Good-deal bounds for option prices under value-at-risk and expected shortfall constraints
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Desmettre, Sascha; Laudagé, Christian; Sass, Jörn Article Good-deal bounds for option prices under value-at-risk and expected shortfall constraints Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Desmettre, Sascha; Laudagé, Christian; Sass, Jörn (2020) : Good-deal bounds for option prices under value-at-risk and expected shortfall constraints, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 8, Iss. 4, pp. 1-22, https://doi.org/10.3390/risks8040114 This Version is available at: https://hdl.handle.net/10419/258067 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/
risks Article Good-Deal Bounds for Option Prices under Value-at-Risk and Expected Shortfall Constraints Sascha Desmettre 1, Christian Laudagé 2,* and Jörn Sass 3 1Institute of Financial Mathematics and Applied Number Theory, Johannes Kepler University Linz, Altenbergerstraße 69, 4040 Linz, Austria; sascha.desmettr[email protected] 2Department of Financial Mathematics, Fraunhofer Institute for Industrial Mathematics ITWM, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany 3Department of Mathematics, University of Kaiserslautern, Erwin-Schrödinger-Straße, 67663 Kaiserslautern, Germany; [email protected] *Correspondence: [email protected].de Received: 29 September 2020; Accepted: 22 October 2020; Published: 30 October 2020 Abstract: In this paper, we deal with the pricing of European options in an incomplete market. We use the common risk measures Value-at-Risk and Expected Shortfall to define good-deals on a financial market with log-normally distributed rate of returns. We show that the pricing bounds obtained from the Value-at-Risk admit a non-smooth behavior under parameter changes. Additionally, we find situations in which the seller’s bound for a call option is smaller than the buyer’s bound. We identify the missing convexity of the Value-at-Risk as main reason for this behavior. Due to the strong connection between good-deal bounds and the theory of risk measures, we further obtain new insights in the finiteness and the continuity of risk measures based on multiple eligible assets in our setting. Keywords: good-deal bounds; risk measures; multiple eligible assets; Value-at-Risk; Expected Shortfall 1. Introduction Pricing in incomplete financial markets is a demanding task for which various approaches are considered in the finance literature. On the one hand, no arbitrage pricing theory leads to wide price ranges. On the other hand, equilibrium theory could give unique prices, but these are not very robust with respect to (w.r.t.) model parameter changes. The theory of good-deal bounds is a compromise between these two concepts. Essentially, good-deals provide portfolios, which are acceptable in terms of the risk measure under consideration, but which have price zero. So, a good-deal bound not only excludes prices that lead to arbitrage opportunities, but also good-deals and thus prices that are viewed as too favorable for one of the market participants. The important point is the way of quantifying a good-deal. The seminal paper of Cochrane and Saa-Requejo (2000) uses Sharpe-ratios to define good-deals. This approach is extended in ˇ Cerný (2003) and Björk and Slinko (2006). Bernardo and Ledoit (2000) look at the gain-loss ratio. Another approach are test measures and floors, see for example, Carr et al. (2001). This idea is extended to a dynamic framework in Larsen et al. (2005). ˇ Cerný and Hodges (2002) put together all these ideas and present a unifying theory for good-deal bounds. They also draw parallels to the classical arbitrage pricing theory. Arai (2011) uses risk measures based on shortfall risk to determine good-deal bounds. The strong connection between good-deal bounds and risk measures is presented for the first time in Jaschke and Küchler (2001). The extension of their results to a non-coherent set of good-deals can be found in Staum (2004). Different representation results for convex risk measures in a pricing setup are given in Arai and Fukasawa (2014) and Farkas et al. (2015). Risks 2020,8, 114; doi:10.3390/risks8040114 www.mdpi.com/journal/risks
Risks 2020,8, 114 2 of 22 To the best of our knowledge, there is no comprehensive analysis of good-deal bounds characterized by acceptance sets based on Value-at-Risk and on Expected Shortfall. We would like to fill this gap in a special but standard market setting. Therefore, we use the concept of risk measures based on multiple eligible assets as for example, given in Scandolo (2004), Artzner et al. (2009) and Farkas et al. (2015) . They are an extension of risk measures based on a single eligible asset, see Artzner et al. (1999). Inspired by Cochrane and Saa-Requejo (2000), our focus lies on the pricing of European type options in a Black-Scholes market setup without intermediate trading. The restriction to a one-period market model is justified by the following reasons: First of all, for discrete time models the problem of pricing in an incomplete market is rather the rule than the exception. Second, recent works in the field of risk measures are restricted to such kind of models, see for example, Liebrich and Svindland (2017), Liebrich and Svindland (2019) and Baes et al. (2020). Our contributions are divided into three main parts, corresponding to Sections 3–5. In Section 3we examine risk measures based on multiple eligible assets. In our one-period market model, we find an equivalent condition to the so-called absence of acceptability arbitrage condition in Artzner et al. (2009) . If this condition is fulfilled, the multi-asset risk measure is finite. For the coherent acceptance set based on the Expected Shortfall this is a consequence of the results in Farkas et al. (2015). In Section 3.2 we also use their results to prove lower semicontinuity for the multi-asset risk measure w.r.t. the non-coherent acceptance set based on the Value at Risk. But the results in Farkas et al. (2015) do not apply in our concrete setting for showing that the multi-asset risk measure