Jan 18, 1994 - possibilities (~double- decker'...). The inclusion of transaction costs leads ..... express the total variation of the bank's wealth. AW between t = 0.
J.
Phys.
I
FFance
4
(1994)
863-881
3UNE
1994,
PAGE
863
Classification
Physics
Abstracts
01.75
02.50
01.90
The Black-Scholes option pricing problem generalization and extensions for finance: of stochastic processes Jean-Philippe
Bouchaud
Didier
(~) and
Sornette
Condensé, CEA-Saclay~ 91191 Condensée, Université
de la
(Received
accepted
16
February 1994)
quantifier
le
coût
January1994,
L'aptitude
Résumé.
gestion
de
portefeuille
à
dans
l'Etat
Gif
Matière
2,
Cedex
un
a
mathematical large class
(~)
(~) Service de Physique de de Physique (~) Laboratoire 70, Parc Valrose, 06108 Nice 18
in
de
Cedex, France Antipolis, B-P.
Yvette
sur
Nice-Sophia
France
marché
du
aléatoire
risque
et
à
constitue
définir la
base
stratégie
une
de
la
théorie
optimale
de
moderne
de
finance. considérons d'abord le problème le plus simple de ce type, à savoir celui de Nous l'option d'achat ~européenne', qui a été résolu par Black et Scholes à l'aide du calcul stochastique d'lto appliqué aux marchés modélisés Log-Brownien. Nous présentons un par un processus formalisme simple et puissant qui permet de généraliser l'analyse à une grande classe de processus stochastiques, tels que les processus ARCH, de Lévy et ceux à sauts. Nous étudions également le des Gaussiens corrélés, dont nous qu'ils donnent bonne description montrons cas processus une (MATIF, CAC40, FTSEIOO). Notre de trois indices boursiers résultat principal consiste en (fonctionnelle) d'une1nini1nisation l'introduction du concept de stratégie optimale dans le sens du portefeuille d'actions. annulé du risque en fonction Si le risque peut être pour les processus 'quasi-Gaussien' non-corrélés, dont le modèle de Black et Scholes est un exemple, cela n'est plus d'options "corrigés". vrai dans le cas général, le risque résiduel des coûts permettant de proposer du marché telles les de En présence de très grandes fluctuations décrites que processus par critères fixer rationnellement le prix des options nécessaires Lévy, de et sont nouveaux pour méthode à d'autres que'asiatiques', discutés. Nous appliquons types d'options, telles sont notre ~américaines', et à de nouvelles options que nous introduisons les 'options à deux étages'... comme la
L'inclusion
temps
des
frais
caractéristique
Abstract.
The
de de
ability
dans
transaction
le
formalisme
conduit
à
l'introduction
naturelle
d'un
transaction.
to
puce
nsks
and
devise
optimal
investment
strategies
m
the
presence
theory. We first consider an the simplest such problem of a so-called "European call option" initially solved by Black and markets modelled by a log-Brownian stochastic Scholes stochastic calculus for process. using Ito formalism allows generalize the analysis to a A simple and powerful presented which to is us such as ARCH, jump or Lévy We also address large class of stochastic processes. processes, Gaussian description of three the correlated which is shown of to be a good processes, case dilferent market indices (MATIF, CAC40, FTSEIOO). Our result is the introduction of the main of
uncertain
"random"
market
is the
cornerstone
of
modem
finance
JOURNAL
864
concept the
to
of an optimal portfolio. If the stochastic
Gaussian' for
general
more
concept
The
value
for of
inclusion
The
In
N°6
the
for and
presence
transaction
leads
costs
to
this
is
risk is
obtained
of
large
deviations
the
are
of
a
respect
'quasithe
case
suggests
such
apply
also
natural
the
Lévy
in
as
possibilities
new
appearance
and We
discussed.
discuss
with
longer
no
residual very
risk
uncorrelated
continuous
model),
Scholes
of the
of the
minimization
particular
fixing of the option prices 'Asian', 'American', and
national
types of options,
other
decker'...). trading
criteria
new
to
Black
option prices.
