Journal of Mathematical Finance
Vol.06 No.04(2016), Article ID:71225,18 pages
10.4236/jmf.2016.64043
On the Solution of the Multi-Asset Black-Scholes Model: Correlations, Eigenvalues and Geometry
Mauricio Contreras1, Alejandro Llanquihuén2, Marcelo Villena1
1Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago, Chile
2Facultad de Ciencias Exactas, Universidad Andrés Bello, Santiago, Chile
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: August 23, 2016; Accepted: October 11, 2016; Published: October 14, 2016
ABSTRACT
In this paper, the multi-asset Black-Scholes model is studied in terms of the im- portance that the correlation parameter space (equivalent to an N dimensional hypercube) has in the solution of the pricing problem. It is shown that inside of this hypercube there is a surface, called the Kummer surface, where the determinant of the correlation matrix
is zero, so the usual formula for the propagator of the N asset Black-Scholes equation is no longer valid. Worse than that, in some regions outside this surface, the determinant of
becomes negative, so the usual pro- pagator becomes complex and divergent. Thus the option pricing model is not well defined for these regions outside
. On the Kummer surface instead, the rank of the
matrix is a variable number. By using the Wei-Norman theorem, the pro- pagator over the variable rank surface
for the general N asset case is computed. Finally, the three assets case and its implied geometry along the Kummer surface is also studied in detail.
Keywords:
Multi-Asset Black-Scholes Equation, Wei-Norman Theorem, Correlation Matrix Eigenvalues, Kummer Surface, Propagators
1. Introduction
Since the seminal work of Black, Scholes and Merton on option pricing, see [1] [2] , an important research agenda has been developed on the subject. This research has mainly centered in extending the basic Black and Scholes model to well known empirical regularities, with the hope of improving the predicting power for the famous formula, see for example [3] - [6] . An interesting extension has been the modeling of many underlying assets, which has been called the multi-asset Black-Scholes model [3] [7] . In this case, the option price satisfies a diffusion equation considering many related assets. The first work addressing this problem in the literature was Margrabe (1978), see [8] . The Margrabe formula considered an exchange option, which gives its owner the right, but not the obligation, to exchange b units of one asset into a unit of another asset at a specific point in time. Specifically, Margrabe derived a closed-form expression for the option by taking one of the underlying assets as a numeraire and then applying the Black and Scholes standard formulation. Later Stulz [9] found analytical formulae for European put and call options on the minimum or the maximum of two risky assets. In this particular case, the solution is expressed in terms of bivariate cumulative standard normal distributions, and when the strike price of the option is zero the value reduces to the Margrabe pricing. Other interesting papers that follow in this literature are [10] - [15] . The numerical implementation of the solution of the multi-asset Black-Scholes model is increasingly difficult for models with more that three assets, see for instance [16] - [18] . One important point, that has been missed in the literature, is that in all of the multi-asset Black-Scholes models mentioned above, the relationship between assets is modeled by their correlations, and hence it is implicitly assumed that a well behaved multivariate Gaussian distribution must exist in order to have a valid solution.
In this paper, the multi-asset Black-Scholes model is studied in terms of the im- portance that the correlation parameter space (which is equivalent to an N dimensional hypercube) has in the solution of the option pricing problem. It is shown that inside of this hypercube there is a surface, called the Kummer surface [19] - [22] , where the determinant of the correlation matrix
is zero, so over
the usual formula for the propagator of the N asset Black-Scholes equation is no longer valid. Worse than that, outside this surface, there are points where the determinant of
becomes negative, so the usual propagator becomes complex and divergent. Thus the option pricing model is not well defined for some regions outside
. On
the rank of
matrix is a variable number, depending on which sector of the Kummer surface the correlation parameters are lying. By using the Wei-Norman theorem [23] - [26] , the propagator along the Kummer surface
, for the N assets case is found. This expression is valid whatever the value of the
matrix ranks over
.
This paper is organized as follows. Section 2 describes the traditional multi-asset Black-Scholes model. In Section 3, the problem is formulated as a N dimensional diffusion equation. In Section 4, the implied geometry of the correlation matrix space is analyzed, specially when its determinant is zero, which coincides with a Kummer surface in algebraic geometry. The Kummer surface and its geometry are reviewed for the particular case of three assets in Section 4.1. In Section 5, by using the Wei-Norman theorem the propagator over the variable rank surface for a general N asset case is computed. Finally, some conclusions and future research are presented in Section 6.
