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THE EXISTENCE OF THE RISK-EFFICIENT OPTIONS
THE EXISTENCE OF THE RISK-EFFICIENT OPTIONS
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2014. Nov, 21(4): 307-316
Copyright © 2014, Korean Society of Mathematical Education
  • Received : September 06, 2014
  • Accepted : November 13, 2014
  • Published : November 30, 2014
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JU HONG KIM

Abstract
We prove the existence of the risk-effcient 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 risk-effcient option.
Keywords
1. INTRODUCTION
Let
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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
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to avoid the arbitrage opportunities [4] .
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)
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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
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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
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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
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where the liability L is a random variable bounded below by a constant at time T ,
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and ρ ( L XT ) is a final risk.
Assumption 1.5. The convex risk measure ρ satisfies the Fatou property
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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
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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
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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
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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 [7] 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
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−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.,
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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 ( [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 Neyman-Pearson problem (2.1) .
Lemma 2.2 ( [6] ) . 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
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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
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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
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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
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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
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Thirdly, let’s prove the translation invariance.
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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
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Theorem 2.4. If x ∈ (0, K ), then there exists H * ∈ [0, K ], E [𝜑 H *] = x such that
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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.
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Now we give bounded conditions to x for the E [𝜑 H *] = x to be a no-arbitrage price. Xu [7] defined the selling price SP and the buying price BP of the option H (≥ 0) as
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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
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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 [5] ) . Let ( ξn ) n≥1 be a sequence in
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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
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Define
  • 𝒳(x,b) = {X|X∈ 𝒳(x) andXT≥x − b}.
Then we have
Theorem 3.2 ( [7] ) . Under two assumptions (1.3) and (1.4) and
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, there exists an optimal admissible hedging portfolio X* ∈ 𝒳( x , b ) which is the solution of the minimal risk problem
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for any b ∈ ℝ + and x ∈ ℝ.
Let H be a payoff function of an option, x ∈ ℝ + , and let
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be fixed.
Lemma 3.3. There exists
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−measurable H* and
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∈ 𝒳( x , b ), depending on H* such that EQ [ H *] = x ,
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Proof . By Theorem 3.2, for each H there exists
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such that
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Choose the sequences Hn and
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satisfying
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Then Lemma 3.1 implies that there exist the sequences
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such that
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The sequence
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can be expressed as the convex combination
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Set
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in which is the sequence Hi in the chosen pair Hi and
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It is easy to see
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If we apply the Lebesgue Dominated Convergence Theorem to the equation (3.2), then there exists H * such that
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Q -a.s., and EQ [ H *] = 𝑥.
So we have
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By applying the Fatou property to
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and also using the inequality (3.3), we have
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Since EQ [ H *] = x and
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we have
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Theorem 3.4. Let p ( H ) = EQ [ H ] be the pricing rule of the option H for a fixed
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. Let x 0 be an initial capital. Then there exists a risk-efficient option H* satisfying
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where L is the initial liability uniformly bounded below by cL .
Proof . Let
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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
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such that
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Let ∊ > 0. Then since
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there exists a large nonnegative integer N ∈ ℤ + satisfying
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The equation (3.4) and Lemma 3.3 imply the following inequality
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So we have
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and so
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On the other hand, since
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we have the inequality
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and by letting b go to infinity we get
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By the inequalities (3.5) and (3.6), we get
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The theorem has been proved.
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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.
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