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A P P L I C A T IO N O F T H E O P T IO N M A R K E T P A R A D I G M T O
T H E S O L U T I O N O F I N S U R A N C E P R O B L E M S
M I C H A E L G . W A C E K
Abs t r ac t
T h e B l a c k - S c h o l e s o p ti on p r i c in g f o r m u l a f r o m f i -
nance theory i s cons i s t en t wi th the assumpt ion tha t the
m a r k e t p r i c e o f the unde r l y ing as s e t a t any f u t ur e da t e
i s lognormal ly d i s t r ibu ted wi th t ime-dependent param-
e ters and can b e show n to be a spec ia l case o f bo th
a m or e ge ne r a l op t ion m o de l an d a f am i l i a r ac t uar ia l
fun c t io n used in excess o f los s appl ica t ions . Th i s in -
s igh t l eads to an un ders tanding o f the s imi lar it y be -
tween opt ions and cer ta in insurance concept s . Because
insurance an d f ina nc e have deve lop ed separate ly d i f-
f e r e n t par ad i gm s ar e u s e d by the p r ac t it ione r s in e ac h
f ie ld . W hen these para digm s are shared a new per -
s pe c ti v e on ri sk m anag e m e n t p r oduc t de v e l opm e n t
an d pr ic ing espec ia l ly o f insurance and re insurance
emerges .
1. R E L A T I O N S H I P O F T H E B L A C K S C H O L E S F O R M U L A A N D T H E
A C T U A R I A L E X C E S S O F L O S S F U N C T I O N
I n 1 9 7 3 , F i s c h e r B l a c k a n d M y r o n S c h o l e s p u b l i s h e d t h e i r
n o w c la s si c p a p e r e n t it le d T h e P r ic i n g o f O p t i o n s a n d C o r p o r a te
L i a b i l i ti e s , [ 1 i n w h i c h t h e y d e r i v e d t h e o p t i o n p r i c i n g fo r m u l a
t h a t b e a r s t h e i r n a m e . G e r b e r a n d S h i u [ 2 ] d e s c r i b e d t h a t p a p e r
a s p e r h a p s t h e m o s t i m p o r t a n t d e v e l o p m e n t i n t h e t h e o r y o f f i-
n a n c ia l e c o n o m i c s i n t h e p a st t w o d e c a d e s . T h e a d v e n t o f th e
m o d e r n d e r iv a t i v e s m a r k e t i s g e n e r a l l y tr a c e d b a c k t o t h e i n tr o -
d u c t i o n o f e x c h a n g e - t r a d e d e q u i t y o p t i o n s i n th e U .S . ( 1 9 7 3 ) a n d
t h e d e v e l o p m e n t o f t h e B l a c k - S c h o l e s m o d e l [3].
701
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702 APPLICATION OF THE OPTION MARKET PARADIGM
B l a c k a n d S c h o l e s s h o w e d t h a t u n d e r c e r t a i n c o n d i t i o n s t h e
c u r re n t p u r e p r e m i u m , l
c t ( S ) ,
f o r a c a l l o p t i o n t o b u y a p a r t i c -
u l a r a s s e t f o r p r i c e S , a t , a n d o n l y a t , t i m e t ( w h e r e t i s t h e t i m e
t o e x p i r y ) i s
c , ( S ) = P o N ( d l ) - S e - r t N ( d 2 ) , w h e r e
l n P 0 / S ) + r + 0 . 5 c r 2 ) t
d ~ = ~ r v q ( 1 . 1 )
d 2 = l n ( P o / S ) + ( r - 0 . 5 a 2 ) t
a v ' 7
a n d w h e r e P 0 i s t h e c u r r e n t m a r k e t p r i c e , r i s t h e r i s k - f r e e f o r c e
o f i n t e r e s t , a i s a m e a s u r e o f a n n u a l i z e d p r i c e v o l a t i l i t y , a n d N
is t h e c u m u l a t i v e d i st r i b u t io n f u n c t i o n o f t h e s t a n d a rd n o r m a l
d i s t r i b u t i o n .
T h i s i s a d a u n t i n g f o r m u l a , a n d i n t h i s f o r m i t p r o v i d e s l i t t l e
i n s i g h t i n t o t h e u n d e r l y i n g o p t i o n s p r i c i n g p r o b l e m . O n e o f t h e
k e y p o i n t s o f th i s p a p e r is th a t F o r m u l a 1 .1 , t h e B l a c k - S c h o l e s
f o r m u l a , i s a c t u a l l y a s p e c i a l c a s e o f a f a m i l i a r a c t u a r i a l f u n c t i o n
w r i t t e n in a n u n f a m i l i a r f o r m . T h i s w i l l l e a d u s t o s o m e i m p o r t a n t
i n s i g h ts a b o u t b o t h o p t i o n s a n d i n s u ra n c e .
C o n s i d e r t h a t t h e p u r e p r e m i u m o f a c a l l o p t i o n e x e r c i s a b l e
o n l y o n t h e .e x p ir y d a t e ( a E u r o p e a n o p t i o n ) d e p e n d s o n th e
m a r k e t ' s c u r r e n t o p i n i o n a b o u t t h e p ro b a b i l it y d i s t ri b u t io n o f th e
m a r k e t p r i c e o f t h e u n d e r l y i n g a s s e t o n t h e e x p i r y d a t e . I f t h e
o p t i o n e x e r c i s e p r i c e i s S , t h e o p t i o n w i l l o n l y b e e x e r c i s e d i n
t h e e v e n t t h e m a r k e t p r i c e a t e x p i r y e x c e e d s S . I ts v a l u e i n t h e s e
c i r c u m s t a n c e s w i l l b e t h e a m o u n t b y w h i c h t h e m a r k e t p r i c e
e x c e e d s S . I n o t h e r c i r c u m s t a n c e s , t h e o p t i o n h o l d e r w il l le t th e
IFinancial economists use the term price or premium. However, to make clear to
actuarial readers that there is no embedded charge for risk or expenses in the Black-
Scholes valuation, we shall use the actuarial term pure premi um.
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o p t i o n e x p i r e u n e x e r c i s e d a n d , i f h e w a n t s to o w n t h e a ss et , b u y
t h e a s s e t a t t h e m a r k e t p r i c e . T h e o p t i o n t o b u y t h e a s s e t a t a
h i g h e r t h a n m a r k e t p r i c e w i ll b e w o r t h l e ss . T h e v a l u e o f t h e o p -
t i o n a t e x p i r y i s t h e p r o b a b i l i t y - w e i g h t e d a v e r a g e o f a l l p o s s i b l e
e x p i r y s c e n a r i o s .
S u p p o s e t h e p r o b a b i l it y d i s t r i b u t io n o f m a r k e t p r i c e s a t e x p i r y
is r e p r e s e n t e d b y t h e r a n d o m v a r ia t e x . T h e n t h e e x p e c t e d v a l u e
o f t h e o p t i o n a t e x p i r y i s
s
~ X - S )
u t u r e V a l u e [ c t S ) ] = f x ) d x . 1 .2 )
T h e e x p r e s s i o n o n t h e r ig h t h a n d s i d e o f F o r m u l a 1 .2 is
t h e f u t u r e
v a l u e
o f th e o p t i o n p u r e p r e m i u m , s i n c e x is d e f i n e d f o r t h e
e x p i r y d a t e , w h i c h i s i n t h e f u t u r e . I t s p r e s e n t v a l u e , d i s c o u n t e d
a t th e r i sk - f re e in te re s t ra te , 2 is
s
~ x S) 1 .3 )
t S ) = e - r t _
f x ) d x .
N o w c o m p a r e F o r m u l a s 1 . 1 a n d 1 . 3 . F o r m u l a 1 . 1 , t h e B l a c k -
S c h o l e s f o r m u l a , d e p e n d s o n t h e a s s u m p t i o n t h a t m a r k e t p r i c e s
a r e l o g n o r m a l l y d is tr ib u t e d . F o r m u l a 1 .3 is m o r e g e n e r a l a n d h a s
n o e m b e d d e d d i s tr i b u ti o n a l a s s u m p t i o n . H o w e v e r , if t h e v a r ia t e
x i n F o r m u l a 1 . 3 i s a s s u m e d t o b e l o g n o r m a l a n d t h e c o r r e c t
2 T h i s i s j u s t i f ie d o n t h e b a s i s t h at u s i n g a n y o t h e r r at e w o u l d o p e n t h e d o o r t o r i s k - f r e e
a r b i t r a g e p r o f i t s . I t i s p o s s i b l e t o c r e a t e a r i s k l e s s p o r t f o l i o b y h e d g i n g a l o n g p o s i t i o n
i n t h e u n d e r l y i n g a s s e t b y s e l l i n g s h o r t a n a p p r o p r i a t e n u m b e r o f c a l l o p t i o n s o n t h e
u n d e r l y i n g a s s e t . B e c a u s e it is r i sk l e s s , t h i s h e d g e d p o r t f o li o m u s t e a r n t h e r i s k - f re e r a t e
o f r e t u r n . H o w e v e r , f o r t h i s t o b e t r u e ( a n d i t m u s t b e t r u e t o a v o i d r i s k - f r e e a r b i t r a g e
p r o f i t o p p o r t u n i t i e s ) , i t t u r n s o u t t h a t t h e i n t e r e s t r a t e f o r d i s c o u n t i n g t h e e x p e c t e d v a l u e
o f t h e c a ll o p t i o n a t e x p i r y m u s t a ls o b e t h e r i s k - f r e e r a te . T h e f i n a n c e l i t e r a tu r e r e f e r s
t o t h i s p h e n o m e n o n a s r i s k - n e u t r a l v a l u a t i o n a n d i t a p p l i e s t o v a l u a t io n o f al l f i n a n c ia l
d e r i v a t i v e s o f a s s e t s w h e r e s u i t a b l e c o n d i t i o n s f o r h e d g i n g e x i s t . F o r f u r t h e r d i s c u s s i o n
o f r i s k - n e u t r a l v a l u a t i o n a n d r i s k - f r e e d i s c o u n t i n g , s e e H u l l [ 7 ].