w.r.t. the Value-at-Risk is finite and not globally upper semicontinuous. We prove this in Proposition 3and Theorem 2. As a concrete application, we choose an European option payoff as future risky position. For its risk calculation we develop numerical approaches to determine the value and the optimal solution of these risk measures. The benefits of fast algorithms is for example, pointed out in (Björk and Slinko 2006, Section 5): “[. ..] the problem of numerically determining good-deal bounds is relatively complex in terms of CPU time, so there is a very clear need to develop fast, approximative good-deal pricing algorithms”. In Section 4we find new relations between the absence of acceptability arbitrage condition and the absence of good-deals of the first kind in Jaschke and Küchler (2001). Working under the latter condition, the consideration of the Value-at-Risk and the Expected Shortfall allows us to demonstrate the differences between non-convex and convex pricing bounds. From this we obtain that the seller’s good-deal bound based on the Value-at-Risk does not behave smooth if we vary the strike of an option. This behavior is triggered by jumps in the optimal solution of the underlying optimization problem. Additionally, in the Value-at-Risk case the seller’s bound could become smaller than the buyer’s bound even if there are no good-deals of the first kind on the market. This is not the case for the Expected Shortfall due to a separation argument, as we show in Section 5. Our results allow to conclude that pricing w.r.t. the non-convex bounds is problematic, whereas the good-deal bounds based on Expected Shortfall provide a reasonable way for option pricing in incomplete markets. The paper is organized as follows: In Section 2.1, we introduce the financial market and repeat some well-known results within this market. In Sections 2.2 and 2.3, we recall the definition of risk measures using multiple eligible assets. Sections 3–5include our findings for good-deal bounds based on Value-at-Risk and Expected Shortfall risk measures as outlined above. Throughout the whole manuscript we use the following standard notations: For two sets A and B , A⊂B denotes that A is a subset of B . Moreover, A(B describes that A is a proper subset of B . The complement of a set A is written as A{ . For n∈N the standard scalar product of two points x , y∈Rn is denoted by hx,yi . The Euclidean norm of a vector x∈Rn is written as kxk2 . The unit sphere is defined by ∂BRn:={x∈Rn| kxk2=1}. The positive cone of a partially ordered vector space X is denoted by X+ . The linear span of a set of vectors S⊂ X is written as span S . The epigraph of a function f:X → [−∞,∞] is defined as epi (f):={(X,α)∈ X × R|f(X)≤α}, see for example, (Aliprantis and Border 2006, Section 1.7).
Risks 2020,8, 114 3 of 22 For a probability space (Ω,F,P) and p∈[1, ∞) the space of equivalence classes of p -integrable random variables is denoted by Lp(Ω,F,P) or Lp for short. In the following, we always equip them with the usual Lp -norm and the P -almost sure (a.s.) order, that is, every space Lp is a Banach lattice, see (Aliprantis and Border 2006, Theorem 13.5). The distribution function of a random variable X is denoted by FX . The corresponding upper quantile function is q+ X , see for example, (Föllmer and Schied 2016, Section A.3). 2. Multi-Asset Risk Measures 2.1. Financial Market We use a Black-Scholes market setup with one risk-free and one risky asset. We consider financial positions up to a time horizon T∈R>0 . A filtered probability space Ω,F,(Ft)t∈[0,T],P with FT=F is used. The financial market consists of a bank account B and a stock price process S . The bank account has initial price B0∈R>0 and interest rate r∈R . The stock price is modeled via a geometric Brownian motion with initial price S0∈R>0 , trend b∈R , volatility σ∈R>0 and Brownian motion W. The values of the bank account and the stock price at time t∈[0, T]are Bt=B0exp (rt),St=S0exp b−1 2σ2t+σWt. (1) Remark 1. The option pricing results in this manuscript rely on the use of this financial market model. Nevertheless, some of our results could be extended to a financial market model with multiple stocks modeled as a multi-dimensional geometric Brownian motion. When discussing these results in Remarks 5and 7we denote by ˆ S a d-dimensional geometric Brownian motion with d ∈N. Now we need to describe the relevant spaces for our pricing problem. They are used in the next subsection to introduce risk measures based on multiple eligible assets. Definition 1 (Marketed space) . The space of financial payoffs is X:=L1(Ω,F,P) . The marketed space is M:=span {BT,ST} . The pricing functional π:M → R is defined such that (s.t.) for each ϕB , ϕS∈R it holds that πϕBBT+ϕSST:=ϕBB0+ϕSS0. Further, for x ∈Rwe set Mx:={Z∈ M|π(Z)=x}. Remark 2. Our results hold for every choice of Lp -space with p∈[1, ∞) . For the sake of simplicity, we use the concrete choice p =1. As mentioned before, we only allow for trading at time 0, that is, no intermediate trading is allowed. We call this the one-period Black-Scholes model (one-period BSM). In a discrete time setting, an incomplete market is the rule rather than the exception. As it is pointed out in (Föllmer and Schied 2016, Section 5.5) , under an additional homogeneity assumption the binomial model in Cox et al. (1979) is the only complete financial market model in discrete time. We repeat the well-known finding that the one-period BSM is incomplete, and hence the problem of pricing an option is a demanding task. Lemma 1. In the one-period BSM the following statements hold: (i) The market is incomplete. (ii) A European call option with strike K ∈R>0is not attainable. Proof. (i) The probability space (Ω,F,P) supports the random variable ST which has a continuous distribution. By (Föllmer and Schied 2016, Proposition A.31) the probability space (Ω,F,P) is atomless. The first claim follows by (Föllmer and Schied 2016, Corollary 1.42). (ii) The arbitrage-free price range of the European call option is (max {0, S0−K},S0) . The second result is then a consequence of (Föllmer and Schied 2016, Corollary 1.35).
