In this work, two models for legal and illegal financiers are presented. The aim of the financiers are different: a bank try to minimize the defalt probabilityof the funded company, while the illegal financier aims to bring the company to bankruptcy and, at the same time, to obtain the maximum level of the firm's guarantee wealth. A couple of stochastic dynamic optimization problems are solved. The illegal case let intervene a numerical analysis of the microeconomic situation of the firm, strating fromreal data and writing new simulation�procedure in Matlab and GAMS. The legal case has been solved in closed-form, by using stochastic control theory.

Anna Grazia Quaranta

Raffaella Barone

Roy Cerqueti

English

Research Papers in Economics

Universitµ a degli Studi di Macerata
Dipartimento di Istituzioni Economiche e Finanziarie
A stochastic model for ¯nanciers
Ra®aella Barone, Roy Cerqueti, Anna Grazia Quaranta
Temi di discussione n. 42
Maggio 2007A stochastic model for ¯nanciers
Ra®aella Barone, Roy Cerqueti, Anna Grazia Quaranta
Abstract
In this work, two models for legal and illegal ¯nanciers are presented. The aim of the
¯nanciers are di®erent: a bank try to minimize the default probability of the funded
company, while the illegal ¯nancier aims to bring the company to bankruptcy and, at
the same time, to obtain the maximum level of the ¯rm's guarantee wealth. A couple
of stochastic dynamics optimization problems are solved. The illegal case let intervene a
numerical analysis of the microeconomic situation of the ¯rm, starting from real data and
writing new simulation procedure in Matlab and GAMS. The legal case has been solved
in closed-form, by using stochastic control theory.
Ra®aella Barone, Universitµ a di Lecce.
E-mail: r.barone@economia.unile.it.
Roy Cerqueti, Universitµ a di Roma "La Sapienza".
E-mail: roy.cerqueti@uniroma1.it.
Anna Grazia Quaranta, Universitµ a di Camerino.
E-mail: annagrazia.quaranta@unicam.it.Introduction
The aim of this paper is to determine the level of the interest rate that allow
to maximize the target function of both the legal and the illegal function.
We propose two models. The ¯rst of these determines the interest rate that
maximize the probability of redeeming the loan when the contract is made
whit a bank. The second one maximize the probability of the borrower's
default when he/she has recourse to the usury market [6]. Italian law de¯nes
granting of loans at an interest rate higher than the last one. Such a de¯-
nition has little economical content. As a matter of fact, the rate at which
the loan is released may not be higher than the legal one, but it may even
be more convenient, just because often, behind the usury credit - an illegal
activity by its very nature - there is hidden a further illegal activity, namely
the ¯nancial money laundering. According to a de¯nition based on the lit-
erature, money laundering may be the described as an economical activity
that has the function of transforming a potential purchasing °ow, which is
the result of illegal activities, and, as such, not directly usable as choices of
consume and of investment, into an e®ective purchasing power. In fact a
subject who has at his/her disposal an income from illegal activities cannot
use it without facing himself/herself the problem of avoiding to increase the
probability of being predicted because of his/her illegal activities. For this
reason the above subject has to decide whether and in which percentage of
their illegal money it is suitable to cleaning. Among the most used cleaning
techniques there is the granting of loans at no interest, whose usefulness for
the criminal is made evident by the analysis of at last two factors: on the
