Munich Personal RePEc Archive
An efficient lattice algorithm for the
libor market model
Tim, Xiao
Risk Analytics, Capital Markets Risk Management, CIBC, Toronto,
Canada
18 June 2011
Online at
https://mpra.ub.uni-muenchen.de/32972/
AN EFFI CI EN T LATTI CE ALGORI TH M FOR TH E
LI BOR M ARKET M OD EL
Tim Xiao1
Risk Analyt ics, Capital Market s Risk Managem ent , CI BC, Toront o, Canada
ABSTRACT
The LI BOR Market Model has becom e one of t he m ost popular m odels for
pricing int erest rat e product s. I t is com m only believed t hat Mont e- Carlo sim ulat ion is
t he only viable m et hod available for t he LI BOR Market Model. I n t his art icle,
however, we propose a lat t ice approach t o price int erest rat e product s wit hin t he
LI BOR Market Model by int roducing a shift ed forward m easure and several novel fast
drift approxim at ion m et hods. This m odel should achieve t he best perform ance
wit hout losing m uch accuracy. Moreover, t he calibrat ion is alm ost aut om at ic and it is
sim ple and easy t o im plem ent . Adding t his m odel t o t he valuat ion t oolkit is act ually
quit e useful; especially for risk m anagem ent or in t he case t here is a need for a
quick t urnaround.
Ke y W or ds: LI BOR Market Model, lat t ice m odel, t ree m odel, shift ed forward
m easure, drift approxim at ion, risk m anagem ent , calibrat ion, callable exot ics, callable
bond, callable capped float er swap, callable inverse float er swap, callable range
accrual swap.
1
The views expressed here are of t he author alone and not necessarily of his host inst itut ion.
Address correspondence t o Tim Xiao, Risk Analyt ics, Capit al Markets Risk Managem ent , CI BC,
The LI BOR Market Model ( LMM) is an int erest rat e m odel based on evolving
LI BOR m arket forward rat es under a risk-neut ral forward probabilit y m easure. I n
cont rast t o m odels t hat evolve t he inst ant aneous short rat es ( e.g., Hull-Whit e,
Black-Karasinski m odels) or inst ant aneous forward rat es ( e.g., Heat h- Jarrow- Mort on (HJM)
m odel) , which are not direct ly observable in t he m arket , t he obj ect s m odeled using
t he LMM are m arket observable quant it ies. The explicit m odeling of m arket forward
rat es allows for a nat ural form ula for int erest rat e opt ion volat ilit y t hat is consist ent
wit h t he m arket pract ice of using t he form ula of Black for caps. I t is generally
considered t o have m ore desirable t heoret ical calibrat ion propert ies t han short rat e
or inst ant aneous forward rat e m odels.
I n general, it is believed t hat Mont e Carlo sim ulat ion is t he only viable
num erical m et hod available for t he LMM ( see Pit erbarg [ 2003] ) . The Mont e Carlo
sim ulat ion is com put at ionally expensive, slowly converging, and not oriously difficult
t o use for calculat ing sensit ivit ies and hedges. Anot her not able weakness is it s
inabilit y t o det erm ine how far t he solut ion is from opt im alit y in any given problem .
I n t his paper, we propose a lat t ice approach wit hin t he LMM. The m odel has
sim ilar accuracy t o t he current pricing m odels in t he m arket , but is m uch fast er.
Som e ot her m erit s of t he m odel are t hat calibrat ion is alm ost aut om at ic and t he
approach is less com plex and easier t o im plem ent t han ot her current approaches.
We int roduce a shift ed forward m easure t hat uses a variable subst it ut ion t o
shift t he cent er of a forward rat e dist ribut ion t o zero. This ensures t hat t he
dist ribut ion is sym m et ric and can be represent ed by a relat ively sm all num ber of
discret e point s. The shift t ransform at ion is t he key t o achieve high accuracy in
relat ively few discret e finit e nodes. I n addit ion, we present several fast and novel
drift approxim at ion approaches. Ot her concept s used in t he m odel are probabilit y
dist ribut ion st ruct ure exploit at ion, num erical int egrat ion and t he long j um p t echnique
This m odel is act ually quit e useful for risk m anagem ent because norm ally
full-revaluat ions of an ent ire port folio under hundreds of t housands of different fut ure
scenarios are required for a short t im e window. Wit hout an efficient algorit hm , one
cannot properly capt ure and m anage t he risk exposed by t he port folio.
The rest of t his paper is organized as follows: The LMM is discussed in Sect ion
I . I n Sect ion I I , t he lat t ice m odel is elaborat ed. The calibrat ion is present ed in
Sect ion I I I . The num erical im plem ent at ion is det ailed in Sect ion I V, which will
enhance t he reader’s underst anding of t he m odel and it s pract ical im plem ent at ion.
The conclusions are provided in Sect ion V.
I .
LI BOR M ARKET M OD EL
Let (,F ,
Ft t0,P ) be a filt ered probabilit y space sat isfying t he usualcondit ions, where denot es a sam ple space, F denot es a - algebra, P denot es
a probabilit y m easure, and
Ft t0 denot es a filt rat ion. Consider an increasingm at urit y st ruct ure 0T0T1...TN from which expiry- m at urit y pairs of dat es
(Tk1,Tk) for a fam ily of spanning forward rat es are t aken. For any tim e tTk1, we
define a right - cont inuous m apping funct ion n(t) by Tn(t)1tTn(t). The sim ply
com pounded forward rat e reset at t for forward period (Tk1,Tk) is defined by
1
) , (
) , ( 1 ) , ; ( : )
( 1
1
k k k k k k
T t P
T t P T
T t F t F
( 1)
where P(t,T) denot es t he t im e t price of a zero- coupon bond m at uring at t im e T and
) , ( : k 1 k
k T T
is t he accrual fact or or day count fract ion for period (Tk1,Tk) .
