THE EXISTENCE OF THE RISK-EFFICIENT OPTIONS
THE EXISTENCE OF THE RISK-EFFICIENT OPTIONS
The Pure and Applied Mathematics. 2014. Nov, 21(4): 307-316
• Received : September 06, 2014
• Accepted : November 13, 2014
• Published : November 30, 2014 PDF e-PUB PubReader PPT Export by style
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JU HONG, KIM

Abstract
We prove the existence of the risk-effcient options proposed by Xu  . The proof is given by both indirect and direct ways. Schied  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 risk-effcient option.
Keywords
1. INTRODUCTION
Let PPT Slide
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be a complete filtered probability space. Let S =( St ) 0≤tT be an adapted positive process which is a semimartingale. It is assumed that the riskless interest rate is zero for simplicity and PPT Slide
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to avoid the arbitrage opportunities  .
Definition 1.1. A self-financing strategy ( x , ξ ) is defined as an initial capital x ≥ 0 and a predictable process ξt such that the value process (value of the current holdings) PPT Slide
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is P -a.s. well-defined.
The set of admissible self-financing portfolios 𝒳( x ) with initial capital x is defined as PPT Slide
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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 PPT Slide
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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 ρx (·) with initial capital x is defined as the risk PPT Slide
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where the liability L is a random variable bounded below by a constant at time T , PPT Slide
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and ρ ( L XT ) is a final risk.
Assumption 1.5. The convex risk measure ρ satisfies the Fatou property PPT Slide
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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 PPT Slide
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Lemma 1.7 (  ) . The minimal risk defined as (1.2) is a convex risk measure. Moreover, the translation invariance property satisfies the following relations PPT Slide
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Lemma 1.8 (  ) . Let L be the initial liability bounded below by a constant and H be the positive payoff function. Then for any fixed number x PPT Slide
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The risk-effcient options are defined as the options having the same selling price, which minimize the risk. That is, the risk-effcient 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  defined such risk-effcient 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 risk-effcient options. Every contingent claim is replicable, i.e., perfectly hedged in a complete market. We should consider risk-effcient options in an incomplete market.
This paper is structured as follows. We prove the existence of risk-effcient options by using Schied’s result in Section 2. We prove it by finding the sequences converging to the risk-effcient option in Section 3.
2. INDIRECT PROOF
In this section, we assume that 𝜌 is convex risk measure satisfying Fatou property and H is PPT Slide
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−measurable contingent claim which is bounded. Xu  treated option H which is positive.
Schied  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., PPT Slide
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where the price density 𝜑 is a P −a.s. strictly positive random variable with E [𝜑] = 1. The problem is called the Neyman-Pearson problem for the risk measure ρ .
Lemma 2.1 (  ) . 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 Neyman-Pearson problem (2.1) .
Lemma 2.2 (  ) . Any solution H* of the Neyman-Pearson 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 PPT Slide
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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 risk-effcient option H * satisfying PPT Slide
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where L is the initial liability uniformly bounded below by cL , 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 PPT Slide
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Then η is well defined by Assumption 1.6.
Lemma 2.3. η (− H ) is a convex risk measure and law-invariant .
Proof . First, let’s prove the convexity. Let H 1 , H 2 and H be PPT Slide
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-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 PPT Slide
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Thirdly, let’s prove the translation invariance. PPT Slide
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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 PPT Slide
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Theorem 2.4. If x ∈ (0, K ), then there exists H * ∈ [0, K ], E [𝜑 H *] = x such that PPT Slide
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Proof. η ( H ) is a convex risk measure by Lemma 2.3. By Lemmas 2.1 and 2.2, it is proved. PPT Slide
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Now we give bounded conditions to x for the E [𝜑 H *] = x to be a no-arbitrage price. Xu  defined the selling price SP and the buying price BP of the option H (≥ 0) as PPT Slide
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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 x0x ( 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 PPT Slide
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Thus for the E [𝜑 H *] = x to be a no-arbitrage price of H *, it should satisfy the inequalities
• SP(H) ≤E[𝜑H*] =x≤BP(H).
3. DIRECT PROOF
In this section, wefind the sequences converging to the risk-effcient option for the proof of its existence.
Lemma 3.1 (Föollmer and Schied  ) . Let ( ξn ) n≥1 be a sequence in PPT Slide
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such that sup n | ξn | < +∞ P-a.s .. Then there exists a sequence of convex combinations
• ηn∈con𝜐{ξn,ξn+1,…}
which converges P-a.s. to some PPT Slide
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Define
• 𝒳(x,b) = {X|X∈ 𝒳(x) andXT≥x − b}.
Then we have
Theorem 3.2 (  ) . Under two assumptions (1.3) and (1.4) and PPT Slide
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, there exists an optimal admissible hedging portfolio X* ∈ 𝒳( x , b ) which is the solution of the minimal risk problem PPT Slide
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for any b ∈ ℝ + and x ∈ ℝ.
Let H be a payoff function of an option, x ∈ ℝ + , and let PPT Slide
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be fixed.
Lemma 3.3. There exists PPT Slide
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−measurable H* and PPT Slide
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∈ 𝒳( x , b ), depending on H* such that EQ [ H *] = x , PPT Slide
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Proof . By Theorem 3.2, for each H there exists PPT Slide
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such that PPT Slide
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Choose the sequences Hn and PPT Slide
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satisfying PPT Slide
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Then Lemma 3.1 implies that there exist the sequences PPT Slide
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such that PPT Slide
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The sequence PPT Slide
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can be expressed as the convex combination PPT Slide
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Set PPT Slide
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in which is the sequence Hi in the chosen pair Hi and PPT Slide
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It is easy to see PPT Slide
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If we apply the Lebesgue Dominated Convergence Theorem to the equation (3.2), then there exists H * such that PPT Slide
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Q -a.s., and EQ [ H *] = 𝑥.
So we have PPT Slide
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By applying the Fatou property to PPT Slide
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and also using the inequality (3.3), we have PPT Slide
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Since EQ [ H *] = x and PPT Slide
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we have PPT Slide
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Theorem 3.4. Let p ( H ) = EQ [ H ] be the pricing rule of the option H for a fixed PPT Slide
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. Let x 0 be an initial capital. Then there exists a risk-efficient option H* satisfying PPT Slide
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where L is the initial liability uniformly bounded below by cL .
Proof . Let PPT Slide
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be fixed. Since ρ x+x0 ( L + H ) = ρ x ( L + H ) − x 0 , we need only to consider
• ρx(L + H).
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 well-defined for all X ∈ 𝒳( x ).
By Theorem 3.2, for each H there exists PPT Slide
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such that PPT Slide
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Let ∊ > 0. Then since PPT Slide
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there exists a large nonnegative integer N ∈ ℤ + satisfying PPT Slide
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The equation (3.4) and Lemma 3.3 imply the following inequality PPT Slide
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So we have PPT Slide
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and so PPT Slide
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On the other hand, since PPT Slide
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we have the inequality PPT Slide
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and by letting b go to infinity we get PPT Slide
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By the inequalities (3.5) and (3.6), we get PPT Slide
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The theorem has been proved. PPT Slide
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For the pricing rule EQ [ H ] = x of the option H to be an no-arbitrage price, it should also satisfy
• SP(H) ≤x≤BP(H),
as we showed the reason in Section 2.
Acknowledgements
This work was supported by the research grant of Sungshin Women’s University in 2014.
References