Risks 2020,8, 114 4 of 22 2.2. Acceptance Sets Assume that the future risky payoff of a market agent is X∈ X . Further, the agent adheres to an acceptability criterion, modeled as a specific set of future payoffs. If X is not contained in this set the agent has to take management actions to ensure the acceptability of the future payoff. Definition 2 (Acceptance set) . An acceptance set is a nonempty subset A(X which is monotone, that is, A+X+⊂ A. The acceptance sets which we consider are typically defined with the help of monetary risk measures. We use the following definition for them, see for example, (Cheridito and Li 2009, Definition 2.1) or (Geissel et al. 2018, Definition 1.1). Definition 3 (Monetary risk measures) . A map ρ:X → (−∞,∞] is called a monetary risk measure on X if it fulfills the following properties: (i) Finiteness at 0: ρ(0)∈R. (ii) Monotonicity: For all X,Y∈ X s.t. X ≤Y P-a.s. it holds ρ(X)≥ρ(Y). (iii) Cash invariance: For all X ∈ X and m ∈Rit holds ρ(X+m)=ρ(X)−m. Further, we say ρis convex if it satisfies: (iv) Convexity: For all X,Y and λ∈(0, 1)it holds ρ(λX+(1−λ)Y)≤λρ (X)+(1−λ)ρ(Y). ρis called coherent if it is convex and it also satisfies: (v) Positive homogeneity: For all X ∈ X and λ∈R≥0it holds ρ(λX)=λρ (X). The acceptance set w.r.t. the monetary risk measure ρis defined by Aρ:={X∈ X |ρ(X)≤0}. (2) Remark 3. For the sake of consistency with Lemma 2, we note that the convexity, respectively positive homogeneity, in Definition 3is equivalent to the convexity, respectively conicity, of the epigraph of the monetary risk measure ρ. As mentioned before, our focus lies on the most prominent monetary risk measures, namely the Value-at-Risk and the Expected Shortfall. Example 1 (Acceptance sets).(i) The Value-at-Risk (VaR) for a random variable X∈ X at level λ∈(0, 0.5)is defined by VaRλ(X):=−q+ X(λ). We obtain the following expression for it: VaRλ(X)=inf {m∈R|P(X+m<0)≤λ}. (3) The VaR acceptance set AVaRλ={X∈ X |VaRλ(X)≤0}, (4) is a cone, but not a convex set in general. (ii) The Expected Shortfall (ES) with level λ∈(0, 0.5) is defined for a random variable X∈ X by ESλ(X):=1 λ λ R0 VaRt(X)dt. The corresponding acceptance set AESλ={X∈ X |ESλ(X)≤0}. (5) is a convex cone.
Risks 2020,8, 114 5 of 22 Remark 4. VaR and ES are in use in several regulatory frameworks, like for example, Basel III/IV or Solvency II. Wang and Zitikis (2020) introduce an intuitive set of four axioms to describe economic intentions in a risk assessment framework, namely monotonicity, law-invariance, prudence and no reward for concentration (NRC). The main intention of the regulator is reflected by the NRC axiom. It states the following: If two payoffs realize large losses in the case of a specific stress scenario, then no diversification effects occur and the capital requirement of the sum of the payoffs is equal to the sum of the individual capital requirements, that is, no capital reduction is possible. A functional ρ:L1→R satisfies the four axioms if and only if it is the ES at a specific level, see for example, (Wang and Zitikis 2020, Theorem 1). Therefore, the ES is the only monetary risk measure satisfying these economic desirable axioms. This is especially one advantage of using ES instead of VaR in a capital adequacy test. In Sections 4and 5we justify that the ES is also preferable to VaR when pricing financial derivatives. 2.3. Multi-Asset Risk Measures The basic idea of risk measures is to buy assets on a financial market s.t. the resulting position becomes acceptable. The agent in the one-period BSM would search for a marketed portfolio Z∈ M s.t. together with the risky position X∈ X it holds that X+Z∈ A. (6) Usually, there are multiple portfolios s.t. this condition is satisfied. Hence, we need an objective function to decide which portfolio should be used. We assume the agent wants to achieve the acceptability constraint at minimal costs. This leads to the definition of risk measures based on multiple eligible assets. Definition 4 (Multi-asset risk measures) . For an acceptance set A ⊂ X , the multi-asset risk measure for a risky position X ∈ X is given by ρA,M,π(X):=inf {π(Z)|Z∈ M,X+Z∈ A}. (7) The single-asset risk measure for a nonzero, positive random variable U ∈ M is ρA,U,π(X):=inf m∈R X+m π(U)U∈ A. (8) The following result recalls well-known properties of a multi-asset risk measure, see for example, Farkas et al. (2015) [Lemma 2]. Lemma 2. For an arbitrary acceptance set A⊂X , the corresponding multi-asset risk measure fulfills the following properties: (i) ρA,M,πis decreasing. (ii) ρA,M,π is M -additive, that is, for all X∈ X and all Z∈ M the translation property ρA,M,π(X+Z)=ρA,M,π(X)−π(Z)holds. (iii) If Ais convex, then ρA,M,πis convex, that is, the epigraph epi (ρA,M,π)is convex. (iv) If Ais a cone, then ρA,M,πis positively homogeneous, that is, the epigraph epi (ρA,M,π)is a cone. 3. Hedging with Multi-Asset Risk Measures In this section we present several properties of the multi-asset risk measures in our concrete market framework and provide explicit formulas for measuring the risk of European options. 3.1. Representation and Absence of Acceptability Arbitrage We start by showing a new representation for multi-asset risk measures. This is used to determine the risk measures in our examples.
