David Lando - Cox Process and Risky Securities.pdf

8/14/2019 David Lando - Cox Process and Risky Securities.pdf

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Rev iew of Derivatives Research , 2, 99--120 (1998)@ 1998 Kluw er Academ ic Publishers, Boston. Manufactured in The Netherlands.

O n o x P r o c e s s es a n d r e d it R i sk y S e c u r it ie sDAVID LAN DODepa rtment o f Operations Research Universityo f Copenhagen Universitetsparken5DK -2100 Copenhagen O. Denm ark

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Abstract A framew ork is prese nted for mo deling defaultable securities and credit derivativeswh ich allows fordependence between m arket risk factors and credit risk. The framework reduces the technical issues of m odelingcredit risk to the sam e issues faced wh en m odeling the o rdinary term structure of interest rates. It is show n howto g eneralize a mo del of Jarrow, Lando and Turub ull (I997) to allow for stochastic transition intensities betw eenrating categories and into defau lt. This g eneralization can hand le contracts with p aym ents explicitly linked toratings. It is also show n how to o btain a term structure mod el for a ll different rating categories simultaneouslyand how to obtain an affine-likestructure. An implementation s g iven in a simple one factor model in which theaffine structure gives do sed form solutions.Keywords: credit risk, C ox pro cess, credit derivatives,ratings.

1 . I n t r o d u c t i o n

T h e a i m o f t h is p a p e r is t o il lu s t ra t e h o w C o x p r o c e s s e s - - a l s o k n o w n a s d o u b l y s t o c h a s t icP o i s s o n p r o c e s s e s - - p r o v i d e a u s ef u l f r a m e w o r k f o r m o d e l i n g p r i ce s o f f i n a n ci a l in s t r u m e n t si n w h i c h c r e d i t r is k i s a s ig n i f i c a n t f a c to r . B o t h t h e c a s e w h e r e c r e d i t r i s k e n t e rs b e c a u s eo f t h e ri sk o f c o u n t e r p a r t y d e f a u l t a n d t h e c a se w h e r e s o m e m e a s u r e o f c r e d i t r is k s u c h a sa c r e d i t s p r e a d i s u s e d a s a n u n d e r l y i n g v a r i a b l e i n a d e r i v a t iv e c o n t r a c t c a n b e a n a l y z e dw i t h i n o u r f r a m e w o r k .

T h e k e y c o n t r ib u t i o n o f t h e a p p r o a c h i s t h e s i m p l i c i ty b y w h i c h i t a l lo w s c r e d i t r i skm o d e l i n g w h e n t h er e is d e p e n d e n c e b e t w e e n t h e d e f a u l t- f r e e t e r m s t r u ct u r e a n d t h e d e f a u l tc h a r a c te r is t ic s o f fi rm s . A s a n i l lu s t ra t io n o f t h e m e t h o d w e p r e s e n t a g e n e r a l i z e d v e r s i o no f a M a r k o v i a n m o d e l p r e s e n t e d i n J a rr o w , L a n d o a n d T u r n b u l l (1 9 9 7 ) . T h i s g e n e r a l i z a t io na l l o w s t h e in t e n s i ti e s w h i c h c o n t r o l t h e d e f a u l t i n t e n s i ti e s a n d t h e t ra n s i t io n s b e t w e e n r a t in g st o d e p e n d o n s t at e v a r i a b l e s w h i c h s i m u l t a n e o u s l y g o v e r n t h e e v o l u t i o n o f t h e te r m s t ru c t u rea n d p o s s i b l y o t h e r e c o n o m i c v a r i a b le s o f i n te r es t. T h i s m o d e l c a n b e u s e d t o a n a l y z e av a r i e t y o f c o n t r a c ts i n w h i c h c r e d i t ra t in g s a n d d e f a u l t a r e p a r t o f t h e c o n t r a c t u a l s e tu p .T w o i m p o r t a n t e x a m p l e s a r e d e f a u l t p r o t e c t i o n c o n t r a c t s w h i c h p r o v i d e i n s u r a n c e a g a i n s td e f a u lt o f a f i rm a n d s o - c a l l e d c r e d i t t r i g g e r s i n s w a p c o n t ra c ts w h i c h t y p i c a l l y d e m a n d t h es e t t le m e n t o f a c o n t r a c t in t h e e v e n t o f d o w n g r a d i n g o f o n e o f th e p a r ti e s t o th e c o n t ra c t . B yi m p o s i n g s o m e a d d i t i o n a l s t r u c tu r e o n t h e m o d e l w e d e r i v e a n e w c l a ss o f m o d e l s w h i c h in as p e c i al s e tu p b e c o m e s u m s o f a f fi n e t e r m s t ru c t u r e m o d e l s . T h e s e m o d e l s s i m u l t a n e o u s l ym o d e l t e r m s tr u c tu r e s f o r d i f f e r e n t r a ti n g s a n d a l l o w s p r e ad s t o fl u c tu a t e r a n d o m l y e v e n i np e r i o d s w h e r e t h e r a ti n g o f t h e d e f a u l ta b l e f ir m d o e s n o t c h a n g e . S u c h m o d e l s a r e u s e f u l f o rm a n a g e r s o f c o r p o r a t e b o n d p o r t f o l io s ( an d r i sk y g o v e r n m e n t d e b t ) w h o a r e c o n s t r a in e d b y


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100 DAVIDL A N D 0

G i v e n s u c h l i ne s t h e e v e n t o f a d o w n g r a d i n g m a y f o r c e th e m a n a g e r t o e i th e r s ell o f f p a r to f t he pos i t i on o r tr y t o o f f - l oad so m e c r ed i t exp osure i n c r ed i t de ri va ti ves m arke ts .