I n a c t u a ri a l a p p l i c a ti o n s i n v o l v i n g i n s u r a n c e c l a i m s ( w h e r e h e d g i n g i s n o t p o s s i b l e ), it
i s s o m e t i m e s i m p l i c i t ly r e c o g n i z e d t h a t t h e r is k - f r e e r a te i s n o t a l a p ro p r ia t e b y d i s c o u n t i n g
a t t h e r i s k - f r e e r a t e , a n d t h e n a d d i n g a r i s k c h a r g e t o t h e d i s c o u n t e d r e s u l t . T h i s i s
e q u i v a l e n t t o d i s c o u n t i n g a t a r a t e l e s s t h a n t h e r i s k - f r e e r a t e . W e h a v e d e l i b e r a t e l y
c h o s e n t o c h a r a c t e r i z e c ~ S ) a s a p u r e p r e m i u m t o l e a v e t h e d o o r o p e n to an a d d i t io n a l
r i sk c h a r g e w h e r e a p p r o p r ia t e .
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distribution parameters are chosen 3 Formula 1.1 can be derived
from Formula 1.3. In other words the Black-Scho les formula is
a special case of Formula 1.3. The proof of this is in Appendix A.
Formula 1.2 which differs from Formula 1.3 only by a present
value factor also defines a familiar actuarial function seen fre-
quently in excess of loss insurance applications. For example if
x is a random variate representing the aggregate value of losses
occurring during an annual period then Formula 1.2 defines the
expected value of losses in excess of an aggregate loss amount of
S. This function is an important tool in pricing aggregate excess
or stop-loss reinsurance covers.
A second example relates to the more common type o f excess
of loss coverage where the excess attachment point S is defined
in terms of individual losses rather than in the aggregate over a
period. In this context if x is a random variate representing the
loss severity distribution with mean M then Formula 1.2 defines
the expected portion of M attributable to losses in excess of S. If
the result of Formula 1.2 equals C then C M is the excess pure
premium factor. If N is the expected number of losses then NC
is the expected value of excess losses.
Let us summarize what we have established. Formula 1.2
defines an important element of excess of loss pricing. It dif-
fers only by a present value factor from Formula 1.3 which de-
fines a general formula for European call option pricing. Form-
ula 1.1 the Black-Scho les formula is a special case of Formula
1.3.
The implication of this is that excess of loss insurance and
call options are essentially the same concepts. The one deals
with insurance claims and the other deals with asset prices but
the pricing mathematics is basically the same.
3 F o r m u l a s 1 .1 a n d 1 .3 p r o d u c e t h e s a m e r e s u l t i f x i s a I o g n o r m a l v a ri a te w i t h p a r a m e t e r s
l n P +rt -O .5a2t , ax/7 , w h e r e t h is c h ar a c te r iz a t io n f o l lo w s H o g g a n d K l u g m a n [ 4] ,
w h o d e f i n e a l o g n o r m a l d i s t r i b u ti o n b y r e f e r e n c e t o th e /~ a n d t r o f t h e r e l a te d n o r m a l
d i s t r ib u t i o n . S e e A p p e n d i x A f o r t h e p r o o f o f t hi s .
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T h i s i n s i g h t is p o t en t i a ll y t r e m e n d o u s l y p o w e r f u l . I f e x c e s s o f
l o s s i n s u r a n c e a n d c a l l o p t i o n s a r e e s s e n t i a l l y t h e s a m e c o n c e p t
i n d i f f e r e n t c o n t e x t s , t h e n i t m u s t b e p o s s i b l e t o t r a n s l a t e i d e a s
f r o m o n e c o n t e x t i n to t h e o t h e r c o n t e x t. I n t h e r e m a i n d e r o f th i s
p a p er , w e w i l l e x p l o r e s o m e o f th e p o t e n t ia l a p p l i c a t io n s o f t h e
o p t i o n s m a r k e t p a r a d i g m t o i n s u r a n c e p r o b l e m s .
2. IMPLICATIONSOF THE EQUIVALENCE OF OPTION AND
ACTUARIAL EXCESS OF LO SS MODELS
T h e m a t h e m a t i c a l e q u i v a l e n c e o f f i n a n c e t h e o r y ' s B l a c k -
S c h o l e s f o r m u l a a n d a n i m p o r t a n t a c tu a r ia l f u n c t i o n u s e d i n e x-
c e s s o f l o s s i n s u r a n c e a p p l i c a t i o n s h a s a n u m b e r o f i m p o r t a n t
i m p l i c a t i o n s f o r t h e c o n v e r g e n c e o f i n s u r a n c e a n d f i n a n c e . I n
t hi s p a p e r w e w i ll e x p l o r e a fe w o f t h e m .
O p t i o n m a r k e t p a r a d i g m s c a n b e u s e d t o t h in k a b o u t i n s u r a n c e
p r o b l e m s ; a n d t h i s m a y w e l l le a d to n e w i n s u r a n c e o r, p e r h a p s
m o r e l i k e l y , r e i n s u r a n c e p r o d u c t s .
T h e m o r e g e n e r a l a c tu a r ia l e x c e ss o f l o ss p a r a d i g m , w h i c h
e n c o m p a s s e s a n d f r e q u e n t l y u s e s d i s t r i b u t i o n s o t h e r t h a n t h e
l o g n o r m a l , c a n b e u s e d to t h i n k a b o u t t h e p r i c in g o f o p t i o n s o n
a s se ts f o r w h i c h m a r k e t p r ic e s a r e n o t l o g n o r m a l l y d i s tr ib u t e d .
T a k i n g t h e t w o p r e v i o u s p o i n t s t o g e th e r , i t is p o s s i b l e t o m o v e
b e y o n d e x i s t i n g o p t i o n s a n d a c t u a r i a l p a r a d i g m s t o s p a w n a
n e w o n e t h a t e n c o m p a s s e s b o t h . T h i s , in t u rn , m a y l e a d to
n e w p r o d u c t o p p o r t u n i t i e s f o r i n s u r e r s , i n v e s t o r s , o r b o t h .
3 . THE OPTION MA RKET PARADIGM
T h e f i n a n c ia l m a r k e t s h a v e b e e n t r e m e n d o u s l y c re a t iv e i n d e -
v i s in g p r o d u c t s a n d t e c h n i q u e s f o r m a n a g i n g f i n a n c ia l ri sk . M o s t
o f th i s a c t iv i ty h a s o c c u r r e d i n w h a t is l o o s e l y c a l l e d th e d e r i v a -
t iv e s m a r k e t . O p t i o n s a re a t t h e c o r e o f t h is m a r k e t , a n d i t is o n
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706 APPLICATION OF THE OPTION MARKETPARADIGM
F I G U R E 1
EXPIRY VALUE PROFILE: CALL OPTION c t S )
xp i ry
Va lue
Under l y ing Asset Pr i ce a t Exp i r y X,
t h is p a r t o f t h e d e r i v a t i v e s m a r k e t t h a t w e w i l l f o c u s o u r a t t e n ti o n .
M a n y d e r i v a t i v e p r o d u c t s a r e b u i l t a r o u n d o p t i o n f e a t u r e s .
Bas ic Op t ions
A E u r o p e a n
call
o p t i o n ,
ct S) ,
r e p r e s e n t s t h e f i g h t b u t n o t
t h e o b l i g a t i o n t o b u y t h e u n d e r l y i n g a s s e t a t , a n d o n l y a t , t i m e
t a t a p r i c e o f S . F o r m u l a 1 .3 d e s c r i b e s t h e p r i c e o f s u c h a c a l l
o p t i o n . F i g u r e 1 s h o w s i ts e x p i r y v a l u e p ro f i le .
A n A m e r i c a n c a l l o p t i o n i n c o r p o r a t e s t h e r ig h t t o e x e r c i s e a t
a n y t im e u p t o a n d i n c l u d i n g t im e t . T h e B l a c k - S c h o l e s f o r m u l a
a p p l i e s t o th e p r i c i n g o f E u r o p e a n c a l ls . In t h is d i s c u s s i o n o u r
r e f e r e n c e s w i l l b e t o E u r o p e a n - s t y l e o p t io n s u n l e ss o t h e r w i s e
s p e c i f i e d .
A E u r o p e a n p u t o p t i o n , pt S ) , r e p r e s e n t s t h e f i g h t b u t n o t
t h e o b l i g a t i o n t o s e l l t h e u n d e r l y i n g a s s e t a t , a n d o n l y a t , t i m e t
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7 7
F I G U R E 2
EXPIRY V ALU E PR OFILE: PU T OPTION, P t ( S )
E x p i r y
V a l u e
0
S
U nde r l y ing As se t P r ice a t Exp i r y X t
a t a p r i c e o f S . F i g u r e 2 s h o w s t h e e x p i r y v a l u e p r o f i l e o f a p u t
o p t i o n .