I nvert ing t his relat ionship ( 1) , we can express a zero coupon bond price in
t erm s of forward rat es as:
k
t n j t n k
t F T
t P T t P
) ( ) (
) ( 1
1 )
, ( ) , (
LI BOR M ar k e t Mode l D yn a m ics
Consider a zero coupon bond num eraire P(,Ti) whose m at urit y coincides wit h
t he m at urit y of t he forward rat e. The m easure i
Q associat ed wit h P(,Ti) is called Ti
forward m easure. Term inal m easure N
Q is a forward m easure where t he m at urit y of
t he bond num eraire P(,TN) m at ches t he t erm inal dat e TN.
For brevit y, we discuss t he one- fact or LMM only. The one- fact or LMM ( Brace
et al. [ 1997] ) under forward m easure i
Q can be expressed as
I f ik,tTi, k k t
k i j
j j
j j j k
k
k dt t F t dX
t F
t F t t
F t t
dF ( ) ( )
) ( 1
) ( ) ( )
( ) ( )
( 1
( 3a)I f ik,tTk1, dFk(t)
k(t)Fk(t)dXt ( 3b)I f ik,tTk1, k k t
i k j
j j
j j j k
k
k dt t F t dX
t F
t F t t
F t t
dF ( ) ( )
) ( 1
) ( ) ( )
( ) ( ) (
1
( 3c)where Xt is a Brownian m ot ion.
There is no requirem ent for what kind of inst ant aneous volat ilit y st ruct ure
should be chosen during t he life of t he caplet . All t hat is required is (see Hull-Whit e
[ 2000] ) :
1
0 2 1
2 1 2
) ( 1
) , ( : )
( Tk k
k k
k
k u du
T
T
( 4)
where k denot es t he m arket Black caplet volat ilit y and denot es t he st rike. Given
t his equat ion, it is obviously not possible t o uniquely pin down t he inst ant aneous
volat ilit y funct ion. I n fact , t his specificat ion allows an infinit e num ber of choices.
People oft en assum e t hat a forward rat e has a piecewise const ant inst ant aneous
volat ilit y. Here we choose t he forward rat e Fk(t) has const ant inst ant aneous volat ilit y
Sh ift e d Forw a r d Me a sur e
The Fk(t) is a Mart ingale or drift less under it s own m easure k
Q . The solut ion
t o equat ion (3b) can be expressed as
ts k t
k k
k t F s ds s dX
F
0 0
2
) ( )
( 2 1 exp ) 0 ( )
( ( 5)
where Fk(0)F(0;Tk1,Tk) is t he current ( spot ) forward rat e. Under t he volat ilit y
assum pt ion described above, equat ion (5) can be furt her expressed as
k t
k k
k t F t X
F
2 exp ) 0 ( ) (
2
( 6)
Alt ernat ively, we can reach t he sam e Mart ingale conclusion by direct ly deriving t he
expect at ion of t he forward rat e ( 6); t hat is
) 0 ( 2
exp 2
1 ) 0 ( 2
) (
exp 2
1 ) 0 (
2 exp 2
exp 2
1 ) 0 ( ) (
2 2
2 2
0
k t t k
t k t k
t t t
k k k
k
F dY t Y
t F
dX t
t X
t F
dX t X X
t t
F t F E
( 7)
where Xt, Yt are bot h Brownian m ot ions wit h a norm al dist ribut ion (0, t) at t im e t,
) | ( : )
( t
t E
E F is t he expect at ion condit ional on t he Ft, and t he variable subst it ut ion
used for derivat ion is
k t
t
X
t
Y
( 8)This variable subst it ut ion t hat ensures t hat t he dist ribut ion is cent ered on zero and
sym m et ry is t he key t o achieve high accuracy when we express t he LMM in discret e
finit e form and use num erical int egrat ion t o calculat e t he expect at ion. As a m at t er of
fact , wit hout t his linear t ransform at ion, a lat t ice m et hod in t he LMM eit her does not
exist or int roduces t oo m uch error for longer m at urit ies.
k t
k k
t k k k
k t F t X F t Y
F
2 exp ) 0 ( 2
exp ) 0 ( ) (
2 2
( 9)
Since t he LMM m odels t he com plet e forward curve direct ly, it is essent ial t o
bring everyt hing under a com m on m easure. The t erm inal m easure is a good choice
for t his purpose, alt hough t his is by no m eans t he only choice. The forward rat e
dynam ic under t erm inal m easure N
Q is given by
t k k N
k j
j j
j j j k
k
k dt F t dX
t F
t F t
F t
dF ( )
) ( 1
) ( )
( )
(
1
( 10)The solut ion t o equat ion (10) can be expressed as
k tk k k
t s k t
k k
k
k t F t ds dX F t t X
F
2 ) ( exp ) 0 ( 2
) ( exp ) 0 ( ) (
2
0 0
2
( 11a)
where t he drift is given by
t N
k
j k j
j j
j j
t N
k
j j k j
k ds
s F
s F ds
s t
0 1
0 1 1 ( )
) ( )
( )
(
(11b)
where
j(s)jFj(s)/
1jFj(s)
is the drift term.
Applying ( 8) t o ( 11a) , we have t he forward rat e dynam ic under t he shift ed
t erm inal m easure as
k t
k k k
k t F t t Y
F
2 ) ( exp ) 0 ( ) (
2
( 12)
D r ift Approx im a t ion
Under t erm inal m easure, t he drift s of forward rat e dynam ics are st at
e-dependent , which gives rise t o sufficient ly com plicat ed non-lognorm al dist ribut ions.