T h e m o d e l i n g f r a m e w o r k i s si m i la r t o th a t o f J a r ro w a n d T u r n b u l l ( 1 9 95 ) , J a rr o w , L a n d oa n d T u r n b u l l ( 1 9 9 7 ), L a n d o ( 1 9 9 4 ) , M a d a n a n d U n a l ( 1 99 5 ) , A r t z n e r a n d D e l b a e n ( 1 9 9 5 ),Du f f le and S ing l e ton (1996 , 1997) , and Duf f le , Sch ro de r and Sk i adas (1996) i n tha t de f au l ti s m o d e l e d t h r o u g h a r a n d o m i n te n s it y o f t h e d e fa u l t ti m e . H o w e v e r , t hi s p a p e r d i s p e n s e sw i t h t h e i n d e p e n d e n c e a s s u m p t i o n e m p l o y e d i n J a rr o w a n d T u r n b u ll ( 1 9 95 ) , Ja rr o w , L a n d oand Turnbu l l ( 1997) , M adan and U na l (1995) . I n t h i s r e spec t ou r approac h r e sem bles tha t o ft he w ork s o f D uf f le and S ing l e ton (1996 , 1997) , and Duf f le , Sch rod e r and Sk i adas (1996) .Th e app roach i n t h is pape r is no t based on t he recu r s ive m e tho ds em ploy ed t he re . Th i ss a c ri fi c es s o m e g e n e r a l i ty b u t t h e s t ru c t ur e o f C o x p r o c e s s e s p r o d u c e s a c l as s o f m o d e l sw h i c h a re s i m p l e t o d e r iv e y e t s e e m s r i c h e n o u g h t o h a n d l e i m p o r t a n t a p p l ic a ti o n s w h e non ly on e - s ided c r ed i t r i sk i s cons ide r ed . Fo r an ana lys i s o f two s ided de fau l t r i sk i n anin t ens i t y -based f r am ework , s ee Duf f l e and Huang (1996) .

T h e p r o s a n d c o n s o f m o d e l i n g d e f a u l t i n a n i n t e n s i t y - b a s e d f r a m e w o r k a s o p p o s e d t ou s i n g t h e c la s s ic a l a p p r o a c h o r i g i n a ti n g i n B l a c k a n d S c h o l e s ( 1 9 7 3 ) a n d M e r to n ( 1 9 7 4 )a r e d i scus sed e l s ewh ere , s ee fo r exam ple D uf f le and S ing l e ton (1996 , 1997) , and the r ev i ewp a p e r s b y C o o p e r a n d M a r ti n ( 1 9 9 6 ) a n d L a n d o ( 1 9 9 7 ). H e r e w e s i m p l y n o t e t h at th isp a p e r a n d t h e w o r k b y D u f f le a n d S i n g le t o n ( 1 9 9 6 ) p o i n t t o o n e i m p o r t a n t a d v a n t a g e o f th ein t ens i t y -based f r am ework , nam e ly t he f ac t t ha t t he i n t ens i t y based f r am ework i s capab l eo f r educ ing t he t echn i ca l d i ff icu l ti e s o f m od e l i ng de f au l tab l e c l a im s t o t he s am e one f ace dw hen m ode l i ng t he t e rm s t ruc tu r e o f non-d e fau l t ab l e bo nd s and r e l a t ed derivatives .

Th e s t ruc tu r e o f t he pape r i s a s f o l l ows : I n s ec t i on 2 w e ou t li ne the bas i c cons t ruc t i ono f a C o x p r o c e s s, c o n c e n t r a t in g o n t h e f ir st j u m p t im e o f s u c h a p r o c es s . K n o w i n g h o w t ocons t ruc t a p roces s m eans k no w ing how to s im ula t e it. H ence t h i s s ec t ion i s im p or t an t i fone wa n t s t o im p lem en t t he m ode l s . Fu r the rm ore , t he exp l ic i t con s t ruc t i on g ives t he key t op r o v i n g t h e r e s u lt s l a te r o n b y - - e s s e n t i a l l y - - u s i n g p r o p e rt ie s o f c o n d i ti o n a l e x p e ct at io n s .

S e c t io n 3 th e n s h o w s h o w t o c a lc u l a te p r i c e s o f t h re e b u i l d i n g b l o c k s f r o m w h i c h c r e d itr i sky bo nd s and de r iva t ives can be cons t ruc ted . Seve ra l exam ples o f c l a im s a r e g iven wh icha re cons t ruc t ed f rom these bu i l d ing b locks . W i th in t h is f r am ew ork we have g r ea t f lex ib il it yi n h o w w e s p e c i f y th e b e h a v i o r o f t h e s ta te v a r i a b le s w h i c h s i m u l t a n e o u s ly d e t er m i n e t h ede fau l t fr ee t e rm s t ruc tu re and t he l i ke l ihood o f de f au lt . W hen d i f fus ions a r e used a s s ta t eva r i ab l e s , t he Fe yn m an -K ac f r am ew ork hand l e s p r i c ing o f c r ed i t r i sky de riva ti ve s ecu r i ti e si n v i rt ua l ly t he s am e w ay a s i t hand l e s de r i va ti ves i n w h ich c r ed i t ri sk is no t p re sen t . Th a tp a r ti c u la r s e tu p i s o n e o f d i f f u s i o n w i th k i l l i n g - - a s p e ci al c a s e o f t h e g e n e ra l fr a m e w o r k .W e a l so ou t l ine ho w the no t i on o f f rac t i ona l r ecove ry , a s i n troduc ed i n Duf f le and S ing l e ton( 1 9 9 6 ), m a y b e v i e w e d i n te r m s o f th i n n i n g o f a p o i n t p ro c e s s .

S e c t io n s 4 a n d 5 p r e s e n t a p p li c a ti o n s o f th e m e t h o d o l o g y . I t i s sh o w n h o w a M a r k o v i a nm o d e l p r o p o s e d b y J a rr o w , L a n d o a n d T u r n b u ll ( 1 9 9 7 ) m a y b e g e n e r a l i z e d t o i n cl u d e tr a n si -t i on r a t e s be tw een c r ed i t r a t ings w h ich d epen d o n t he st at e va ri ab le s . O ne im p or t an t a spec to f t h is gene ra l i za t i on i s t ha t c red i t sp r eads a r e a l l owe d t o f luc tua t e r andom ly even be twe enra t ings tr ans it ions . Sec t i on 6 g ives a conc re t e im p lem en ta t i on o f t he gene ra l i zed M arkov i anm od el by co ns id er ing an a f f lne mo del for bo th the in te rest ra te and the t rans it ion in tens i ti es .I t i s sho w n h ow one c an ca l i b r a te t he m od e l ac ros s a ll c r ed i t c l a s se s s im u l t aneous ly t o fi t


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ON COXPROCESSES 101

initial spot rate spreads and the sensitivities of these spreads to changes in the short treasuryrate. Section 7 concludes.