T h e g e n e r a l f o r m u l a f o r th e p r i c e o f a p u t, p t ( S ) is
P t ( S ) = e - r ( S - x ) . f ( x ) d x . 3 . 1 )
S p r e a d s
T h e c o m b i n a t i o n o f tw o c al l o p t i o n s , o n e b o u g h t a n d o n e s o l d;
e .g . ,
c t ( S , T ) = c t ( S ) - c t ( T ) , w i th T > S 3 . 2 )
i s k n o w n a s a c a l l o p t i o n s p r e a d .
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I n i n s u r a n c e p a r l a n c e ,
ct S , T )
r e f e r s t o a n
excess layer, c t S , T )
is th e p u r e p r e m i u m f o r t h e l a y e r o f T - S e x c e s s o f S .
P u t o p t i o n s p r e a d s c a n b e d e f i n e d i n a s i m i l a r w a y t o c a l l
o p t i o n s p r e a d s . 4
Implications for Insurance Applications
O n c e w e r e c o g n i z e t h a t a c a l l s p r e a d i s t h e s a m e t h i n g a s
a n e x c e s s l a y e r , a n e w w o r l d o p e n s u p . I n t h e o r y , e v e r y o p t i o n
a n d r e l a te d d e r i v a ti v e p r o d u c t m u s t h a v e a n i n s u r a n c e a n a l o g u e
S i n c e t h e d e r i v a t i v e m a r k e t s h a v e b e e n e n o r m o u s l y c r e a t i v e i n
d e v e l o p i n g n e w p r o d u c t id e a s, i t s h o u l d b e p o s s i b l e t o m i n e t h a t
t r o v e o f i d e a s f o r p o t e n t i a l l y i n n o v a t i v e i n s u r a n c e a n d r e i n s u r -
a n c e p r o d u c t c o n c e p t s .
A s a n e x a m p l e o f h o w t hi s c a n b e d o n e , w e w i l l an a l y ze t h e
d e r iv a t iv e s c o n c e p t o f a cylinder. T h e n w e w i l l r e c o n s t r u c t i t a s
a r e i n s u r a n c e p r o d u c t .
I n i ts e x t r e m e f o r m , a z e r o c o s t c y l i n d e r i s c r e a t e d b y t h e s i-
m u l t a n e o u s p u r c h a s e o f a c a ll a n d s a l e o f a p u t ( o r v ic e v e r sa ) o f
e q u a l v a l u e , u s u a l ly a t d i f fe r e n t o u t - o f - t h e - m o n e y e x e r c is e p r i c e s
b u t h a v i n g t h e s a m e e x p i r a t io n d a te . 5 I f th e c y l i n d e r i n v o l v e s a
l o n g c a l l ( i . e . , t h e p u r c h a s e o f a c a l l ) a n d a s h o r t p u t ( i . e . , t h e
s a l e o f a p u t ) , i t s v a l u e i n c r e a s e s w h e n t h e v a l u e o f t h e u n d e r l y -
i n g a s s e t i n c r e a s e s a n d d e c r e a s e s w h e n t h e a s se t v a l u e d e c r e a s e s .
T h i s i s a " b u l l i s h " p o s i t i o n . I f t h e c y l i n d e r i n v o l v e s a s h o r t c a l l
a n d a lo n g p u t , it s v a l u e i n c re a s e s w h e n t h e v a l u e o f t h e u n d e r l y -
i n g a ss e t d e c r e a s e s a n d d e c l i n e s w h e n t h e u n d e r l y i n g a ss e t v a l u e
i n c r e a s e s . T h i s i s a " b e a r i s h " p o s i t i o n .
4 F o r a d e t a i le d d i s c u s s i o n o f t h e m a t h e m a t i c s o f c al l, p u t, a n d c y l i n d e r s p r e a d s , s e e
A p p e n d i x B . T h e r e a r e a l so a n u m b e r o f g o o d r e f e r e n c e b o o k s o n f i n a n c i a l d e r i v a ti v e s ,
i n c l u d i n g R e d h e a d [ 5 ] an d H u l l [ 7 ], th a t p ro v i d e m o r e c o m p r e h e n s i v e t r e a t m e n t o f t h e
s u b j e c t. T h e r e i s a l s o a B r i t is h p a p e r , K e m p [ 8 ] , w h i c h e x a m i n e s t h e s u b j e c t f r o m a m o r e
a c t u a r i a l p e r s p e c t i v e , a l t h o u g h i t i s n o t p a r t i c u l a r l y o r i e n t e d t o w a r d n o n - l i f e i s s u e s .
5 T h i s is t h e e x t r e m e f o r m . N o t e th a t a c y l i n d e r n e e d n o t b e z e r o c o s t . F o r f u r t h e r
d i s c u s s i o n o f c y l i n d e r s a n d o t h e r o p t i o n c o m b i n a t i o n s , s e e [ 5 ].
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APPLICATION OF THE OPTION MARKET PARADIGM 7 9
F I G U R E 3 A
EXPIRY VALUE PROFILE: BULL CYLINDER OPT ION cy l t S , T)
Exp i ry
V a lu e
O - -
U nde r ly ing As se t Price at Expiry X~
B u l l a n d b e a r c y l i n d e r s a r e d e f i n e d a s f o l lo w s :
c y l t S , T ) = c t T ) - p t S ) , T > Po > S
bul l )
- c y l t S , T ) = p t S ) - c t T ) , T > Po > S
b e a r )
a n d t h e i r e x p i r y v a l u e p r o f i le s a r e s h o w n i n F i g u r e s 3 A a n d 3 B .
F o r a n o w n e r o f t h e u n d e r l y i n g a s s e t , e s t a b l i s h i n g a
b e a r
c y l i n d e r p o s i t io n p a r t ia l ly h e d g e s h i s a s s e t p o s i t io n a n d r e d u c e s
i ts v o l a t il it y . S i n c e i n t h e c a s e o f a z e r o c o s t c y l i n d e r t h e v a l u e s
o f t h e s h o r t c a l l a n d l o n g p u t a r e e x a c t l y o f f s e t t i n g , n o m o n e y
c h a n g e s h a n d s a t i n c e p t i o n o f t h i s p o s i t i o n . A t e x p i r a t i o n , i f t h e
v a l u e o f t h e u n d e r l y i n g a s s e t i s X t , t h e v a l u e o f t h e c y l i n d e r
p o s i t i o n i s
- X t
- - T ,
X > T ;
O T > X t > S ;
s - x s >_ x .
8/11/2019 97701
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710 APPLICATION OF THE OPTION MARKETPARADIGM
F I G U R E 3 B
EXPIRY VALUE PROFILE: BEAR CYLINDER OPTION
c y l t S , T )
E x p i r y
V a l u e
O - -
U n d e r l y i n g A s s e t P r ic e a t E x p i r y X t
T A B L E 1
Expiry Value Expiry Value
Asset Price of Cylinder Asset Cylinder
X t >_ T - X , - T ) T
T > X , > S o x ,
S > X , S - X , S
T h e h o l d e r o f th i s p o s i t i o n g a i n s
S - X t
f o r s m a l l v a l u e s o f
X t
a n d l o s e s X t - T f o r l a r g e v a l u e s o f X r F o r m i d d l e v a l u e s o f X t
h e g a i n s o r l o s e s n o t h i n g . H i s h e d g e d p o s i t i o n a t e x p i r y o f t h e
c y l i n d e r i s s u m m a r i z e d in T a b l e 1.
I n w o r d s , t h i s i m p l i e s t h a t t h e h e d g e d p o s i t i o n y i e l d s t h e r e -
t u r n s o f t h e u n d e r l y i n g a s s e t i .e .,
X t - P o ,
b u t s u b j e c t t o a m a x -
i m u m l o s s o f P0 - S a n d a m a x i m u m g a i n o f T - P0.
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A P P LI CA T IO N O F T H E O F F I O N M A R K E T P A R A DI G M 7 1 1
I f , r a t h e r t h a n o w n i n g t h e u n d e r l y i n g a s s e t , a n i n v e s t o r h a s a
s ho r t pos i t i on i n i t ( i . e . , i t i s a l i a b i l i t y ) , he c a n pa r t i a l l y he dge
t h a t p o s i t i o n w i t h a
b u l l
c y l i n d e r .
S u p p o s e t h e u n d e r l y i n g a s s et is t h e r i g h t to r e c o v e r i n s u r a n c e
c l a i m s . T o a n i n s u r e d , t h i s i s a n a s s e t ( a l o n g p o s i t i o n ) . T o a n
i n s u r e r, i t i s a l i a b il i ty ( a s h o r t p o s i t i o n ) . T h e r e f o r e , a n i n s u r e r
c o u l d u s e a b u l l c y l i n d e r t o p a r t i a l l y h e d g e h i s e x p o s u r e .