I
(functional)
vanish
to
(including processes.
risk-corrected
of
processes,
method
processes
of
sense
made
be
may
stochastic
the
in
strategy risk
PHYSIQUE
DE
ouf
(~double-
charactenstic
scale.
time
Introduction.
l.
explaining why physicists might be interested in economy and fithe fact that market exchange is a typical example of complex reason nance. fluctuations the apparently random of market from where result system, puces many causes, and such as non-linear of traders with highly speculators inter-dependent behaviors, response embedded unpredictable way in a somewhat in a time evolving environment. This diiliculty behavior of market fact fundamental predicting the prices has been argued be in to m a propmarkets: obvious predictable opportunity should rapidly be erased by erty of "eilicient" any itself [1]. Present of the market mathematics theory of finance are the and economic response developed from the perspective of stochastic models in which agents can revise their decisions continuously m time as a response to a variety of personal and externat stimuli. It is the that provides the major diiliculty in the complexity of the agent expectations and interactions study of finance. The development of new concepts and tools, in the theory of chaos, complexity and self-organizing (non linear) systems in the physics community in the past decades [2], make thus the modelling of market exchange particularly enticing, especially for physicists, exemplified by quite a number of attempts: see e-g- [3-9] for recent discussions this within as Another, more anecdotic due the is that job and physics crisis, context. to reason, more more students will probably end up working in finance. It is possible (although of course not certain) that this population will bring new concepts and methods, leading to a rapid evolution of the There
various
are
reasons
first
A
lies in
field. There
is
of
ory
also
historical
a
Brownian
Einstein's
motion
classic
1905
laid
the
finance: such
as
nally') [13-16] are
now
In
the
a
upon
dormant
stock
particular,
market
Bachelier
the
idea
that
until
the
sixties,
these
when
Bachelier,
fluctuations
proposed
fluctuations many
a
follow
important
formula a
discovered
[loi,
rive
for
the
years
the
thebefore
price of
an
Brownian
process.
This
and
methods
were
ideas
activity paved the way to the seminal work of Black and Scholes [11] mathematical option pricing theory, which stands as a landmark in the development of shown to be directly useful for an every day financial stochastic calculus activity was option pricing, which was, before Black and Scholes, only empincally land not 'ratiofixed. A considerable development of the field has then followed, both at fundamental commercial levels, and softwares based on the Black-Scholes approach and numerous
developed. on
somewhat
In
mathematician,
French with
connection
paper.
option (see below), based work
the
reason:
in
This
renewed
available.
nutshell,
followmg:
the
simplest option
suppose
that
an
problem (the so called 'European certain to buy a given share, a wants
pncing
operator
call time
options') t
=
T
is
from
VALUATION
OPTION
N°6
(t
now
0),
=
at
865
'striking' price x~. If the share value at t option. His gain, when reselling immediately
fixed
a
PROBLEM
T, x(T),
=
exceeds
xc,
the
price x(T), x(T) dilference x(T) the On if the buy the is thus the does contrary, operator not < xc. xc share. Symmetrically, a European put option gives the owner of a stock the right to self his These possibilities given to the operator by -sayshares at time T at a preassigned price x~. the "bank" options and have obviously themselves a price. What is this price, and what are trading strategy should be followed by the bank between now and T, depending on what the T? share value x(t) actually does between t 0 and t 'exercises'
operator
his
=
the
at
current
=
the option problem can be the elementary building block of the In a sense, considered as general problem of the evaluation of risks associated with market exchanges and more generally with activity. Indeed, an option on a stock can be viewed as an insurance human premium against the risk created by the uncertainty of share prices. When you insure your car against collisions, you are buying from the a'put' option, 1-e- an option to self your insurance company at
car
given price.
a
premium
of it
remains
market
will
almost
no
the
with
primary
The
portfolios. their profits.
their
limits
on
pocket an extra and the speculator. proceed along the Black
assuming
that
Furthermore,
give
to
For
a
Sellers
have
never
investors
and
the
pay
cost, buyers of options can of options who expect little
right the
you
control
some
(you
accident
an
have
you
over
limit
insured
amount.
changes
how
in
prices
market
in
the
placing
with
fosses
change
short, options satisfy the needs of both pricing of much more complex financial
pay the leave what
to
prudent person essentially securities
the
fines.
same
gives
model
share
the
In
premmm.