This m eans t hat an explicit analyt ic solut ion t o t he forward rat e st ochast ic different ial
ways t o approxim at e t he drift , which is t he fundam ent al t rickiness in im plem ent ing
t he Market Model.
Our m odel works backwards recursively from forward rat e N down t o forward
rat e k. The N- t h forward rat e FN(t) wit hout drift can be det erm ined exact ly. By t he
t im e it t akes t o calculat e t he k-t h forward rat e Fk(t), all forward rat es from Fk1(t) t o
) (t
FN at t im e t are already known. Therefore, t he drift calculat ion ( 11b) is t o
est im at e t he int egrals cont aining forward rat e dynam ics Fj(s), for j = k+ 1,…,N, wit h
known beginning and end point s given by Fj(0) and Fj(t). For com plet eness, we list
all possible solut ions below.
Fr oz e n D r ift ( FD) . Replace t he random forward rat es in t he drift by t heir
det erm inist ic init ial values, i.e.,
N
k
j k j
j j
j j
t N
k
j k j
j j
j j
k t
F F ds
s F
s F t
1
0 1 1 (0)
) 0 ( )
( 1
) ( )
(
( 13)
Ar it h m e t ic Aver a ge of t h e For w a r d Ra t es ( AAFR). Apply t he m idpoint
rule ( rect angle rule) t o t he random forward rat es in t he drift , i.e.,
N
k
j k j
j j
j
j j
j
k t
t F F
t F F
t
1 2 1 2 1
) ( ) 0 ( 1
) ( ) 0 ( )
(
( 14)
Ar it h m e t ic Ave ra ge of t he D r ift Te r m s ( AADT). Apply t he m idpoint rule t o
t he random drift t erm s, i.e.,
N
k
j j k
j j
j j
j j
j j
k t
t F
t F
F F t
1 1 ( )
) ( )
0 ( 1
) 0 ( 2
1 )
(
( 15)
Ge om e t r ic Ave r a ge of t he Forw a r d Ra t e s ( GAFR) . Replace t he random
forward rat es in t he drift by t heir geom et ric averages, i.e.,
N
k
j j k
j j j
j j j
k t
t F F
t F F t
1
) ( ) 0 ( 1
) ( ) 0 ( )
(
Ge om e t r ic Aver a ge of t he D r ift Te r m s ( GAD T) . Replace t he random drift
t erm s by t heir geom et ric averages, i.e.,
N
k
j j k
j j
j j
j j
j j
k t
t F
t F
F F t
1 1 ( )
) ( )
0 ( 1
) 0 ( )
(
( 17)Con dit ion a l Ex pect a t ion of t h e For w a r d Rat e ( CEFR) . I n addit ion t o t he
t wo endpoint s, we can furt her enhance our est im at e based on t he dynam ics of t he
forward rat es. The forward rat e Fj(s) follows t he dynam ic ( 9) ( The drift t erm is
ignored) . We can derive t he expect at ion of t he forward rat e condit ional on t he t wo
endpoint s and replace t he random forward rat e in t he drift by t he condit ional
expect at ion of t he forward rat e.
Pr oposit ion 1. Assum e t he forward rat e Fj(s) follows t he dynam ic ( 9) , wit h
t he t wo known endpoint s given by Fj(0) and Fj(t). Based on t he condit ional
expect at ion of t he forward rat e Fj(s), t he drift of Fk(t) can be expressed as
N
k j
t
k j t F F j j
t F F j j
k ds
s F E
s F E t
j j
j j
1 0
) ( ), 0 ( 0
) ( ), 0 ( 0
] |
) ( [ 1
] |
) ( [ )
(
( 18a)
where t he condit ional expect at ion of t he forward rat e is given by
t s t s
F t F F s
F
E j
t s
j j j t F F
j j j
2 ) ( exp ) 0 (
) ( ) 0 ( ] |
) ( [
2 )
( ), 0 ( 0
( 18b)
Proof. See Appendix A.
Con dit ion a l Ex pect a t ion of t h e D r ift Te rm ( CED T) . Sim ilarly, we can
calculat e t he condit ional expect at ion of t he drift t erm and replace t he random drift
t erm by t he condit ional expect at ion.
Pr oposit ion 2. Assum e t he forward rat e Fj(s) follows t he dynam ic ( 9) , wit h
t he t wo known endpoint s given by Fj(0) and Fj(t). Based on t he condit ional
N kj j k
t t F F j j j j k ds s F s F E t j j 1 0 ) ( ), 0 ( 0 ) ( 1 ) ( ) (
( 19a)
where t he condit ional expect at ion of t he drift t erm is given by
) ( ) ( / ) ( 1 1 ) ( 1 ) ( | ) ( 2 ) ( ), 0 ( 0 ) ( ), 0 ( 0 s s s s F s F E s E Cj Cj Cj t F F j j j j t F F j j j j j ( 19b)
t s t s F t F F s j t s j j j j Cj 2 ) ( exp ) 0 ( ) ( ) 0 ( 1 ) ( 2
( 19c)
t s t s t s t s F t F F
s j j
t s j j j j Cj ) ( exp 1 ) ( exp ) 0 ( ) ( ) 0 ( ) ( 2 2 2 2
2
( 19d)
Proof. See Appendix A.
The accuracy and perform ance of t hese drift approxim at ion m et hods are
discussed in sect ion I V.
I I .
TH E LATTI CE PROCED U RE I N TH E LM M
The “lat t ice” is t he generic t erm for any graph we build for t he pricing of
financial product s. Each lat t ice is a layered graph t hat at t em pt s t o t ransform a
cont inuous- t im e and cont inuous-space underlying process int o a discret e-t im e and
discret e-space process, where t he nodes at each level represent t he possible v alues
of t he underlying process in t hat period.