one hand, it allows him/her to avoid having to do with the Institutions, in
particular the banking system, then reducing the probability of the ¯nding
out of both the original crime and the illegal activity; on the other hand, the
1granting of a loan at no interest allows the criminal to pursue his/her ¯nal
goal, namely to enter into possession of what has been o®ered by the bor-
rower - the collateral is the fundamental tool in view of money laundering,
and as such, is endowed with a greater intrinsic value - when the borrower
is not able to pay back a loan. Indeed, what happens in the case of the
borrower's default? In loan granted by a bank the borrower is forced to go
bankrupt, thus losing the pledge o®ered as a guaranty, which has a legal
economical value lower than that attributed to it by the borrower; this is
both because of the monitoring costs of the investment project, which are
charged to ¯nancier, and because of the legal costs incurred in the recovery of
the pledge. The usurer on the contrary o®ers the possibility of renegotiating
the debt. This advantage o®ered by the usurer respect to the bank, renders
the contract initially stipulated by the illegal ¯nancier a loan at no interest,
a 'trap for fools'. In a second moment the usurer may change the contract
conditions in order to minimize the probability of the borrower's default and,
thus, to take possession of the pledge, which usually consist of the ¯nanced
enterprize itself.
In this work, two models for legal and illegal ¯nanciers are presented. The
aim of the ¯nanciers are di®erent: a bank try to minimize the default prob-
ability of the funded company, while the illegal ¯nancier aims to bring the
company to bankruptcy and, at the same time, to obtain the maximum level
of the ¯rm's guarantee wealth. A couple of stochastic dynamics optimization
problems are solved. The illegal case let intervene a numerical analysis of the
microeconomic situation of the ¯rm, starting from real data and writing new
simulation procedure in Matlab and GAMS. The legal case has been solved
in closed-form, by using stochastic control theory.
The paper is organized as follows. In the next section, the evolution equation
2of the company's wealth is described. Sections 3 and 4 are devoted, respec-
tively, to the analysis of the legal and illegal cases. Last section concludes.
1 The evolution equation
We suppose that the wealth of the ¯rm at time t; denoted by X(t); is de-
scribed by a stochastic di®erential equation.
Let us consider a probability space with ¯ltration (­;F;fFtgt¸0;P); where
the ¯ltration Ft is constructed as
Ft = ¾
³
X(s); 0 · s · t
´
[ N; 8 t ¸ 0;
with
N :=
n
A 2 F jP(A) = 0
o
:
The evolution of ¯rm's wealth is
8
> > > <
> > > :
dX(t) = (¹X(t) ¡ ®t)dt + ¾X(t)dW(t);
X(0) = X0
(1)
where
² ¹;¾ 2 R are the related to the drift and di®usion term,
² ®t 2 R is the intensity of the debt's payment at time t;,
² X0 is an integrable random variable in [0;K] with law ¼0 and measur-
able with respect to F0 representing the initial value of the ¯rm,
² W(¢) is a standard 1-dimensional Brownian Motion that is independent
of ¼0.
3We assume the existence of an inverse relation between the risk of the fund
project and the restitution interest rate.1
For sake of simplicity, we assume X0 =: x 2 (0;K) deterministic and
constant.
Remark 1 The boundary values 0 and K represent the absorbing barriers
for the dynamic of the wealth of the ¯rm. When the value 0 is attained, we
are in the situation of the failure of the ¯rm. K represents the case of total
restitution of the debt by the ¯rm.
Remark 2 There exists a unique solution for the controlled equation (1) (we
remind the reader, for example, to Âksendal (1995)).
