No-arbitrage principle in conic finance
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Vazifedan, Mehdi; Zhu, Qiji Jim Article No-arbitrage principle in conic finance Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Vazifedan, Mehdi; Zhu, Qiji Jim (2020) : No-arbitrage principle in conic finance, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 8, Iss. 2, pp. 1-34, https://doi.org/10.3390/risks8020066 This Version is available at: https://hdl.handle.net/10419/258019 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 No-Arbitrage Principle in Conic Finance Mehdi Vazifedan * and Qiji Jim Zhu Department of Mathematics, Western Michigan University, 1903 West Michigan Avenue, Kalamazoo, MI 49008, USA; [email protected] *Correspondence: [email protected] Received: 1 May 2020; Accepted: 9 June 2020; Published: 19 June 2020 Abstract: In a one price economy, the Fundamental Theorem of Asset Pricing (FTAP) establishes that no-arbitrage is equivalent to the existence of an equivalent martingale measure. Such an equivalent measure can be derived as the normal unit vector of the hyperplane that separates the attainable gain subspace and the convex cone representing arbitrage opportunities. However, in two-price financial models (where there is a bid–ask price spread), the set of attainable gains is not a subspace anymore. We use convex optimization, and the conic property of this region to characterize the “no-arbitrage” principle in financial models with the bid–ask price spread present. This characterization will lead us to the generation of a set of price factor random variables. Under such a set, we can find the lower and upper bounds (supper-hedging and sub-hedging bounds) for the price of any future cash flow. We will show that for any given cash flow, for which the price is outside the above range, we can build a trading strategy that provides one with an arbitrage opportunity. We will generalize this structure to any two-price finite-period financial model. Keywords: conic finance; convex optimization; arbitrage pricing 1. Introduction It has been known for a long time (Harrison and Kreps 1979;Harrison and Pliska 1981; Delbaen and Schachermayer 1994 ) that no-arbitrage opportunity in a one price economy is equivalent to the existence of an equivalent martingale measure. This result is usually referred to as the Fundamental Theorem of Asset Pricing (FTAP). Rogers (1994) used a directional derivative argument to provide a discrete-time proof for FTAP. The case of trading continuously, in particular, was discussed by Harrison and Pliska (1981). In practice, the law of one price often does not hold. Recently, the literature on transaction costs and the study of no-arbitrage in the presence of bid–ask spreads has been expanding ( Jouini and Kallal 1995 ;Bion-Nadal 2009;Guasoni et al. 2011). Madan (2012) considers financial equilibrium where there are two separate prices at which one may buy from or sell to the market known as ask and bid prices. This is a realistic situation in financial markets. Inside an efficient market, at a given time, a financial security trades at a unique price. The equilibrium conditions that are needed to provide a unique price all depend on arbitrage opportunities that are quickly exploited. A wide category of financial securities trade in vast and diverse markets and few of them meet the equilibrium conditions we mentioned earlier. Market clearing becomes troubling, which leads to prices that are not unique for equivalent securities in different markets. Whenever the above conditions are not satisfied, the “law of one price“ fails to hold. On the other hand, an incomplete market can also take place even under the assumption of “one price”. Incomplete condition explains that there exists the presence of some residual risks that can not be eliminated regardless of the existence of the best hedging ( Eberlein et al. 2009 ; Jacka 1992 ). In addition, the markets establish a phenomenon that is not anticipated in the one price theory called illiquidity. Illiquidity can be explained as the lack of ability of Risks 2020,8, 66; doi:10.3390/risks8020066 www.mdpi.com/journal/risks
Risks 2020,8, 66 2 of 34 the market to establish a unique price, that is, to eliminate the spread between the bid and ask prices. This happens when there is an absence of information and/or a lack of interested parties. In these situations, the bid and ask prices are the only real market information that can be observed. Thus, the theoretical framework that has been used to explain the one price market is not accurate enough to deal with many situations one faces in practice. We adopt the framework of “conic finance“ in this paper. Some earlier studies that use the theory of conic finance include Madan (2012); Cherny and Madan (2010); Eberlein and Madan (2012); Eberlein and Madan (2009) ;Eberlein et al. (2011). The direction of trade is what distinguishes the theory of two-price economies from that of one price economy. Now there are two types of prices, one price is used for buying from the market called the ask price, and the other price is used for selling to the market called the bid price. In the situation of one price economy, the market plays the role of an auctioneer, who clears the trades and decides the prices. However, in the two-price economy, the financial market plays a role of a passive counterparty to all transactions, who buys at the ask price and sells at the bid price. The spread that is between the bid and ask prices becomes a measure of illiquidity. It also evaluates the principal required to support a position and the cost of unraveling a position. We also find it necessary to mention that the one-period case discussed in this paper was included in the book by Carr and Zhu (2018) with reference to this paper (a working paper back then). 