There are t wo prim ary t ypes of lat t ices for pricing financial product s: t ree
lat t ices and grid lat t ices ( or rect angular lat t ices or Markov chain lat t ices) . The t ree
lat t ices, e.g., t radit ional binom ial t ree, assum e t hat t he underlying process has t wo
possible out com es at each st age. I n cont rast wit h t he binom ial t ree lat t ice, t he grid
lat t ices (see Am in [ 1993] , Gandhi-Hunt [ 1997] , Mart zoukos- Trigeorgis [ 2002] ,
process t o change by m ult iple st at es, are built in a rect angular finit e difference grid
( not t o be confused wit h finit e difference num erical m et hods for solving part ial
different ial equat ions) . The grid lat t ices are m ore realist ic and convenient for t he
im plem ent at ion of a Markov chain solut ion.
This art icle present s a grid lat t ice m odel for t he LMM. To illust rat e t he lat t ice
algorit hm , we use a callable exot ic as an exam ple. Callable exot ics are a class of
int erest rat e deriv at ives t hat have Berm udan st yle provisions t hat allow for early
exercise int o various underlying int erest rat e product s. I n general, a callable exot ic
can be decom posed int o an underlying inst rum ent and an em bedded Berm udan
opt ion.
We will sim plify som e of t he definit ions of t he universe of inst rum ent s we will
be dealing wit h for brevit y. Assum e t he payoff of a generic underlying inst rum ent is
a st ream of paym ent s Zii
Fi(Ti1)Ci
for i= 1,…,N, where Ci is t he st ruct uredcoupon. The callable exot ic is a Berm udan st yle opt ion t o ent er t he underlying
inst rum ent on any of a sequence of not ificat ion dat es
t
1ex,
t
2ex,...,
t
exM. For anynot ificat ion dat e ex
j
t
t , we define a right - cont inuous m apping funct ion n(t) by
) ( 1
)
(t nt
n t T
T . I f t he opt ion is exercised at t, t he reduced price of t he underlying
inst rum ent , from t he st ruct ured coupon payer’s perspect ive, is given by
N
t n i
N i
i i i i t N
t n i
N i
i t
N PT T
C T F E T
T P
Z E T
t P
t I t
I () 1
)
( ( , )
) ( )
, ( )
, (
) ( : ) (
~
( 20)
where t he rat io ~I(t) is usually called t he reduced value of t he underlying inst rum ent
or t he reduced exercise value or t he reduced int rinsic value.
Lat t ice approaches are ideal for pricing early exercise product s, given t heir
“ backward- in-t im e” nat ure. Berm udan pricing is usually done by building a lat t ice t o
st andard. The lat t ice m odel described below also uses backward induct ion but
exploit s t he Gaussian st ruct ure t o gain ext ra efficiencies.
First we need t o creat e t he lat t ice. The random process we are going t o m odel
in t he lat t ice is t he LMM ( 12) . Unlike t radit ional t rees, we only posit ion nodes at t he
det erm inat ion dat es (t he paym ent and exercise dat es) . At each det erm inat ion dat e,
t he cont inuous-t im e st ochast ic equat ion (12) shall be discret ized int o a discret e-t im e
schem e. Such discret ized schem es basically convert t he Brownian m ot ion int o
discret e variables. There is no rest rict ion on discret izat ion schem es. At any
det erm inat ion dat e t, for inst ance, we discret ize t he Brownian m ot ion t o be equally
spaced as a grid of nodes yi,t, for i = 1,…,St. The num ber of nodes St and t he space
bet ween nodes t yi,t yi1,t at each det erm inat ion dat e can vary depending on t he
lengt h of t im e and t he accuracy requirem ent . The nodes should cover a cert ain
num ber of st andard deviat ions of t he Gaussian dist ribut ion t o guarant ee a cert ain
level of accuracy. We have t he discret e form of t he forward rat e as
k it
k t i k k
t i
k t y F t y t y
F ,
2 , ,
2 ) , ( exp ) 0 ( ) ;
( ( 21)
The zero- coupon bond (2) can be expressed in discret e form as
k
t n j
t i j j t
i t n t
i k
y t F y
T t P y T t P
) (
, ,
) ( ,
) ; ( 1
1 )
; , ( ) ; , (
( 22)
We now have expressions for t he forward rat e (21) and discount bond (22) ,
condit ional on being in t he st at e yi,t at t im e t, and from t hese we can perform
valuat ion for t he underlying inst rum ent .
At t he m at urit y dat e, t he value of t he underlying inst rum ent is equal t o t he
payoff, i.e.,
) ( ) ,
(TN yi,TN ZN yi,TN
The underlying st at e process Xt in t he LMM ( 11) is a Brownian m ot ion. The
t ransit ion probabilit y densit y from st at e ( xi,t, t) t o st at e (xj,T , T ) is given by
) ( 2 ) ( exp ) ( 2 1 ) , ; , ( 2 , , , , t T x x t T T x t x
p it jT jT it
( 24)
Applying t he variable subst it ut ion ( 8) , equat ion ( 24) can be expressed as
) ( 2 ) ( exp ) ( 2 1 ) , ; , ( 2 , , , , t T t T y y t T T y t y
p it jT jT it T t
( 25)
Equat ion ( 20) can be furt her expressed as a condit ional value on any st at e
( yi,t, t) as:
j j j j j T N t n j j t j T t i T T N j T j j t i N t i dy t T t T y y y T T P y Z t T y T t P y t I
) ( 2 , , , ) ( 2 ) ( exp ) ; , ( ) ( ) ( 2 1 ) ; , ( ) ; ( (26)
This is a convolut ion int egral. Som e fast int egrat ion algorit hm s, e.g., Cubic
Spline I nt egrat ion, Fast Fourier Transform ( FFT) , et c., can be used for opt im izat ion.