Scholes'
and
if you
destroyed, to buy it
if your car is which is obliged
insurance
can
worthless
But
of options is thus
function
affect
option will be
That
nothing).
collect
and
both
to
answer
an
follows
value
the
above
log-Brownian process, buyer's portfolio, in
a
questions (price and strategy): they construct strategy by a such a way that, for the bank,
exactly duplicate the the whole Tisk fTee (the precise is will be discussed below). statement process meamng of this This translated construction is stochastic calculus [17] dilferential using lto's into a partial equation, the solution of which containing both the option price and the bank trading strategy (see Appendix A). bank
the
which
The
of
aim
(at
least
formulae
can
way
coTTelated those
this in
readily
which
could
risk
is
even
allows
us
to
The
interest
obtained
for
processes. risk in
the
allow zero
reformulate
to
is
eyes).
our
be
minimizing
residual
paper
Brownian
product, which
can
We to
one
Scholes
We
discuss
particular taken
into
We
also
dilferent
several
'American'), how
or
characteristic
a
of
extensions
include
to
option
prices.
cost
trading
time
find
We
that
stochastic
Iog-Browman model in our general extremely strongly fluctuating cases,
this
processes,
formalism. new
criteria
introduced.
approach,
our
the
flexible
that
(continuous 'quasi-Gaussian')
Lévy processes are also considered. For these for rationally fixing the option prices must be
l'Asian',
with
and
transparent
more
'generalized' Black-Scholes stochastic induding general processes, Black-Scholes generalized strategies as the value of the residual nsk as a byis
Tisk-coTTected
propose
and
of
obtain
sense
particular
Black
include
dass
to
propose
functional
a
for
our
a
m
formulation
large
a
problem
the
of
of
scale
in
particular
transactions
naturally
to
other
types of options
(market friction). appears
when
this
We
show
'friction'
in is
account.
give
evidence,
CAC40, FTSEIOO), that which justifies the pTocess,
based
the
on
interest
of
(MATIF, analysis of three dilferent indices BTownian quite well described by a coTTelated are Black-Scholes formula. generalization of the
statistical
a
fluctuations our
JOURNAL
866
Solution
2.
of
ization
option
trie
of
Black
and
problem
PHYSIQUE
DE
using
risk
a
I
N°6
procedure
minimisation
general-
:
Scholes.
appendix 1, we give a brief summary of the results and method used by Black and Scholes, subsequent workers in this field. This may be useful in order to connect our approach most mathematical with those developed in the As already stated, their approach finance literature. calculus heavily stochastic straightforward relies Ito the understand to [17] and it is not on
In
and
underlying principles at the origin of their completely dilferent point of view and a basic knowledge in probability theory. shall
We
x(t)
colt
(x(t))t=0,T
as
a
t, knowing
at
the
express the account
a
method
that
by
a
it
was
y
at
of the
variation
which
time t, and
at
probability
certain
t'
t,
x~, T) for a call option on a share of initial price xo starting at t 0, of striking price xc and maturing at time T; 2) the potential loss equal to -(x(T) x~) if x(T) > x~ (stemming from the fact that T) and zero otherwise; 3) the gain or the bank must, in this case, produce the share at t of stock incurred the time period extending from t loss due to the variation during 0 puces This last depends on the number of shares ç$(x,t) at time t (at which the T. term to t The fact that this third share price is x) held by the bank. be taken into term account must considering the simply be illustrated by of the following that the scenario price suppose con and t
taking
T
=
into
existence
of the
=
=
=
=
with certainty between t T. Then, it is dear 0 and t advantage m buying a share at t 0 (ç$(xo>t 0) 1) and 1), at which time, it will give it to the buyer until t T (ç$(x,0 < t < T) This simple example suggests that, generally for an arbitrary realization more xc. of shares prior to the exercice time can be price x(t), holding a certain amount share
increase
to
was
would
then
=
=
have
=
=
=
=
=
that
bank
the
holding it for the price
in
of the
share
for
favorable
ç$(x, t) has the meamng of the number of shares per option, taken in the limit where large number of options and shares are traded simultaneously. It is of course in this limit a trading makes Then, if the bank has, at time t, ç$(x,t) shares, the that continuous sense. bank,
the
of its
variation
true
fluctuations
of the
of
conversion
butions
is
total
for
with
9(u) last
and t
=
=
term
T.