2 C o n s t r u c t io n o f a C o x P r o c e s s

Before setting up our model of credit risky bonds and derivatives, we give both an intuitiveand a formal description of how the default process is modeled. The formal descriptionis extremely useful when performing calculations and simulations of the model, whereasthe intuitive description is helpful when the economic content of the models are to beunderstood.

Recall that an inhomogeneous Poisson process N with non-negative) intensity functionl .) satisfies

(:: )u ) d uP Art - N , ----- = ( f ). exp - l u ) d u k = O , 1 . . . . .In particular, assuming No = 0, we have

P N t = O ) = e x p - f o t l u ) d u ) .A way of simulating the first jump r of N is to let E1 be a unit exponential random variableand define { / 0 }= inf t : l u ) d u > E 1 2.1)

A Cox process is a generalization of the Poisson process in which the intensity is allowedto be random but in such a way that i f we condition on a particular realization l -, o)) ofthe intensity, the jump process becomes an inhomogeneous Poisson process with intensityl s , ~ o ) .

In this paper we will write the random intensity on the forml s , o )) = X X s )

where X is an Ra-valued stochastic process and ~. : R a --~ [0, oo) is a non-negative,continuous function. The assumption that the intensity is a function of the current levelof the state variables, and not the whole history, is convenient in applications, but it is notnecessary from a mathematical point of view.

The state variables will include interest rates on riskless debt and may include time, stockprices, credit ratings and other variables deemed relevant for predicting the likelihood ofdefault. Intuitively, given that a firm has survived up to t ime t, and given the history of Xup to time t , the probability of defaulting within the next small time interval At is equal to) ~ X t ) A t + o At).

Formally, we have a probability space f2, ~, P) large enough to support an Ra-valu edstochastic process X = { X t : 0 < t < T f } which is r ight-continuous with left limits and a


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1 0 2 D A V I D L A N D O

u n i t e x p o n e n t i a l r a n d o m v a r i a b l e E 1 w h i c h i s i n d e p e n d e n t o f X . G i v e n a l s o is a f u n c t io n3. : R e ---> R w h i c h w e a s s u m e i s n o n - n e g a t i v e an d c o n t in u o u s . F r o m t h e s e t w o i n g r e d i e n t sw e d e f in e t h e d e f a u lt ti m e z a s f o l l o w s :

{ f 0 Iin f t : ~ . ( X s ) d s >_ E 1 9 2 .2 )T h i s d e f a u l t t i m e c a n b e t h o u g h t o f a s t h e fi rs t j u m p t im e o f a C o x p r o c e s s w i t h i n t e n s i typ ro ce ss ~ . X s ) . N o t e t h a t t h is i s a n e x a c t a n a l o g u e t o e q u a t i o n 2 . 1 ) w i t h a r a n d o m i n t e n s i t yr e p l a c i n g t h e d e t e r m i n i s t i c i n te n s i ty f u n c t i o n .

W h e n ) ~ ( X s ) i s l a r g e , t h e i n t e g r a t e d h a z a r d g r o w s f a s t e r a n d r e a c h e s t h e l e v e l o f t h ei n d e p e n d e n t e x p o n e n t i a l v a r ia b l e f a st e r, a n d t h e r e fo r e t h e p r o b a b i l i t y t h at r i s s m a l l b e c o m e sh i g h e r . S a m p l e p a t h s f o r w h i c h t h e i n t e g r a l o f t h e i n t e n s i t y i s in f i n i te o v e r t h e i n t e r v a l [0 , T f ]a r e p a t h s f o r w h i c h d e f a u l t a lw a y s o c c u r s.

F r o m t h e d e f i n it io n w e g e t th e f o l l o w i n g k e y r e l a ti o n s h i p s :

( f 0 )( z > t ] ( S ~ ) o < , < _ t ) = exp - L ( g D d s t 9 [0 , TI ] 2 . 3 )( f 0 )r > t ) = E e x p - - , k X s ) d s t 9 T f ] 2.4)W h a t w e h a v e m o d e l e d a b o v e , i s o n l y th e f ir st j u m p o f a C o x p r o c e s s . W h e n m o d e l i n g a

C o x p r o c e s s p a s t t he f ir st j u m p - - s o m e t h i n g w e w i l l n ee d i n S e c t i o n 4 - - o n e h a s to i n s u ret h a t t h e i n t e g r a t e d i n t e n s i t y s t a y s f in i t e o n f i n i te i n t e r v a l s i f e x p l o s i o n s a r e t o b e a v o i d e d .O n e t h en p r o c e e d s a s f o l l o w s : G i v e n a p r o b a b i l i t y s p a c e f 2, f , P ) l a r g e e n o u g h to su p p o r ta s t a n d a r d u n i t r a t e P o i s s o n p r o c e s s N w i t h N o = 0 a n d a n o n - n e g a t i v e s t o c h a s t i c p r o c e s s~ . t ) w h i c h i s in d e p e n d e n t o f N a n d a s s u m e d t o b e r i g h t - c o n t in u o u s a n d i n t e g r a b l e o n f in i tein t e rva l s , i .e .

f tt ) : = ~ . ( s ) d s < o ~ t E [0 , T]T h e n A 0 ) = 0 a n d A h a s n o n - d e c r e a s i n g r e a l iz a t i o n s . N o w d e f in i n g

2Qt : = N A t ) )w e h a v e a C o x p r o c e s s w i t h i n t en s i ty m e a s u r e A . F o r t h e t e c h n i c a l c o n d i t i o n s w e n e e d t oc h e c k t o s e e t h a t t h is d e f i n it io n m a k e s s e n s e , s e e G r a n d e l l 1 9 7 6 , p p . 9 - 1 6 ) . T h i s a p p r o a c hi s a ls o r e le v a n t w h e n e x t e n d i n g t h e m o d e l s p r e s e n t e d i n t h is p a p e r t o c a s e s o f r e p e a t e dd e f a u l ts b y t h e s a m e f i rm .