We use t he Trapezoidal Rule I nt egrat ion in t his paper for ease of illust rat ion.
I n com ple te inf or m a t ion h a n dlin g. Convolut ion is widely used in Elect rical
Engineering, part icularly in signal processing. The im port ant part is t hat t he far left
and far right part s of t he out put are based on incom plet e inform at ion. Any m odels
t hat t ry t o com put e t he t r ansit ion values using int egrat ion will be inaccurat e if t his
problem is not solved, especially for longer m at urit ies and m ult iple exercise dat es.
Our solut ion is t o ext end t he input nodes by padding t he far end values on each side
and only t ake t he original range of t he out put nodes.
Next , we det erm ine t he opt ion values in each final not ificat ion node. On t he
last exercise dat e, if we have not already exercised, t he reduced opt ion value in any
st at e yi,M is given by
,0
Then, we conduct t he backward induct ion process t hat is perform ed by
it erat ively rolling back a series of long j um ps from t he final exercise dat e ex
M
t across
not ificat ion dat es and exercise opport unit ies unt il we reach t he valuat ion dat e.
Assum e t hat in t he previous rollback st ep ex
j
t , we calculat ed t he reduced opt ion
value: ( , , )/ ( , N; i,j)
ex j j i ex
j y P t T y
t
V . Now, we go t o ex
j
t1. The reduced opt ion value at texj1 is
) ; , ( ) , ( , ) ; , ( ) , ( max ) ; , ( ) , ( 1 , 1 1 , 1 1 , 1 1 , 1 1 , 1 1 , 1 j i N ex j j i ex j c j i N ex j j i ex j j i N ex j j i ex j y T t P y t V y T t P y t I y T t P y t V ( 28a)
where t he reduced cont inuat ion value is given by
j ex j ex j ex j j ex j j j i j j N ex j j ex j ex j ex j j i N ex j j i ex j c dy t t t t y y y T t P y t V t t y T t P y t V
) ( 2 ) ( exp ) ; , ( ) , ( ) ( 2 1 ) ; , ( ) , ( 1 2 1 1 1 , 1 1 , 1 1 ,1
( 28b)
We repeat t he rollback procedure and event ually work our way t hrough t he
first exercise dat e. Then t he present value of t he Berm udan opt ion is found by a final
int egrat ion given by
1 1 2 1 1 1 1 1 1 1 1 2 exp ) ; , ( ) , ( 2 1 ) , 0 ( ) 0 ( dy t t y y T t P y t V t T P pv ex ex N ex ex ex N Bermudan
( 29)
The present value or t he price of t he callable exot ic from t he coupon payer’s
perspect ive is:
) 0 ( ) 0 ( ) 0
( Bermudan underly_instrument
payer pv pv
pv ( 30)
This fram ework can be used t o price any int erest rat e product s in t he LMM
set t ing and can be easily ext ended t o t he Swap Market Model (SMM) .
I I I . Ca libr a t ion
First , if we choose t he LMM as t he cent ral m odel, we need t o price int erest
rat e derivat ives t hat depend on eit her or bot h of cap and swapt ion m arket s. Second,
reasonable expect at ion t hat t he calibrat ed m odel we int end t o use t o price our
exot ic, will at least correct ly price t he m arket inst rum ent s t hat we int end t o hedge
wit h. Therefore, in an exot ic derivat ive pricing sit uat ion, recovery of bot h cap and
swapt ion m arket s m ight be desired.
The calibrat ion of t he LMM t o caplet prices is quit e st raight forward. However,
it is very difficult , if not im possible, t o perfect ly recover bot h cap and swapt ion
m arket s. Fort unat ely for t he LMM, t here also exist ext rem ely accurat e approxim at e
form ulas for swapt ions im plied volat ilit y, e.g., Rebonat o's form ula.
We int roduced a param et er and set i i
where i
denot es t he m arket
Black caplet volat ilit y. One can choose different for different i. For sim plicit y we
describe one sit uat ion here. By choosing
1, we have perfect ly calibrat ed t heLMM t o t he caplet prices in t he m arket . However, our goal is t o select a t o
m inim ize t he sum of t he squared differences of t he volat ilit ies derived from t he
m arket and t he volat ilit ies im plied by our m odel for bot h caps and swapt ions
com bined.
I n t he opt im izat ion, we use Rebonat o’s form ula for an efficient expression of
t he m odel swapt ion volat ilit ies, given by
Re
2, 2 1
, 2
,
2 1
, 2 0
, 2
,
) 0 (
) 0 ( ) 0 ( ) 0 ( ) 0 (
) ( ) ( )
0 (
) 0 ( ) 0 ( ) 0 ( ) 0 ( 1
bonato j
i
j j j i j i j i
T
j i ij j i j i LMM
S F F w w
dt t t S
F F w w
T
( 31a)
where
ij= 1 under one- fact or LMM. The swap rat e S,(0) is given by
, (0) i 1wi(0)Fi(0)
S ( 31b)
1 1
1 1
1
) 0 ( 1
) 0 ( 1 )
0 (
k
k
j j j
k
j j j
i
i
F F
Assum e t he calibrat ion cont aining caplet s and G swapt ions. The error
m inim izat ion is given by
G j
swn N j bonato
N j M
i i i 1
2 , Re
, 1
2
min
( 32)
where swn
N j,
denot es t he m arket Black swapt ion volat ilit y. The opt im izat ion can befound at a st at ionary point where t he first derivat ive is zero; t hat is,
G j
bonato N j M
i i
G j
swn N j M
i i
1 Re
, 1
1 ,
1
( 33)
I n t erm s of forward volat ilit ies, we use t he t im e- hom ogeneit y assum pt ion of
t he volat ilit y st ruct ure, where a forward volat ilit y for an opt ion is t he sam e or close
t o t he spot volat ilit y of t he opt ion wit h t he sam e t im e t o expiry. The t im
e-hom ogeneous volat ilit y st ruct ure can avoid non-st at ionary behavior.
