Journal of Applied Mathematics and Stochastic Analysis
9, Number 3, 1996, 271-280.
A STOCHASTIC MODEL FOR THE FINANCIAL
MARKET WITH DISCONTINUOUS PRICES
LEDA D. MINKOVA
Technical University of Sofia
Institute of Applied Mathematics and Informatics
P.O. Box 38, 1000 Sofia, Bulgaria
(Received May, 1994; Revised January, 1996)
ABSTRACT
This paper models some situations occurring in the financial market. The
asset prices evolve according to a stochastic integral equation driven by a Gaussian martingale. A portfolio process is constrained in such a way that the wealth
process covers some obligation. A solution to a linear stochastic integral equation
is obtained in a class of cadlag stochastic processes.
Key words: Contingent Claim Valuation, Representation of Martingales,
Stochastic Integral Equation, Option Pricing, Portfolio Processes.
AMS (MOS) subject classifications: 60H20, 60H30.
1. Introduction
In the present paper we model investments of an economic agent whose decisions cannot
affect market prices (a "small investor").
Karatzas and Shreve in [7] considered a market model in which prices evolve according to a
stochastic differential equation, driven by Brownian motion. Aase [1] and M. Picqu and M.
Pontier [9] studied a more general model in which the evolution of asset prices is a combination of
a continuous process based on Brownian motion (a semimartingale) and a Poisson point process.
The security price model that we use is a linear stochastic equation driven by a Gaussian
martingale. This is a natural generalization, because the market is not continuous and the
Brownian motion cannot model jump processes. Moreover, the instants of jumps of a Gaussian
martingale are nonrandom.
The techniques we use include the martingale representation theorem and the Girsanov’s type
theorem. We also find a solution to a linear stochastic integral equation.
2. The Model
We consider a model of a security market where an economic agent is allowed to trade continuously up to some fixed planning horizon 0 <_ T < oc. We shall denote by X the wealth of this
1This work
MM-440/94.
Printed in the U.S.A.
was supported by the National Science Foundation of Bulgaria,
()1996 by North
Atlantic Science Publishing Company
Grant No.
271
LEDA D. MINKOVA
272
agent at time t. Let the process M-
(Mt, Ft, O <_ <_ T) be a
{Ft, 0 <_ t <_ T}
Gaussian martingale on a fixed
be the augmentation under P of
O<_t<c. F o contains the null sets of P and F is
[0, c) is the square characteristic of U.
probability space (f, F, P)
a natural filtration FtM=r(Ms,0<_s<_t),
right continuous. (M)t EM2t, t E +
and the filtration F
Let us suppose that the agent invests in two assets (or "securities"). One of the assets, called
bond, has a finite variation on [0, T], and its price model is
f
Po(t)
/ Po(s- )r(s- )d(M)s
P0(0)- P0, 0 _< t <_ T.
(0,t]
The other one, called stock, is "risky". Its price is modeled by the linear stochastic equation
P(t)
f
f
(0,t]
(0,t]
/ P(s- )A(s- )d(M)s + / P(s- )(r(s- )dMs,
P(0)- p.
Here the interest rate process r(t) > 0, 0 <_ t < oc of the bond, the appreciation rate process A(t)
of the stock, and volatility process r(t) > 0, 0 _< t < c will all be nonrandom, F-predictable processes such that
f r2(s-)d(M)s<oo, f A2(s-)d(M)s
<oo,
(1)
(o, oo)
(o, oo)
er(s- )d(M)s < cxa,
P-a.s.
(0,)
In addition, A(t- )A(M)t + r(t- )AM > 1, t e (0, T], to ensure a limited liability of the stock.
Let re(t) denote the number of stocks held at time t. Then the amount invested in the stocks
is
m(t)P(t).
II(,)
(H(t),Ft) 0 <_t<_ T describes the investment policy and will be called a portfolio
process. It is assumed to be measurable, Ft-predictable and
The process
H2(s_)d(M)s < c,
P-a.s.
(2)
(o,T]
for every finite number T
stock short.
> 0. Note that H(t)
On the other hand, C(t), 0 <_ t _< T is
nondecreasing and Ft-predictable such that
a
can be negative, which amounts to
selling the
non-negative consumption process, assumed to be
/C(s- )d</)s <
oo, P-a.s.
(3)
(o,T]
for every finite number T
> 0.
The quantity
no(*)
x,-
is invested in the bond at any particular time and may also become negative.
interpreted as borrowing at the interest rate r(t).
This is to be
A Stochastic Model for the Financial Market with Discontinuous Prices
>_ 0, and the wealth
We
assume now that the investor starts with some initial wealth x
satisfies the linear stochastic equation
X
/ II(s- )(s- )dM + / II(s- )[A(s- )- r(s- )]d(M)s
273
at time t
s
(0,t]
+
(0,t]
/(0,t] [X s_r(s-)-c(s-)]d(M)s
(4)
O<t<_T;
x(0)-
Conditions (1), (2), and (3) ensure that the stochastic equation
the class of cadlag adapted processes (see Section 5 and Theorem 3).