3 T h r e e B a s i c B u i ld i n g B l o c k sT h e m o d e l f o r t h e d e f a u l t - f r e e te r m s t r u c t u r e o f i n t e r e s t r a t e s i s g i v e n b y a s p o t r a t e p r o c e s s ,a m o n e y m a r k e t a c c o u n t a n d a n e q u i v a l en t m a r t i n g a l e m e a s u r e t o w h i c h a l l e x p e c t a t i o n s i n


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ON COX PROCESSES 103

this paper refer:

( f 0 )( t ) = exp r s d sp ( t , T ) = E \ ~ - ~ I .T ,

The default time z is modeled using the framework described in the previous section.The intensity process is denoted L. Hence we may write the information at time t as

~ , = g , v 7 6where processes relevant in determining values of the spot rate and the hazard rate of defaultare adapted to ~ and

7 - / t = a { l { r _ < l : 0 < s < t }holds the information of whether there has been a default at time t. Most applications Willbe built around a state variable process X, in which case we write R ( X t ) for the spot ratert at time t and the informational setup may be stated as

9 t ~ G r {X s : 0 < S < t }7/ t = a{ l{~


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104 DAVID L N D O

N o t e t h a t th e r a n d o m v a r ia b le X a n d t h e p r o c e s s Y c a n b e t h o u g h t o f a s p ro m i s e d p a y m e n t s ,w h e r e a s t h e p a y m e n t c a p t u r e d b y Z is a n a c tu a l p a y m e n t a t t h e ti m e o f d e fa u lt .

T he k ey ad van t age o f the s e tup is t ha t t he ca l cu l a t ion o f p r i ce s bec om es s imi l a r to o rd ina ryt e rm s t ruc tu r e mode l i ng :P r o p o s i t io n 3 . 1 A s s u m e h a t h e e x p e c ta t io n s

E I T I Y s l e x p ( - I S r u d u ) d s a n d

a r e a l l i n i te . T h e o l lo w in g d e n t it ie sh o l d o r th e th r e e c l a i m s i s t e d a b o v e :E ( e x p ( - - I r r s d s ) X l { ~ > r } .T t )

= l{ ~ > ,} E e x p ( - I r ( r s + X s ) d s ) X G t ) (3.1)E ( / r y s l l ~ > s t e x p ( - I ~ r u d u ) a s . T t )

= l I r > t l E ( I r Y s e x p ( - I ~ ( r u + L u ) d u ) d s ~ , ) (3.2)E ( e x p ( - I ~ r s d s ) Z r F t )

= l{ ~ > t jE I r z s X s e x p ( - I S ( r u + X u ) d u ) d s G t ) (3 .3)Proof .

EIn t he

0 on t heE

Fi r s t w e show tha tv n , ) = l > , e xp - s ds 3 . 4 )

fo l l ow ing com puta t i ons w e use t he f ac t tha t t he cond i t iona l expec t a t i on i s c l ea r l yset {r _< t} and that the s et { r > t } i s an a tom o f 7 - (t. Th e re fo re( l{~ >r l l~ r v 7 /t) = l{~> / le ( l l~ > r t l6 r v 7-/ ,)

P ( { r > T } A { r > t } I 6 r )= l{r>t} P ( r > t i e r )P ( z > T t ~ r )= ll~>'J P (r > t ] G r )


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ON COX PROCESSES 10 5

e x p e s s )= lib>t}

w hic h prov es 3 .4).N ow cons ide r 3 .1 ).

= E ( E ( e x p ( Z f r r , d s ) X l , r > r , G T V ~ t ) ' . F t )= E ( e x p ( - - f T r s d s ) X E ( I { ,> T } I G T V T ~ t ) .F t )

(e x ,, - I , )N o w w e w a n t to re p l a ce th e c o n d i t io n i n g o n ~ t w i t h c o n d i t io n i n g o n F t. R e c a l l t h a tE1 i s a un it exponen t ia l r ando m va r i ab l e w h ich i s i ndepen den t o f t he s igm a fi e ld f i r . I n

par t icu la r , E1 is inde pen den t o f the s igm a f ie ld tr ex p ( - f t r ( r s + ~ - s ) d s ) X ) v ~ t a n dthe re fo re s ee W i ll iams 1991 9 .7 k ))

E e x p - f r r ~ + Z s d s ) X I G v a E O )= E e x p - f r r , + Z s d s ) X ~ , ) . 3.5)

B u t w e a l so h a v e t h e f o l l o w i n g in c l u s io n a m o n g s i g m a f i el ds :~ t C G t V ~ /t C ~ t V o ( E 1 ) . 3 .6 )

C o m b i n i n g 3 .4 ) a n d 3 .6 ) w e c o n c l u d e t h at

E e x p - f r r s + ~ s d s ) X G v ~ t ) = E e x p - f r r ~ + Z s d s ) X G ) .w h i c h i s w h a t w e w a n t e d t o s h ow .

Th e p ro of o f 3 .2 ) p roc eeds ex ac t l y a s abov e by f irs t con d i t i on ing on GT v 7 -/t.Fo r t he p ro of o f 3 .3 ) no t e t ha t con d i t i ona l l y on G r and fo r s > t t he dens i t y o f t he de f au l t

t im e i s g iven by

~ ( I )s P r _ < s i r > t , G r ) = ~ . s e x p - ~ .u d u .


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1 0 6 DAVID LAN DO

H e n c e w e f in dE ( e x p ( - f ~ r s d s ) Z ~ .T zt)

= E ( E ( e x p ( - f ~ r s d s ) Z r GT v 7 Lt) [.Trt)

= l l r > o E ( ~ r Z s ) ~ s e x p ( - f S ( r u + 3 . u ) d u ) d s G , )w h e r e i n t h e l a s t l i n e w e h a v e u s e d t h e s a m e a r g u m e n t a s a b o v eby Gt .

t o r e p l a c e ~ t

W e now tu rn t o exam ples an d i ll u s tr a ti ons o f t h i s f r ame w ork .Example 3 . 2 ( O p t i o n o n c r e d i t s p r e a d ) I n th e e x am p l e s co n s i d e re d a b o v e t he p ri m a r yfo cu s is secur i t i es w i th c redi t r isk . N ote , how ever , tha t a m od el o f the defau l t in tens i ty )~p r o d u c e s a m o d e l f o r th e y ie l d s p re a d f o r b o n d s o f a ll m a t u r it ie s . H e n c e o u r f r a m e w o r kw i l l e a s i ly p r o d u c e m o d e l s f o r o p t io n s o n y i e l d s p re a d s a s w e l l .E x a m p l e 3 . 3 ( T h e independence case A p r i m a r y f o c u s o f th is p a p e r is t o a l lo w f o rdep end enc e be tw een de fau l t i n tens i ti e s and s ta te va r iab l e s . N o te t ha t t he i ndep end encecase i s ea s i l y cap tu r ed a s we l l . I n t he s ta t e va r iab l e fo rmu la t i on w e wo u ld s im ply le tX = (X 1 , X 2) w i th X 1 and X 2 i ndepen den t p roces ses an d de f ine t he spo t r a t e a s t he p roces sr (X 1 ) and t he de f au l t i n t ens it y p roc es s a s ~ .(X2). O f cour se , e it he r o f t hese p roce s ses m aybe i nc luded a s a s ta t e va r iab l e t hem se lves .