u
for
u
m
the
e.:
i
> 0
=
and
r-h-s-
x~,
e
process
(x(t))
(Ù(u)
not
taking
ojxjT)
otherwise
equation
but
reverse,
bank
x~) is
and
and
t + dt is
that
the
term
+
equal
il) quantifies
the
a
into a
only ~~~~'~~
t
between
dt
the
T)
assets)
~~. (Note ç$(x, t)
of the
of the
zero
of
or
wealth
cjxo,
other
+
t
assets
of
realisation
àw
The
$
other
into
variation
given
a
(shares
W
price,
share
shares
the
Hence
wealth
real
change
account
given
the
the
to
describes
x
dt
of
due
wealth).
three
above
contri-
strategy ç$(x, t):
/~ ijx, t))dt to
u
elfect
times
of the
the
ii) Heaviside
trading
function).
between
t
=
0
OPTION
N°6
INDEPENDENT ~~ local slopes dt
2.1
the
shall
INCREMENTS
x(t).
of
the
t)(~~ ),
~~ because
[18], (~~
case
(AW)
C(xo>
xc,
=
T)
/
cc
times.
In
ail
expression
an
the
use
of the thus
and
for
suppose
us
however
postenor,
gives the
0
and
realisations
0, reads C(xo> xc,T)
=
dt
we
always
is
dt
shall
dilferent
over
dilferent
problem,
discretized
average
condition
for
867
Let
VALUE.
independent
convenience,
For
PROBLEM
SHARE
THE
time
a
dt
game'
'fair
unbiased
the
in
ç$(x,
e
to
xi).
~b(xj)(~z+i means Then, denoting by (..) fact
dt The
OF
statistically
are
always implicitly refer
(ç$(x, t)~~)
VALUATION
while
a
follows,
that
like ç$(x,
continuum
process
in
dt notation.
value
the
European option pricing formula, =
(9(x(T)
xc
has
one
to
which
)),1-e-
dx'(x'- xc)P(x', T[xo> 0)
=
we
~~
t)
(x(t)),
uncorrelated
that
(2)
~~
Scholes pricing formula (see Eq.(A2) in Appendix always explicited for the Log-Brownian is 1) case, for which P(x,T[xo> 0) is given by (Al of Appendix that this formula (2) holds genIt is important equation 1 [11, 13]. to note erally, without any assumption on the specific form of P(x, T[xo> 0) (although the assumption below, Eq. (12)). The interpretation of equation of independent increments is important see (2) is straightforward: this formula simply says that the theoretical option puce C is such risk-free profit should that, on average, the total expected gain of both parties is zero: not The expected gain of the operator is indeed given by the right hand side of equation (2), exist. x'- xc if x' > x~ and zero otherwise, which be weighted by the since its gain is either must corresponding probability. Note that the option price is independent of the portfolio strategy This is in of the bank, 1-e- of the specific choice of the number çi(~, t) of shares per option. Scholes and subsequent authors, for whom the with the point of view of Black and contrast option price is deeply linked with the underlying strategy. This
recovers
the
well-known
Black
and
which
In
Gaussian
the
P(x,T[xo>0)
case,
reads:
ÀÎ
~~~~'~~~°'~~ where
[28].
"volatility"
D is the
Introducing
~~~°'~~'~~
underlying
of the
complementary
the
~~~~~~
to
the
means
that,
greater
than
other
limit.
which
as
the
~~
~~
/%
=
diffusion
erfc(u)
/
coe~fcient ~
e
~~~
é
du
of the
price)
stock
exp(-u~),
finds
one
~~~
jj~~~j/Î~
followmg asymptotic -
+cc)
and
much
function
error
~"~~~ ~~
C(Xc C(Xc
Ii.e,
stock
2à~
u
CjXc) Equation (4) Ieads
~~ ~~~
-
-cc)
t
m
j4)
XcerfcjXc)j
behaviour:
C(0)
=
(2r)~~/~,
(2r)~~/~X/~ exp(-X))
-VSX~
+
(8r)~~/~X/~ exp(-X))
expected, C(xo>~c,T) is of order /% for ~c m xo> very initial price xo> and equal to the quasi certain gain of
small xo
xc
if xc m
is
the
JOURNAL
868
PHYSIQUE
DE
I
N°6
optimal portfolio strategy of the bank, 1-e- the best function ç$* ix, t) between 0 and T? A first plausible per option as a function of time profit, i-e- find ç$(x,t) such that AW given by equation il) be idea would be to maximize However, only the third term m the r-h-s- of equation (j) depends on ç$(~, t), which maximum. What
should
giving the
Iinear
be
cannot
The
equivalent
is
correlations
to
that
between
the
of
ç$(x,
t)~~dt.
knowing
without
and
thus
~~ increments
in
for
strategy
the
cannot
the
bank is to attempt to mmimize measured by the fluctuations of AW
its
t).
