Download Fractional Differo-Integral Calculus for Finance: some Results

Fractional Differo-Integral Calculus:
towards a Theory of Fractal Financial Laws
Marco CORAZZA
Department of Applied Mathematics
University "Ca' Foscari" of Venice (Italy)
Carla NARDELLI
Department of Studies on Resources, Firm, Environment, and Quantitative Methodologies
University of Messina (Italy)
1. Introduction
Since the pioneer works of Bachelier and Working, the classical approach concerning the
analysis of the financial asset returns has been characterized by the assumptions of
independent and identical distribution of probability for the random variables
ln Pt  t   ln Pt , with t  0, 1, , N  t ,
where
P t  is the financial asset price at time t.
In particular, in the modern economic-financial theories (like, for instance, the one on
the impossibility of making arbitrage, and the one on the derivative pricing) the specific
hypothesis of independent probability distribution has became a paradigm.
Nevertheless, an increasing number of empirical analysis has well set in evidence the
partial inadequacy of the assumption of the latter hypothesis, because of the constant
presences of auto-dependence both in the short-middle term and in the long one.
Starting from these peculiarities, several Authors have rejected the hypothesis of
efficiency in the weak form for the financial markets and They have conjectured the non
markovianity for the stochastic processes generating the variations of the financial asset
prices.
Such empirical results have favored an increasing interest towards the stochastic
processes known as "fractional Brownian motions" because of their capability of
representing Gaussian stochastic processes whose generic increases are stocastically
dependent one each other. In approximative terms, such stochastic processes can be
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interpreted as a properly weighted summation of the past realizations of a standard
Brownian motion; it is possible to formalize as follows this characterization of the fractional
Brownian motion by means of the stochastic differo-integral calculus:
0
1
H  0.5
H  0.5
 t  s
BH t  
  s
dB s 
  H  0.5 
 


t

H  0.5
t  s
dB s


0

where
BH t  is the fractional Brownian motion at time t, with H  0, 1 ,
  is the gamma function, and
B t  is the standard Brownian motion at time t.
In this work we determine the financial laws following which the riskless component
of a risky portfolio must evolve in order of avoid the possibility of arbitrages under the
assumption following which the dynamics of the stochastic component of the same portfolio
is driven by a fractional Brownian motion.
More in detail, in order to describe such a dynamics of the riskless component of the
considered risky portfolio,
 firstly, we part from the following fractional differential equation that generalizes the
classical model of the compounded interest financial law:
d
Ct     Ct ,
dt 
where
  0, 1  Q is the fractional order of integro-differentiation,
C  is the riskless component of the immunized portfolio, and
  0 is a constant,
 then, by means of a proprer application of the differo-integral calculus (which is
sketched in the next section), we give the solution of the latter fractional differential
equation, and
 finally, we present some deepenings concerning the role played by the arbitrary
constants of this solution in order to avoid, or less, the possibility of arbitrages.
2. The solving approach: a sketch
Step 0: one sets   0, 1  Q , that is 0    m n  1 , with m, n  N  and m and n
without common divisors; then
d
dm n
C t   m n C  t     C  t  ;
dt 
dt
2
dm n
step 1: one applies the operator m n  to both the terms of the differential equation, that is
dt
dm n
dt m n
 dm n
 dm n
C
t
 dt m n    dt m n   Ct  ,


dm n
 (first term), and by the application
dt m n
of the second method of Liouville (second term), one obtaines:
from which, by the property of the operator
d 2m n
dm n
 m n 1
C
t



;


2m n
m n Ct   c1  t
dt
dt
dm n
Ct     Ct  , and subsituting it in step 1, one
step 2: recalling from step 0 that
dt m n
obtaines
d 2m n
Ct       Ct   c1  t m n1 ;
2m n
dt
step 3: iterating other n  2 times step 1-step 2, one has


 
d m n  d m n  d 2m n

C
t
   n  Ct  





mn
mn
2m n
dt
dt
dt
 


 
d
c j   n 1 j  m n  t m n 1 ;
dt
j 1
n 1

n 1 j  m n
n  2 times
step 4: it is possible to prove that:
d  n1 j  m n    m n 1  1
t

dt  n1 j  m n 
  m n  n  j   1   m n  n j 1
t
;
 m n  1
 n 1 j  m n 
moreover, we define
 n1 j  m n 
1
 m n  n  j   1   m n  n j 1

;
kj 
t
m n  1
def
step 5: finally, substituing step 4 in step 3 one obtaines the following integer-order linear
differential equation:

n 1
dm
 m n  n  j 1
n
C
t



C
t

cj  k j  t     ;




m
j 1
dt
step 6: the solution (that is, the fractional compounded interest rate financial law) is:
Ct  
m ~


n 1
 c
c j  exp  j  t 
j  k j  expt  

t
0
 m n  n  j 1
exp y  y     dy ,
j 1
j 1

 


solut. of the omog. diff. eq.
solution of the particular differential equation
where
~
c j and c j are m  n  1 arbitrary constants, and
 j are the solution of the equation m   n  0 .
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