply from the pricing
of Forward Rate Agreements (FRAs) in a noarbitrage setting.
Second, multifactor versions of the LMM are difficult to implement, especially for the pric
ing of Bermudanstyle swaptions. We employ the recombining binomialtree methodology
of Ho, Stapleton and Subrahmanyam (1995) (HSS) to construct a swaption pricing model,
which does not have to rely on MonteCarlo simulation or the lowerbound approximations
commonly employed. As an illustration, we implement a twofactor example of the LMM
and price various swaptions and other interestrate sensitive contracts.
2
Main Features of LIBOR Market Models
The LMM is a termstructure model which recovers caplet and floorlet values that are
consistent with the market practice of applying the Black model to price options on interest
rates, defined on a LIBOR basis. Many market participants build LMMs and use them
to price pathdependent interestrate derivatives such as Bermudanstyle swaptions. The
main assumption of the model is that the forward rate is conditional lognormal under the
riskneutral measure. Since in the model the drift of the forward rate is stochastic, the
LIBOR is not unconditional lognormal. If it was, then as in the Black and Karasinski
(1991) spotrate model, the Black model would not hold for caplets and floorlets, due to
the effects of stochastic discounting
1
.
The Black model for a caplet is given by:
1
Pricing a caplet (floorlet) using the BK model leads to caplet (floorlet) prices which are less than (greater
than) those obtained from the Black model.
1
Definition 1 [The Black Model: InterestRate Caplet]
caplet
t
=
A
1 + f
t,t+T
d d[f
t,t+T
N (d
1
)  kN (d
2
)]B
t,t+T
where
d
1
= ln(
f
t,t+T
k
) + s
2
(T )/2
s
v
T
d
2
= d
1
 s
v
T
where
A is the principal value of the caplet.
B
t,t+T
is the value at t of a zerocoupon bond paying 1 unit of currency at t + T .
d is the interestrate reset interval (ex. 3 months) as a proportion of a year.
k is the strike rate.
f
t,t+T
is T period forward LIBOR at time t.
s is the volatility of T period LIBOR.
The LIBOR Market Model is a model of the stochastic evolution of interest rates that is
consistent with the above formula holding for all caplets, with maturities T = 1, 2, ..., N .
The model allows other interestrate dependent contingent claims, such as Europeanstyle
and Bermudanstyle swaptions, to be priced in a way that is consistent with the pricing of
the caplets.
The LMM is rooted in financial theory. In particular it uses the ideas of the riskneutral
measure, forward parity, and noarbitrage asset pricing relationships. These ideas are well
documented in texts such as Pliska (1997). For convenience, we restate the most important
results here. Since we will be concerned with the pricing of zerocoupon bonds, the relevant
ideas concern zerodividend paying assets. We have the following results.
For a zerodividend paying asset:
2
1. The noarbitrage spot asset price is given by
2
S
t
= B
t,t+1
E
t
[B
t+1,t+2
E
t+1
[B
t+2,t+3
[....E
t+T 1
[B
t+T 1,t+T
S
T
]]]],
where the expectation is taken under the periodbyperiod riskneutral measure.
2. The T period forward price of the asset is given by
F
t,t+T
= S
t
/B
t,t+T
3. The expected oneperiodahead forward price of the asset is
3
E
t
(F
t+1,t+T
) = F
t,t+T
 cov
t
(F
t+1,t+T
, B
t+1,t+T
) B
t,t+1
B
t,t+T
.
Since forward rates are closely related to forward prices of zerocoupon bonds, and since we
will be interested in the drift of forward rates, we now apply these results to price forward
contracts on zerocoupon bonds.
Lemma 1 (ZeroCoupon Bond Forward Prices) When expectations are taken under
the riskneutral measure:
1. The T period forward price of a oneperiod maturity zerocoupon bond is
B
t,t+T ,t+T +1
= E
t
(B
t+1,t+T ,t+T +1
) + B
t,t+1
B
t,t+T
cov
t
(B
t+1,t+T ,t+T +1
, B
t+1,t+T
)
2. The oneperiod ahead forward price of a long maturity bond is:
B
t,t+1,T
= E
t
(B
t+1,t+2
B
t+1,t+2,T
)
Lemma 1 is directly useful if we are building a stochastic process of forward prices. It shows
how the drift of the forward price depends upon the covarariance of the forward prices. A
similar effect is found in the drift of forward rates. In the following section we use a corollary
of the lemma in the analysis of the forward rate drift.
2
See, for example Pliska (1997), chapter 2
3
This follows from taking expectations and using the definition of covariance.