2. The Multi-Asset Black-Scholes Model
Consider a portfolio consisting of one option and N underlying assets. Let be the price processes for the assets;
where each asset satisfies the usual dynamic
(1)
and the
Wiener processes
are correlated according to
(2)
where is the symmetric matrix
(3)
so
(4)
If the price process for the option is, the value V of the portfolio is given by
(5)
where are the shares of each asset in the portfolio. The self-financing portfolio condition ensures that
(6)
and applying It Lemma for one gets
(7)
According to [4] , for a free arbitrage set of N assets, the return of the portfolio is
(8)
and from Equations (7) and (8) one has
(9)
Collecting and
terms in the above equation one gets:
(10)
and
(11)
From Equation (11), and given the independence of the for
(12)
or equivalently
(13)
so one arrives at the multi-asset Black-Scholes equation
(14)
which must be integrated with the final condition
for constant r, ,
and a simple contingent claim
.
3. The Multi-Asset Black-Scholes Equation as a N Dimensional Diffusion Equation
Here, some transformations are developed, which maps the multi-asset option pricing equation in a more simpler diffusion equation. If one makes the change of variables
(15)
in (14), one can map this equation to
At least if one defines as
(16)
then satisfies the equation
Now, by defining the variables
(17)
the above equation can be written as
And finally, by defining the forward time coordinate
(18)
one arrives at
(19)
Now performing the transformation
(20)
one can change the variables to the
coordinates that diagonalizes the
matrix
(21)
where
(22)
and U is the change basis matrix, with,
. In this diagonal coordinate system, the diffusion equation read finally
(23)
Now this equation is studied in terms of the behavior of the eigenvalues.
4. The Geometry of the r Matrix
The matrix in (3) can be characterized completely for the
dimen-
sional vector
(24)
which lies inside of an M dimensional hypercube centering in the origin and of length 2. Thus, the matrix is a function of
:
. Note that, for some point
inside of the hypercube, the determinant of the
matrix vanishes. For example, for the vertex
(25)
In fact, exists a whole surface inside the hypercube, where the determinant of vanishes. This surface, called Kummer surface
in algebraic geometry [19] - [22] , is defined by the equation
(26)
In fact, one can think of the hypercube as the disjoint union of the subset of point or surfaces of constant C determinant value:
(27)
Let an arbitrary vector in
and let
the determinant of
in each point, that is
. Note that
is a polynomial function in terms of the
coordinates.
The vector given by the M dimensional gradient
is perpendicular to the level surfaces
and gives the direction for greater growth of the function
. Note also that the components of this vector are also polynomial functions of the
coordinates, so
is a continuous vector function.
Consider now a point, that is,
. As
and
are continuous, there is a neighbor of
on
, such that for
the vector
with
, whereas the vector
with
, due to the
function growths along the
direction. Thus, the Kummer surface
separates spacial regions with positive
determinant from that with negative
determinant.
In its diagonal form, Equation (26) is
(28)
where the, that is
(29)
Note that Equation (29) implies that there is at least one eigenvalue that is zero over all the Kummer surface. But on other eigenvalues can also become null. Thus, the Kummer surface is a variable
rank surface.
As is equal to
, the vector
can be written as
(30)
Let say that is the zero eigenvalue over all Kummer surface. Then over
, the vector
is given by
(31)
If is the subregion of
over which there are
null eigenvalues, then by (31)
(32)
Thus higher order rank subregions of the Kummer surface are characterized by the fact that the
vector vanishes on them.
Consider now, the origin where
. It is easy to show that for points
near to the origin, the function
goes as
by expanding
in Taylor series around the origin and keeping the least order terms in the expansion. The
vector near the origin is then
and its an inward radial vector. So near the origin, the constant determinant surfaces
are given approximately by M di- mensional spheres and
growths inward to the origin.