I n t he LMM, forward swap rat es are generally not lognorm al. Such deviat ion
from t he lognorm al paradigm however t urns out t o be ext rem ely sm all. Rebonat o
[ 1999] shows t hat t he pricing errors of swapt ions caused by t he lognorm al
approxim at ion are well wit hin t he m arket bid/ ask spread. For m ost short m at urit y
int erest rat e product s, we can use t he lat t ice m odel wit hout calibrat ion (33) .
However, for longer m at urit y or deeply in t he m oney (I TM) or out of t he m oney
( OTM) exot ics we m ay need t o use t he calibrat ion and even som e specific skew/ sm ile
adj ust m ent t echniques t o achieve high accuracy.
I V.
N U M ERI CAL I M PLEM EN TATI ON
I n t his sect ion, we will elaborat e on m ore det ails of t he im plem ent at ion. We
will st art wit h a sim ple callable bond for t he purpose of an easy illust rat ion and t hen
callable range accrual swap. The reader should be able t o im plem ent and replicat e
t he m odel aft er reading t his sect ion.
Ca lla ble Bon d
A callable bond is a bond wit h an opt ion t hat allows t he issuer t o ret ain t he
privilege of redeem ing t he bond at som e point s before t he bond reaches t he m at urit y
dat e. For ease of illust rat ion, we choose a very sim ple callable bond wit h a one-year
m at urit y, a quart erly paym ent frequency, a $100 principal am ount (A) , and a 4%
annual coupon rat e ( t he quart erly coupon C1) . The call dat es are 6 m ont hs, 9
m ont hs, and 12 m ont hs. The call price (H) is 100% of t he principal. The bond spread
(
) is 0.002. Let t he valuat ion dat e be 0. A det ailed descript ion of t he callable bondand current (spot ) m arket dat a is shown in Exhibit 2.
For a short - t erm m at urit y callable bond, our lat t ice m odel can reach high
accuracy even wit hout calibrat ion ( 33) and incom plet e inform at ion handling.
Therefore, we set 1 and i i
. The valuat ion procedure for a callable bond
consist s of 4 st eps:
St e p 1: Creat e t he lat t ice. Based on t he long j um p t echnique, we posit ion
nodes only at t he det erm inat ion ( paym ent / exercise) dat es. The num ber of nodes and
t he space bet ween nodes at each det erm inat ion dat e m ay vary depending on t he
lengt h of t im e and t he accuracy requirem ent . To sim plify t he illust rat ion, we choose
t he sam e set t ings across t he lat t ice, wit h a grid space ( space bet ween nodes)
2 / 1
, and a num ber of nodes S= 7. I t covers
(S1)3 st andard deviat ions for ast andard norm al dist ribut ion. The nodes are equally spaced and sym m et ric, as shown
in Exhibit 3.
St e p 2: Find t he opt ion value at each final node. At t he final m at urit y dat e
4
H A C
yT V
Vi,4: ( 4, i)min , ( 34)
where A denot es t he principal am ount , C denot es t he bond coupon, and H denot es
t he call price. The opt ion values at t he m at urit y are equal t o t he payoffs as shown in
Exhibit 3.
St e p 3: Find t he opt ion value at earlier nodes. Let us go t o t he penult im at e
not ificat ion dat e T3. The opt ion value in any st at e yi is given by
H V C
yT V
Vi,3: ( 3, i)min , i,c3 ( 35)
Equat ion (35) can be furt her expressed in t he form of reduced value as
) ; , ( , ) ; , ( min ) ; , ( : ~ 4 3 3 , 4 3 4 3 3 , 3 , i c i i i i i y T T P C V y T T P H y T T P V
V ( 36a)
where ,3/ ( 3, 4; i)
C
i PT T y
V denot es t he reduced cont inuat ion value in st at e yi at T3 given
by
3 4 2 3 3 4 4 1 1 4 7 2 3 4 2 3 3 4 4 4 3 4 3 4 3 4 2 3 3 4 4 4 3 4 3 4 4 3 3 , 2 ) ( exp ) , ( 2 ) ( exp ) , ( 2 2 ) ( exp 2 ) ( exp ) , ( 2 ) ( exp ) ; , ( T T T T y y y T V T T T T y y y T V T T T T dY T T T T y Y Y T V T T T T y T T P V i j j j i j j i i c i (36b)
where denot es t he bond spread. Sim ilarly we can com put e t he reduced callable
bond values at T2. All int erm ediat e reduced values are shown in Exhibit 3.
St e p 4: Com put e t he final int egrat ion. The final int egral at valuat ion dat e 0 is
399 . 80 2 ) ( exp ) ; , ( ) , ( 2 ) ( exp ) ; , ( ) , ( 2 2 exp ) , 0 ( 2 ) ( exp ) ; , ( ) , ( 2 exp ) , 0 ( ) 0 ( 2 2 2 2 1 1 4 2 1 2 7 2 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4
T T y y T T P y T V T T y y T T P y T V T T T P dY T T Y Y T T P Y T V T T T P V j j j j j j j ( 37)Moreover, we need t o add t he present value of t he coupon at T1 int o t he final
price. The final callable bond value is given by
398 . 81 ) , 0 ( ) exp( ) 0 ( ) 0
( V T1 P T1 C
V ( 38)
The pseudo- code is supplied in Appendix B for t he im plem ent at ion program .