I f t h e r e w e r e a n e s t a b l i s h e d d e r i v a ti v e s m a r k e t t r a d i n g o p t i o n s
o n i n s u r a n c e c l a i m s , a s th e r e is f o r a n u m b e r o f o t h e r fi n a n c i a l
a ss et s, a n i n s u r e r w o u l d b e a b l e to h e d g e its e x p o s u r e b y b u y i n g
a n y o f a v a r ie t y o f p r o d u c t s ; e . g ., c a ll o p t i o n s , c a ll s p r e a d s , b u l l
c y l i n d e r s , o r b u l l c y l i n d e r s p r e a d s . A t p r e s e n t , t h e r e i s o n l y a l i m -
i te d d e r iv a t iv e s m a r k e t f o r o p t i o n s o n i n s u r a n c e c l a i m s ( n a m e l y ,
t h e e x c e s s o f l o s s r e i n s u r a n c e m a r k e t ) a n d , b r o a d l y s p e a k i n g , i t
o f fe r s o n l y o n e p r o d u c t : t h e c a ll s p r e a d . 6 O n e o f t h e k e y t h e m e s
o f th i s p a p e r i s t h a t c o n c e p t u a l l y t h e r e is n o r e a s o n w h y t h e r e in -
s u r a n c e m a r k e t c o u l d n o t o f f e r s i m i l a r p r o d u c t s t o t h o s e f o u n d
i n t h e b r o a d e r d e r i v a t i v e s m a r k e t .
N o w le t u s c o n s i d e r h o w t h e c y l in d e r c o n c e p t , w h i c h h a s t h e
a d v a n t a g e o f lo w e r in it ia l c o s t t o th e b u y e r c o m p a r e d t o a s i m p l e
c a ll o p t i o n , m i g h t b e t r a n s l a te d i n t o a r e i n s u r a n c e p r o d u c t . T o
i l l u s t r a t e o n e w a y t h i s m i g h t w o r k , f i r s t i m a g i n e a h i g h l e v e l
e x c e s s o f lo s s l a y e r w i t h a re t e n t i o n o f T a n d a li m i t o f
T 2
T1.
T h e m a r k e t p r e m i u m , i g n o r i n g a l l e x p e n s e s , f o r c o n v e n t i o n a l
c o v e r a g e is c t ( T 1 7 2).
T o c r e a t e t h e c y l i n d e r t y p e s t r u c t u r e , w e n e e d t o i n t r o d u c e a
f e a t u r e e q u i v a l e n t t o t h e s a l e o f a p u t . C o n s i d e r a s e c o n d , u n -
r e i n s u r e d , l a y e r o f S 1 - S 2 e x c e s s o f S 2 w i t h i n t h e c o m p a n y ' s
r e i n s u r a n c e r e te n t i o n , w h i c h w i ll f o r m t h e b a si s o f t h e r e q u i r e d
p u t s p r e a d . L e t
p t ( S 1 , S 2 )
d e n o t e t h e v a l u e o f th i s p u t s p r e a d .
6 A t t h e ti m e t h i s p a p e r w a s w r i tt e n , th e C h i c a g o B o a r d o f T r a d e s e f f o r t s to c r e a t e a
m a r k e t f o r o p t i o n s o n U . S . c a t a s t r o p h e l o s s e s h a d n o t y e t p r o d u c e d s i g n i f i c a n t c a p a c it y .
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7 2
PPLIC TION OF THE OPTION M RKET P R DIGM
A r e i n s u r a n c e c y l i n d e r s p re a d c a n b e c r e a t ed b y t h e p u r c h a s e
b y a c e d i n g c o m p a n y o f t h e h i g h le v e l e x c e s s o f l o ss l a y e r a t a
c o s t o f
c t ( T , T 2 )
a n d t h e e q u i v a l e n t o f t h e s a l e o f a p u t s p r e a d
o n t h e l o w e r l a y e r a t a p r i c e o f p t S i , S 2 ) . ( T h i s i s n o t n e c e s -
s a ri ly a z e ro c o s t cy l i n d e r. ) T h e p r e m i u m o u t l a y o f t h e c e d -
i n g c o m p a n y a t t h e b e g i n n i n g o f t h e c o n t r a c t w o u l d t h e n b e
c t T 1 T 2 ) - P t S 1 , 2 ) . S i n ce t h e re i n su r e r m a y r e q u i re a m i n i m u m
i n it ia l p r e m i u m o f M > 0 , i t m a y b e n e c e s s a r y t o a l l o w t h e r a ti o
o f p u t s t o c a ll s t o b e d i f f e r e n t f r o m o n e . I f th i s r at io i s r e p r e s e n t e d
b y Q , t h e i n i t i a l p r e m i u m i s g i v e n b y
M = c t ( T 1
T2) -
Q p t ( S 1
, 2).
U n d e r t h is s t ru c t ur e , th e p r e m i u m o f
c t ( T i , T 2 )
b u y s e x a c t l y t h e
s a m e e x c e ss p r o t e c t i o n a g a i n st l a rg e c l a i m s a s t h e c o n v e n t i o n a l
r e i n s u r a n c e p r o v i d e s . T h e p r e m i u m c r e d i t o f
Q p t ( S 1 , S 2 )
e m b e d -
d e d i n t h e i n i t i a l p r e m i u m r e p r e s e n t s t h e s a l e o f a p u t s p r e a d
o n t h e l o w e r l a y e r b y t h e c e d i n g c o m p a n y t o t h e r e i n s u r e r , t h e
f in a l v a l u e o f w h i c h w i l l b e s e tt le d a s a n a d d i t io n a l p r e m i u m o f
m i n ( Q ( S l - X t ) , Q ( S I
- 2 ) )
w h e n c l a i m e x p e r i e n c e i s k n o w n .
L e t u s n o w p u t s o m e n u m b e r s t o i t . L e t
c , T ~ , T 2 ) = 2 , 5 0 0 , 0 0 0 ,
p t ( S i , S 2 )
= 3 , 8 8 9 , 0 0 0 ,
Q = 4 5 % ,
S l = 1 5 , 0 0 0 , 0 0 0 , a n d
S 1 - S z
= 5 , 0 0 0 , 0 0 0 .
T h e n t h e i n i t i a l p r e m i u m i s c a l c u l a t e d a s f o l l o w s :
M = c t ( T , T2 ) - Q . p , ( S 1
2)
= 2 , 5 0 0 , 0 0 0 - ( . 4 5 ) ( 3 , 8 8 9 , 0 0 0 )
= 7 5 0 , 0 0 0 .
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APPLICATION OF THE OPTION MARKET PARADIGM
T A B L E 2
713
Initial Addit ional Total
Claims X Premium Premium Premium
Xt < S 2 750 2,250 3,000
S 2 < X < S 1 750 (45%)( 15,000 - X:) Slides 750 to 3,000
S 1 < X, 750 0 750
Note
Premium figures in thousands.
A t e x p i r y o f th e c o n t r a c t ( o r a t s u c h t i m e a s a g r e e d ) , a n a d d i t i o n -
a l p r e m i u m , A , e q u a l to t h e e x p i r y v a l u e o f t h e p u t s p r e a d is
d u e :
A = m i n [ Q ( S l -
X t ) , Q S l -
$2)]
= l e ss e r o f : ( . 4 5 ) ( $ 1 5 , 0 0 0 , 0 0 0 -
x t )
a n d ( . 4 5 ) ( $ 5 , 0 0 0 , 0 0 0 ) .
T h e t ot al p r e m i u m u n d e r a c y l i n d e r r e i n s u r a n c e s t ru c t u re d e -
p e n d s o n t h e f in a l c o s t o f c l a i m s , X , a s s h o w n i n T a b l e 2 .
T h i s c o m p a r e s t o t h e f ix e d p r e m i u m o f $ 2 , 5 0 0 ,0 0 0 u n d e r t h e
c o n v e n t i o n a l c o n t r a c t a n d i s s h o w n g r a p h i c a l l y o n F i g u r e 4 . I n
t he c y l i n d e r s t r u c tu r e , th e c e d i n g c o m p a n y p a y s a h i g h e r p r e -
m i u m f o r its c o v e r a g e o f T2 - T e x c e s s o f T1 w h e n t h e c l a im e x -
p e r i e n c e i n t h e r e t a i n e d s u b l a y e r o f S 1 - S 2 e x c e s s o f S 2 is g o o d
( u p to $ 3 ,0 0 0 , 0 0 0 v e r s u s $ 2 ,5 0 0 , 0 0 0 ). I t p a y s a l o w e r p r e m i u m
w h e n c l a i m e x p e r i e n c e in th a t l a y e r is b a d ( $ 7 5 0 , 0 0 0 v e r s u s
$ 2 ,5 0 0 ,0 0 0 ) . I n o t h e r w o r d s , t h e c o m p a n y p a y s m o r e w h e n its
n e t c l a i m s e x p e r i e n c e i s r e l a t i v e l y g o o d a n d i t c a n a f f o r d h i g h e r
r e i n s u r a n c e p r e m i u m s , a n d l e s s w h e n i t s n e t i s p o o r a n d i t c a n
l ea s t a f f o r d t h e b u r d e n o f e v e n n o r m a l r e i n s u r a n c e p r e m i u m s .
T h i s is il l u s tr a t e d g r a p h i c a l l y i n F i g u r e 5 in t e r m s o f t h e e f f e c t
o n u n d e r w r i t in g p r o fi t. T h i s p r e m i u m s t r u c tu r e is m o r e e f f e c t i v e
i n r e d u c i n g t h e v o la t il i t y o f a c e d i n g c o m p a n y ' s n e t u n d e r w r i t i n g
r e s u l t th a n t h e c o n v e n t i o n a l s tr u c t u r e . B e c a u s e o f t h is s t a b il it y ,
i t m i g h t a p p e a l t o r e i n s u r a n c e b u y e r s w h o u s e e x c e s s o f l o ss
c o v e r a g e t o r e d u c e u n d e r w r i t i n g v o l a t i l i t y .