Let a curve that starts in the origin and that is normal to all
surfaces, that is, its tangent vector is parallel to the
vector in each point. Because, near the origin the vector
is radial, one can reach any point of the space starting from the origin using such a curve. Moving along
in the outer direction, the
function always decreases from its initial value 1. Thus, at some point
in
, the
function vanishes. Thus means that the Kummer surface
must contain a closed subsurface
that enclosed the origin. Then inside of this closed subsurface
the determinant of the
matrix must be positive and outside
there are points where the determinant of the correlation matrix is necessarily negative. Note that
can be contained totally inside the hypercube or can cut it in different regions with positive or negative determinant values respectively.
Thus, outside there are regions where the determinant
(33)
so at least one of the eigenvalues must be negative outside. Inside
however
(34)
This implies that pairs of eigenvalues can be negative. But inside the eigenvalue cannot be negative. To prove that, consider the origin
where all eigenvalues
are equal to one. When
moves outward along a curve
that start at the origin, each eigenvalue
will change its value from its initial positive value 1, but cannot become negative. If
for some points
along
inside of
, then there is a point
where
. This implies that the vector
would cross the surface
, but it is impossible because
is inside of
where
. Then inside the surface
all eigenvalues of the correlation matrix are positive.
In order to grasp the above ideas in detail the case of three assets is studied in the next sub section.
The Geometry of the N = 3 Assets Case
The matrix, for the three assets case, is equal to
(35)
where the vector is written as
. For this parameteri- zation the determinant of the
matrix is
The constant determinant surfaces
in the interior of the hypercube are shown in Figure 1, for some positive values between
. Instead, in Figure 2, some surfaces for negative C values are displayed with
.
The Kummer surface is given by the condition
, that is
(36)
Figure 1. (a), (b)
, (c)
, (d)
, (e)
.
From (36) one found that the Kummer subsurface inside the hypercube is given by the parametric equations
(37)
Figure 3 shows the Kummer superior subsurface given by
, the Kummer inferior subsurface
given by
and the complete Kummer subsurface
.
Because separates a region with
from that with
and due to the origin
the determinant is one, then inside of
the determinant of the
matrix must be positive, which is consistent with Figure 1. The region situated between
and the cube has negative determinant in this case.
In terms of its diagonal form, the matrix inside or outside
where
, is
Figure 2. (a), (b)
, (c)
, (d)
, (e)
.
(38)
where the three eigenvalues,
and
when
.
On the Kummer superior subsurface, the diagonal form of the
matrix is
(39)
where
(40)
Figure 3. (a) Kummer superior subsurface, (b) Kummer in- ferior subsurface
, (c) complete Kummer subsurface
. Note that the Kummer subsurface
is closed and its is completely inside the hypercube in this case. Thus the region between
and the hypercube has negative
determinant for the three assets system.
and
(41)
Figure 4 gives the eigenvalues and
as functions of x and y.
For the Kummer inferior subsurface, the diagonal form of the
matrix is instead
(42)
Figure 4. (a), (b)
.
where
(43)
and
(44)
Figure 5 gives the eigenvalues and
as functions of x and y.
Note that the eigenvalues and
are always greater than zero, but
and
are zero for the extreme values of the correlation parameter
and
. Figure 6 shows both eigenvalues
and
in the same graph. It is possible to see clearly that the
proper value becomes equal to zero only for the extreme correlations value cases
(45)
which are the vertexes of the Kummer subsurface in Figure 3 or the four base points of Figure 6.
Thus, depending on which region of the three dimensional cube the vector is lying, the correlation matrix
has two null eigenvalues, one null eigenvalue or it can be invertible. Thus the rank of the
matrix changes when
moves along the Kummer surface.
5. Pricing, the Wei-Norman Theorem, Propagators and SK
The problem of pricing the multi-asset option is now tackled by taking into account the geometrical properties of the correlation
matrix analyzed in the Section 3. In order to do that one needs first to solve the Equation (23). For this, the Wei-Norman theorem [23] - [26] is applied. In this particular case this theorem estab- lishes that the solution of (23) can be writing as
(46)
Figure 5. (a), (b)
.
