THE EXISTENCE OF THE RISK-EFFICIENT OPTIONS

Abstract

We prove the existence of the risk-efficient 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-efficient option.

1. INTRODUCTION

Let be a complete filtered probability space. Let S=(St)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 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)

is P-a.s. well-defined.

The set of admissible self-financing portfolios 𝒳(x) with initial capital x is defined as

Let L0 be the set of all measurable functions in the given probability spaces.

Definition 1.2. A coherent measure of risk ρ : L0 → ℝ∪{∞} is a mapping satisfying the following properties for X, Y ∈ L0

(1) ρ(X + Y ) ≤ ρ(X) + ρ(Y ) (subadditivity),(2) ρ(λX) = λρ(X) for λ ≥ 0 (positive homogeneity),(3) ρ(X) ≥ ρ(Y ) if X ≤ Y (monotonicity) ,(4) ρ(Y + m) = ρ(Y ) − m for m ∈ 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 : L0 → 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

where the liability L is a random variable bounded below by a constant at time T,

and ρ(L − XT ) is a final risk.

Assumption 1.5. The convex risk measure ρ satisfies the Fatou property

Assumption 1.6. ρ : L0 → 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 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 ρx0+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, x0 is the initial capital, and ρx0+α(L + H) is the minimal risk obtained by optimal hedging with capital x0 + 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 −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 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[𝜑H*] = 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 x0, we show that in Theorem 2.4 there exists a risk-effcient option H* satisfying

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

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 H1, H2 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 H1 ≤ H2. Then

Thirdly, let’s prove the translation invariance.

So η is a convex risk measure.

Last, let’s prove η(−H1) = η(−H2) whenever H1 = H2 P−a.s.. Let H1 = H2 P−a.s.. Then we have L + H1 = L + H2 P−a.s.. Since ρ(L + H1) = ρ(L + H2), we get

Theorem 2.4. If x ∈ (0, K), then there exists H* ∈ [0, K], E[𝜑H*] = 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[𝜑H*] = 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

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 ρx0+x(L + H) and ρx0−x(L − H) do not exceed the initial risk ρx0(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[𝜑H*] = 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 such that supn|ξn| < +∞ P-a.s .. Then there exists a sequence of convex combinations

ηn ∈ con𝜐{ξn, ξn+1,…}

which converges P-a.s. to some

Define

𝒳(x, b) = {X | X ∈ 𝒳(x) and XT ≥ 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 EQ[H*] = x,

Proof. By Theorem 3.2, for each H there exists such that

Choose the sequences Hn 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 Hi in the chosen pair Hi 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 EQ[H*] = 𝑥.

So we have

By applying the Fatou property to and also using the inequality (3.3), we have

Since EQ[H*] = x and we have

Theorem 3.4. Let p(H) = EQ[H] be the pricing rule of the option H for a fixed . Let x0 be an initial capital. Then there exists a risk-efficient option H* satisfying

where L is the initial liability uniformly bounded below by cL.

Proof. Let be fixed. Since ρx+x0(L + H) = ρx(L + H) − x0, 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 ρx(L + H) is well-defined 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 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.