(4)
has a unique solution in
3. Characterization of the Portfolio Process
If
A(t)
r(t) for every t e [0, c),
the drift
II(s- )[A(s- )- r(s(0,t]
vanishes from the right-hand side of
measure P which removes this drift.
Let us denote by
(4). When A(t) r(t)
we introduce a new probability
Ct the solution of the equation
Ct-1-
O<_t <_T,
] Cs-O(s-)dMs,
(0,t]
where
A(t)- r(t)
O(t)
From
adapted to
A, r, and r, it follows that
Then the exponential supermartingale
our assumptions on
{F t- }.
O(t)
is bounded, measurable and
is actually a martingale, where
a(t-)
A(t- r(t-
AMt Mt- MtHere
+.
for0<t<T.
M and (MClt are the continuous parts of the processes M
We define the
new probability measure
P (A)
and
(M}t
respectively, for t E
P"
E(TIA) A F T on (, F).
The probability measures P and P are mutually absolutely continuous on F T.
The process
t
is a P-Gaussian martingale
[8],
M t+
and
/ O(s-)d(M)s
O<_t<_T,
(0,t]
((M)t,P)
((M)t,P),
0
<_ t _< T.
(6)
LEDA D. MINKOVA
274
(4)
With respect to a new probability measure, equation
II(s )r(s )dM s +
X
(o,t]
can be rewritten as
C(s )]d(M)s, 0 < t _< T,
[X s r(s
(7)
(o,t]
x(0)and the solution
(see
X
(I)(t--- -twhere
(o,t]
5) for 0 _< t _< T, leads to
C(s-)
x/
Section
O(s- )[1 + r(s-)&(M)s d(M)
(o,t]
is a unique strong solution of the homogeneous equation corresponding to
(P(t)
If we suppose that 1
impossible if
r(s)
1
(7)"
/ ((s- )r(s- )d(M)s.
(o,t]
< 0 for some s
1
But
e S, then A(M)s < (s-)"
Consequently, 1 + r(s- )A(M)s > 0 for every s e +.
+ r(s-)A(M}s
is nonnegative.
+
(8)
(s )[1 + r (s )A(M)s
this is
Let us notice also that
inf
t[+
I(I)(t)l >0.
The right-hand side of (8) is a P-local martingale. If (H, ) is an admissible pair (i.e., X >_
_< t _< T a.s.), the left-hand side is nonnegative, consequently it is a nonnegative supermartingale under P. From the supermartingale property we obtain that
0, 0
E
IXT
(I)(T) +
/ ((s-)[1 C(s-)
+ r(s- )A(M)s d(/17/>sl<x,
(10)
(0,T]
where E denotes the expectation operator under measure P.
This condition is also sufficient for the admissibility in the sense of the following theorem.
Theorem 1:
such that
Suppose that
E
x
>_ 0
BT
(I)(T) +
(0, T]
and B T is a nonnegative
FT-measurable
random variable,
C(s-)
< x.
(I)(s-)[1 + r(s- )A(M)s d()s
Then there exists a portfolio process II such that the pair
dowment x and the terminal wealth X T is at least B T.
(H,C)
Proof: It is obvious that we can assume equality to hold in
Let us define the nonnegative process
,,- E B r +
C(s-)
+
(O,T]
is admissible
(11)
for
the initial en-
(11).
d(/l>s F
(12)
#0
x,
A Stochastic Model for the Financial Market with Discontinuous Prices
275
which is a P-martingale and has "cadlag" paths.
_< t _< T by
Define the process #t, 0
#twhere Ct is the density
and (it) ().
/ -ld(s- , }s,
fit +
(13)
(o,t]
It is well known that the process #t is a P-martingale
(5).
Now by the martingale representation theorem [8], if (M, #) is
an
[5], #o- o,
a Gaussian process, there exists
Ft-predictable measurable process h(s), such that
/ h2(s-)d(M)s <
for every finite T
P-a.s.
(O,T]
> 0 and
#t--#o+
] h(s-)dMs,
O <_t <_ T.
(14)
(o,t]
The process
(13)
can be
represented as
tt-
#t
f
O(s- )h(s- )d(M}s
!
(o,t]
O <_ t <_ T.
(15)
From equalities (6), (14), and (15)it follows that
f
#t +
t
I O(s- )h(s- )d(M)s
(o,t]
#o +
/ h(s
(o,t]
)[dffI s O(s )d(M>s +
/ O(s
(o,t]
)h(s )d(M>s
(16)
/ h(- )d/.
o+
(o,t]
Now
II(t-)-h(t-)((t-)[l+r(t-)A(M)t
(t-)
0<t<T
(17)
is a well-defined portfolio process.