3
3
The LIBOR Market Model
The LIBOR Market Model is constructed by forming a process for the evolution of forward
rates of various maturities. We have to be careful with the notation. We adopt the following:
The LMM: Notation
• B
t,t+d
= Value at t of a zerocoupon bond paying 1 unit of currency at t + d.
• d = Interestrate reset interval (ex. 3 months) as a proportion of a year
• B
t,t+T
= Value at t of a zerocoupon bond paying 1 unit of currency at t + T .
• x
t
= Cash flow paid at time t
• S
0
= Value at t = 0 of a cash flow paid at time t
• f
t,t
= Spot LIBOR at time t:
B
t,t+d
=
1
1 + f
t,t
d
• f
t,t+T
= T period forward LIBOR at time t
• B
t,t+T ,t+T +t
= Forward price at t for delivery of a zerocoupon bond (with maturity
t ) at T .
We begin the derivation of the LMM by defining the following standard contract: A Forward
Rate Agreement (FRA) on LIBOR, with maturity T , has a payoff
(f
t+T ,t+T
 k)d
1 + f
t+T ,t+T
d
at date t + T . Note that the definition assumes that the contract is settled at time t + T on
a discounted basis at time t. Also, for simplicity we assume that d is a constant. In practice
the precise payoff depends on the day count. Hence the above contract can be thought of
as a theoretical, or idealised FRA. We now have the following result:
FRA Pricing and the Drift of the Forward rate
We begin by stating the following corollary of Lemma 1:
4
Corollary 1 One and TwoPeriod Forward Rates
From Forward Parity:
B
t,t+1,t+3
= B
t,t+1,t+2
B
t,t+2,t+3
Using
B
t,t+1,t+3
= E
t
(B
t+1,t+2
B
t+1,t+2,t+3
)
It follows that
E
t
1
1 + f
t+1,t+1
1
1 + f
t+1,t+2
=
1
1 + f
t,t+1
1
1 + f
t,t+2
.
We first price a oneperiod FRA. Since the FRA must have a zero value:
E
t
(f
t+1,t+1
 f
t,t+1
) d
1 + f
t+1,t+1
d
= 0,
where expectations are taken under the riskneutral measure. It follows that
E
t
f
t+1,t+1
1 + f
t+1,t+1
d
=
f
t,t+1
1 + f
t,t+1
d .
Also
E
t
(f
t+1,t+1
)  f
t,t+1
= cov
f
t+1,t+1
,
1
1 + f
t+1,t+1
d
(1 + f
t,t+1
d) = 0
The difference between the expected forward rate at t + 1 and the forward rate at t, i.e.
the drift of the forward rate over the first period, depends on the covariance of the rate
with the bond price. In the case of the two period forward rate we have a similar, but more
complicated result, which we state in the following proposition:
Proposition 1 Drift of TwoPeriod Forward
Since a twoperiod FRA has a zero value:
E
t
f
t+1,t+2
 f
t,t+2
1 + f
t+2,t+2
d
1
1 + f
t+1,t+1
d
= 0.
5
It follows that
E
t
(f
t+1,t+2
)f
t,t+2
= cov f
t+1,t+2
,
1
1 + f
t+1,t+2
d ·
1
1 + f
t+1,t+1
d (1 + f
t,t+1
d) (1 + f
t,t+2
d)
For small changes,
dx
x
= d ln x + k, and
cov (· · ·) = cov ln f
t+1,t+2
, ln
1
1 + f
t+1,t+2
d
1
1 + f
t+1,t+1
d
f
t,t+2
(1 + f
t,t+1
d) (1 + f
t,t+2
d)
Hence
E
t
[f
t+1,t+2
]  f
t,t+2
=
cov ln f
t+1,t+2
, ln
1
1 + f
t+1,t+2
d
1
1 + f
t+1,t+1
d
f
t,t+2
=
cov ln f
t+1,t+2
, ln
1
1 + f
t+1,t+2
d
f
t,t+2
 cov ln f
t+1,t+2
, ln
1
1 + f
t+1,t+1
d
f
t,t+2
Proposition 1 shows that the drift of the two period ahead forward rate depends on the two
covariance terms. We now evaluate these covariance terms using a well known property of
normally distributed variables. First, we state the following modification of Stein’s Lemma:
Lemma 2 Stein’s Lemma for Lognormal Variables
Assume that x and y are lognormal variables. Then
cov ln x, ln
1
1 + y
= E
y
1 + y cov (ln x, ln y)
Hence,
cov ln f
t+T ,t+T
, ln
1
1 + f
t+T ,t+T
d
=
E
f
t+T ,t+T
d
1 + f
t+T ,t+T
d
var(ln f
t+T ,t+T
)
=
f
t,t+T
1 + f
t,t+T
d
var(ln f
t+T ,t+T
)
6
Stein’s lemma, applied above to the case of lognormal variables, provides the key to evaluat
ing the covariance terms that determine the drift of the forward rates. It turns covariances
in to logarithmic covariances. We now state the drift terms, in the general case, assuming
first for simplicity that d = 1. We have, given Stein’s Lemma, for small changes in forward
rates
f
t,t+2
= E
t
[f
t+1,t+2
]  f
t,t+2
f
t,t+2
1 + f
t,t+2
d s
2,2
 f
t,t+2
f
t,t+1
1 + f
t,t+1
d s
1,2
which states that the drift is dependent, as in HJM, on a series of discounted covariances.