We
gain (AW) is zero, the risk R is (AW~), and thus exphcitly depends on R i-estrategy ç$*(~, t) so that the risk is functionally
strategy ç$(x,
the
=
times,
dilferent
at
is
term
specific realization of ~(t) ion average, be performed either on the average).
advance
maximization
This
dt dt
raturai
next
AW
of
absence
in
maximized
vamshes
term
of
maximization
#(~, t) and,
in
this
the
of shares
the
that
means
be
number
risk.
around
Since its
its
average
(zero)
average,
determine
optimal
the
mimmized:
à«
ôij~,t)"='" independent
For ~~
dt
~~
[t
dt
Ît,
(but
necessarily Gaussian)
not
t'),
D(x)à(t
"
R
xi where
Rc
is
the
risk
"bare"
Rc
Î
+
Î
-2
T
+co CXJ
dt
T
such
increments
Îc
dz'
~~ )~~ which
D(X)P(I,
dX +co
co
dt
that
Îm
dz
t(X0> 0)çi~(X,
ç$(x,t)(x'
T)P(x, t[xo 0)P(x', T[x, t)
would
prevail
in
the
absence
)j
zc)~P(x, T[xo>
t)
xc)
t)_~~,
/~ dx(z
=
i~~
finds:
one
Rc
"
°
=
of
[C(xo
(6)
trading (ç$(x, t) xc,
0):
+
ii)
T)]~
~~
The
term
(~~)~~ dt
condition
ix, t)
mcrement
(~~ dt
T) is
t~_~~,
and
final
a
the
Equation (6) contains a positive showing that an optimal solution one). The general solution reads:
4*lI, t)
It is
average term
exists,
non-vanishing, contrary
interest
leads
by
rate
proportional and
conditioned
increment
instantaneous
mean
ix', T).
condition
(neglecting
0
=
the
'
'
to
to
ç$ a
~
to the
and
reduced
a
negative nsk
unconditionned
change
suitable
a
of
term
(compared
frame
[18])
linear
in
the
to
/~ dz'l)1(~,t)-(~,,T) ~~((~~ PII', TII, t)
=
initial
the
to
ç$,
bare
18)
and
R*
=
/~ dt /~~
Rc
o
which
time,
are
valid
processes.
for
an
The
arbitraTy process
(9)
-cc
uncoTTelated can
dzD(z)P(z, t[0, 0)ç$*~ ix, t)
furthermore
stochastic
be
including 'jump', time-dependent, with
pTocess,
explicitly
or
a
discrete-
variance
OPTION
N°6
Dix, t)
which
which
is
are
Regressive
(8-9)
D
will be
cc
=
can
independent
identical
random
delays
t, and
T
variables
=
~
Brownian ~~
and
g(x)q(t),
for ail
that
and
g(x)
~ dt
=
Indeed, is
This
the
zero
~~
mean
and
then
reads:
that
the
increments
given by
variance
(The
D
dt case
z)P(x',T[x,t)
z~)(z'
are
where
(8')
variables,
=
expression (3) will exactly hold for ail time
then
(8)
equation
/°'dxtjxt
into
x~)
:
~~Gi(>
~"~>~~
j8tt)
an
were
z(t)
arbitrary aiming
of the
and
more
function
at
find
to
was
prices.
stock
and q
generally
We
Gaussian
a
a
strategy
do
recover
approach
risk
R~
=
for
D
that
the
risk
result
for
the
exactly Brownian,
be
Namely,
models.
/~ /~Î~ dzPG(z,t[0, 0)ç$à~(z, t)
we
find
Gaussian
/Î
not
processes,
da~PGia~> tia~o>
vanish
in
(9')
dt
PG(xi> T[xo>0)à(xi
will
such
this
'quasi-Gaussian'
for
due
to
the
0) ~~~~~ii~"~
x2)
following identity: ~~
~~~~~ii~"~
general.