Exam ple 3 .4 ( F e y n m a n - K a c r e p r e s e n t a t i o n ) A n i m p o rt an t s p ec ia l c a se is w h e nGt = cr{Xs : 0 < s < t} (3.7 )

w he re X i s an Rd-v a lued d i f fus ion p roce s s w i th d i ffus ion ma t r ix a ( x , t ) an d d r i f t vec to rb ( x , t ) and i n f i n it e s ima l gen e ra to r1 a a dO 2 F ( x ) O F ( x ). m t F ( x ) : = - ~ Z L a i k ( t , x ) ~ + ~ - ~ b i ( t , x ) - F E C 2 ( R )i l k=l i= OXi

L et T = Tf and le t f : R d ---> R , g : [0, T] x R a - -~ R and k : [0 , T] x R d - + [0, c


9/22


H e n c e , u n d e r s u i t a b l e r e g u l a r i t y c o n d i t i o n s2 o n t h e d i f f u s i o n c o e f f i c i e n t s a n d o n t h ef u n c t i o n s f , g , k , t h e e x p e c t a t i o n i n 3 . 8 ) i s a l s o a s o l u t i o n s a t i s f y i n g a n e x p o n e n t i a l g r o w t hc o n d i t i o n ) t o t h e C a u c h y p r o b l e m

3 v- - - + k v = . A t v + g [ O , T ] x R dO tv T , x ) = f x ) x q R aa n d w e s e e t h a t t h e c r e d i t r i s k y s e c u r it i e s c a n b e m o d e l e d u s i n g e x a c t l y t h e s a m e t e c h n i q u e sa s d e f a u l t - f r e e s e c u r i t i e s .

E x a m p l e 3 .5 F r a c t i o n a l r e c o v e r y a n d t h i n n i n g I n D u f f l e a n d S i n g l e t o n 1 9 9 6 ) , a n dD u f f le , S c h r o d e r a n d S k i a d a s 1 9 9 6 ) a n o t i o n o f f r a c t i o n a l r e c o v e r y is c o n s i d e r e d w h i c ha l l o w s n o t o n l y t h e i n t e n s i t y b u t t h e r e c o v e r y r a t e to e n t e r i n t o t h e e x p r e s s i o n f o r t h e d e f a u l ta d j u s t e d s p o t r a te p r o c e s s . H e r e w e s k e t c h h o w t h is m a y b e g i v e n a s i m p l e in t e r p r e ta t i o n i nt h e C o x p r o c e s s f r a m e w o r k t h r o u g h t h e n o t i o n o f t h i n n i n g . R e c e i v i n g a fr a c t i o n ~bt o f p r e -d e f a u l t v a l u e i n t h e e v e n t o f d e f a u l t o f a c o n t r a c t i s e q u i v a l e n t , f r o m a p r i c i n g p e r s p e c t i v e ,t o r e c e i v i n g th e o u t c o m e o f a l o t t e r y in w h i c h t h e f u l l p r e - d e f a u l t v a l u e i s r e c e i v e d w i t hp r o b a b i l i t y u n d e r t h e m a r t i n g a l e m e a s u r e ) q~t a n d 0 i s r e c e i v e d w i t h p r o b a b i l i t y 1 - ~b t, i .e .t h e e v en t o f d e f a u lt h a s b e e n r e t a i n e d w i t h p r o b a b i l i t y 1 - S t . T h i s i n tu r n m a y b e v i e w e da s a d e f a u l t p r o c e s s i n w h i c h t h e r e i s 0 r e c o v e r y b u t w h e r e t h e d e f a u l t i n t e n s i ty h a s b e e nt h i n n e d u s i n g t h e p r o c e s s ~b, p r o d u c i n g a n e w d e f a u l t i n t e n s i t y o f ~ . 1 - ~ b). I n t h e c a s e o f ac l a i m w i t h p r o m i s e d p a y m e n t X r a n d f r a c ti o n a l r e c o v e r y a t t h e d e f a u lt d a t e o f ~b, o n e t h e no b t a i n s t h a t w i t h Z ~ = q ~ c ~ _ , t h e p r i c e o f th i s c l a i m a t t im e t i s g i v e n a s

C t --~- E e x p - - f t r r s d s ) X l { , > T } + e x p - - f r r s d s ) Z r l{ t< r < _ T } ] .T t)= l { ~ > o E e x p - / r r s + 3 . s 1 - C g s ) ) d s ) X G t ) .

4 . A G e n e r a l iz e d M a r k o v i a n M o d e lI n t h is s ec t io n , a s a c o n c r e t e a p p l i c a t i o n o f t h e fr a m e w o r k p r e s e n t e d a b o v e , w e s h o w h o w t og e n e r a l i z e t h e m o d e l d e s c r i b e d in J a rr o w , L a n d o a n d T u m b u l l 1 9 9 7 ) f o r p r i c i n g d e f a u l t a b l eb o n d s . I t i s s h o w n h o w t h e m o d e l c a n b e g e n e r a l i z e d to i n c o r p o r a t e s t at e d e p e n d e n c e int r a n s i t i o n r a t e s a n d r i s k p r e m i a a l l o w i n g f o r s t o c h a s t i c c h a n g e s i n c r e d i t s p r e a d s e v e nb e t w e e n r a t i n g s t ra n s i t i o n s .