We are now in a position to state the drift in the BGM version of the LMM. First we make
an additional assumption. We assume that the covariances (of the logarithms of the forward
rates) depend only on the maturity of the forward rates, i.e. they are not time dependent.
We then can establish:
Proposition 2 The BGM Model
Given intertemporal stability of the covariances, if f
t,t
is the mmonth LIBOR rate then
B
t,t+m
=
1
1 + df
t,t
,
d ˜ m
12
If the period length t to t + 1 is also m months, then
E [f
t+1,t+T
]  f
t,t+T
= df
t,t+T
f
t,t+T
1 + df
t,t+T
s
T ,T
+ df
t,t+T 1
f
t,t+T
1 + df
t,t+T 1
s
T 1,T
+ · · ·
In the LMM, the drift of the forward rate at a point in time depends upon the level of
the rate. It also depends on the sum of a series of discounted covariances. Since, the drift
depends on the forward rate, it is stochastic. This is the property that causes the problem in
implementing the model. The stochastic drift can be handled by a MonteCarlo simulation.
But, to avoid this brute force approach we show in the following section that we can model
the stochastic drift using adjusted conditional probabilities.
7
4
The HSS Recombining Node Methodology and the LIBOR
Market Model
Ho, Stapleton and Subrahmanyam (1995) [HSS] suggest a general methodology for cre
ating a recombining multivariate binomial tree to approximate a multivariate lognormal
process. An adaptation of this methodology has been used by Peterson, Stapleton and
Subrahmanyam (2003) [PSS] to build a twofactor spot rate model of the termstructure.
In this section we show that a similar application can be made in the case of the LMM.
There are some differences in this case however. First, we will assume in this version of the
LMM that the stochastic factors driving the term structure are independent logBrownian
motions. Hence, there is no mean reversion or correlation in the factors.
We assume that the proportionate up and down movements in the logbinomial process due
to factor i = 1, 2 are u(T )(i) and d(T )(i) respectively, for the forward rate with maturity
T . This imples that the up and down moves, depend on the maturity of the forward, but
not on the time t.
In the HSS method, the T period forward rate at time t, in state r, s, [after r downmoves
in factor 1 and s downmoves in factor 2] is given by
f
t,t+T ,r,s
= f
t1,t+T
[u
T
(1)]
tr
[d
T
(1)]
r
[u
T
(2)]
ts
[d
T
(2)]
s
(1)
where
d
T
(i) =
2
1 + e
2s
T
(i)
v
d
u
T
(i) = 2  d
T
(i),
for
t = 1, 2, ..., N
T = 0, 1, ..., N  t.
Here we have assumed that volatilities are time independent (i.e. they are dependent only
on maturity T )
4.1
Forward Rate Drifts and HSS Probabilities
In this section we describe the computation of the drifts of the forward rate process and
the HSS conditional probabilities.
8
Let m
t,t+T
(i) denote the drift per period of the T period forward rate at time t due to factor
i. In general, the drift of the forward rate at time t is
m
t,t+T ,r,s
(i)
=
d
df
t,t+1,r,s
1 + df
t,t+1,r,s
s
0,T
(i) +
df
t,t+2,r,s
1 + df
t,t+2,r,s
s
1,T
(i) + ... +
df
t,t+T ,r,s
1 + df
t,t+T ,r,s
s
T ,T
(i)  [s
T ,T
(i)]
2
2
and the probability of an up move is
q
t,t+T ,r,s
(i)
=
[m
t,t+T ,r,s
(i) + (t  r) ln u
T +1
(i) + r ln d
T +1
(i)  (t  r) ln u
T
(i)  r ln d
T
(i)

ln d
T
(i)]/[ln u
T
(i)  ln d
T
(i)].