Risks 2020,8, 114 6 of 22 Proposition 1 (Representation in terms of stock holdings) . Assume a monetary risk measure ρ . For every X∈ X we obtain: ρAρ,M,π(X)=inf ϕ∈RϕS0+ρ(ϕST+X) BT B0. (9) Proof. The claim follows from the cash invariance of the monetary risk measure and the relation ρAρ,M,π(X)=inf nϕBB0+ϕSS0ϕB,ϕS∈R,ρϕSST+X≤ϕBBTo. So, the minimum for given ϕS=ϕis attained for ϕB=ρ(ϕST+X)/BT. Before we focus on acceptance sets based on VaR and ES, we examine an expression which is helpful to determine finiteness and continuity properties of multi-asset risk measures. It is a characterization of the so-called absence of acceptability arbitrage condition in (Farkas et al. 2015, Section 3). If this condition is not satisfied, it is possible to construct acceptable positions with arbitrary negative costs, see Example 2. This would destroy any reasonable pricing and lead to infinite capital requirements. Therefore, in the following we discuss the important concept of the absence of acceptability arbitrage opportunities, which is closely related to the absence of good-deals of the first kind, which is introduced rigorously in Definition 5below. Theorem 1 (Absence of acceptability arbitrage) . Assume a positive homogeneous monetary risk measure ρ s.t. ρ(ST)and ρ(−ST)are finite. In the one-period BSM it holds that ρAρ,M,π(0)=0, if BT B0 ∈−ρ(ST) S0 ,ρ(−ST) S0, (10) and ρAρ,M,π(0)=−∞ otherwise. The condition (10) is equivalent to the existence of a value m∈R s.t. Aρ∩{Z∈ M|π(Z)≤m}=∅, which is the classical definition for absence of acceptability arbitrage. Proof. By positive homogeneity, we have ϕS0+ρ(ϕST+0) BT B0= ϕS0+ρ(ST)B0 BT,ϕ>0, 0 , ϕ=0, ϕS0−ρ(−ST)B0 BT,ϕ<0. This yields for ρ∗,1 =inf ϕ>0ϕS0+ρ(ϕST+0) BT B0,ρ∗,2 =inf ϕ<0ϕS0+ρ(ϕST+0) BT B0, that ρ∗,1 =−∞ if BT B0 <−ρ(ST) S0 and 0 otherwise, as well as, ρ∗,2 =−∞ if BT B0 >ρ(−ST) S0 and 0 otherwise. This yields the claim, since by Proposition 1we get ρAρ,M,π(0)=min n0, ρ∗,1,ρ∗,2o. The second part is a direct consequence of (Farkas et al. 2015, Lemma 4). Remark 5. (i) Consider a level λ∈(0, 0.5) , initial prices S0=B0= 1, zero interest rate and −ESλ(ST)≤BT< −VaRλ(ST). In such a model, VaR leads to acceptability arbitrage opportunities, while ES does not.
Risks 2020,8, 114 7 of 22 (ii) Assume that there are more risky assets ˆ S in the market. In an analog manner to the proof of Theorem 1 we could show that the multi-asset risk measure of the zero payoff is zero iff for every point in the unit sphere ϕ∈∂BRd it holds that DBT B0 ˆ S0,ϕE≤ρ−ˆ ST,ϕ . Furthermore, we obtain a representation result by taking the infimum over all elements in Rd and replacing products with the scalar product in (9) if necessary. Example 2 (Acceptability arbitrage opportunities) . If acceptability arbitrage exists, that is, for every m∈R , Aρ∩{Z∈ M|π(Z)≤m}6=∅ or equivalently BT B0/∈h−ρ(ST) S0,ρ(−ST) S0i by Theorem 1, then there exists a sequence of acceptable marketed portfolios which falls below every negative bound of costs. For instance, if BT B0 <−ρ(ST) S0, we build long-positions in the stock and short-positions in the bank account, s.t. we obtain the sequence of buy and hold trading strategies ϕB n,ϕS nn∈N=−nρ(ST) B0ρ(ST)+S0BT ,−nBT B0ρ(ST)+S0BTn∈N . Such a strategy corresponds by positive homogeneity to portfolios with ρϕB nBT+ϕS nST= 0and prices πϕB nBT+ϕS nST=−n. The previous example shows that if there exists an acceptability arbitrage opportunity, one can construct portfolio payoffs which have an arbitrary low price but are still acceptable. If there are no such opportunities one can hope for reasonable prices following the idea of acceptability. This is also strongly related with finiteness and continuity of the associated multi-asset risk measures as we analyze in detail in the following. 