T h e r e , t h e ti m e t o d e f a u l t i s m o d e l e d a s th e a b s o r p t i o n t i m e i n s ta t e K o f a M a r k o v c h a i nw i t h g e n e r a t o r m a t r i x

- - ) L 1 ~ - 1 2 . ~ . 1 3 ~ I K ~

= . . . . .~ ' K - I . 1 ~ . K - 1 . 2 . . . . ~ K - 1 ~ . K - I . K0 0 . . . . . . 0

4 .1 )


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108 DAVID LANDO

and)~ij > 0 for i j

KX i = ~ X i j , i = l . . . . . K - 1 .j = l , j iIn the study of Fons and Kimball (1991) it is documented that default rates of lower ratedfirms show significant time variation and even for the top rated firms for which default isextremely rare the standard deviations of default rates may still be significant in relativeterms. Also, Duffee (1996) shows that noncallablebond yield spreads fall when the Treasuryterm structure yield increases. Whether this variation can be captured through variation inthe state variables or whether a parametrization in terms of constant intensities as in Jarrow,Lando and Turnbull (1997) performs equally well is an interesting empirical issue whichwe are currently checking. However, it is necessary with an extension if one is to take intoaccount the random fluctuations on credit spreads for constant time to maturity bonds.

Here, we will show how the Markovian model can be generalized to incorporate randomtransition rates and random default intensities. We will start with a fairly general modeland then impose special structure to gain computational advantages.

First, define the matrix--~-1 X s) ~-12 X s) ~,13 X s) - . . ~ . I K ( X s )x21(Xs) -z2(XD Z23(XD . .- X2r (Xs)

x ( s ) = : . . . . .) ~ K - I , I (X s ) ~ . K - 1 .2 ( S s) . . . . ~ . K _ l ( X s ) ~ , K - 1 .K ( X s )

0 0 . . . . . . 0

and assume thatK

)~i (Xs) : ~ ~ , i j (X s) , i = 1 . . . . . K - 1.j= l , j~ iIt is not essential that the random entries A x ( s ) a r e written as functions of state variables,

but we choose this formulation since this we believe will be the interesting one to workwith in practice. Also, we could of course include time as one of the state variables.Heuristically, we think of, say, ~.l(Xs)At as the probabili ty that a firm in rating class 1 willjump to a different class or default within the (small) time interval At. We now give theprecise mathematical construction and note again that this is essential if one wants to runsimulations with the model:

Let Ell . . . . . EIK, E21 . . . . . E 2 K . . . . be a sequence of independent unit exponential ran-dom variables)

Assume that a firm starts out in rating class 7?0. Define

{ I }~o,i = inf t : ~ . ~ o , i ( X s ) d s > E l i i = 1 . . . . . K


11/22


a n d l e tto o = m in ~ o o , ii~7o

H e r e , Zoo m o d e l s t h e t i m e t h e r a t i n g c h a n g e s f r o m t h e i n i t i a l l e v e l O o a n d t h i s o c c u r s t h ef ir st t i m e a s e t o f c o m p e t i n g r i s k s w h i c h t r ie s to p u l l t h e ra t i n g a w a y f r o m Oo e x p e r i e n c e sa j u m p . T h e n e w r a t i n g l e v e l m a y t h e n b e w r i t te n a s

r/1 = a rg m in r~o i .i ~ O o 'a n d i f t h e v a l u e o f 01 is K t h i s c o r r e s p o n d s t o d e f a u l t. N o w l e t

v ~ , i = i n f t : ) ~ , i X s ) d s > E 2ia n d

i = 1 . . . . . K

to1 = m in zol i .i ~H e n c e t h e n e x t j u m p o c c u r s a s t i m e Z Ol, t h e n e w r a t i n g l e v e l i s

~2 ----a r g r a i n z o ~i ~ r h 'a n d s o f o rt h . W e u s e r t o d e n o t e t h e t i m e o f d e f a u l t , i. e.

= i n f { t : ~ t = K }W i t h t hi s c o n s t r u c t io n w e o b t a i n a c o n t i n u o u s ti m e p r o c e s s w h i c h c o n d i t i o n a l l y o n t he

e v o l u t i o n o f th e s t at e v a ri a b l e s is a n o n - h o m o g e n e o u s M a r k o v c h a i n . C o n d i t i o n a l l y o n t hee v o l u t i o n o f t h e s t a te v a r i a b l e s , t h e t r a n s i t i o n p r o b a b i l i t i e s o f t h i s M a r k o v c h a i n s a t i sf y

O P x s , t )- - - - . A x s ) P x s , t ) .OsI t i s w o r t h n o t i n g t h at t h e s o l u t i o n o f th i s e q u a t i o n i n th e t i m e - i n h o m o g e n e o u s c a s e i s

t y p i c a l l y n o t e q u a l t oP x s , t ) = e x p f s t A x u ) d u )

w h i c h i s m o s t e a s i ly s e e n b y r e c a l l i n g t h at w h e n A a n d B a r e s q u a re m a t r ic e s , w e o n l y h a v ee x p ( A + B ) = e x p ( A ) e x p ( B )

w h e n A a n d B c o m m u t e .F o r a w a y o f w r i t i n g t h e s o l u t i o n u s i n g p r o d u c t i n t e g r al n o ta t i o n , s e e f o r e x a m p l e G i l l

a n d J o h a n s e n ( 1 9 9 0) .


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1 i D A VI D L A N D O

5 P r i c in g B o n d s a n d D e r i v a t i v e s in t h e G e n e r a l iz e d M a r k o v M o d e lA g e n e r a l b o n d p r i c i n g f o r m u l a f o r d e f a u lt a b le b o n d s w i th z e r o r e c o v e r y is th e n g i v e n i nt he f o l l o w i n gP r o p o s i t i o n 5 .1 C o n s i d e r a z e r o c o u p o n b o n d m a t u r i n g a t t i m e T i s su e d b y a f i r m w h o s er a t in g a t t i m e t i s i a n d w h o s e c r e d i t r a t in g a n d d e f a u l t i n t e n si ty e v o l v e a s d e s c r i b e d b y t h ep r o c e s s q a b o v e. T h e n g i v e n z e r o r e c o v e r y in d e fa u lt , t h e p r i c e o f th e b o n d i s g i v e n a s

v i t , T ) = E e x p - / r R X , ) d s ) l - P x t , T ) i , K )t )w h e r e P x ( t , T ) ;, K i s t h e ( i, K ) t h e l e m e n t o f t he m a t r ix P x ( t , T ) .P r o o f :

v i ( t , T ) l{r>r}B t)Ei- B - ~ . S t J= E i ( l { r > r } e x p ( - - f t r R ( X s ) d s ) . s t )

w here w e use t he s ame a rgum en t a s ea rl ie r t o r ep l ace .S t by ~ t .