By Remark 2, and ¯xed x 2 (0;K) and ® 2 R, we denote the unique
solution of (1) as X®
x(¢).
Moreover, let us denote with T the set of the stopping times in [0;+1),
i.e.
T := f¿ : ­ ! [0;+1]jf¿ · tg 2 Ft; 8t ¸ 0g; (2)
and let us de¯ne the exit time ¿(0;K) of the dynamic from (0;K) as
¿(0;K) := inf
n
t ¸ 0jX
®
x(t) = 2 (0;K)
o
; (3)
Remark 3 Standard results provide the measurability of ¿(0;K) with respect
to the ¾-algebra F. Indeed, it results ¿(0;K) 2 T .
1We remind to [13], extending to the risk-neutral agents the Assumption of Stiglitz and
Weiss on the risk-adverse behavior of the funded ¯rms with respect to the interest rates
applied to the ¯nanciers.
42 The optimization problem [legal case]
In the legal case the guarantees are irrelevant. In fact, in presence of bankruptcy,
the ¯nancier loses completely the credit grant. Hence, we assume in this con-
text that g(t) ´ 0, for each t.
A company is funded by a bank only under obvious pro¯t conditions.
So we write the value function V as
V (x) := min
®2A
P(X
®
x(¿(0;K)) = 0); (4)
where the admissible region of the problem is
A :=
n
® : [0;+1) £ ­ ! [±1;±2] ½ R
+ such that
K1 +
hZ ¿(0;K)
0
®te
¡±tdt
i
¸ K2; ®t 2 Ft; 8t ¸ 0
o
: (5)
K1 and K2 are nonnegative constants: K1 is the amount of the original debt
of the ¯rm while K2 is the threshold representing the minimal pro¯t that
the ¯nancier aims to obtain. ± is the discounted factor, while ±1 and ±2 are,
respectively, the lower and the upper bounds for the interest rate.
2.1 Hamilton-Jacobi-Bellman Equation
The following result holds.
Theorem 4 (HJB Equation) Suppose that V 2 C2((0;K)) \ C0([0;K]).
Then
sup
a2[±3;±4]
n
(¹x ¡ a)V
0(x) +
1
2
¾
2x
2V
00(x)
o
=: sup
a2[±3;±4]
Ha(x) = 0; 8 x 2 (0;K);
(6)
with the boundary conditions
V (0) = 0; V (K) = H; (7)
5where H is a positive constant relative to the income of the ¯nancier, when
the debt is completely paid.
In the next result we analyze the problem of existence, uniqueness and
closed form for the solution of the Hamilton Jacobi Bellman Equation.
Theorem 5 The following conditions hold.
1. There exists a¤ 2 [±1;±2] such that
Ha¤(x) = sup
a2[±1;±2]
Ha(x);
for each x 2 [0;K].
2. There exists an unique solution of the equation (6)-(7).
3. The function
V (x) = C
Z x
0
exp
n
¡
2a¤
¾2t
¡
2¹
¾2logt
o
dt; (8)
with
C = H ¢
³Z K
0
exp
n
¡
2a¤
¾2t
¡
2¹
¾2logt
o
dt
´¡1
;
is the solution of the Hamilton Jacobi Bellman Equation (6)-(7).
Proof.
1. Fixed x 2 [0;K], then Ha(x) is a continuous function of the variable
a in the compact set [±1;±2]. By Weierstrass' Theorem, we obtain the
thesis.
2. The proof comes out from the standard existence and uniqueness the-
orem for systems of ¯rst order ordinary di®erential equations.
63. Let us consider a¤ the same as in the ¯rst point of this theorem. (6)
can be written as
8
> > > <
> > > :
(¹x ¡ a¤)°(x) + 1
2¾2x2°0(x) = 0; x 2 (0;K);
°(x) = V 0(x); x 2 (0;K):
(9)
By separating the variables, the ¯rst order di®erential equation in (9)
becomes
d°
°
=
2(a¤ ¡ ¹x)
¾2x2 dx;
and then
log°(x) = ¡
2a¤
¾2x
¡
2¹logx
¾2 + C1; C1 2 :
Therefore we obtain
°(x) = C2x
¡
2¹
¾2e
¡ 2a¤
¾2x C2 2
+:
These computations provide
V (x) =
Z x
0
°(t)dt = C2
Z x
0
t
¡
2¹
¾2e
¡ 2a¤
¾2t + C3; C3 2 :
By imposing the boundary conditions (7), we get the thesis.
Corollary 6 The value function V is strictly increasing in (0;K).
Proof. By the previous theorem, we have that V is twice di®erentiable
in (0;K). Moreover, it results:
V
0(x) = Cx
¡
2¹
¾2e
¡ 2a¤
¾2x > 0;
being
C = H ¢
³Z K
0
exp
n
¡
2a¤
¾2t
¡
2¹
¾2logt
o
dt
´¡1
> 0:
That is what we want to prove.
72.2 Optimal strategies
Our starting point, for this section, is the Hamilton-Jacobi-Bellman Equation
stated in the previous sections.