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714 P P L I C T I O N O F T H E O P T I O N M R K E T P R D I G M
FIGURE
ILLUSTRATION OF CYLINDER REINSURANCE PREMIUM
STRUCTURE
~ ' 3 . 5
C
0
= 3
E
E_ 2,5
. m
E
2
t
1 5
m
_ 1
1
t ~ 0 , 5
R
w
L
O . . . . . . . . . . . .
L L ;
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C o n v e n t i o n a l
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FIGURE 5
ILLUSTRATION OF CYLINDER REINSURANCE EFFECT ON
UNDERWRITING PROFIT
20
c -
o
, D
E
0 .
O ~
0
t -
1 5
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m
5 1 0 1 5
C a t a s t r o p h e L o s s ( m i l l i o n s )
C o n v e n t i o n a l
Cylinder
,
20
8/11/2019 97701
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APPLICATION OF THE OPTION MARKET PARADIGM 7 1 5
T h e f l i p s i d e o f t h i s i s t h a t t h e r e i n s u r e r ' s v o l a t i l i t y i s i n -
c r e a s e d . W h y w o u l d a r e i n s u r e r b e w i l l i n g t o o f f e r s u c h a s t r u c -
tu re , w h i c h r e d u c e s p r e m i u m s w h e n c l a im s a re h i g h e r ? T h e a n -
s w e r i s t h a t , i n t h e c o n t e x t o f a r e i n s u r e r ' s d i v e r s i f i e d p o r t f o l i o ,
t h e i n c r e m e n t a l v o l a t i l i t y w i l l b e s m a l l , w h i l e t h e e x t r a b e n e -
f it t o th e r e i n s u r e r ' s c u s t o m e r m a y w e l l s t r e n g t h e n t h e o v e r a ll
r e i n s u r a n c e re l a ti o n s h ip . T h e r e i n s u r a n c e m a r k e t h a s s o m e t i m e s
b e e n c r it ic i z e d f o r s e l li n g o f f t h e s h e l f p r o d u c t s t h a t i t w a n t s
t o s e l l , r a t h e r t h a n w h a t c e d i n g c o m p a n i e s a c t u a l l y w a n t t o b u y .
I n c la s se s o f r e in s u r a n c e w h e r e r e i n s u r e r s c a n s e l l a s m u c h o f f-
t h e - s h e l f p r o d u c t a s t h e y w a n t , t h e r e e x i st s l it tl e o r n o p r e s s u r e
f o r t h e m t o i n t r o d u c e i n n o v a t i v e s t r u c t u r e s l i k e t h e f o r e g o i n g
e x a m p l e . H o w e v e r , t o t h e e x t e n t s o m e r e i n s u r e r s w a n t t o p u r -
s u e a m o r e c u s t o m e r - f o c u s e d s tr a te g y o r s i m p l y f e el c o m p e t i t iv e
p r e s s u r e , p r o d u c t i n n o v a t i o n w i l l i n c r e a s i n g l y b e g i n t o e m e r g e .
I n d e e d , t h e a u t h o r i s a w a r e o f a t l e a s t o n e m a j o r r e i n s u r e r t h a t
h a s d e v e l o p e d a p r o d u c t t h a t h a s f e a t u r e s s i m i l a r t o t h is e x a m p l e .
T h e c y l i n d e r i s o n l y o n e e x a m p l e . T h e r e a r e u n d o u b t e d l y
m a n y o t h e r p r a c t i c a l i n s u r a n c e a n d r e i n s u r a n c e p r o d u c t s w a i t -
i n g t o b e d i s c o v e r e d b y e x p l o r i n g t h e d e r i v a t i v e s p r o d u c t p a r a -
d i g m .
4 P R I C I N G O P T I O N S W H E N F U T U R E P R I C E S A R E N O T
L O G N O R M A L
T h e B l a c k - S c h o l e s m o d e l r el ie s o n t he a s s u m p t i o n th a t m a r-
k e t p r i c e c h a n g e s o v e r a n y f i n i t e t i m e i n t e r v a l ( e x p r e s s e d b y t h e
r a t i o P n / P ~ _ I ) a r e l o g n o r m a l l y d i s t r i b u t e d . S i n c e t h e p r o d u c t o f
l o g n o r m a l v a r ia t e s is a l s o l o g n o r m a l , th is a s s u m p t i o n l e a d s t o t h e
c o n v e n i e n t c o n c l u s i o n t h a t f u t u r e m a r k e t p r i c e s a r e a l s o s o d i s -
t r i b u te d w i th p r e d i c t a b le t i m e - d e p e n d e n t p a r a m e t e r s . T h e b e a u t y
o f th is i s t h at th e s a m e f r a m e w o r k c a n b e u s e d t o d e t e r m i n e th e
p u r e p r e m i u m p r i c e fo r a o n e m o n t h , six m o n t h , o r o n e y e a r
o p t i o n , o r o n e f o r a n y o t h e r t i m e p e r i o d .
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716 APPLICATION OF THE OPTION MARKET PARADIGM
O t h e r s t o c h a s ti c p r i c e m o v e m e n t m o d e l s h a v e b e e n d e s c r i b e d
b y o t h e r s [ 2] . L i k e B l a c k - S c h o l e s , t h e y s u p p o r t t h e p r i c i n g o f
o p t i o n s o f a n y m a t u ri ty . H o w e v e r , f o r a ss e ts s u b j e c t to s u d d e n o r
e x t r e m e p r ic e m o v e m e n t s , o r w h i c h a r e h i g h l y i l l iq u i d , a re a li s t ic
s t o ch a s t i c p r i c e m o v e m e n t m o d e l m a y n o t e x is t. I n d e e d , s o m e
a n a l y s t s e . g ., P e t e r s [6 ] ) a r g u e t h a t
l l
s u c h m o d e l s a r e f l a w e d
s in c e t h ey r e ly o n t o o m a n y a s s u m p t i o n s th a t m a r k e t e x p e r i e n c e
h a s s h o w n t o b e u n r e a l i s t i c . ) T h i s d o e s n o t m e a n t h a t o p t i o n s
c a n n o t b e p r i c e d f o r s u c h a s se ts , b u t w e n e e d a d i f f e r e n t m o d e l . 7
T o p r i c e a c a l l o p t i o n e x e r c i s a b l e a t t i m e t , w e n e e d a n e s t i -
m a t e o f th e p r o b a b i l i ty d i s t ri b u t i o n o f th e u n d e r l y i n g a s s e t p r i c e
a t t i m e t a s v i e w e d f r o m t h e v a n t a g e p o i n t o f to d a y . I f i t is p o s -
s i b l e t o e s t i m a t e t h i s p r i c e d i s t r i b u t i o n , i t i s p o s s i b l e t o p r i c e
a n o p t i o n . P r i c i n g o p t i o n s o f d i f f e r e n t m a t u r i ti e s c o n s i s t e n t l y is
m o r e d i ff ic u l t w i t h o u t a p r i c e m o v e m e n t m o d e l , b e c a u s e i t r e-
q u i r e s s e p a r a t e e s t i m a t e s o f t h e p r i c e d i s t r i b u t i o n f o r e a c h e x e r -
c i s e d a t e ; b u t i t c a n b e d o n e .
F o r m u l a 1 .3 , w i t h o u t th e r e q u i r e m e n t th a t x b e l o g n o r m a l , c a n
b e u s e d t o p r i c e a n y o p t i o n i n t h i s w a y . O f c o u r s e , i f t h e a s s e t
p r i c e a t t i m e t is n o t lo g n o r m a l , t h e ca ll o p t i o n p u r e p r e m i u m
d e r i v e d u s i n g F o r m u l a 1 . 3 i s n o t e q u i v a l e n t t o B l a c k - S c h o l e s .
A s w i t h t h e e s t i m a t i o n o f lo s s d is t ri b u t io n s , d e t e r m i n a t i o n o f th e
p r i c e d i s t ri b u t i o n o f a n a s s e t m a y b e m a d e d i f f ic u l t b y s p a r s e n e s s
o f d a t a.
5. COMBIN ING THE OPTION AND ACTUAR IAL PARADIGMS
S e c t i o n 1 e s t a b l is h e d t h a t o p t i o n p r i c i n g is a n a l o g o u s t o e x -
c e s s o f l o ss i n s u r a n c e p r ic i n g . S e c t io n 3 s h o w e d h o w n e w i n-
s u r a n c e i n n o v a t i o n s c a n b e d e v e l o p e d u s i n g t h e o p t i o n m a r k e t
p r o d u c t p a r a d i g m . S e c t i o n 4 d i s c u s s ed h o w to p r i c e o p t io n s o u t -
7Even for the pricing of options on equities, for which Black-Scholes is widely used,
traders recognize its imperfections. Fischer Black even wrote a paper entitled "How to
Use the Holes in Black-Scholes," reprinted in [3]
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PPLIC TION OF THE OPTION M RKET P R DIGM 7 7
s i d e t h e B l a c k - S c h o l e s f r a m e w o r k . T h i s s e c t i o n w i l l i l l u s t r a t e
h o w t h e s y n t h e s is o f th e s e id e a s c a n l e a d to n e w p r o d u c t c o n c e p t s
o u t s i d e t h e c u r r e n t s c o p e o f a n y t h i n g w i d e l y o f f e r e d in e i t h e r th e
f i n a n c i a l o r i n s u r a n c e m a r k e t t o d a y .