From (12), (16), and (17),
E
t
we
get
(T) +
(0, T]
(s- )[1 2i;- )A{M}s
x
s
(18)
+
h(s-)dM s.
(o,t]
By using (18) and (8),
fit
we obtain
,c,
x(I)(t---+ / (I)(s-)[1 c(-)
+ r(s- )A<M>s d<>
(o,t]
(19)
LEDA D. MINKOVA
276
where
XtI1’ c,x is a solution of equation (7) for the pair (II, C) and the initial capital x >_ 0.
Now, from (18) and (19), it follows that
(20)
(t,T]
Consequently,
Xt
c,x is nonnegative and (II, C) is
an admissible
strategy.
4. Valuation of Contingent Claim
Definition: A contingent claim is a nonnegative
FT-measurable random variable B that satis-
fies
<x.
0<E
The hedging price of this contingent claim is defined by
vd---efinf{x > 0, =l (II, C)
Theorem 2: The value
Proof:
of the
Let us suppose that
admissible, such that
X’ c, > B P-a.s. }.
contingent claim is attained and
XF’c’’>_ B
a.s. for some value of x
> 0 and
a suitable pair
(II, C). Then from (10)it follows that
T) -<
-<
_< U.
Consequently, z- E
Let us define the nonnegative random process
where
ffh- E
F is
a P-Gaussian martingale, such that
ff0- E
Analogously to the proof of the Theorem 1, we can apply the generalized Girsanov’s theorem
and the martingale representation theorem.
By comparing the processes
Xo(t) =z+
and
x’’x
we obtain that
O(t)
(I) (t)
/ h(s-)dM
s
(o,t]
Xo(t)-
xn, ’,z,
o < t < T.
(21)
Consequently, z >_ U.
Remark 1: Let us note that
(21)
yields
Xo()-
x
’’z-
,
.s.,
i.e., the contingent claim is attained with the initial capital U, portfolio II, and zero consumption.
This fact could be used as a starting point for solving appropriate optimal problems.
A Stochastic Model for the Financial Market with Discontinuous Prices
277
Pmark 2: If (M}t t, we have (M,P)and (M,P) (standard) Wiener processes, and
empty. Then Theorem 1 and Theorem 2 reduce to the results of Karatzas and Shreve [7] and
Cvitani and Karatzas [2].
Corollary:
Let C(t)= 0 and let the agent invest in
representations hold:
((t). E
X
x
[P(T)
(P(T) Ft
(0,t]
Then, the following
<_ t <_ T;
(i)
(- )d() +
(- )dM-
.xp
0
one stock asset.
(0,t]
( )d(M )
(o,t]
[1 + (- )() + (- )a],
0
t
__
T.
(i) follows from (20) when C(t) O.
be proved that P(T) is a P-Gaussian martingale and
(i.T)
Proof: Representation
By Ito’s rule it can
of the following stochastic equation:
P(T)
(I)(T)
p+
[ P(s(I)(s-)
(ii)
is a unique solution
r(sdI s T > O.
[1 + r(s- )A{M)s
j
(0,T]
X
0
Consequently, from representation (i) it follows that
o--’
gale and it yields representation (ii).
t
T is
a P-Gaussian martin[]
5. A Linear Stochastic Integral Equation
In this section
X --X 0
0
_< t < c,
-
we will obtain a solution of the stochastic equation
/(0,t] [S(s)Xs_ + a(s)]dM + / [A(s)Xs_ + a(s)]db(s),
s
(22)
(0,t]
which is a more general than we anticipate.
Let M (Mr, Ft) M o 0, F
r(M s, s <_ t), t e N + [0, (x), be a cadlag Gaussian martingale, b- b(t), t E N +-nonrandom, right-continuous function with finite variation on each finite
interval. Suppose the function b(t) be a real-valued deterministic function, absolutely continuous
and
with respect to (M)t-
EM2t
/ 7sd(M)s,
b(t)
teN+,
(0,t]
where 7--(/t, Ft_), Ft_- r(Ms, O < s < t) is a F-predictable function, Xo-(Xo, Fo) be a
Gaussian random variable, independent of M. A(t), r(t), a(t), and S(t)are nonrandom F-predictable functions, such that
I a2(s)d<M)s
< c,
f A2(s)db(s)<
(o,o)
(o,)
S2(s)db(s)
and
(0,)
We will find
a solution of equation
P-a.s.