~~ =
PG(xi, T[xo>0)PG(x2> T[xo>0)
show that the integral usmg the identity il 0), one can exactly equal to R~, thus leading to a zeTo Tesidual Tisk equality (10) will however net hold for an arbitrary
residual
noise.
reads:
risk
exactly
vanishes
is
Scholes
R*
(9')
Auto-
~~
transform
may
models
residual
the
which
[16]),
for
Suppose first
cases.
/~ dz'(z'
Gaussian
realizations
Log-Brownian
the
e-g-
is
where
Black
What zero,
(see, stands
formalism
log-Brownian
=
processes
'ARCH'
exactly the result obtained by Black and Scholes il1, 13] using a completely (see Appendix 1), when replacing PG(x',T[x,t) in equation (8") by the expression (Al). In fact, one can show that equation (8") holds both for the log-Brownian models and more generally for 'quasi-Gaussian' models such that
expression
dilferent
dt
of
~/
ii ix, t) This
869
Dit) [19]. I'ARCH'
studied
variance
below). Equation (8)
are
one
several
in
~~ dt
much
the
t-dependent
a
simplified
be
addressed
furthermore
for
as
PROBLEM
Heteroscedasticity.)
ç$*(z,t)
If
with
processes
Conditional
Formulae
of time,
function
a
Gaussian
VALUATION
This
is
the
main
in the
(10)
right hand side of equation 'quasi-Gaussian models'.
for all
stochastic
dilference
process
between
[23] and thus the present
subsequent workers 1) we find that a vanishing residual risk be achieved in the general case; 2) however, this does not imply that an cannot optimal strategy does not exist. We have indeed found an optimal ç$* ix, t) which the minimize risk, given by expression (9) and which is a simple generalization of Black and Scholes result. relevant to various These findings are In particular: situations. concrete ("leptokurtosis") are expected when the a) strong Gaussian behaviour deviations from a formula (9) allows one to the residual time delay T estimate t is not large. In this case, our and the accordingly. risk option puce correct and
that
of Black
and
Scholes
and
JOURNAL
870
b)
importantly, the Rather, trading
More
with
time.
using the
estimated
integral
and
finite
its
be
The
r.
risk
R*
that
/
P
of
below:
us
be
con
hold value that More
end
if
we
this
made
replaces
one
z(T): ~§(z(T))
for is
C~ (ID
=
a
into
zo
9(z(T)
Gaussian
T)
process,
dt
o
precisely,
dt for
+m
C~ (ID
T)
xc,
Îm
=
between
continuous
a
Ill)
of "
(z(T)).
determine
to
option.
of the
realization
Ill)
Formula
consistently
self
Note
be
will
of
optimal
an
remarkable
of the This
result
'Black-Scholes'
the
risk
would
still
that
result
z~) by an arbitrary function ~§(z(T)) of the final it is always possible to choose C~ and tp(z, t) so
/~ tp(z, t) ~~
+
continuously separated by fact, it can be
account.
by stressing the implications 'quasi-Brownian' processes.
a
z~,
z~
can
taken
is
occurrences, case; in
dilference
the
when
=
that
in
trade
to
O(r~)
+
used
of
zero
probability
be
prevents
number
total
the
is
how it
function
the
(z(t))
if
finite
will
o-S, 1-e-
=
show
friction
section
vanish
to
P
shall
market
when
r
for
maximum
is
importance Let
dx'PG lx', T[xo>0)
~~
R*
value
cc
a
to
)P(1- P)
=
N°6
(market friction)
costs
restricted
R*
where
I
again be non Euler-McLaurin formula, which gives approximant. We find: sum
interval
time
non-zero
a
transaction
will
PHYSIQUE
DE
realisation
any
of the
random
(z(t)).
series
dz'~§ ix') PG lx', T[zo 0)
and
wlz, t)
=
/~~° dz'~llz') ~~~~~)~~'~~ -cx~
These
allow
results
variable
z(T)
as
CORRELATED
2.2
Gaussian
process,
a
one
to
the
express
path-independent GAUSSIAN
such
that
arbitrary (non-linear)
sum
PROCESS.
dilference
the
Vit
over
Let
variance =
us
z(t)
t'), where Vi. is a given D[r[; the generalization to a power VIT) Mandelbrot and Van Ness (see [24] b) under 'persistent' or 'anti-persistent' way to model and
past
values
of
now
assume
z(t')
is
function.
