We prove the existence of the riskeffcient options proposed by Xu
[7]
. The proof is given by both indirect and direct ways. Schied
[6]
showed the existence of the optimal solution of equation (2.1). The one is to use the Schied’s result. The other one is to find the sequences converging to the riskeffcient option.
1. INTRODUCTION
Let
be a complete filtered probability space. Let
S
=(
S_{t}
)
_{0≤t≤T}
be an adapted positive process which is a semimartingale. It is assumed that the riskless interest rate is zero for simplicity and
to avoid the arbitrage opportunities
[4]
.
Definition 1.1.
A
selffinancing
strategy (
x
,
ξ
) is defined as an initial capital
x
≥ 0 and a predictable process
ξ_{t}
such that the value process (value of the current holdings)
is
P
a.s. welldefined.
The set of admissible selffinancing portfolios 𝒳(
x
) with initial capital
x
is defined as
Let
L
^{0}
be the set of all measurable functions in the given probability spaces.
Definition 1.2.
A
coherent measure of risk ρ
:
L
^{0}
→ ℝ∪{∞} is a mapping satisfying the following properties for
X
,
Y
∈
L
^{0}

(1)ρ(X + Y) ≤ρ(X) +ρ(Y) (subadditivity),

(2)ρ(λX) = λρ(X) for λ ≥ 0 (positive homogeneity),

(3)ρ(X) ≥ρ(Y) ifX≤Y(monotonicity) ,

(4)ρ(Y+m) =ρ(Y) −mform∈ ℝ (translation invariance).
The conditions of subadditivity (1) and positive homogeneity (2) in Definition 1.2 can be relaxed to a weaker quantity, i.e., convexity
Convexity means that diversification does not increase the risk. Also refer to the papers
[1
,
3]
for coherent or convex risk measures.
Definition 1.3.
A map
ρ
:
L
^{0}
→ ℝ is called a
convex risk measure
if it satisfies the properties of convexity (1.1), monotonicity (3) and translation invariance (4).
Definition 1.4.
The
minimal risk ρ^{x}
(·) with initial capital
x
is defined as the risk
where the liability
L
is a random variable bounded below by a constant at time
T
,
and
ρ
(
L
−
X_{T}
) is a final risk.
Assumption 1.5.
The convex risk measure
ρ
satisfies the
Fatou property
Assumption 1.6.
ρ
:
L
^{0}
→ ℝ satisfies
ρ
(
X
) =
ρ
(
Y
) whenever
X = Y P
− a.s. and for the positive payoff function
H
, the bounded conditions
Lemma 1.7
(
[7]
)
.
The minimal risk defined as (1.2) is a convex risk measure. Moreover, the translation invariance property satisfies the following relations
Lemma 1.8
(
[7]
)
.
Let L be the initial liability bounded below by a constant and H be the positive payoff function. Then for any fixed number x
The riskeffcient options are defined as the options having the same selling price, which minimize the risk. That is, the riskeffcient options are the
H
that minimizes
ρ
^{x0+α}
(
L + H
) with the constraint
p
(
H
) =
α
, where
p
(
H
) is the selling price of the option
H
,
L
is the initial liability,
x
_{0}
is the initial capital, and
ρ
^{x0+α}
(
L + H
) is the minimal risk obtained by optimal hedging with capital
x
_{0}
+
α
as defined in (1.2). Here
ρ
is a risk measure. Xu
[7]
defined such riskeffcient options and asked a question of their existence. The option seller could get the same minimal risk even though he or she choose any one of available riskeffcient options. Every contingent claim is replicable, i.e., perfectly hedged in a complete market. We should consider riskeffcient options in an incomplete market.
This paper is structured as follows. We prove the existence of riskeffcient options by using Schied’s result in Section 2. We prove it by finding the sequences converging to the riskeffcient option in Section 3.
2. INDIRECT PROOF
In this section, we assume that 𝜌 is convex risk measure satisfying Fatou property and
H
is
−measurable contingent claim which is bounded. Xu
[7]
treated option
H
which is positive.
Schied
[6]
supposes an agent wishes to raise the capital 𝜐(≥ 0) by selling a contingent claim and tries to find a contingent claim such that the risk of the terminal liability is minimal among all claims satisfying the issuer’s capital constraints, i.e.,
where the price density 𝜑 is a
P
−a.s. strictly positive random variable with
E
[𝜑] = 1. The problem is called the
NeymanPearson problem
for the risk measure
ρ
.
Lemma 2.1
(
[6]
)
.
Assume that the conditions of convexity (1.1), monotonicity in Definition 1.2 and Fatou property (1.3) hold. Then there exists a solution to the NeymanPearson problem (2.1)
.
Lemma 2.2
(
[6]
)
.
Any solution H* of the NeymanPearson problem (2.1) with capital constraint
𝜐 ∈ [0,
K
]
satisfies E
[𝜑
H
*] = 𝜐.
In terms of liabilities −
X
and −
Y
, the properties of convexity (1.1), monotonicity (3) and translation invariance (4) in Definition 1.2 are respectively expressed as
The properties of (2.2), (2.3) and (2.4) can be easily derived by taking
ρ
(−
X
) = 𝜓(
X
) for a convex risk measure 𝜓(
X
).
For the option payoff function
H
and an initial capital
x
_{0}
, we show that in Theorem 2.4 there exists a
riskeffcient option H
* satisfying
where
L
is the initial liability uniformly bounded below by
c_{L}
, and the price density 𝜑 is a
P
−a.s. strictly positive random variable with
E
[𝜑] = 1.
In a term of liability −
H
, define
η
as
Then
η
is well defined by Assumption 1.6.
Lemma 2.3.
η
(−
H
)
is a convex risk measure and lawinvariant
.
Proof
. First, let’s prove the convexity. Let
H
_{1}
,
H
_{2}
and
H
be
measurable payoff functions and λ ∈ [0, 1], m ∈ ℝ.