Options on Reinsurance Premiums
C o n s i d e r t h e f o l lo w i n g . A r e i n s u r a n c e c o n t r a c t c a n b e t h o u g h t
o f a s a n a s s e t , n a m e l y t h e r i g h t t o r e c o v e r t h e m o n e t a r y v a l u e o f
q u a l i f y i n g i n s u r a n c e c l a i m s f r o m a r e i n s u r e r .
T h e p r i c e o f a r e in s u r a n c e c o n t r a c t is n o r m a l l y n e g o t ia t e d i n
t h e tw o o r t h re e m o n t h s p r i o r to t h e i n c e p t i o n o r a n n i v e r sa r y o f
t h e c o n tr a c t. S o m e t i m e s t h e r e is s i g n i f ic a n t u n c e r t a i n t y a b o u t t h e
f i n a l p r i c e u n t i l t h e c o m p l e t i o n o f t h e n e g o t i a t i o n s b e t w e e n t h e
c e d i n g c o m p a n y a n d r e i n s u r e r s . U n d e r c e r t a i n c i r c u m s t a n c e s , i t
m i g h t b e v a l u a b l e t o a c e d i n g c o m p a n y to f ix t h e c o s t o f i ts
r e i n s u r a n c e c o v e r a g e a t a n e a r l i e r d a t e , o r a t l e a s t e s t a b l i s h a n
u p p e r b o u n d . U s i n g t h e o p t i o n p r i c i n g p a r a d i g m , i t i s p o s s i b l e
t o e s t a b l i s h a w a y t o p r i c e s u c h a c a p .
S i n c e th e r e in s u r a n c e p r e m i u m , prem t f o r c o v e r a g e i n c e p t i n g
a t t im e t > 0 ( w h e r e t i m e 0 w o u l d b e t o d a y ) i s n o t k n o w n w i t h
c e r t a i n t y t o d a y , i t i s a r a n d o m v a r i a b l e . T h e p u r e p r e m i u m o f a
c a l l o p t i o n o n pre m t c a n t h e r e f o r e b e c a l c u l a t e d u s i n g F o r m u l a
1 .3 L e t u s u s e a n e x a m p l e t o i l lu s t r a t e t h is .
S u p p o s e t h e r a t e o n l i n e ( i . e . , t h e p r e m i u m d i v i d e d b y t h e
l i m i t ) o f a c a t a s t r o p h e r e i n s u r a n c e c o n t r a c t c u r r e n t l y i n f o r c e i s
2 0 % . I t is s ix m o n t h s i n t o t h e y e a r a n d t h e r e h a s b e e n a t o ta l
l o s s t o t h e l a y e r . T h e r e w a s a l s o a t o t a l l o s s t h r e e y e a r s a g o .
I n l ig h t o f th is e x p e r i e n c e , t he p r e m i u m f o r re n e w a l w i ll p r o b -
a b ly b e i n c re a s e d , r e f l e c ti n g a n u p w a r d r e a s s e s s m e n t b y r e i n -
s u re r s o f th e e x p o s u r e t o lo s s. T h e c e d i n g c o m p a n y w i ll a l so
p r o b a b l y b e w i ll i n g t o p a y a s o m e w h a t i n c r e a s e d r a te to b e g i n t o
" p a y b a c k " r e i n s u r e r s . H o w e v e r , t h e n e w r a t e w i l l n o t b e e s t a b -
l i s h e d u n t i l c l o s e r t o t h e r e n e w a l d a t e . I n t h e m e a n t i m e , f o r t h e
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7 8 P P L IC T I O N O F T H E O P T I O N M R K E T P R D I G M
next several months the premium the cedant faces for renewal is
unknown and uncertain.
Suppose the market rate on line for renewal, viewed from
the point six months prior to renewal, has a mean of 30 and
is lognormally distributed with parameters (-1.20,.125). This
implies that a rate increase of some size is nearly certain. It also
implies about a 10 chance of a price of 35 or greater and
about a 1 chance of a renewal price over 40 .
Formula 1.3 can be used to determine the pure premium of a
call option to buy the reinsurance at renewal at a 30 rate on line
(or any other price). If r = 5 and t = .5 (= 6 months), Formula
1.3 implies an option pure premium of (.975) (1.5 )= 1.46
rate on line, or 4.9 of the strike price of 30 rate on line.
If the ceding company were to buy this call option, it would
be certain that the total cost of renewal would be no more than
31.46 rate on line (30 + 1.46 ), and it might be less, since if
the reinsurance market quotes less than 30 , the cedant would
let the option expire unexercised.
Is this reinsurance premium call option a financial derivative
or a reinsurance premium? The answer is, it could be either. In
the way it was described above, it has the form of a derivatives
market instrument. But the concept can also easily be incorpo-
rated into a reinsurance contract. Let us assume the renewal date
is January 1. The option to buy the 12 months coverage incept-
ing next January 1 can be embedded in a reinsurance contract
with a premium payment warranty. If a certain required premium
payment is not received before inception, the contract does not
come into force.
In periods of significant reinsurance pricing uncertainty, pur-
chasing a premium option will reduce that uncertainty and fa-
cilitate a ceding company's reinsurance planning and budgeting
process. The specialist reinsurance market for this type of cov-
erage historically has been largely found in London.
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PPLIC TION OF THE OPTION M RKET P R DIGM
7 9
Rate uarantees
The option parad igm can also be used to think properly about
multi-year rate guarantees in the primary insurance market. In-
sureds sometimes seek to negotiate a fixed rate for several years
or a limit on future rate increases. In these cases the insured is
seeking, in effect, to secure a call option, or series of options, on
future rate levels.
Suppose the insured wants a three-year rate guarantee for cov-
erage that would normally be subject to an annual rate review.
The current rate (which is guaranteed) is denoted by Ro. The
market rates for coverage renewing one year and two years from
now, respectively, are r andom variables R l and R 2. If the dis-
tributions of R 1 and R 2 can be estimated, it is possible to price
the call options the insured is seeking. Then the insured can be
charged for the options. Alternatively, the insurer may decide
not to charge for the options, and merely use the options pricing
exercise to determine the effective rate decrease the three-year
guarantee represents.
If the options cannot be priced because the distributions o f R 1
and R 2 cannot be estimated with sufficient confidence , perhaps
it would be unwise for the insurer to agree to the rate guarantee
At the time this paper was being prepared, multi-year con-
tracts were beginning to appear in the reinsurance market as
well. Obviously the same thought process applies to both in-
surance and reinsurance.
6. CONCLUSION
This paper has sought to demonstrate the value of the options
market paradigm in thinking about and developing new insur-
ance solutions. As the relationship between Formulas 1.1 and
1.3 makes clear, the underlying mathematics of insurance and
the broader financial markets is the same. Apart from potential
regulatory constraints, there is no logical reason why we should
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72 P P L I C T I O N O F T H E O P T I O N M R K E T P R D I G M
n o t s e e a c o n v e r g e n c e o f i n s u r a n c e a n d o t h e r f in a n c i a l s e rv i c e s in
t h e c o m i n g y e a r s . T h i s i s e s p e c i a l l y l i k e l y a t t h e w h o l e s a l e l e v e l
e . g . , r e i n s u r a n c e ) , w h e r e t h e r e l a t i v e i m p o r t a n c e o f d i s t r i b u t i o n
s y s t e m s a n d c u s t o m e r i n te r fa c e re c e d e s a n d t h e i m p o r t a n c e o f
p u r e r is k c h a r a c t e r is t ic s i n c r e a s e s .
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APPLICATION OF THE OPTION MARKET PARADIGM 72
R E F E R E N C E S
[1 ] B lack , F ischer and M yron Scholes , Th e Pr ic ing o f Op t ions
and Corporate Liabi l i t ies , Journal of Political Economy 81
May-June 1973, p . 637.
[2] Gerber , Ha ns and El ias Shiu , M art ingale A ppro ach to Pr ic-
ing Perpe tua l Amer ican Opt ions ,
ASTIN Bulletin
24, 2,
November 1994, p . 195.
[3] Ko nishi , Atsuo and Ravi D at ta treya,
The Handbook of
Derivative Instruments
Chicago, Probus Publ ishing Co. ,
1991.
[4] H ogg, R obert V. and Stuart A. K lugm an, Loss Distributions
New York, John Wiley & Sons, 1984.
[5] Redhead, Keith,
Introduction to Financial Futures and Op-
tions
Cambr idge , England , Woodhead-Fau lkener L imi ted ,
1990, pp. 98-102, 161-162.
[6] Peters, Edgar,
Fractal Market Analysis
New York , John W i-
ley & Sons, 1994.
[7] Hull , John,
Options Futures and Other Derivatives
T hi rd
(Internat ional) Edi t ion, London, Prent ice Hal l In ternat ional ,
Inc., 1997.
[8] Kem p, M. H. D. , A ctuar ies and D erivat ives ,
British Actu-
arial Journal 3, Part I, 1997, p. 51.
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7 APPLICATION OF THE OPTION MARKETPARADIGM
APPENDIX A
DERIVATION OF THE BLAC K S CHOL ES OPTION PRICING
FORMULA FROM A LOGNORMAL ASSET PRICE ASSUMPTION
L e t
eo
l =
F =
t h e c u r r e n t m a r k e t p r i c e o f t h e s e cu r i ty u n d e r l y i n g
t h e o p t i o n ,
t i m e ( i n y e a r s ) t o o p t i o n e x p i r y ,
t h e r i s k - f r e e i n t e r e s t r a te u s e d f o r c o n t i n u o u s
c o m p o u n d i n g ( i . e ., t h e f o r c e o f i n t e r e s t ) ,
x -- a r a n d o m v a r i a b l e f o r t h e fu t u r e m a r k e t p r i c e o f t h e
s e c u r i t y u n d e r l y i n g t h e o p t i o n , a t t i m e t ( e x p i r y ) .