In
a
the
law
behaviour
the
name
function
~§(z(T))
of the
random
z(t). that
we
Gaussian case
of'fractional
a
general
variable
of
VIT)
have
uncorrelated
Dr~~ =
with
was
Browman
correlated zero
mean
increments,
proposed by motion', as a
evolution of share prices. This is outside case validity of the Black and Scholes formalism, whereas it tums out to be exactly approach. The interest of this model is also motivated soluble for arbitrary Vi.) within our statistical analysis of some charts, which we now descnbe. We first determined the by the z(to)]~)to, where (...)to denotes a sliding function VIT) by computing the average ([z(to + r) the choice of the 'initial' time to and z(t) is the daily (closing) value of the London average over CAC-40 and the French MATIF, in the period 1987-1992, coTTected by FTSE-100, the Paris behaviour of the function VIT) is reproduced in figures la, b. Quite The the aueTage tTend. VIT) with until first grows linearly standard uncorrelated Gaussian interestingly, process r as a departures found be observed. reaches r~, beyond which strong 100, 350, 250 to are t rc is r days for the FTSE, CAC-40, and MATIF respectively (250 days correspond to a year in real around time). Then, VIT) saturates and, for the FTSE and MATIF, decTeases to a minimum IA cycle of period statistical pseudo-cycle days, corresponding 500 700 true to a two-year r 0). We have checked that a certain degree of stationarity correspond to V(T*) T* would valid on the restricted periods 87-90, 89-92. similar conclusions holds: are the
domain
of
=
=
OPTION
N°6
VALUATION
PROBLEM
871
FTSE-100 CRC-40 Fit / 500 /
'~
400
,/
'
~
'
,/
_
~
'~
300
>
~
, ,
200
ioo
0 0
200
400
600
800
1000
~
a)
~MATIF
'
.-"
,Î.~'
~t
-,---Fit
Z
.,-"
$
;"
5
o o
soc
iooo
isoo
~
b)
fias a function of the time delay r a) Behaviour of the root mean fluctuations Fig. l. square (in days) for the FTSE and the CAC-40. Note that the initial part of the curve is very well fitted by V(r) cc r, which corresponds to an walk, before a uncorrelated random 'Saturation' rather regime is des Agents sharply reached. b) Same as figure la, but for the MATIF. The CAC-40 (CAO: compagnie FTSE-100) is a French (resp. British) market index calculated from a weighted de Change) (resp. (resp. 100) MATIF stands of stock of the of activity of the 40 country. sectors puces main average and options exchange). International de France (French futures for Marché à Terme
Next, the the 50
histogram and
800
approximate of the
days.
Gaussian
rescaled
As
shown
character
excursion in
of
x(to -x(to +r)
~~~°~ y
fi
=
figure 2,
~~~° ~
ail
the
~~,
histograms
was
for
established dilferent
are
(within
by constructing
values
The
of ail
these
between
r
errors)
statistical
is reproduced in figure 3; we average curves fluctuations long as the large the probability described is quite well not too are for Gaussian, although a slightly 'fatter' tait larger deviations. Thus, appears model be of the for correlated good description charts small to not too seems a
superimposed.
of
find
by
that a
as
simple
our
Gaussian
time
delays la
JOURNAL
872
PHYSIQUE
DE
N°6
I
0.4
P
÷
0 3
, i
'
i
/
:
0 2
=1 1'
, i
,_
Î..
.~'. ~
o
0
8
4
12
16
20
4x/QJ(~) Fig. T
=
Histograms 800. 50,100,150,.
2.
shown
is
for
companson
to
now
calculation
As for the fact
that
given by
Although in
thicker
time
to
variable these
noisy,
fine
a
y
(x(to)
=
dilferent
x(to
curves
r)(il@,
+
for
dilferent
superimpose
satisfactonly.
larger
the
values
of
Gaussian
A
(see Fig 3).
fourth
rescaled
moment
than
Gaussian
value 3, is
delays).