η(λ(−H1) + (1 − λ)(−H2)) =ρx+x0(L+ λH1+ (1 − λ)H2)

=ρx+x0(λ(L+H1) + (1 − λ)(L+H2))

≤ λρx+x0(L+H1) + (1 − λ)ρx+x0(L+H2)

= λη(−H1) + (1 − λ)η(−H2).
Secondly, let’s prove the monotonicity. Let
H
_{1}
≤
H
_{2}
. Then
Thirdly, let’s prove the translation invariance.
So
η
is a convex risk measure.
Last, let’s prove
η
(−
H
_{1}
) =
η
(−
H
_{2}
) whenever
H
_{1}
=
H
_{2}
P
−a.s.. Let
H
_{1}
=
H
_{2}
P
−a.s.. Then we have
L
+
H
_{1}
=
L
+
H
_{2}
P
−a.s.. Since
ρ
(
L
+
H
_{1}
) =
ρ
(
L
+
H
_{2}
), we get
Theorem 2.4.
If x
∈ (0,
K
),
then there exists
H
* ∈ [0,
K
],
E
[𝜑
H
*] =
x such that
Proof. η
(
H
) is a convex risk measure by Lemma 2.3. By Lemmas 2.1 and 2.2, it is proved.
Now we give bounded conditions to
x
for the
E
[𝜑
H
*] =
x
to be a noarbitrage price. Xu
[7]
defined the selling price
SP
and the buying price
BP
of the option
H
(≥ 0) as
respectively.
By the translation invariance relation (1.5), the equations (2.6) and (2.7) become