With the Veri¯cation Theorem, we want to formalize the presence of optimal
strategies for this particular control problem. We enunciate it. For the proof,
we remind the reader to Fleming and Soner, 1993.
Theorem 7 (Veri¯cation Theorem) Assume that u 2 C0([0;K])\C2((0;K))
be a solution of (6)-(7).
Then it results
² (a) u(x) ¸ V (x); 8x 2 [0;K].
² (b) Let us consider (®¤;x¤) an admissible couple at x such that
®
¤ 2 argmaxa
n
(¹x
¤(t) ¡ a)V
0(x
¤(t)) +
1
2
¾
2(x
¤(t))
2V
00(x
¤(t))
o
:
Then (®¤;x¤(t)) is optimal at x and it results u(x) = V (x); 8x 2 [0;K]
Now we can describe the optimal strategies associated to this control
problem. The optimal controls ®¤ are bang-bang controls. We get, by the
analysis of HJB Equation,
®
¤(x) =
8
> > > > > > > > > <
> > > > > > > > > :
±2 for x j V 0(x) < 0
±1 for x j V 0(x) > 0
arbitrary for x j V 0(x) = 0
(10)
Since V is strictly increasing, we have that
¡ := fx 2 [0;K]jV
0(x) · 0g = ;:
8Therefore,
®
¤(x) = ±1;
for each x 2 [0;K].
3 The optimization problem [Illegal case]
A key role in the usury model is played by the guarantee g: The guarantee is
related ti the wealth of the ¯rm and takes into account the eventual income
obtained by the ¯nancier in company's bankruptcy case. Therefore it seems
to be obvious to assuming the guarantee nonnegative. We introduce it. We
de¯ne
g : [0;+1) £ ­ ! [0;G];
through a stochastic di®erential equation as follows:
8
> > > <
> > > :
dg(t) = °1(g(t);X(t);®t)dt;
g(0) = g0
(11)
where
² °1 is a real functions,
² g0 > 0 is deterministic.
gt is the subjective wealth of the ¯rm evaluated at time t by the ¯nancier.
It is assumed to evolve deterministically. °1 in (11) is increasing with respect
to X(t) and g(t); and it is decreasing with respect to ®(t):
We assume that the nonnegative dynamic g of the guarantee process is
bounded by a deterministic threshold G: The objective of the illegal ¯nancier
9is to maximize the default probability of the company and, contemporane-
ously, the probability that the guarantee reach the highest level G: Therefore,
the value function is
V (x) := max
®2A
[P(X
®
x(¿(0;K)) = 0) + P(g
®
g0(¿(0;K)) = GjX
®
x(¿(0;K)) = 0)]; (12)
where the admissible region of the problem is
A :=
n
® : [0;+1) £ ­ ! [±3;±4);such that ®t 2 Ft; 8t ¸ 0
o
: (13)
Although it is generally the upper bound of the usury interest rate in¯nity,
empirical evidence shows that ±4 = 500% with only the 9% of the event that
overcome such high threshold2. The lower bound is assumed to be ±3 = 0;
in agreement with the purposes of the illegal ¯nancier (to construct a trap
for the company, to reinvest illegal money). We have built the ® range of
variation on such values and we have added this further constrain to the
forthcoming maximization model, that has been implemented using GAMS.
Our goal is to derive the average interest rate ® that can maximize the
objective function V .
The ¯rst step is to generate randomly a vector ® of interest rates from a
Poisson distribution with parameter ¸ = 4;3 in agreement with the previous
literature.
Then, with the same language, for all generated value ®i we derive by equa-
tion (1), actually with a Monte Carlo simulation, the trajectory of X
®i
t . At
2CENTRO STUDI E RICERCHE SULLA LEGALITA' E CRIMINALITA' ECONOM-
ICA, L'usura tra vecchi con¯ni e nuovi mercati, Roma - 2002.
3To this scope, we have used the MatLab generator; the vector have n = 1000 compo-
nents.
10this aim we ¯x the values ¹ =; ¾; X0 referring to real data4. We discretize
as usual the Brownian Motion dW(t) = ¤¤
p
¢t; where ¤ is a random num-
ber extracted by a centered normal distribution.
For each X
®i
t we derive the time ¿®i in which, for the ¯rst time, the trajectory
of X
