Mathematical Problems of Computer Science 49, 97–102, 2018.

Existence of Maximum Entropy Problem Solution in a

General N-Dimensional Case

Ruben A. Gevorgyan and Narek D. Margaryan

Yerevan State University

ruben−gevorgyan@yahoo.com, narek−margaryan@outlook.com

Abstract

In the following paper, we will define conditions, which need to be satisfied in
order for the maximum entropy problem applied in European call options to have a
solution in a general n-dimensional case. We will also find a minimum right boundary
for the price range in order to have at least one risk neutral measure satisfying the
option pricing formula. The results significantly reduce the computational time of
optimization algorithms used in maximum entropy problem.

Keywords: Entropy, Boundary, Distribution, Options.

1. Introduction

The maximum entropy methodology has recently started to become a quite popular tool
with a huge potential of application in different fields [1, 2, 3, 4]. The core of the theory is
based on Shannon’s classical definition of information entropy [5], which is a crucial foun-
dation in information theory. The maximum entropy approach has been broadly studied
for its application in finance and financial extrapolation [6], and there have been significant
contributions to its development since then, including the application of Legendre transforms
[7], partially finite convex programming [8], the employment of risk neutral moments [9], as
well as the application of the problem as a non-parametric approach in American options
pricing [10]. By theory, in the discrete case, the price of a European call option should be
equal to the mathematical expectation of future pay-offs’ discounted value, thus lying in
their convex hull. In reality, actual market prices may be biased from the theoretical ones
[11] and lie out of the convex hull. We will concentrate on the derivation of conditions for
the existence of solution which will not only reduce the computational time but will also
result in an automated distribution recovery process [12, 13, 14] and will later allow us to
develop algorithmic trading strategies that train on huge data sets.

2. Outline of the Problem

Consider having European call options for n different strike prices. Let us denote the vector
of strike prices with K and the vector of future states with X. X and K needn’t be of the

97



98 Existence of Maximum Entropy Problem Solution in a General N-Dimensional Case

same dimension. Maximum entropy methodology seeks a risk neutral probability measure
p, such that

Ap = b, (1)
n∑

i=1

pi = 1, pi ≥ 0, (2)

S(p) =
n∑

i=1

piln(pi) is maximal, (3)

where b is the vector of current option prices’ future values for each strike and A is
(X1 − K1)

+ (X2 − K1)+ . . . (Xn − K1)+
...

...
...

...
(X1 − Kn)+ (X2 − Kn)+ . . . (Xn − Kn)+


 , (4)

where (x)+ = max(x, 0). The probability vector p and the vector of future states X have
the same dimension, in fact pi is the probability mass assigned to the future state Xi. The
distribution of future states will change as we change the state vector X. The question
that interests us is what kind of state vector should be considered in order for a probability
measure satisfying (1), (2) to exist in the first place. It is obvious that the greater the
number of A’s linearly independent columns is, the bigger will their convex hull be, and so
the more p vectors may exist satisfying (1), (2). So first of all we will consider the state
vector (K1, . . . , Kn, Kn + t) for some arbitrary t. Matrix A will now have the form below.


0 K2 − K1 . . . Kn − K1 Kn − K1 + t
0 0 . . . Kn − K2 Kn − K2 + t
...

...
...

...
...

0 0 . . . Kn − Kn−1 Kn − Kn−1 + t
0 0 . . . 0 t


 . (5)

We will denote A’s columns by a0, a1, . . . , an. Let α(t) denote the angle between an and I,
where I is the unit vector (1, . . . , 1). It is easy to show that limt→∞ cos α(t) = 1, so in order
to see if any t exists, s.t. (1), (2) are satisfied, we will consider I instead of an, assuming
that the angle between b and I isn’t 0 (this assumption holds throughout the text). Let’s
consider the following n + 1 hyperplane - vector pairs (we denote hyperplanes by hp(·)).



hp(a1, a2, . . . , an−1, I), a0

hp(a0, a2, . . . , an−1, I), a1
...

hp(a0, a1, . . . , an−2, I), an−1

hp(a0, a1, . . . , an−2, an−1), I

. (6)

For each hyperplane above, we will denote by Ni its normal “pointing” in the direction
of the associated vector ai (note that a0 = (0, . . . , 0))


⟨N0 − a1, a0 − a1⟩ ≥ 0
⟨N1, a1⟩ ≥ 0
...

⟨Nn, an⟩ ≥ 0

, (7)



R. Gevorgyan and N. Margaryan 99

where ⟨·, ·⟩ denotes the scalar (dot) product.

3. Existence of Solution

The following proposition is obvious.

Proposition 1. There exists a finite t, s.t. (1), (2) are satisfied if and only if the following
inequalities take place.



⟨N0 − a1, b − a1⟩ ≥ 0
⟨N1, b⟩ ≥ 0
...

⟨Nn, b⟩ ≥ 0

. (8)

We now proceed to finding a minimal value for t, s.t. conditions (1) and (2) are satisfied.
b represents the vector of prices and, thus its components are non-negative. Assume that
(8) takes place. If the last component of b, bn is 0, then the minimal value of t for which
b ∈ conv(a0, a1, . . . , an−1, an) is 0 (conv(·) denotes the convex hull). In case bn is greater
than 0, we will use the following lemmas (note that an = an−1 + tI).

Lemma 1. ∃µ > 0, s.t. ∀t for which b ∈ conv(a0, . . . , an−1, an), t ≥ µ > 0.