3.2. Value-at-Risk and Expected Shortfall In this subsection we give basic finiteness and continuity properties for multi-asset risk measures based on VaR and ES acceptance sets. The analysis in this section is inspired by Farkas et al. (2015). Not all of the results in Farkas et al. (2015) are applicable in our concrete setting in combination with VaR acceptance sets. Therefore, Proposition 3and Theorem 2show that the risk measure based on the VaR acceptance set is finite and not globally upper semicontinuous. We start with the easier case of an ES acceptance set. The coherence of the acceptance set allows us to apply some of the findings in Farkas et al. (2015) in order to show that the multi-asset risk measure is finite and continuous. Proposition 2 (Finiteness and continuity of ES) . Assume a level λ∈(0, 0.5) and let the absence of acceptability arbitrage condition be fulfilled, that is, ρAESλ,M,π(0)>−∞ . Then ρAESλ,M,π is finite and continuous. Proof. Note that the strictly positive elements of L1 are all random variables which are P -a.s. strictly positive. Hence, ST is a strictly positive element of M . Further, by (Farkas et al. 2014, Lemma 4.3), we see that ST∈int AESλ , that is, int AESλ6=∅ . Therefore, all conditions in (Farkas et al. 2015, Proposition 2) are satisfied. This leads to the claim. In our setup we cannot employ the results of Farkas et al. (2015), to say something about finiteness and continuity of the multi-asset risk measure w.r.t. VaR. Due to the missing convexity, we could not apply (Farkas et al. 2015, Proposition 2) for convex or (Farkas et al. 2015, Proposition 3) for coherent acceptance sets. Moreover, since the space L1 does not admit any order unit, we are also not allowed to apply (Farkas et al. 2015, Proposition 1). Therefore, we present new results for finiteness and continuity of VaR in our concrete setting.
Risks 2020,8, 114 8 of 22 Proposition 3 (Finiteness and lower semicontinuity of VaR) . Let λ∈(0, 0.5) . If BT B0∈ −VaRλ(ST) S0,VaRλ(−ST) S0then the following two statements hold: (i) ρAVaRλ,M,πis finite. (ii) ρAVaRλ,M,πis lower semicontinuous. Proof. (i) As a direct consequence of Proposition 1we have ρAVaRλ,M,π<∞ . Without loss of generality (w.l.o.g.) assume that S0=B0= 1. Let X∈L1 and e∈R>0 s.t. BT∈q+ ST(λ)+e,q+ ST(1−λ)−e . Choose an arbitrary sequence of real numbers (ϕn)n∈N. For each n∈Nwe obtain: P(X≤nST+ϕn(BT−ST)) ≥ P{X≤nST}∩nST<q+ ST(λ)+eo ,ϕn≥0, P{X≤nST}∩nST>q+ ST(1−λ)−eo ,ϕn<0. This implies that there exists n∗∈N s.t. for every sequence (ϕn)n∈N and every n>n∗ we obtain P(X≤nST+ϕn(BT−ST)) >λ. This implies ρAVaRλ,M,π>−∞. (ii) It holds that AVaRλ is closed (see (Farkas et al. 2014, Section 4.1)), 0 ∈ AVaRλ and M is finite-dimensional. Further, from our assumption we obtain AVaRλ∩ M0={0} . Therefore, every condition in (Farkas et al. 2015, Proposition 5) is fulfilled. This implies that AVaRλ+M0 is a closed set. The lower semicontinuity then follows from (Farkas et al. 2015, Proposition 4). Remark 6. The condition BT B0∈−VaRλ(ST) S0,VaRλ(−ST) S0is further characterized in Section 4. To show that ρAVaRλ,M,π is not upper semicontinuous, we start by identifying the interior of the set AVaRλ+M0. In view of (Farkas et al. 2014, Section 4.1) we see that AVaRλ+M0=X∈L1 ∃ϕ∈R:PX+ϕS0 B0 BT−ϕST<0≤λ. (11) The following result characterizes int AVaRλ+M0 , the interior of AVaRλ+M0 . We omit the proof because it is analog to the one in (Farkas et al. 2014, Lemma 4.1). Lemma 3 (Interior augmented acceptance set).For a level λ∈(0, 0.5)we obtain: int AVaRλ+M0=X∈L1 ∃ϕ∈R:PX+ϕS0 B0 BT−ϕST≤0<λ. (12) To complete the analysis, we show that VaR is not globally continuous in our setting. Therefore, we use (Farkas et al. 2015, Proposition 4) to create a counterexample for upper semicontinuity, that is, we try to find X∈ X and Z∈ M with X+Z/∈int AVaRλ+M0but ρAVaRλ,M,π(X)<π(Z). The proof then demonstrates the difficulty in characterizing continuity properties, if we use a non-convex acceptance set in combination with multiple eligible assets. The reason for this is the complex structure of the augmented acceptance set AVaRλ+M0 , which makes it harder to characterize the points in it. Theorem 2 (Absence of upper semicontinuity of VaR) . Let λ∈(0, 0.5) and assume that there are no acceptability arbitrage opportunities, that is, BT B0∈h−VaRλ(ST) S0,VaRλ(−ST) S0i . This implies that ρAVaRλ,M,π is not (globally) upper semicontinuous.