A s s u m e x i s l o g n o r m a l l y d i s tr i b u t e d w i th p a r a m e t e r s l n P0 +
r t -
0 . 5 ~ 2 t a n d a v e , a n d m e a n E ( x ) = Pt = e x p ( ln P 0
+ r t ) .
T h i s i m -
p l i e s P t = P 0 ' e f t .
X t = th e a c t u a l f u t u r e m a r k e t p r i c e o f t h e s e c u r i t y
u n d e r l y i n g t h e o p t i o n , a t e x p i r y .
c t S )
= t h e c u r r e n t p u r e p r e m i u m ( i . e . , i g n o r i n g t r a n s a c t i o n
c o s t s a n d r i s k ) f o r a n o p t i o n t o b u y t h e u n d e r l y i n g
s e c u r i t y a t a p r i c e o f S a t t i m e t. T h i s i s k n o w n a s a
c a l l o p t i o n w i t h a s t ri k e p r i c e o f S . B e c a u s e o f i t s
f e a t u r e o f e x e r c i s e a t o n l y o n e d a t e , i t i s k n o w n a s a
E u r o p e a n o p t i o n .
T h e c a l l o p t i o n
c t S )
w i l l h a v e n o i n t r i n s i c v a l u e a t e x p i r y i f
t h e m a r k e t p r i c e , X t , o f t h e s e c u r i t y is b e l o w t h e s t r i k e p r i c e , S . I n
t h a t c a s e , it i s c h e a p e r t o b u y t h e s e c u r i t y d i r e c t l y a t p r i c e
X t
t h a n
t o e x e r c i s e t o o p t i o n t o b u y a t e x p i r y p r i c e S . N o r a t i o n a l i n v e s t o r
w o u l d p a y a n o n - z e r o p r e m i u m f o r su c h a n o p t i o n ; h e n c e i ts n i l
v a l u e .
c t S )
w i l l h a v e i n t r i n s i c v a l u e o f X t - S a t e x p i r y i f t h e m a r k e t
p r i c e X~ e x c e e d s t h e s t r i k e p r i c e S . A n i n v e s t o r w o u l d b e i n d i f -
f e r e n t t o b u y i n g t h e s e c u r i t y d i r e c t l y a t p r i c e
X t
a n d b u y i n g t h e
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APPLICATION OF THE OPTION MARKET PARADIGM
7 3
c a l l o p t i o n
c t ( S )
a t a p r i c e o f
X t - S
f o r im m e d i a t e e x e rc i s e a t
p r i c e S .
T h e p u r e p r e m i u m o f
c t ( S )
i s t h e p r o b a b i l i t y w e i g h t e d m e a n
o f a l l p o s s i b l e i n t r i n s i c v a l u e s a t e x p i r y , d i s c o u n t e d t o r e f l e c t
p r e s e n t v a l u e . 8
I f t h e c o r r e c t i n t e r e s t ra t e f o r d i s c o u n t i n g i s th e r i s k - f r e e r a t e ,
t h e p u r e p r e m i u m i s e x p r e s s e d a s :
= e - r t . ~ S (x - - S ) . f ( x ) d x
(A . 1)
t ( S )
i s )
e - r t x f ( x ) d x - S ( x ) d x ( A . 2 )
/ : s
= e - f t . x . f ( x ) d x - x . f ( x ) d x
I n g e n e r a l , t h e f i r s t m o m e n t d i s t r i b u t i o n
A X f ( x )
d x
E x)
o f a l o g n o r m a l v a ri a te x w i th p a r a m e t e r s ( , a ) is a ls o l o g n o r m a l
w i t h p a r a m e t e r s (
+ 0 - 2 , o ) .
I n t h e p r e s e n t c a s e , x i s l o g n o r m a l ( l n P0 + r t - O . 5 0 . E t , 0 . x / t )
a n d its f ir st m o m e n t d i s tr i b u t io n h a s p a r a m e t e r s ( ln P0 + r t +
0 . 5 0 . 2 t , 0 . x / t ) . A c c o r d i n g l y , t h e s e c o n d t e r m w i t h i n t h e m a i n
SThe justification for use of the risk free rate is described in footnote 2 in the body of
the paper.
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7 4 P P L I C T I O N O F T H E O P T I O N M R K E T P R D I G M
b r a c k e t s o f F o r m u l a A . 3 c a n b e re s t a t e d a s f o l l o w s :
f o S x f ( x ) d x = E ( x ) ' N ( l n S - ( I n P + r t + O '5cr2t)c r y / ~
= p t . N ( l n S - ( l n P o + r t + O .5 ~ 2 t) )
a v q
w h e r e N is t h e c u m u l a t i v e d i s tr i b u ti o n f u n c t i o n o f t h e st a n d a r d
n o r m a l d i s t r i b u t i o n .
E v a l u a t i o n o f t h e o t h e r t e r m s o f F o r m u l a A . 3 i s s t ra i g h t fo r -
w a r d , a n d t h i s f o r m u l a c a n n o w b e r e w r i t t e n a s :
- S e - r t ' ( 1 - N ( ln s - ( ln P + r t- O '5 a 2 t) ) ~ r v ~
=P(1-N(lnS-lnP-(r+O'5cr2)t))~--v~
-Se- r t ' (1 -N( lnS- lnP- ( r -O '5cr2) t ) )~v /7
=Po(1-N(ln(S/P)--(r+O'5cr2)t))\ c ry /~
7 , ~ J / ;
A . 4 )
a n d , s i n c e 1 - N z ) = N - z ) ,
c t ( S ) = P N ( l n ( P / S ) + ( r + O '5cr2)tC r y ' 7
_ S e - r t . N ( l n ( e o / S ) + ( r - O .5 cr2 ))
7~ 7- . (A .5)
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PPLIC TION OF THE OPTION M RKET P R DIGM
7 5
L e t
a n d
d i = l n ( P o / S ) + ( r +
0.5cr2)t
~ v q '
d 2 = l n ( P o / S ) + ( r -
0 .5a2 ) t
T h e n F o r m u l a A . 5 c a n b e r e s t a t e d a s
c t ( S ) = P o N ( d l ) - S e - r t
N(d2)
T h i s i s t h e B l a c k - S c h o l e s o p t i o n p r i c i n g f o r m u l a .
(A .6 )
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7 6
PPLIC TION OF THE OPTION M RKET P R DIGM
PPENDIX B
V A L U A T I O N O F C A L L , P U T , A N D C Y L I N D E R S P R E A D S
C a l l S p r e a d s
T h e v a l u e o f a
c a l l s p r e a d c t ( T
, T 2 ) w i t h T2 > T a n d t i m e t t o
e x p i r y i s g i v e n b y
c t ( T l
,T2) =
c t ( T ~ - c t (T 2 )
= e rt
= e rt
( x - T I ) . f ( x ) d x - . ( x -
T 2) .
f ( x ) d x
[ f ~ ; Z ( x - T , ) f ( x ) d x + f T T ( X - T l ) f ( x ) d x
- 7 x - T 2 )- x ) d x I
N o t e t h e s i m i l a r i t y t o t h e f o r m u l a s u s e d t o w o r k w i t h e x c e s s
l a y e r s i n i n s u r a n c e a p p l i c a t i o n s .
I f t h e a c t u a l p r i c e o f t h e u n d e r l y i n g a s s et a t e x p i r y o f t h e
o p t i o n is X t, t h e v a l u e o f t h e l o n g c a ll s p r e a d p o s i t i o n a t e x p i r y
i s g i v e n b y
T 2 - T1 X _> T2;
X - T I T 2 > X > 5 ;
O , ~ > _ x , .
T h i s i s s h o w n g r a p h i c a l l y i n F i g u r e B - 1 .
= e - r t ( x - T 1 ) - f ( x ) d x + ( T 2 - T 1 ) . f ( x ) d x .
B . 1 )
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APPLICATION OF THE OPTION MARKET PARADIGM 727
F I G U R E B - 1
EXPIRY VALUE PROFILE: CALL OPTION SPREAD c t (T1 , T2)
E x p i r y
V a l u e
T,.-T~
T T 2
U n d e r l y i n g A s s e t P r ic e a t E x p i r y X t
P u t S p r e a d s
T h e v a l u e o f a p u t s p r e a d P t S 1 , S 2 )
w i t h S > S 2
a n d t i m e t t o
e x p i r y i s g i v e n b y
P t S I , 2) = P t S 1 ) - P t S 2 )
[ / o / o
e - r t (S 1 - x ) . f ( x ) d x - (S 2 - x ) . f ( x ) d x
[ / o Z ? ~ s ,
e - r t
(S 1
- x ) f ( x ) d x + - x ) . f ( x ) d x
- f o s ~ S 2
x ) f ( x ) d x ]
e / o ]
- x ) . f ( x ) d x ( S 1 - 2 ) . f ( x ) d x .
( B . 2 )
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7 8 APPLICATION OF THE OPTION MARKETPARADIGM
F I G U R E B 2
EXPIRY VALUE PROFILE: PUT OPTION SPREAD
Pt Si ,S2)
E x p i r y
V a l u e
S -S=
2 ST
U nd er l y i ng As s e t P r ic e a t Ex p i r y X
T h e v a l u e o f th e l o n g p u t s p r e a d p o s i t i o n a t e x p i r y i s g i v e n b y
O x > s1 ;
S 1 - x t S 1 > x t 2 > 3 2 ;
S 1 - 8 2 S 2 ~_~ X .
