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/
MPRA Paper No. 32972, posted 26 Aug 2011 01:58 UTC
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, 161 Bay St reet , 12t h Floor, Toront o, ON M5J 2S8, Canada; em ail: Tim .Xiao@CI BC.com
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 ( we only posit ion nodes at t im es when decisions need t o be m ade) .
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 usual condit 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 increasing m 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 P tT
T t T P
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 t j n
k Pt T F t
T t
P () ()
) ( 1 ) 1
, ( ) ,
( ( 2)
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 Qi associat ed wit h P(,Ti) is called Ti forward m easure. Term inal m easure QN 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 Qi can be expressed as
I f ik,tTi, kj i k k t
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 regardless of t ( see Brigo- Mercurio [ 2006] ) .
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 Qk. The solut ion t o equat ion (3b) can be expressed as
k
t k
t k sk t F s ds s dX
F 0 0
2 ( )
) 2 (
exp 1 ) 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 k k t
k t F t X
F
exp 2 ) 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 (
2 exp ) 1 0 2 (
) exp (
2 ) 1 0 (
exp 2 exp 2
2 ) 1 0 ( ) (
2 2
2 2 0
k t t k
t k t k
t t t
k k k
k
F t dY Y t
F t dX
t X t
F
t dX 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.
Aft er applying t his variable subst it ut ion (8) , equat ion ( 6) can be expressed as
k k k t k k k t
k t F t X F t Y
F
exp 2 ) 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 QN is given by
t k k N
k j
j j
j j j k
k
k dt F t dX
t F
t t F
F t
dF ( )
) ( 1
) ) (
( )
( 1
( 10)The solut ion t o equat ion (10) can be expressed as
k k
t k
t k s k k k k tk 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
t N j
k
j j k j
k ds
s F
s ds F
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 k k k t
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 equat ions cannot be obt ained. Therefore, m ost work on t he t opic has focused on
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
t N j
k
j k j
j j
j j
k t
F ds F
s F
s t F
0 1 11 (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 t F
1 21
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
t F
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 ) (
(
( 16)
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
t F
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
Nj k t j k
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 expect at ion of t he drift t erm j, t he drift of Fk(t) can be expressed as
N
k
j j k
t
t F j F
j j j
k ds
s F
s E F
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 E F
s E
Cj Cj Cj t
F j F
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] , Hagan [ 2005] , and Das [ 2011] ) shown in Exhibit 1, which perm it t he underlying
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 ured coupon. 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 est
1ex, t
2ex,..., t
exM. For any not ificat ion dat e ttexj , 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 E F
T T P E Z T
t P
t t I
I () () 1
) , (
) ( )
, ( )
, (
) : ( )
~(
( 20)
where t he rat io ~() t
I 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 carry out a dynam ic program m ing calculat ion via backward induct ion and is
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 k it k k it
t i
k t y F t y t y
F ,
2 ,
, ) (0)exp (, ) 2
;
( ( 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 P t T y F t 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
I ( 23)
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
t j i N
t
i dy
t T
t T y
y y
T T P
y Z t y T
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
)
; , (
)
; max (
)
; , (
) ,
( , ,
ex M i ex M ex
M i ex M
y T t P
y t I y
T t P
y t V
( 27)
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 texM 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 texj , we calculat ed t he reduced opt ion value: V(texj ,yi,j)/P(texj ,TN;yi,j). Now, we go t o texj1. 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
ex j 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
t dy t
t t
y y y
T t P
y t V t
y t 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 exp 2
)
; , (
) , ( 2
) 1 , 0 ( ) 0
( dy
t t y y
T t P
y t V t T P
pv ex
ex
N ex
ex N ex
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, we will undoubt edly use various swapt ions t o hedge a callable exot ic. I t is a
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 ii 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 he LMM 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,
22 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 LMM i
S F F w w
dt t S t
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
w ( 31c)
Assum e t he calibrat ion cont aining caplet s and G swapt ions. The error m inim izat ion is given by
iM1 i i 2 Gj1 Rebonatoj,N swnj,N 2min ( 32)
where swnj,N denot es t he m arket Black swapt ion volat ilit y. The opt im izat ion can be found 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 m ove on t o som e t ypical callable exot ics, e.g., callable capped float er swap and
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 bond and 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 a st 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 T4, t he payoff of t he callable bond in any st at e yi is given by