Risks 2020,8, 114 15 of 22 prices are given. Note that the no arbitrage price range is (max {0, S0−K},S0) . The plots are inspired by Cochrane and Saa-Requejo (2000) [Figure 1]. Figure 3. Good-deal bounds depending on the stock price. Compared with the Black-Scholes and subhedging price the good-deal bounds w.r.t. the ES do not admit an unexpected behavior. For the VaR the buyer good-deal bounds seem plausible. What is not directly intuitive is that the seller’s bound is sometimes lower than the buyer’s bound. For instance, this is the case for stock prices in the interval [1.11, 1.15]. For a specific ϕ∈[0, 1]and any λ∈(0, 0.5)we obtain: ϕS0+VaR1−λϕS1−(S1−K)+≤ϕS0+VaRλϕS1−(S1−K)+. Nevertheless, we are not able to conclude that the seller’s bound is always greater or equal than the buyer’s bound as our counterexample shows. This undesirable behavior occurs if there does not exist an equivalent martingale measure s.t. the extended market is free of good-deals. We will discuss this fact in detail in Section 5.2 below. To provide more intuition on the shape of the seller’s good-deal bound based on VaR in Figure 3, in Figure 4 we plot the objective function from the optimization problem of the seller using VaR as preference criterion, that is, we plot the function ϕ7→ ϕS0+VaRλϕS1−(S1−K)+ . Each plot corresponds to a specific stock price S0 . We see that the minimum for rising stock prices goes hand in hand with a rising number of stock shares. For the stock prices of 0.9 and 0.95 there are one local and one global minimum. For increasing stock prices in
Risks 2020,8, 114 16 of 22 the interval (0.9, 0.95) the stock position belonging to the global minimum changes in a discontinuous manner, because at some stock price local and global minima change their roles. This is the reason for the kink of the seller good-deal bound in Figure 3. Figure 4. Seller’s objective function for different stock prices.
Risks 2020,8, 114 17 of 22 Finally, we would like to elaborate on the level λ . Therefore, we alter the previous parameters in the following way: T=1, B0=1, S0=1.12,r=0, b=5%, σ=20%, K=1. In Figure 5we illustrate the good-deal bounds depending on the level λ . For the VaR acceptance set we see that only for high levels the seller’s price is lower than the buyer’s price. Figure 5. Good-deal bounds depending on the level λ. 4.3. Limiting Behavior For increasing stock prices the seller’s good-deal bound in Figure 3tends to the subhedging price. In this short subsection we give the reason for this behavior in the VaR-case. There is an intuitive explanation. If the stock price goes up the call option is deeper in-the-money. This results in the following two effects: • It becomes more likely that the option is exercised, resulting in a loss for the seller. • Higher losses for the seller become more probable. Therefore, it is more expensive to hedge the short call position. As we have seen in Figure 4, the seller buys one stock share for high enough stock prices. Even if the seller goes short in the bank account, the future loss is bounded from below. The resulting seller good-deal bound is S0+VaRλS1−(S1−K)+. (20) If λ>FS1(K), then S0+VaRλS1−(S1−K)+=S0−K, (21) that is, the seller’s bound is equal to the subhedging price. For large enough stock prices the condition is fulfilled, because of lim S0→∞FS1(K)=lim S0→∞ Φ ln (K/S0)−b−σ2/2 σ!=0. (22) An analog explanation holds for the case if the stock price tends to zero.
Risks 2020,8, 114 18 of 22 5. Extension of the Basis Market In this section we explain why it can happen that the seller’s good-deal bound is smaller than the buyer’s good-deal bound using a VaR acceptance set. This is due to the failure of the so-called extension theorem for VaR. Subsequently, we prove the validity of the extension theorem for an ES acceptance set, showing that such a situation cannot happen for ES. 5.1. Duality Relations We develop a duality relation for the seller’s good-deal bound w.r.t. an acceptance set based on a positive homogeneous monetary risk measure ρ . Let Pρ(C) be the set of prices for a financial payoff C∈ X s.t. there are no good-deals of the first kind in the extended market, that is, for p∈ Pρ(C) there do not exist ϕS,ϕC∈Rexcluding the case ϕS=ϕC=0 s.t. ρϕSST−S0 BT B0+ϕCC−pBT B0≤0. (23) By positive homogeneity of ρ we obtain that L1 +⊂ A . This implies that every arbitrage opportunity is also a good-deal. Hence, the set Pρ(C) is a subset of the no arbitrage price range of C . Therefore, the set of no good-deal prices in the extended market can be characterized as a subset of all equivalent martingale measures. This subset of equivalent martingale measures is denoted by P∗ ρ(C). In the following we assume that there exists at least one price s.t. there are no good-deals in the extended market, that is, we work under the assumption P∗ ρ(C)6=∅. (24) Remark 9. This condition implies that the basis market M satisfies the condition of the absence of good-deals of the first kind. Now we are able to present the duality result. It is analog to the corresponding result for the superhedging price, see for example, (Föllmer and Schied 2016, Theorem 1.32). Proposition 4. The seller’s good-deal bound for a P -a.s. non-negative random variable C in the one-period BSM is given by ρ(−C)=inf ϕ∈RϕS0+ρ(ϕST−C) BT B0=sup P∗∈P∗ ρ(C) EP∗(C). (25) Proof. From the discussion before we obtain that sup P∗∈P∗ ρ(C) EP∗(C)=sup Pρ(C) . Assume there exists ϕ∈R s.t. for a value m∈R it holds that ρmBT B0+ϕST−S0BT B0−C≤ 0. From cash invariance we obtain B0 BTρϕST−S0BT B0−C≤m . Therefore, every p≥m cannot be a price in the extended market s.t. there does not exist a good-deal. This implies ρ(−C)≥sup Pρ(C). It is left to show that for every m>sup Pρ(C) it holds that m≥ρ(−C) . Therefore, assume m>sup Pρ(C). There are ϕS,ϕC∈Rexcluding the case ϕS=ϕC=0 such that ρϕSST−S0 BT B0+ϕCC−mBT B0≤0.