®i
t hits the barrier f0;Kg. The threshold K is de¯ned by the existing
empirical relation K = ¯X0
5.s
Then, we calculate ¿¤(0;K) = 1
n
Pn
i=1 ¿®i. For the illegal ¯nancier it is rea-
sonable to suppose that there is an inverse relation between G; such as how
much he estimates the wealth of the ¯rm at the beginning of the contract,
and X(0) ([6, 3]); due to this reason, as G ¡! 1, then X(0) ¡! 0: On the
other side, it is well known that X(0) =
X(t)
(1+®)t; so it is easy to obtain the
objective function that we have to maximize to realize the illegal ¯nancier
goal (12).
We approximate V (X) of the illegal ¯nancier as
V (x) = max
®2A
(1 + ®)t
X(t)
:
The set of constraints is
8
> > > > <
> > > > :
G =
(1 + ®t)t
X(t)
;
X(t) = X(t ¡ 1) + f[¹X(t ¡ 1) ¡ ®t]¢t + ¾X(t ¡ 1) ¢ ¤
p
¢tg:
We consider ¿¤(0;K) to de¯ne the ¢t value for each time.
As a result we have obtained that the average level ® of the interest rate
need to obtain the purpose of the illegal ¯nancier coincides with the upper
4In particular, we consider ¹ = 1 + ½ = 1:001; where ½ is the revaluation rate of the
company, ¾ = 0:01 and X0 = 950000:
5¯ comes out from the the ratio between G and X0. The empirical mean value of X0
is 950000. Therefore, G = 1325000 and ¯ = 1:42. The data are available under request.
The Source does not want to be nominated in the present paper.
11bound of its variation range; in particular we have found that the level of ®
increases while ¢t decreases (see Figure).
Sample of the relationship between the interest 
rate and exit time
0
200
400
600
800
1000
1200
0 500 1000 1500
tau
a
l
f
a
Serie1
4 Conclusions
In this work the best interest rate, that a ¯nancier has to apply to a funded
company to maximize an objective function, has been explored. The dis-
tinguishing between legal and illegal ¯nancier has been proposed. At this
aim we construct a couple of models. The scope of the legal ¯nancier is to
maximize the probability that the funded ¯rm will restitute totally its debt,
while the illegal ¯nancier try to maximize at the same time the default prob-
ability of the company and the guarantee related to company's wealth. The
solution strategies of the models are di®erent: in the legal case, an analytical
closed form solution is showed; the illegal problem has been solved by using a
numerical approach. The numerical approach is due to a consistent and not
12easy formalization of the usury-type situation, that does not allows a better
tractability.
The best interest rate is the maximum one in the illegal case and the mini-
mum one in the legal ¯nancier model. In the former model the interest rate is
as close as possible to the maximum one as well as an eventual renegotiation
date approaches the mean ¿¤.
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Brownian motion and stochastic calculus,

Colletarel in Credit Rationing in Markets with Imperfect Information: Note, The American Economic Review,

Il vantaggio di bussare due volte: contratti bancari ed usura, diritti di proprietµ a, valore della garanzia e della rinegoziazione, Economia Politica,

Implementing Money Laundering (Chapter 4), in Masciandaro

Money Laundering Regulation: The Microeconomics,

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Social Choice and Multicriterion Decision Making,

Social multi-criteria evaluation: Methodological foundations and operational consequences,

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Success or failure of a ¯rm under di®erent ¯nancing policies: a dynamic stochastic model,

The Role of Collateral in a Model of Debt Renegotiation,

Why usury can be cheaper?,

A stochastic model for financiers

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