The convergence result s shown in Exhibit 4 indicat e what occurs for a given grid
space when we increase t he num ber of nodes S. The speed of convergence is very
fast , ensuring t hat a sm all num ber of grids are sufficient . All calculat ions are
converged t o 100.7518. One sanit y check is t hat t he callable bond price should be
close t o t he st raight bond price if t he call prices becom e very high. Bot h of t hem are
com put ed as 103.3536.
Ca lla ble ca ppe d floa t e r sw a p
A callable capped float er swap has t wo legs: a regular float ing leg and a
st ruct ured coupon leg. The st ruct ured coupon rat e of t he j -t h period (Tj1,Tj) is given
by } ], ), ( max{min[ 1 F j C j j j j j j j
j A F T K K
C ( 39)
where Aj is t he not ional am ount ,
C j
K is t he rat e cap, F
j
K is t he rat e floor,
j is t hespread and
j is t he scale fact or. For
j> 0, it is called a callable capped float erWe choose a real m iddle life t rade wit h m ore t han 10 years rem aining in it s
lifet im e. The float ing leg has a quart erly paym ent frequency wit h st ep- down
not ionals and st ep-up spreads. The st ruct ured coupon leg has a sem i- annually
paym ent frequency wit h varying not ionals, spreads, scales, rat e caps, and rat e
floors. The call schedule is sem i- annual.
Ca lla ble r a n ge a ccr u a l sw a p
A callable range accrual swap has t wo legs: a regular float ing leg and a
st ruct ured coupon leg. The st ruct ured coupon rat e of t he j -t h period (Tj1,Tj) is given
by
j
j i
T
T t
j i j j j
M RI A
C 1
1
( 40a)
where
otherwise K t
t t F K
if
Ii j i i i j
0
) , ; (
1 min max
( 40b)
where R is t he fixed rat e, Kjmin and Kjmax are t he accrual range of t he j - t h period,
) , ; (ti ti ti
F is t he LI BOR rat e,
is t he range accrual index t erm , Mj is t he t ot alnum ber of t he business days in t he j -t h period.
We choose a real 10 years m at urit y t rade. The float ing leg has a quart erly
paym ent frequency and t he st ruct ur ed coupon leg has a sem i- annually paym ent
frequency wit h varying accrual ranges. I t st art s wit h t he first call opport unit y being
in 3 years from incept ion, and t hen every year unt il t he last possibilit y being 9 years
from incept ion. The range accrual index t erm is 6 m ont hs.
The lat t ice im plem ent at ion procedure for a callable capped float er swap or a
callable range accrual swap is quit e sim ilar t o t he one for a callable bond except t he
The convergence diagram s of pricing calculat ions are shown in Exhibit s 5 and
6. Each curve in t he diagram s represent s t he convergence behavior for a given grid
space as nodes are increased. All of t he lat t ice result s are well converged. I f t he grid
space is sm aller, t he algorit hm has bet t er convergence accuracy but a slower
convergence rat e, and vice verse.
We benchm arked our m odel under different drift approxim at ion m et hods wit h
several st andard m arket approaches, e.g., t he regression- based Mont e Carlo in t he
full LMM and t he HJM t rinom ial t ree. The m odel com parisons for t he accuracy and
speed are shown in Exhibit s 7 and 8. Wit h regards t o accuracy, as expect ed, t he FD
perform s very badly. AAFR and GAFR do a lit t le bet t er but errors go in different
direct ions. The sam e conclusions can be drawn for AADT and GADT. Bot h CEFR and
CEDT are t he best . I n t erm s of CPU t im es, FD, AAFR, AADT, GAFR and GADT are t he
sam e. But CEFR and CEDT are slower, especially in t he callable range accrual swap
case.
V.
CON CLU SI ON
I n t his paper, we proposed a lat t ice m odel in t he LMM t o price int erest rat e
product s. Conclusions can be drawn, support ed by t he previous sect ions. First , t he
m odel is quit e st able. The fast convergence behavior requires fewer discret izat ion
nodes. Second, t his m odel has alm ost equivalent accuracy t o t he current pricing
m odels in t he m arket . Third, t he im plem ent at ion of t he m odel is relat ively easy. The
calibrat ion is very sim ple and st raight forward. Finally, t he perform ance of t he m odel
is probably t he best am ong all known approaches at t he t im e of writ ing.
We use t he following t echniques in our m odel: shift ed forward m easure, drift
approxim at ion, probabilit y dist ribut ion st ruct ure exploit at ion, long j um p, num erical
t echniques, t he m odel achieves sufficient accuracy in relat ively few t im e st eps and
discret e nodes, which m akes it a very efficient m et hod.
For ease of illust rat ion, we present t he lat t ice m odel based on t he Trapezoidal
Rule int egrat ion. A bet t er but slight ly m ore com plicat ed solut ion is t o spline t he
payoff funct ions. The cubic spline of t he opt ion payoffs can achieve higher accuracy,
especially for Greeks calculat ions, and higher speed. Alt hough cubic spline t akes
som e t im e, t he lat t ice will require m uch fewer nodes (23 ~ 28 nodes are good
enough) and can perform a m uch fast er int egrat ion. I n general, t he spline m et hod
can provide a speedup fact or around 3 ~ 5 t im es.
We have im plem ent ed t he lat t ice m odel t o price a variet y of int erest rat e
exot ics. The algorit hm can always achieve a fast convergence rat e. The accuracy,
however, is a bit t rickier, depending on m any fact ors: drift approxim at ion
approaches, num erical int egrat ion schem es, volat ilit y select ions, and calibrat ion, et c.
Som e work, such as calibrat ion, is m ore of an art t han a science.
REFEREN CE
Am in, K. “ Jum p diffusion opt ion valuat ion in discret e t im e.” Journal of Finance, Vol.