Proof. Assume the opposite, then ∀ϵ > 0 ∃t0 < ϵ, s.t. ∃γ0, . . . , γn, γi ≥ 0,
∑n

i=0 γi = 1, for
which γ0a0 + . . . + γnan = b. Let

r = inf
q∈conv(a0,...,an−1)

ρ(b, q),

ϵ =
r

ρ(b, I)
,

where ρ is the Euclidean distance. For the ϵ above there exists 0 < t0 < ϵ, s.t. γ0a0+. . .+
γnan = b ⇔ γ0a0 + . . . + (γn−1 + γn)an−1 + t0γnI = b ⇒ ρ(γ0a0 + . . . + (γn−1 + γn)an−1, b) =
ρ(b − t0γnI, b) ≤ t0ρ(I, b) < r, resulting in a contradiction.

Lemma 2. If for some t0 b ∈ conv(a0, . . . , an), then this also holds for any t > t0.

Proof. Let t > t0, γ
′
n−1 = γn−1 + γn

t−t0
t
, γ

′
n = γn

t0
t
, then γ

′
n−1 + γ

′
n = γn−1 + γn and

γ0a0 + . . . + γ
′
n−1an−1 + γ

′
nan = γ0a0 + . . . + γnan = b

We now know that the set T of all possible t’s for which b ∈ conv(a0, . . . , an) is bounded
from below by a positive number and unbounded from above. The next lemma proves that
for t = inf T b is again in the convex hull conv(a0, . . . , an).

Lemma 3. Let T be the set of all t’s, s.t. b ∈ conv(a0, . . . , an), then t = inf T ∈ T.

Proof. Let’s assume the opposite. As t is the infimum of T, then for ∀ϵ > 0 ∃t0 ∈ T, s.t.
0 < t0 − t < ϵ. Let

r = inf
q∈conv(a0,...,an−1,an−1+tI)

ρ(b, q),



100 Existence of Maximum Entropy Problem Solution in a General N-Dimensional Case

ϵ =
r

ρ(b, I)
.

By the assumption there exists a t0 < t+ϵ, s.t. b = γ0a0 +. . .+γnan. Let a
′
n = an−1 +tI,

then ρ(γ0a0 + . . . + γn−1an−1 + γna
′
n, b) = ρ(b + (t − t0)I, b) ≤ (t0 − t)ρ(I, b) < r, resulting in

a contradiction.

Based on the lemmas we may now formulate the main theorem of the article.

Theorem 1. If condition (8) is satisfied, the angle between b and I isn’t 0, then b ∈
conv(a0, . . . , an), where an = an−1 + tI and γn−1 = 0 in the linear representation of b by
vectors a0, . . . an. The minimal value of t, t is given by

t =
bn(Kn − Kn−1)

bn−1 − bn
. (9)

Proof. We only need to show that γn−1 = 0. Assume it’s not, then

b = γ0a0 + . . . + γnan = γ0a0 + . . . + γn−1an−1 + γn(an−1 + tI) =

γ0a0 + . . . + (γn + γn−1)an−1 +
tγn

γn + γn−1
(γn + γn−1)I =

γ0a0 + . . . + (γn + γn−1)(an−1 +
tγn

γn + γn−1
I).

As γn > 0, then

tγn
γn + γn−1

< t,

Which is a contradiction. Having known that an−1 doesn’t ”participate” in the linear
representation of b, we only need to find the value of t, s.t. b ∈ hp(a0, . . . , an−2, an). For
that we will find the normal N of the hyperplane and solve ⟨N, b⟩ = 0 for t. We find N by
observing the determinant of the following matrix based on the vectors from the hyperplane.



K2 − K1 0 . . . . . . 0
K3 − K1 K3 − K2 0 . . . 0

...
Kn−1 − K1 Kn−1 − K2 . . . 0 0
Kn − K1 + t Kn − K2 + t . . . Kn − Kn−1 + t t

e1 e2 . . . en−1 en



. (10)

The determinant is

(−1)2n−1t(K2 − K1) . . . (Kn−1 − Kn−2)en−1+

(−1)2n(Kn − Kn−1 + t)(K2 − K1) . . . (Kn−1 − Kn−2)en.

So N = (0, . . . , 0, −t, Kn − Kn−1 + t), and therefore

⟨N, b⟩ = 0 ⇔ t =
bn(Kn − Kn−1)

bn−1 − bn
.



R. Gevorgyan and N. Margaryan 101

4. Conclusion

As a result we obtained a way of checking whether a solution to the maximum entropy
problem applied in European call options exists, before starting the optimization. If (8)
takes place, then in order for the solution to exist, the right bound of future states vector
must be greater than or equal to the value of t described in the theorem above. Checking
the existence of solution prevents the user from unknowingly proceeding to the stage of
entropy maximization over an empty set of discrete probability distributions, which would
yield unpredictable results.

References

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[9] L. S. Rompolis, “A new method of employing the principle of maximum entropy to
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[10] Yu. Xishen and Li Yang, “Pricing american options using a nonparametric entropy
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[12] N. D. Margaryan “Assessment of asset price distributions using maximum entropy
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[13] N. D. Margaryan “An algorithmic approach to solving the maximum entropy problem”,
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[14] N. D. Margaryan “A boundary for the existence of solution to the maximum entropy
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52, no 1, pp. 3-7, 2018.

Submitted 18.10.2017, accepted 12.02.2018.



1 0 2 Existence of Maximum Entropy Problem Solution in a General N-Dimensional Case

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