W h i l e th i s a p p r o a c h p r o d u c e s m o d e l s t h a t c a n e a s i ly b e s i m u l a te d u s i n g t h e c o n s t r u c ti o n o ft he p roces s rl g iven abo ve , i t is o f in t e re s t t o de r i ve m ode l s w h ich , a l t hou gh m ore r e s tr ic t ivein the i r a s sum pt ions , o f f e r com puta t i ona l t r ac tab i l it y and poss ib ly even ana ly t ica l so lu ti ons .O ne wa y t o do t h is i s t o imp ose m ore s t r uc tu r e o n t he cond i t i ona l i n t ens i ty m a t r ix su ch t ha tt he expres s ion fo r P x ( s, t ) simpl i f ies :L e m m a 5 . 2 F o r i = 1 . . . . . K - 1 , l e t ] s : R a ~ [0 , ~ b e a n o n - n e g a t i v e f u n c t i o nd e fi n ed o n t h e s ta t e s p a c e o f X a n d a s s u m e t h a t f o r a l m o s t e v e r y s a m p l e p a t h o f X , w e h a v e

o /~i Xs)ds ~ .L e t I z ( X ~ ) d e n o t e t h e K x K d i a g o n a l m a t r i x d i a g ( I z l ( X s ) . . . . . IZ K -1 ( X s ) , 0) . F o r e a c h

a t h o f X a s s u m e t h a t t h e t im e d e p e n d e n t g e n e r a t o r m a t r i x h a s t h e re p r e s en t a ti o nA x ( s ) = B l z ( X s ) B - 1 ,

w h e r e B i s a K x K m a t r i x w h o s e c o l u m n s c o n s i s t o f K e i g e n v e c t o r s o f A x (s ) . A l s o , d e f in et h e d i a g o n a l m a t r i x


13/22

ON COX P ROCE SS E S 1 1 1

E x ( s , t ) = e x p 1xu ~ i0 . . . 0

T h e n w i th P x ( s , t ) = B E x ( s , t ) B - 1, P x ( s , t ) s a t i s f ie s K o l m o g o r o v s b a c k w a r d e q u a t i o na P x ( s , t )

Os- - = - A x ( s ) P x ( s , t )

a n d P x ( s , t ) i s t h e t r an s i ti o n p r o b a b i l i ty o f a n ( in h o m o g e n e o u s ) M a r k o v c h a in o n {1 . . . . . . K }.Proof S i n c e

A x ( s ) B = B l z x ( s )w e s e e t h a t

a P x ( s , t )a s - - B ( - l Z x ( s ) ) E x ( s , t ) B -1= - A x ( s ) B E x ( s , t ) B -1= - A x ( s ) P x ( s , t )

w h i c h s h o w s t ha t P x ( s, t ) i s i n d e e d a s o l u t i o n t o t h e b a c k w a r d e q u a t i o n . F r o m s e c t i o n 4 .4i n G i l l a n d J o h a n s e n ( 1 9 9 0 ) w e k n o w t ha t u n d e r o u r a s s u m p t i o n s o n A x ( s ) t h i s e q u a t i o nh a s a u n i q u e s o l u t i o n , a n d t h e s o l u t i o n is a t r a n s it i o n m a t r i x f o r a n i n h o m o g e n e o u s M a r k o vc h a i n w i t h th e d e n s i t y o f t he ' i n t e n s i t y m e a s u r e ' g i v e n b y A x ( s) . H e n c e P x ( s, t ) d e f i n e s aM a r k o v c h a in , a s w a s t o b e s h o w n . 9

T h e a s s u m p t i o n t ha t th e e i g e n v e c t o r s o f A x ( s ) e x i s t a n d a r e n o t t i m e v a r y i n g is a s t ro n ga s s u m p t i o n . T h e s i m p l i f i c a t i o n i s, h o w e v e r , p r o f o u n d a s w e l l a s w e s e e i n t h e n e x t r e s u ltw h i c h d e li v e rs a b o n d p r i c i n g m o d e l w h i c h m o d e l s b o n d s o f a l l c r ed i t ca t e g o ri e s s i m u l t a -n e o u s l y :Proposition 5 . 3 A s s u m e t h a t t r a n s i ti o n s b e t w e e n r a t in g c la s s e s c o n d i t i o n a ll y o n X a r eg o v e r n e d b y t h e g e n e r a t o r m a t r i x

A x ( s ) = B t x ( X s ) B - 1w h e r e B a n d l z ( X s ) a r e d e f in e d a s in L e m m a 5 .2 . A s s u m e i n a d d it io n , t h a t t h e K t h r o w o fA x ( s ) i s 0 ( c o r r e s p o n d i n g t o t h e s t a t e o f d e f a u l t b e i n g a n a b s o r b i n g s t a t e) . T h e s p o t r a t ep r o c e s s f o r d e f a u l t f r e e d e b t i s g i v e n b y R ( X ) . T h e p r i c e a t ti m e t o f a z e r o c o u p o n b o n dw i t h m a t u r i t y T , i s s u e d b y a f i r m i n ra t i n g c l a s s i a n d ( fo r s i m p l i c i t y ) w i t h z e r o r e c o v e r y in


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11 2 DAVID LANDO

default is given byv i ( t , T ) = j ~ = l ~ i j E e x p ( I z j ( X u ) - R ( X u ) d u ) ~ t

w here the coefficients flij are d efined below.P r o o f : C o n d i t i o n a l l y o n X t h e p r o b a b i l i t y o f d e f a u l ti n g b e f o r e T g i v e n a c re d i t r a ti n g o f ia t ti m e t ( a n d h e n c e n o p r e v i o u s d e f a u l t s i n c e d e f a u l t i n th i s m o d e l i s a n a b s o r b i n g s ta t e ) i se q u a l t o th e ( i , K ) t h e n t ry o f th e m a t r ix Px ( t , T ) . T h i s e n t r y h a s t h e f o rm