Risks 2020,8, 114 19 of 22 The basis market does not admit good-deals of the first kind. This implies ϕC6= 0. If ϕC< 0, we obtain from the positive homogeneity of ρthat ρmBT B0 −ϕS ϕCST−S0 BT B0−C≤0. It remains to show that the case ϕC> 0 is not feasible. We know that there exists a price p∈ Pρ(C) for each a,b∈Rexcluding the case a=b=0 and we have ρaST−S0 BT B0+bC−pBT B0>0. At the same time positive homogeneity of ρimplies ρϕS ϕCST−S0 BT B0+C−mBT B0≤0. These two statements together with monotonicity of ρand m>pimply 0≥ρϕS ϕCST−S0 BT B0+C−mBT B0>ρϕS ϕCST−S0 BT B0+C−pBT B0>0, which is a contradiction. Remark 10. A corresponding result holds for the buyer’s good-deal bound: ρ(−C)=sup P∗∈P∗ ρ(C) EP∗(C)≥inf P∗∈P∗ ρ(C)EP∗(C)=−ρ(C). (26) 5.2. Failure of the Extension Theorem If we go back to Figure 3, we are now able to answer the question why the seller’s bound w.r.t. the VaR is lower than the buyer’s bound. If we recall Remark 10, then we know that this is only possible if P∗ VaRλ(ST−K)+=∅. (27) This means that there is no price s.t. the extended market is free of good-deals of the first kind. This must also hold if there are no good-deals of the first kind on the basis market M . From Lemma 4, we obtain that the absence of good-deals on the basis market is equivalent to BT B0 ∈ q+ ST(λ) S0 ,q+ ST(1−λ) S0!. (28) This condition is independent of the stock price and is in particular fulfilled in Example 5. The situation that every possible option price leads to an extended market which is not free of good-deals means that for every price p∈Rthere exists θ∈Rsuch that θS0+VaRλθST+(ST−K)+BT B0 ≤ −p, (29) or θS0+VaR1−λθST+(ST−K)+BT B0 ≥ −p. (30)
Risks 2020,8, 114 20 of 22 In contrast to the basis market the absence of good-deals of the first kind depends on the stock price. For θ= 1, the earlier argument that the seller’s good-deal bound converges to the subhedging price, could be used in an analog way to show that for high enough stock prices both VaR expressions in the previous inequalities are equal to the subhedging price S0−K . Hence, one of the above conditions is fulfilled for large enough stock prices. Remark 11. The extension theorem as described in ( ˇ Cerný and Hodges 2002, Section 2.1) states, that a basis market which is free of good-deals could be extended s.t. the extended market, is free of good-deals of the first kind. But this would be a contradiction to our Value-at-Risk example, that is, a contradiction to P∗ VaRλ(ST−K)+=∅ . Therefore, a sufficient assumption in the extension theorem is not satisfied. In fact, in our setting the VaR acceptance set is not boundedly generated 2,3 . This is intuitive, because the proof of the extension theorem is based on a separation argument. Such a separation is likely to fail if the acceptance set is not convex, like in the VaR-case. The previous remark raises the question if the extension theorem holds for the ES acceptance set. This is indeed the case. The reason is that the convexity of the acceptance set allows for a separation argument to extend the pricing functional to the whole space of contingent claims. The arguments in the following proof are in line with the ones in the Proof of Theorem 2.3 in ˇ Cerný and Hodges (2002) . Nevertheless, our setup differs slightly from the one in ˇ Cerný and Hodges (2002), for example, the intersection of M0 and A could contain the zero vector, which is not the case in ˇ Cerný and Hodges (2002). Furthermore, our proof does not explicitly use the condition of a boundedly generated set and we apply the Interior Separating Hyperplane Theorem instead of relying on the Hahn-Banach Theorem. Theorem 4 (Expected Shortfall extension theorem) . Assume the one-period BSM model and good-deals defined by the ES acceptance set w.r.t. a level λ∈(0, 0.5) . Assume that there are no good-deals of the first kind on the basis market M . In this setup, there exists an extension of the pricing functional π:M → R to the whole space of financial payoffs Xs.t. there are no good-deals in the extended market. Proof. From (Farkas et al. 2014, Lemma 4.3) we obtain int AESλ={X∈ X |ESλ(X)<0}. This shows that 0 /∈int AESλ , which in turn implies that int AESλ\{0}=int AESλ . Again, from (Farkas et al. 2014, Lemma 4.3) we obtain ST∈int AESλ . A straightforward calculation gives that AESλ\{0} is convex. The assumption of no good-deals of the first kind on the basis market states that AESλ∩ M0\{0}=∅ . Summarizing, AESλ\{0} and M0 are two non-empty, disjoint, convex sets for which AESλ\{0} contains an interior point. Therefore, by the Interior Separating Hyperplane Theorem (see e.g. (Aliprantis and Border 2006, Theorem 5.67)) there exists a non-zero continuous functional pwhich properly separates these two sets. Since M0 is a subspace, we obtain for each x∈ M0 that p(x)= 0 and additionally we could assume that for each x∈ AESλ\{0} it holds that p(x)≥ 0. Since the separation is proper, there exists y∈ AESλ\{0}with p(y)>0. Note, that every x∈ AESλ\{0} obtains a representation with α∈R , s∈[p=0] s.t. x=αy+s . This implies p(x)=αp(y) . The scalar α cannot be zero ( x/∈ M0 ) and also not negative ( p(x)≥ 0). 2 Let K be a convex set of claims disjoint from the origin. It is called boundedly generated if there exists a closed bounded subset B⊂K s.t. any point in K can be regarded as a scalar multiple of a point in B , see ( ˇ Cerný and Hodges 2002, Definition 2.2). 3Nevertheless, the VaR acceptance set is closed in Lp, see (Munari 2015, Proposition 2.4.5).