T h i s is s h o w n g r a p h i c a l l y i n F i g u r e B - 2 .
P u t - C a l l P a r i t y
T h e r e is an i m p o r t a n t r e l a t i o n s h i p b e t w e e n t h e v a l u e o f c a l ls
a n d p u t s k n o w n a s p u t - c a l l p a r it y . C o n s i d e r t w o p o r t f o l i o s . T h e
f ir s t c o n s i s t s o f a n a s s e t w i t h a v a l u e o f P0 a n d a r e la t e d p u t o p t i o n
w o r t h p t T l ) . T h e s e c o n d c o n s i s t s o f a T - b i l l v a l u e d a t T1 e - r t a n d
a c a l l o p t i o n o n t h e a s s e t i n t h e f ir s t p o r t f o l i o , v a l u e d a t c t ( T1).
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P P L IC T I O N O F T H E O P T I O N M R K E T P R D I G M
7 9
T h e s e t w o p o r t f o l i o s h a v e i d e n ti c a l e x p i r y v a l u e p r o f i l e s n a m e l y ,
m a x T 1 , P t) ), s o u n l e s s t h e r e a r e o b s t a c l e s t o a r b i t r a g e t r a d i n g ,
t h e y m u s t h a v e e q u a l m a r k e t v a l u e s f o r a n y T1 _> 0 :
P o + p t ( T I ) = T1 e - r t + c t (T 1 ) .
B . 3 )
W e c a n u s e p u t - c a l l p a r i t y t o d e r i v e t h e a n a l o g o u s r e l a t i o n s h i p
b e t w e e n p u t a n d c a l l s p r e a d s :
S i n c e
T1 e - n = Po + P t ( T I ) - G ( T I )
a n d
T 2 e - r t = e o + p t ( T 2 ) - - c t ( T 2 ) ,
t h e n
T 2 - T l ) e - r t = p t T 2 ) - c t T 2 ) - p t T l ) + c t T 1 )
= c t ~ , T : ) + p t T z , T ~ ) .
B . 3 a )
A b r i e f a n a ly s i s o f F o r m u l a B . 3 a s h o w s t h a t i t is c o n s i s t e n t w i t h
u s i n g t h e r i s k - f r e e r a t e f o r d i s c o u n t i n g E u r o p e a n o p t i o n p u r e
p r e m i u m s . I f w e r es ta te F o r m u l a B . 3 a i n t e r m s o f i n te g r al s a n d
t re a t t h e i n te r e s t ra te t o b e u s e d f o r d i s c o u n t i n g t h e r i g h t s i d e o f
t h e e q u a t i o n a s a n u n k n o w n , i , w e o b t a i n :
T - T ) e - r '
= e - i t ( f T T 2 ( x - - T 1 ) ' f ( x ) d x
/ ( / o
( T2 - T 1 . f ( x ) d x + ( T2 - x ) f ( x ) d x
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7 3 0 APPLICATION OF THE OPTION MARKET PARADIGM
T A B L E 3
E x p i r y L o n g C a l l S h o r t P u t C a s h S h o r t P u t +
Pr ice Va lue Va lue Va lue Cash Va lim
X , >_ T2 T2 - T t 0 T2 - T T2 - T
T2 > X , > T X , - T , - (T 2 - X ) T2 - T X t - T ,
T , > X , o - ( r 2 - T ) T 2 - T o
= e - i t f o T 2 x - T l ) f x ) d x .
J ?
( T 2 - T 1 ) -
f x ) d x +
( T 2 - x )
f x ) d x
= e - i t ( f o T 2 ( T 2 - T l ) f ( x ) d x + / : 7 ( T 2 - T l ) f ( x ) d x,
= e - i ( T - T11- f ( x ) d x
, 0
= e - i t T 2 _ T 1 ) ,
w h i c h i m p l i e s i = r.
F o r m u l a B . 3 a a l s o i m p l i e s a d e f i n i t i o n f o r a c a l l s p r e a d i n
t e r m s o f a p u t s p r e a d a n d T - b i ll s: 9
Ct (TI ,T2) = ( T 2 - T I ) e - rz - p t ( T z , T l ) . ( B . 3 b )
T h i s m e a n s t h at i t i s p o s s i b l e t o a c h i e v e a s y n t h e t i c c a ll s p r e a d
p o s i t i o n u s i n g p u t s p r e a d s a n d v i c e v er s a. I n p a r t i c u la r , F o r m u l a
B . 3 b s a y s t h a t s e l l i n g a p u t s p r e a d , p t ( T z , T i ) , a n d h o l d i n g t h e
p r e s e n t v a l u e o f T2 - T i n T - b i ll s is e q u i v a l e n t t o b u y i n g a c a l l
s p r e a d , q ( T l , T 2 ) . T o s e e t hi s, T a b l e 3 c o m p a r e s t h e e x p i r y v a l u e s
o f t h e s e t w o p o s i t i o n s .
9 N o t e t h a t f o r m u l a s B . 3 a a n d B . 3 b i m p l y a p u t - c a l l p a r i t y r e l a t i o n s h i p f o r s p r e a d s t h a t ,
u n l i k e t h e o r d i n a r y p u t - c a l l p a r i t y f o r m u l a , h a s n o r e f e r e n c e t o P 0
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APPLICATIONOF THE OPTION MAR KET PARADIGM 731
T A B L E 4
Expiry Lon g Put Shor t Cal l Cash Shor t Ca l l+
Price Value Value Value Cash Value
x , > r ~ o - ( r ~ - r , ) T ~ - r, 0
r2 > x, > r, r2 - x ,
- < x , - r 0 r 2 - r
T2 - X ,
T I > X ` T t - T ~ 0 T 2 - T I r 2 - r
A l t e r n a t i v e l y , s i n c e
P t T 2 , T 1 ) = T2 - T l ) e - r ` _ c , T I , T 2 ) ,
b u y i n g a p u t s p re a d
p t T 2 , T l )
i s e q u i v a l e n t t o s e l l i n g a c a l l s p r e a d
c t T 1 , T 2 )
a n d h o l d i n g t h e p r e s e n t v a l u e o f T2 - T l i n T - b i l l s, a s
s h o w n i n T a b l e 4 .
C y l i n d e r S p r e a d s
T h e b u l l c y l i n d e r s p r e a d , c y l t S I , S z ; T I , T 2 ) , c r e a t e d f r o m t h e
c a l l a n d p u t s p r e a d s d e f i n e d a b o v e , w h e r e T2 > T > S l > 2 , ha s
t h e f o l l o w i n g v a lu e :
cy l (S1 , S 2 ; T I , T 2 ) = c t T 1 T 2 ) - p t S l , 2 )
e - r t x - - T 1 ) . f x ) d x + T - T 1 ) . f x ) d x
_ t j S , S I _ x ) . f ( x ) d x
J S 2
- - f 0 S 2 s 1 - S 2 ) . f x ) d x
] .
B .4 )
T h e v a l u e o f c y lt (S 1 ,S a ;T 1 ,T 2 ) d e p e n d s o n t h e c h o i c e s o f S l , S 2,
T1 a n d T2 . T h e s e p a r a m e t e r s c a n b e c h o s e n t o c r e a t e a c y l i n d e r
s t r u c tu r e t h a t p r o d u c e s t h e d e s i r e d c y l i n d e r v a lu e a t t i m e t to
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7 3 2 A P P L IC A T IO N O F T H E O P T IO N M A R K E T P A R A D IG M
F I G U R E B - 3
EXPIRY VALUE PROFILE: BULL CY LINDER OP TION SPREAD
cy lt ( 1 , 2, TI ,T2)
E x p i r y
V a l u e
T z T 1
- S , - S 2 )
2 1 TI T2
Un de r l y ing Asse t P r i ce a t Exp i r y , X ,
e x p i r y . A d d i t i o n a l f l e x i b i l i t y c a n b e i n t r o d u c e d i n t h e c y l i n d e r
s t r u c t u r e b y r e l a x i n g t h e r e q u i r e m e n t t h a t t h e s a m e n u m b e r o f
c a l l a n d p u t sp r e a d s a r e u se d . I f Q i s d e f i n e d a s t h e r a t i o o f
t h e n u m b e r o f p u t s t o t h e n u m b e r o f c a l l s , t h e n t h e v a l u e o f
c y l t ( S 1 , S 2 ; T 1
,T2) i s g iven by
cy l( S l , 2; T], T2)
= e - r t [ / T ( 2 ( x - T 1 ) f ( x ) d x + [ ( T 2 - T l )
- Q . - x ) . f x ) d x
: 2
(S 1 - 2) . f ( x ) d x .
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A P P LI C A TI O N O F T H E O P T IO N M A R K E T P A R A D IG M 7 3 3
At expiry the value of the bul l cy l inder spread pos i t ion is g iven
b y
~ - ~
x t ___ ~
x t - ~ vz > x ___ T~ ;
O , I 1 > x , > s ~ ;
- Q . ( S 1 - x t ) ,
S 1 ~ S t ~
S 2 ;
- Q . S 1 - S 2 ,
$2 ~> X t.
Th is i s i l lus t ra ted for Q = 1 in F igu re B 3 .