Figure 6. The eigenvalue as function of
.
where
(47)
with
(48)
and
(49)
that is
(50)
by inserting N one dimensional Dirac’s deltas, one can write the above equation as
(51)
or as
(52)
where the propagator is defined by
(53)
with the N dimensional Dirac’s delta. Now using the Fourier expansion
(54)
the propagator can be written finally as the product
(55)
The Propagator Inside S0
When is inside of
, all eigenvalues
are positive, so the N integrations in (55) can be performed to give [27] [28]
(56)
or
(57)
By using transformations (15), (16), (17) and (18) one can write the propagator for the option price in the
space as
(58)
with
(59)
which is the usual form of the propagator in the S space (see for example [3] [7] ). Note this form of the propagator is valid only when. So (58) can be applied inside the closed subsurface
or some region between
and the interior of the hypercube that verifies
and have only positive eigenvalues.
6. The Propagator for the Kummer Surface SK
In this section, an expression for the propagator over the Kummer surface is obtained. It is assumed that a region
of
that has
non zero eigenvalues and
null eigenvalues. Due to it is on the
surface, the Equation (26) implies that one of the coordinates of the
vector, is determined by the other
coordinates. These independent coordinates are called
. Thus in this section, the vector
is an M dimensional vector that depends on
in- dependent coordinates. In this situation the propagator in (55) gives
(60)
By performing the integrations
(61)
If the N dimensional vector is separated in two parts as
(62)
the above propagator can be written in a more compact form as
(63)
where
(64)
is the reduced diagonal matrix on the Kummer surface
. If one separates the vector
in A and B components as
(65)
then relation (20) induces the transformation
(66)
where,
,
and
are the matrices that result from sectioning
into A and B components.
The quadratic term in the exponential of (61) can be expressed in the and
components as
(67)
Now, from (66)
(68)
The Dirac’s delta in (63) implies that
(69)
The above equation permits writing the vector in terms of
as
(70)
replacing in (67) one can write the quadratic term as
(71)
where is defined by
(72)
From (66)
(73)
Using (68) and (71) in (52), the option price can be written as
(74)
Integrating over gives
(75)
where must be evaluated from (70) in terms of
and
as
(76)
where the rectangular matrix
is defined by
(77)
It must be noted that, the eigenvalues
, and the rectangular matrix
are functions of the vector
that lies on the null surface
. Thus the option price is also a function of
. Using (15), (16), (17) and (18) one can write the option price in the
space as
and is given by
(78)
where the components of the are given by
(79)
and the components of the vector are given in terms of
,
and
according to
(80)
with the components of the rectangular matrix
(81)
When moves over the Kummer surface
, the rank of the
matrix can change, so the dimensions of
and
also change, but Equation (78) is always valid.
7. The Propagator Outside S0
When the vector is lying outside the Kummer subsurface
, there are regions where the determinant of the correlation matrix is negative. This implies that the propagator given in (58) becomes complex. But, worse than that, in this case one of the eigenvalues
is negative, so the propagator given in (57) generates an exponential growth in the associated
coordinate. Then the convolution in (52) is not well defined. Thus, one cannot price the option in regions outside the Kummer subsurface
that have negative
determinant.
8. Conclusions and Further Research
In this research, the existence of the solution of the multi-asset Black-Scholes model has been analyzed in detail. It has been shown that the correlation parameter space, which is equivalent to an N dimensional hypercube, limits the existence of a valid solution for the multi-asset Black-Scholes model. Particularly, it has been demonstrated that inside of this hypercube there is a surface, called the Kummer surface, where the determinant of the correlation matrix
is zero, the usual formula for the propagator of the N asset Black-Scholes equation is no longer valid. In particular, the case for three assets and its implied geometry has been studied in detail when the determinant of the correlation matrix is zero. Finally, by using the Wei-Norman theorem, the propagator over the variable rank surface
for the general N asset case has been computed, which is applicable over all the Kummer surface, whatever be the rank of the
matrix. This formulation corrects the past solution of this problem and its extensions.
As future research, most of the papers related to the multi-asset Black-Scholes model must be revisited in line of our results, as well as others where it is implicitly assumed that a well behaved multivariate Gaussian distribution must exist, as is the case of the stochastic volatility family (see for instance [29] [30] ).
Cite this paper
Contreras, M., Llanquihuén, A. and Villena, M. (2016) On the Solution of the Multi-Asset Black-Scholes Model: Correlations, Eigenvalues and Geometry. Journal of Mathematical Finance, 6, 562-579. http://dx.doi.org/10.4236/jmf.2016.64043
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