SP(H) = min{x:ρx0(L + H) −ρx0(L) ≤x}

=ρx0(L + H) −ρx0(L),

BP(H) = max{x:x≤ρx0(L) −ρx0(L−H)}

=ρx0(L) −ρx0(L−H)
respectively. Since the final risk exposure both
ρ
^{x0+x}
(
L + H
) and
ρ
^{x0−x}
(
L − H
) do not exceed the initial risk
ρ
^{x0}
(
L
), i.e.,

ρx0(L + H) −x=ρx0+x(L + H) ≤ρ𝑥0(L),

ρx0(L − H) +x=ρx0−x(L − H) ≤ρx0(L),
we have
Thus for the
E
[𝜑
H
*] =
x
to be a noarbitrage price of
H
*, it should satisfy the inequalities
3. DIRECT PROOF
In this section, wefind the sequences converging to the riskeffcient option for the proof of its existence.
Lemma 3.1
(Föollmer and Schied
[5]
)
.
Let
(
ξ_{n}
)
_{n≥1}
be a sequence in
such that
sup
_{n}

ξ_{n}
 < +∞
Pa.s .. Then there exists a sequence of convex combinations
which converges Pa.s. to some
Define

𝒳(x,b) = {XX∈ 𝒳(x) andXT≥x − b}.
Then we have
Theorem 3.2
(
[7]
)
.
Under two assumptions (1.3) and (1.4) and
≠
, there exists an optimal admissible hedging portfolio X*
∈ 𝒳(
x
,
b
)
which is the solution of the minimal risk problem
for any b
∈ ℝ
^{+}
and x
∈ ℝ.
Let
H
be a payoff function of an option,
x
∈ ℝ
^{+}
, and let
be fixed.
Lemma 3.3.
There exists
−measurable H* and
∈ 𝒳(
x
,
b
),
depending on H* such that E^{Q}
[
H
*] =
x
,
Proof
. By Theorem 3.2, for each
H
there exists
such that
Choose the sequences
H_{n}
and
satisfying
Then Lemma 3.1 implies that there exist the sequences
such that
The sequence
can be expressed as the convex combination
Set
in which is the sequence
H_{i}
in the chosen pair
H_{i}
and
It is easy to see
If we apply the Lebesgue Dominated Convergence Theorem to the equation (3.2), then there exists
H
* such that
Q
a.s., and
E^{Q}
[
H
*] = 𝑥.
So we have
By applying the Fatou property to
and also using the inequality (3.3), we have
Since
E^{Q}
[
H
*] =
x
and
we have
Theorem 3.4.
Let p
(
H
) =
E^{Q}
[
H
]
be the pricing rule of the option H for a fixed
.
Let x
_{0}
be an initial capital. Then there exists a riskefficient option H* satisfying
where L is the initial liability uniformly bounded below by c_{L}
.
Proof
. Let
be fixed. Since
ρ
^{x+x0}
(
L + H
) =
ρ
^{x}
(
L + H
) −
x
_{0}
, we need only to consider
For
X
∈ 𝒳(0), by Assumption 1.6 and translation invariance property, the following both inequality and equality

ρ(L + H−XT) ≥ρ(cL+ 0 −XT) ≥cL+ρ(−XT)

≥cL+ρ0(0) > − ∞, and

ρx(L + H) =ρ0(L + H) −x
imply that
ρ
^{x}
(
L + H
) is welldefined for all
X
∈ 𝒳(
x
).
By Theorem 3.2, for each
H
there exists
such that
Let ∊ > 0. Then since
there exists a large nonnegative integer
N
∈ ℤ
^{+}
satisfying
The equation (3.4) and Lemma 3.3 imply the following inequality
So we have
and so
On the other hand, since
we have the inequality
and by letting
b
go to infinity we get
By the inequalities (3.5) and (3.6), we get
The theorem has been proved.
For the pricing rule
E^{Q}
[
H
] =
x
of the option
H
to be an noarbitrage price, it should also satisfy
as we showed the reason in Section 2.
Acknowledgements
This work was supported by the research grant of Sungshin Women’s University in 2014.
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