European call option problem for such a process, one can perform the above, although the presence of correlations slightly complicates the matter. feature uncorrelated from the case, we start comes agam from equation il ). The novel
Tuming same
small
for
rescaled
corresponding
large kurtosis, observed
of the
one
can
the
as
show
that,
even
m
the
)(2rV(t))~~/~ /~ dt'~~~l'~ dt' o+
case
~~'
(~~
=
dt
dz
0, (ç$(x,
~~~~'~~ ôx
exp
-[
t)~~ dt
is
~~
2V(t) ].
no
vanishing.
more
Hence, holding
a
certain
_~
~
portfolio of
correlated
shares
leads
to a
non-zeTo
average
gain (or loss) Çj
dt(ç$(x, t)
e
It is
) ). t
stock prices is thus given by Cc C Çj, where C is price C~ of the option on the correlated still given by equation (2). The price of the option thus depends on the optimal strategy, to be After through the of the risk. manipulations of Gaussian integrals, determined minimization ix, t), determming the optimal precisely integral its obtain equation strategy or more we an ç$* The
=
OPTION
N°6
VALUATION
Averaged
PROBLEM
873
histogram
rescaled
6
A
4
8
$ ~ ~
~
o
'N
~
z
,,
',~~,
,----..___
,~-
',
~FT5E ---".CAC
4
"
, ,
6 0
100
50
150
200
300
250
350
Fig.
Average
3.
MATIF
which
of
should
be
straight
a
plotted
have
We
line
#* là, t)
transform
Fourier
histograms
the
results).
similar
gives
here
3
where
this
(and once
of
kernel
the
RARE
Îm dxe~~~#* ix, t).
the
so
and
K
equation
This
would
crucial
AND
LÉVY
strong
that
Such
has
F(À, t)
=
FTSE as
and
(the
CAC
function
a
of
following
the
y~,
form:
H(À, t)C~
Appendix
in
transform a
back
procedure con for long
important
rather
be
given
are
Fourier
to
self-consistently. and
becomes
are
F, H
functions
invert
to
determined,
EVENTS
fluctuations
and
determined
thus C~) is
V(r)
correlations
2,3
K
the average
(12)
-cc
requires
equation
of this
cc
=
/~ dt' /~ dÀ'#*(À', t')K(À, t, À', t') o
for
both
T,
over
logarithm
the
distributions.
Gaussian
for
figure
in
shown
400 y..z
16
that solving note #* ix, t), with Çj. implemented numerically options, where the elfect
2.
Let
us
obtain
to
be term
(see Figs. la, b). There
PROCESSES.
the
of
notion
variance
might be interesting (or even average) loses
where
the
meaning
at
cases
its
formally. This is the case of Lévy processes, which have been argued by many authors [24] to be adequate models for short enough time lags, when the kurtosis is large. Suppose distribution of the stock price x at time T is given by: then that the probability least
l~
Pjz,Tjzo,o)
=
IZT)?
L~
is the
uncorrelated a
typical Brownian
~t-dependent
equation
(2),
excursion
number. in
the
limit
of the
share
Note
that
process.
(cf., xc
e-g. xn
»
C
time
L~(u) decays, interesting
(ZT)à.
One
~~~~ =
11
L~(u)
process, which
price dunng
[25] ). It is
j13)
IZT)?
~
where ~t < 2 is the characteristic exponent of the Lévy and Z the generalization of the 'volatility', distribution
(ZT))
j~)
to
for
one
T, just u
discuss
corresponding Lévy 'hypervolatility':
the
might
-
as
cc,
the
colt
/fi
in
[f
as u
the
case
where H
of
C~
an
is
,
option pricing formula,
finds:
/~Î
~~ U~
(14)
JOURNAL
874
One
distinguish
thus
must
two
N°6
I
and ~t > 1.
~t < 1
cases:
PHYSIQUE
DE
integral in equation (14) diverges, and hence case, one option pricing impossible ? Yes and no: if the in that cc case process was really described by a Lévy distribution, even far in the taris, the notion of average would be meaningless when ~t < 1, and the price of the option should be fixed using a dilferent criterion. A possibility would be, for example, to demand that a loss (for the bank) greater than a certain acceptable Ievel £ should have a small probability p to occur, giving: i)
~t
that
C
In
1.