4.2
Bond Prices
First compute the forward, oneperiod bond prices B
t,t+T ,t+T +1,r,s
. These are given by
B
t,t+T ,t+T +1,r,s
=
1
1 + df
t,t+T ,r,s
,
for T = 0, 1, 2, ..., N  1 [when T = 0, these are spot prices] Then the long bond prices are
given by:
B
t,t,t+T +1,r,s
= B
t,t,t+T ,r,s
B
t,t+T ,t+T +1,r,s
for t = 0, 1, 2, ..., N  1 and T = 0, 1, 2, ..., N  1. Define the spot prices by
B
t,t,t+T ,r,s
= B
t,t+T ,r,s
4.3
Caplet Prices
The Europeanstyle Caplet is priced using the equations:
cap(t, t, r, s)
=
{q
t,t,r,s
(1)q
t,t,r,s
(2)cap(t, t + 1, r, s)
+
q
t,t,r,s
(1)[1  q
t,t,r,s
(2)]cap(t, t + 1, r, s + 1)
+
[1  q
t,t,r,s
(1)]q
t,t,r,s
(2)cap(t, t + 1, r + 1, s)
+
[1  q
t,t,r,s
(1)][1  q
t,t,r,s
(2)]cap(t, t + 1, r + 1, s + 1)}
1
1 + f
t,t,r,s
d
9
where
cap(t, t, r, s) = max[f
t,t
 k, 0]d
1
1 + f
t,t,r,s
d
4.4
Swaption Prices
Lemma 3 The expected nperiod swap rate is given by
E(s
n,t
) =
n
T =1
a
T
E(f
t,t+T
)
(2)
where
a
T
= 1  B
t,t+T
n
T =1
B
t,t+n
The swap rate at time t, for a swap with maturity n, is given by
s
t,n,r,s
=
1  B
t,t+n,r,s
d
n
t =1
B
t,t+t
(3)
The payoff, at time t on a (payer) swaption with strike rate k, is
swn
t,n,r,s
= max(s
t,n,r,s
 k, 0)d[
n
t =1
B
t,t+t,r,s
]
(4)
The value of the swaption at time t is
max(swn
t,n,r,s
, Swn
t,n,r,s
)
(5)
where
Swn
t,n,r,s
= E
t
(Swn
t+1,n1,r,s
)B
t,t+1,r,s
(6)
10
5
Calibration to Swaption Volatilities
First compute the swap weights: Using the bond prices B
0,t,0,0
= B
0,t
, we have
?
1,n
=
B
0,1
n
t =1
B
0,t
The volatility of the swap rate is then given by
swapvol
t,n
=
n
t =1
n
T =1
?
t
?
T
?
t,T
s
t,t+t
s
t,t+T
0.5
The model can be calibrated to selected swaption vbolatilities by minimising the distance,
measured by the sum of squared differences, between the computed swaption vols and the
market vols. For example, if equal weight is given to each of the swaption vols, the model
is calibrated by selecting the caplet vols so as to solve:
min
{s
1,1+T
}
swapvol
t,n
 swapvol
m
t,n
2
where swapvol
m
t,n
refer to the (Black model) market quotes for swaptions with option ma
turity t and underlying swap maturity n.
Swaption Volatility Quotes
The forward swap rate at time 0, for delivery at t , for a swap with final maturity T , is
given by
s
0,t,T
=
1  B
0,t,T
m[B
0,t,t +1
+ B
0,t,t +2
+ .... + B
0,t,T
]
(7)
The Black model for swaption quotes is
A
0,t,T
B(s
0,t,T
, t, s
t,T t
, k) = swn
0,t,T
(8)
11
where A
0,t,T
is the forward swap annuity:
A
0,t,T
= B
0,t,t +1
+ B
0,t,t +2
+ .... + B
0,t,T
(9)
and where s
t,T t
is the implied volatility of the swaption with maturity t and swap maturity
T  t .
swn
t,t,T ,i,j
=
B
t,t+1,i,j
{q
x
(t + 1, i, j)[q
y
(t + 1, j)swn
t+1,t,T ,i,j
(10)
+
(1  q
y
(t + 1, j))swn
t+1,t,T ,i,j+1
]
+
(1  q
x
(t + 1, i, j))[q
y
(t + 1, j)swn
t+1,t,T ,i+1,j
+
(1  q
y
(t + 1, j))swn
t+1,t,T ,i+1,j+1
]}
where
swn
t,t,T ,i,j
= swn
t,T ,i,j
(11)
6
Some Preliminary Tests of the Model
In this section we present results based on small scale (16 period and 24 period) versions of
the model. We first show that the model is capable of accurately repricing caplets, when
compared to the Black model. We then proceed to price a variety of swaption contracts:
Europeanstyle, Fixedtail Bermudanstyle and Americanstyle swaptions. Where possible,
our prices are compared with those recorded in the literature for similar contracts and data
input.