Risks 2020,8, 114 21 of 22 Hence, it holds that p(x)> 0 and the usage of p as pricing functional in the extended market would exclude good-deals. It also would exclude arbitrage opportunities, because of L1 +⊂ AESλ . But, we are not allowed to use p directly as extended pricing functional. This is due to the fact that p could price the basis assets in a different way than before, that is, pϕBBT+ϕSST6=ϕBB0+ϕSS0 . We obtain a suitable functional ˜ p by rescaling p with p(BT)/B0 , that is, for each x∈L1 we set ˜ p(x)=p(x) p(BT)B0 . 6. Conclusions and Outlook We study good-deal bounds based on Value-at-Risk and Expected Shortfall acceptance sets. Based on a benchmark example we demonstrate that the Value-at-Risk good-deal bounds for a European type call option behave non-smooth under varying the underlying stock price. We find that the reason for this is a jump in the optimal hedging strategy. The application of the Expected Shortfall does not admit this discontinuous behavior. Additionally, in the Value-at-Risk case, the seller’s bound is sometimes smaller than the buyer’s bound. To explain this, we show that the extension theorem in ˇ Cerný and Hodges (2002) is not applicable in the non-convex Value-at-Risk case. In addition, we present new finiteness and continuity results for risk measures based on multiple eligible assets in our concrete setup. These results depend on the one-period Black-Scholes market model. Therefore, a future direction of research could be to incorporate a market with more than one risky asset, for which we gave a short outlook in the Remarks 5and 7. Also the use of intermediate trading strategies is a natural extension of our setup. Author Contributions: C.L. developed the concept and the methodology. C.L. is responsible for the implementation in R Core Team (2020). This manuscript is planned to be part of C.L.’s doctoral thesis. J.S. is his supervisor and S.D. supports this doctoral project. S.D. and J.S. discussed the presented research with C.L. and gave essential suggestions and hints to bring the manuscript into the final form. All authors have read and agreed to this version of the manuscript. Funding: C.L. thanks Fraunhofer Institute for Industrial Mathematics ITWM for its financial support. S.D. is supported by the Austrian Science Fund (FWF) project F5507-N26, which is part of the Special Research Program Quasi-Monte Carlo Methods: Theory and Applications. Acknowledgments: This paper was finalized during a research stay of C.L. at the Institute of Financial Mathematics and Applied Number Theory of the Johannes Kepler University Linz. C.L. gratefully acknowledges the possibility of this stay and thanks the FWF Special Research Program Quasi-Monte Carlo Methods: Theory and Applications for the invitation. The authors also thank three anonymous referees for their helpful comments and suggestions. Conflicts of Interest: The authors declare no conflict of interest. References Aliprantis, Charalambos D., and Kim C. Border. 2006. Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd ed. Berlin/Heidelberg: Springer. [CrossRef] Arai, Takuji. 2011. Good Deal Bounds Induced by Shortfall Risk. SIAM Journal on Financial Mathematics 2: 1–21. [CrossRef] Arai, Takuji, and Masaaki Fukasawa. 2014. Convex Risk Measures for Good Deal Bounds. Mathematical Finance 24: 464–84. [CrossRef] Artzner, Philippe, Freddy Delbaen, Jean-Marc Eber, and David Heath. 1999. Coherent Measures of Risk. Mathematical Finance 9: 203–28. [CrossRef] Artzner, Philippe, Freddy Delbaen, and Pablo Koch-Medina. 2009. Risk Measures and Efficient use of Capital 1. ASTIN Bulletin: The Journal of the IAA 39: 101–16. [CrossRef] Baes, Michel, Pablo Koch-Medina, and Cosimo Munari. 2020. Existence, uniqueness, and stability of optimal payoffs of eligible assets. Mathematical Finance 30: 128–66. [CrossRef] Bernardo, Antonio E., and Olivier Ledoit. 2000. Gain, Loss, and Asset Pricing. Journal of Political Economy 108: 144–72. [CrossRef] Björk, Tomas, and Irina Slinko. 2006. Towards a General Theory of Good-Deal Bounds. Review of Finance 10: 221–60. [CrossRef]
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