48, No. 5 ( 1993) , pp. 1833- 1863.
Brace, A., D. Gat arek, and M. Musiela. “ The m arket m odel of int erest rat e dynam ics.”
Mat hem at ical Finance, Vol. 7, No. 4 (1997) , pp. 127- 155.
Brigo, D., and F. Mercurio. “ I nt erest Rat e Models – Theory and Pract ice wit h Sm iles,
Das, S. “ Random lat t ices for opt ion pricing problem s in finance.” Journal of
I nvest m ent Managem ent , Vol. 9, No.2 ( 2011) , pp. 134- 152.
Gandhi, S. and P. Hunt . “ Num erical opt ion pricing using condit ioned diffusions,”
Mat hem at ics of Derivat ive Securit ies, Cam bridge Universit y Press, Cam bridge, 1997.
Hagan, P. “ Accrual swaps and range not es.” Bloom berg Technical Report , 2005.
Hull. J., and A. Whit e. “ Forward rat e volat ilit ies, swap rat e volat ilit ies and t he
im plem ent at ion of t he Libor Market Model.” Journal of Fixed I ncom e, Vol. 10, No. 2
( 2000) , 46- 62.
Mart zoukos, H., and L. Trigeorgis. “ Real (invest m ent ) opt ions wit h m ult iple sources
of rare event s.” European Journal of Operat ional Research, 136 (2002) , 696-706.
Pit erbarg, V. “ A Pract it ioner’s guide t o pricing and hedging callable LI BOR exot ics in
LI BOR Market Models.” SSRN Working paper, 2003.
Rebonat o, R. “ Calibrat ing t he BGM m odel.” RI SK, March ( 1999) , 74-79.
APPEN D I X A:
Pr oof of Proposit ion 1. We rewrit e ( 9) as
2 ) 0 (
) ( ln 1 ) (
2
t
F t F t
Y j
j j
j
( A1)
I n t he general Brownian Bridge case when t he Wiener process Y(t) has Y(t1)= a and
) (t2
) ( ) )( ( ) ( , ) ( ) )( ( ) ( ~ ) ( 1 2 2 1 1 2 1 t t t t t t t t t a b t t a t N t
Y
Y
Y ( A2)I n our case:
t
1
0
,t
2
t
, a= 0, b=Y(t), s(0,t), t hus ( A2) can be expressed as
t
s
t
s
s
t
Y
t
s
s
N
s
Y
(
)
~
Y(
)
(
),
Y(
)
(
)
( A3)Let ( ) ( ) 2 /2
s s Y s
Aj j j . According t o t he linear t ransform at ion rule, Aj(s) is
a norm al given by
t s t s s s F t F t s s s s s
A Aj j Y j
j j j Y j Aj j ) ( ) ( ) ( , ) 0 ( ) ( ln 2 ) ( ) ( ~ ) ( 2 2 2
( A4)
Let Bj(s)exp
Aj(s)
. By definit ion, Bj(s) is a lognorm al given by
( ), ( )
~ )
(s LogN s s
Bj Aj Aj . According t o t he charact erizat ions of t he lognorm al
dist ribut ion, t he m ean and variance of Bj(s) are
t s t s F t F s s B E s j t s j j Aj Aj j Bj 2 ) ( exp ) 0 ( ) ( 2 ) ( exp ) ( ) ( 2 0 ( A5a)
t s t s F t F t s t s s s s s j t s j j j Aj Aj Aj Bj ) ( exp ) 0 ( ) ( 1 ) ( exp ) ( ) ( 2 exp 1 ) ( exp ) ( 2 2 2 ( A5b)We have t he condit ional expect at ion of t he forward rat e Fj(s) as
t s t s F t F F s B E F s F E j t s j j j j j t F Fj j j
2 ) ( exp ) 0 ( ) ( ) 0 ( ) ( ) 0 ( | ) ( 2 0 ) ( ), 0 ( ( A6)
Pr oof of Pr oposit ion 2. Let Cj(s)1jFj(s)1jFj(0)Bj(s) where Bj(s) is
defined above. According t o t he linear t ransform at ion rule, Cj(s) is a lognorm al
t s t s F t F F s F s j t s j j j j Bj j j Cj 2 ) ( exp ) 0 ( ) ( ) 0 ( 1 ) ( ) 0 ( 1 ) ( 2
(A7a)
t s t s F t F t s t s F s F s j t s j j j j j Bj j j Cj ) ( exp ) 0 ( ) ( 1 ) ( exp ) 0 ( ) ( ) 0 ( ) ( 2 2 2 2 2 2
2
( A7b)
On t he ot her hand, according t o t he charact erizat ions of t he lognorm al
dist ribut ion, t he m ean and variance of Cj(s) are
2 ) ( ) ( exp )
(s s s
Cj
( A8a)
exp
(
)
1
exp
2
(
)
(
)
)
(
s
s
s
s
Cj
( A8b)Solving t he equat ion (A8a) and ( A8b) , we get
) ( / ) ( 1 ) ( ln ) ( 2 s s s s Cj Cj Cj
( A9a)
)
(
)
(
1
ln
)
(
2s
s
s
Cj Cj
( A9b)We know t he first negat ive m om ent of t he lognorm al is
1( )
exp
( ) ( )/2
s s s
C
E j
and have t he condit ional expect at ion of t he drift t erm as
) ( ) ( / ) ( 1 1 2 ) ( ) ( exp 1 ) ( 1 1 ) ( 1 1 1 ) ( 1 ) ( 2 0 0 ) ( ), 0 ( 0 s s s s s s C E s F E s F s F E Cj Cj Cj j j j t F F j j j j j j
( A10)
where Cj(s), Cj(s) are given by ( A7a) and (A7b) .