6.1
Caplet Pricing
In Table 1, we show the results of pricing caplets, given a volatility structure and the forward
rates at time t = 0. The first and fourth columns show the Black model prices at strike
prices of 5% and 6% respectively. The third and sixth columns show our onefactor model
prices at strike prices of 5% and 6% respectively. The second and fifth columns show our
twofactor model prices at strike prices of 5% and 6% respectively. The volatility structure
is a constant 20% in each case. In the case of the twofactor model, the allocation constants
are a
1,0
= 0.8 and b = 0.7. It is clear from the table that in most cases (excluding the very
short term caplets in the onefactor case) the pricing of the caplets is consistent with the
Black model. This confirms that the drift factors and probabilities are correct.
12
6.2
Swaption Pricing
In Table 2 we show results from using a onefactor version of the model to price European
style and ‘fixedtail’ Bermudanstyle swaptions. The results are directly comparable with
those recorded from the LMM and from Carr and Yang’s Markov model. As in Andersen
(2000), the Bermudanstyle swaptions have a ‘lockout period’ shown in column four of
the table. In most cases our model overprices both Europeanstyle and Bermudanstyle
swaptions by a small number of basis points. This is probably due to using the probability
of the longest forward in the underlying swap. Other tests show that using the probability
of the shortest forward in the swap underprices the swaptions.
13
7
References
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16
8
Appendix: Computational Procedure: Forward Volatili
ties and Covariances
8.1
Inputs
1. Input time 0 structure of forward LIBOR rates
f
0,T
, T = 0, 1, ..., N
2. Input time 0 structure of caplet volatilities, denote these
s
t,t
, t = 1, 2, .., N
8.2
Compute Forward Volatilities
The oneperiod forward volatilities solve the following ’bootstrap’ equations:
s
2
1,1
=
s
2
1,1
s
2
1,2
=
2s
2
2,2
 s
2
1,1
s
2
1,3
=
3s
2
3,3
 s
2
1,1
 s
2
1,2
...
=
...
s
2
1,N
=
N s
2
N,N
 s
2
1,1
 s
2
1,2
 ...  s
2
1,N 1
The twoperiod forward volatilities solve the following ’bootstrap’ equations:
s
2
2,2
=
s
2
2,2
s
2
2,3
=
[3s
2
3,3
 s
2
1,1
]/2
s
2
2,4
=
[4s
2
4,4
 s
2
1,1
 s
2
1,2
]/2
...
=
...
s
2
2,N
=
[N s
2
N,N
 s
2
1,1
 s
2
1,2
 ...  s
2
1,N 1
]/2
17
Also, in general we need for t = 1, ...., N  1:
s
2
t,t
=
s
2
t,t
s
2
t,t+1
=
[(t + 1)s
2
t+1,t+1
 s
2
1,1
]/t
s
2
t,t+2
=
[(t + 2)s
2
t+2,t+2
 s
2
1,1
 s
2
1,2
]/t
...
=
...
s
2
t,N
=
[N s
2
N,N
 s
2
1,1
 s
2
1,2
 ...  s
2
1,N 1
]/t
8.3
Computing Factor Loadings
1. Input constants a
1,0
, ...., a
1,N 1
[for convenience, assume a
1,T
= (a
1,0
)
b(T +1)
, then only
input a
1,0
and b.]
2. Compute the relative factor loadings for factor 2 using:
a
2,T
= (1  a
2
1,T
)
0.5
T = 0, 1, ..., N  1
(12)
3. Compute the absolute factor loadings for factor 1 and 2 using:
ß
1,T
=
a
1,T
s
1,T +1
, T = 0, 1, ..., N  1
(13)
ß
2,T
=
a
2,T
s
1,T +1
, T = 0, 1, ..., N  1
(14)
8.4
Compute Covariances
Compute array of cov
i,t,T
, for factors i = 1, 2 and for t = 0, 1, ..., N 1 and T = 0, 1, ..., N 1,
using
cov
i,t,T
= ß
i,t
ß
i,T
(15)
Also compute the covariance:
cov
t,T
= ß
1,t
ß
1,T
+ ß
2,t
ß
2,T
(16)
18
Figure 1
Recombining Tree: Two Factors
ddd
d
d
ddd
d
ddd
ddd
d
d
d
d
d
d
d
d
d
d
d
d
f
1,1
f
2,2
ddd
t = 1
t = 2
Note: The model produces 4 nodes at t = 1, 9 nodes at t = 2, and so on. The lattice
recombines in two dimensions.
19