(0,)
(22)
in the class of cadlag adapted processes
(i.e., pro-
LEDA D. MINKOVA
278
_
right-continuous paths and finite left limits) and it provides a fairly explicit representation. According to [4], such a solution exists and it is unique in the sense of P-indistinguishability.
cesses with
Recall [6] that the random process M and the deterministic nondecreasing function {M) have
their jumps at the same nonrandom moments of time which form a countable set 5 C
\{0}.
Let us notice now that the function b(t) and the process X have their jumps at the same moments.
N+
Let us suppose that the function b- b(t), t E
jumps {0 _< s o < S 1 <... < Sic <... < OO} S with
Ab(sic)
1
N+
has no more than a countable subset of
+ S(sic)AM
A(sic)
sic, k>_1.
It is obvious that
AXsIc
[A(sic)Xs_ + a(sic)]Ab(sic) + [S(sic)Xs + cr(sic)]AMsic
Xs
Consequently,
-
a(sic)[D, 1 + S(sic)AM
s-
s
]+
a(sk)
Xsic r(sk)AMsic A(sic)[1 + S(sk)AMsic].
_
(23)
We will find a solution of equation (22) on the interval [sic, sic + 1), ] 0, with an initial condition X s independent of increments M
sic <_ t < slc + 1, k >_ O, according to (23) and the
Msic,
conditionsimposed on X 0.
The homogeneous equation corresponding to
(22)
/
is
/
(t, sic)--1+
(s-,sic)A(s)db(s) +
(s-,ic)S(s)dMs,
(sk, t]
(sk, t]
@(sic, sic) 1, k 0
and has a unique solution
(t,
[3]"
sic)--exp{(sic,/ A(s)db(s)+(sic,/
t]
H
S(s)dMs
t]
{1 + A(s)Ab(s) + S(s)AMs}. exp{ A(s)Ab(s)- S(s)AMs};
Sk<S<t
(I’(t, sic)
exp
{(si/c,
H
{1 +A(s)Ab(s)+S(s)AMs}
Sk<S_t
where
bC(t)
Let
_
/ S(s)dMCs-1/2/(,t]S2(s)d(MC)s}
A(s)dbC(s) +
t]
(,t]
is the continuous path of the function
b(t), sic <_ < sic + 1, k
us notice that if
A(t)Ab(t) + S(t)AM
1
O.
(24)
A Stochastic Model for the Financial Market with Discontinuous Prices
on
(sk, s k + 1), from the solution of (24)
it follows
inf
s
with some T E
Let
_(I)(t, sk) > 0
<_ < Sk+ 1 AT
k
[s k, oc), k _> 0.
(I)(t), t E +,
(I)() (I)(t, sk)
now define the function
279
(25)
where
sk
7
< s k + l,
]
0.
It follows from(25) that the function (I)- l(t), G + is correct defined and bounded on every
finite interval [0, T], T G
it holds true that
Consequently, for every t G
+.
+,
2(s)d(M)s
(26)
(0,t]
Theorem 3: The unique solution
of the
equation
(22)
is given by
+
+
(27)
(b(s) d(MC)s
slc
<t<slc+l’
k>O.
(,]
X
Proof: Observe that (25) ensures that the process
is well defined. We will show that the
process k from (27) is a solution of equation (22) over the interval [sk, s k + 1), k _> 0.
Xt
We apply Ito’s rule to (27)
on the interval
J
Xkt Xk +
(stc, s k + 1):
/(,] X
X s_ A(s)db(s) +
(,]
+
1
1
(,t]
+A(s)Ab(s) +S(s)AM s
f
s
+/
k
(,t]
II{Ab s) =o}
X s_A(s)db(s)+
f
/
If s k
< < s k + 1, then
I{Ab(s) 0}]
X s_S(s)dM s
(,t]
/(,t]r(s)dMs+(s,t]/ a(s)db(s).
+
S(s)dM s
LEDA D. MINKOVA
280
] [X- A(s) + a(s)]db(s) + J [X- S(s) + r(s)]dM.
X +(,t]
(,t]
X
D
(23)lead to (22).
coefficients A(t), (t), S(t), and a(t)
The last representation and
of equation (22) are f-predictable funcPemark 3: The
tions. It will be convenient for applications to represent solution (27) in the form
Xt
O(t’sk)
Xsk +
(sk, t]
+
(sk, t]
(,t]
(s- )[1 + A(s)Ab(s) + S(s)AMs]
a(s)
(P(s-)[1 + A(s)Ab(s)+ S(s)AMs] db(s)
(I)(s-)[1 + A(s)Ab(s) + S(s)AMs]
s
t
[Sk,8 k+l), ]- 0.
Acknowledgement
The author would like to thank Professor Svetlozar Rachev from the UC Santa Barbara for
his helpful and stimulating discussions and the referee for constructive comments.
References
[1]
[2]
[3]
[4]
[7]
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random
and
process, Cotoch. Process. Appl. 28 (1988), 185-220.
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process
and
J.
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