P x( t , T )i,K = y ~ b i j e x p Ixj(Xu)du b]-~rj lw h e r e b~-~ d e n o t e s t h e ( j , K ) t h e n t r y o f B - 1 a n d u s i n g t h e f a c t th a t th e r o w s o f A x s u m t o0 a n d t h a t t h e l a s t ro w o f A x i s O , o n e m a y c h e c k t h a t b i K b ~ = 1 . De f in ing /~ i j = -bijb;-t~w e m a y t h e r e f o re w r i t e th e p r o b a b i l it y o f n o d e f a u l t a s

( l )- Px( t , T i ,K = Z l i j e x p #j (Xu)du (5 .1)j lL e t s u p e r s c r i p t i o n t h e e x p e c t a t i o n o p e r a t o r s i g n i f y a t i m e t r a t i n g o f th e i s s u e r o f i . T h es u p e r s c r i p t i s r e m o v e d w h e n t h e e x p e c t a t i o n d o e s n o t d e p e n d o n t h is in f o r m a t io n . N o wc o m b i n e P r o p o s i t i o n 5 .1 a n d ( 5 .1 ) t o o b t a i n

v i ( t , T ) = Z f l i j E e x p ( u j ( X u ) - R ( X , ) d u ) G, 9j l

W e h a v e e x p r e s s e d t h e p r i c e o f b o n d i n c r e d it c la s s i a s a l i n e a r c o m b i n a t i o n o f f u n c ti o n a lso f t h e f o r m

( f0 0N o t e t h a t b y u s i n g t h e r e s u l t s o n a f f i n e t e r m s t r u c t u r e m o d e l s ( l e t t i n g / z a n d R b e a f f i n ef u n c t i o n s o f d i f f u s i o n s w i t h a f fi n e d r if t a n d v o l a t i l i t y ) w e o b t a i n h e r e a c l a s s o f m o d e l sw h o s e b o n d p r i c e s a r e e x p r e s s e d a s su m s o f a f fi ne m o d e l s .

T o k e e p t h e e x p o s i t io n s i m p l e w e h a v e l o o k e d a t z e ro c o u p o n b o n d s w i th z e r o r e c o v e ry .T h e m o d e l s a b o v e , h o w e v e r , a ls o a l lo w u s t o p r ic e m o r e c o m p l i c a t e d c o n t r a ct s w h o s ec o n t r a c t u a l p a y o f f a r e l in k e d t o th e r a t i n g o f t h e i s s u er . E x a m p l e s i n c l u d e c r e d i t o p t i o n sa n d s w a p s w i t h c r e d i t t r ig g e r s . T h i s e x e r c i s e c a n b e c a r r i e d o u t f o r a ll t h e t h r e e b u i l d i n gb l o c k s t h a t w e p r i c e d i n S e c t io n 3. C o n s i d e r a n e x a m p l e l i n k e d to c r e d it i n s u r a n c e in w h i c ht h e h o l d e r o f t h e c o n t in g e n t c l a i m r e c e i v e s a p a y m e n t w h i c h d e p e n d s o n s t a t e v a ri a b l e s a n do n t h e r a t in g o f s o m e c o m p a n y a t a f ix e d ti m e T . H e r e , t h e c o n t r a c t i t s e lf n e e d n o t h a ve a n y


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c r e d i t r i sk i n t ha t i t m a y e a s i l y p a y o u t a n a m o u n t i n t h e e v e n t o f d e f a u lt o f t h e fi rm w h o s er a t in g d e t e r m i n e s t h e p a y o u t .P r o p o s i t i o n 5 . 4 L et the mo del be as in the previous proposition, Consider a contingentclaim w hose pa yo ff at t ime T is given a s

Cr = C(XT, t iT) .As sum e that the rating at time t o f the firm under consideration is i. The pr ice o f this claimat time t is given by

s s l ) r )[ = E bikb-ff~C(Xr, m ) e x p ( lz~ (Xu) - R (X ,) du ) G, 9ra l k lP r o o f : F i r s t , n o t e t h a t

P x( t , T) i m = ~ '-~bikbk 1 e x p I~k(Xu)dum 9k l

a n d t h e r e f o r e ; ) )~ = E i C (X r , m )P x( t , T )i,m e x p - R (X u) du Gtm l

= E ( Km~lC(XT,m ) P x ( t , T ) i . r a e x p ( - - f t T R ( X u ) d u ) ] G t )

2 ; )1 X= E C (X T, m ) ~ bikbkm e p I zk(X , )duk l~ g b ik b k2 C (X T ,m )e x p ( l Z k ( X u ) - R ( X u ) du ) Gtm l k l

w h i c h i s t h e d e s i r e d r e s u lt .N o t e t h a t th i s p r i c i n g f o r m u l a a l l o w s p a y m e n t s t o b e d i r e c t l y li n k e d t o t h e r a t i n g s ta t u s

o f a c e r t a i n r i s k y f ir m .C l e a r l y , t h e p a r a m e t r i c a s s u m p t i o n s w e i m p o s e t o g e t t h e v e r y t r a c t a b l e m o d e l a b o v e

a r e st ro n g . B y s t o c h a s t ic s c a l in g o f th e e i g e n v e c t o r s w e a l l o w s t o c h a s t i c c h a n g e s o f t h eK - 1 ) 2 t r a n s i t io n a n d d e f a u l t i n t e n s i t i e s t o v a r y i n a K - 1 d i m e n s i o n a l s u b s p a c e a n d t h i s

w i l l a l l o w s t o c h a s t i c f l u c t u a t i o n s t o a f f e c t h i g h e r r a t e d a n d l o w e r r a t e d f i r m s d i f f e r e n t l y .E m p i r i c a l a n a l y s i s o f th i s f o r m i s c u r re n t l y b e i n g c a r r ie d o u t i n s e p a r a t e w o r k .


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114 DAVID LANDO

Allowing the generator to be specified more freely will complicate computation butsimulation is by no means the only way out: Consider again the equation

O P x ( s , t )= - A x ( s ) P x ( s , t ) .Os

To compute the matrix P x ( s , t ) , use the approximation based on a partition s = so < sm

David Lando - Cox Process and Risky Securities.pdf

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