1.1. Bid–Ask Spreads A collection of theoretical approaches exists that are trying to model bid–ask spreads. Cherny and Madan (2010) offer a few of the current approaches that have been used to model bid–ask spreads. Copeland and Galai (1983) have discussed the order processing and inventory costs of providers of liquidity. Constantinides (1986) and Jouini and Kallal (1995) investigated the spreads that involve transaction costs of trading in liquid markets. However, the researches mentioned above are comparatively satisfactory to liquid markets where a transaction cost can happen at a price that a contrary trade direction can take place with no price effect. The price-spreads that are related to the theory of two-price economies are discussed in Carr et al. (2001) and Cherny and Madan (2010). These are mostly related to locating actual long-term counterparties that are ready to establish a position for an extended period of time. The spread between the bid and ask prices can be observed as a holding of the charge-exerted while the market does not clear rapidly, as to find a counterparty is going to require time and effort because there is no possibility of doing trades in both directions at any transaction price being observed. Specifically, there is not any possibility of complete replication. In addition, the spread between the bid and ask prices becomes a reflection of the cost of holding the residual risk (Eberlein et al. 2011). Therefore, transactions happen close to or at either the ask price or the bid price, conditional on the direction of the trade. The conic finance is trying to model the bid–ask spread by applying the concept of acceptable cash flows to the market (Madan 2015). However, it is assumed that the market requires a minimal level of acceptability for a position to be profitable. As a result of the market competition, the bid-price is being raised, and the asked-price is being lowered to establish an acceptable position. Therefore the bid–ask spread is tried to be narrowed so that the risk of a position will be minimized. This spread can be seen as the cost of unraveling a position. 1.2. Two-Price Economy In a two-price economy, a comparatively classical view of markets consistent with its role in the traditional competitive breakdown, where markets play the role of counterparties to transactions, is being assumed. The only difference from the traditional aspect is that the length of a trade depends on the direction of the trade, while the market is buying at the bid price and selling at the ask price. Now regard the classical market when trading is performed in either direction at the current price. The market is always willing to sell at a higher price or similarly buy at a lower price and welcomes all zero-cost random cash streams that have a positive expectation under the equilibrium pricing essentials.
Risks 2020,8, 66 3 of 34 Notice that this creates a very large set of market risks that are accepted by the classical financial market within a risk-neutral measure. The two-price financial market is more antagonistic as to which trades it will accept. The collection of zero-cost risks acceptable to the financial market is much smaller as a set. A modeling process of this set of acceptable risks can be observed at Artzner et al. (1999) , that was further expanded in Cherny and Madan (2009) and Cherny and Madan (2010). Especially, the set of zero-cost risks that are acceptable to the financial market as a set of random variables is being modeled as a convex cone, including all the non-negative random variables. The theoretical structure required for supporting the two-price economy has been given a huge amount of attention in the past few years. The theory was popularized into the field of mathematical finance by Constantinides (1986) and was introduced as coherent risk measures by Artzner et al. (1999). The research that connects the theory of two-price economies to concave distortions was performed by Cherny and Madan (2009) and Cherny and Madan (2010). They are the researches that gave a perceptive of how to establish the bid–ask prices admissible to the theory of two-prices. After that, the theory of two-prices was given the name “conic finance”, for example Madan and Schoutens (2011) and Madan and Schoutens (2012). In the following sections, we present the theory of two-prices in an abstract manner as it was set out in Madan (2015). The rest of this paper is structured as follows. In Section 2we introduce the model, starting by definitions of sets of zero cost and attainable cash flows. We also characterize the utility maximization of trading cash flows. In Section 3the Fundamental Theorem of Asset Pricing in conic finance (two-price economy) will be stated and proved (all proofs will be in Appendix A), using the pricing factor generated by a solution of the dual problem. In Section 4we use take advantage of the existence of pricing factors and define price bounds for any attainable cash flow. Additionally, in concrete examples, we will illustrate the trading strategies in the presence of arbitrage opportunities for 1 and 2-period models, along with the super-hedging theorem for the 1-period case. In Lemma 1a computation complexity will be given, in presence of all possible zero cost bonds. Finally in Section 5 the multi-period case will be discussed where only the one-period bonds and assets will be considered through an iterative process. 2. The Model (Multi-Period) In this section, we first define the general multi-period model along with all of the necessary components of the model. We establish the FTAP theorem for the multi-period model in conic finance (two-price economy) by considering a utility maximization problem and its dual. We also explain the relation between solutions to our primal and dual problems, along with the price factors derived from the solution of the dual problem, by considering a utility maximization problem and its dual. We use this price factor to find a super-hedging and sub-hedging for any acceptable cash flow. 2.1. The Model Definitions We start by defining the general model and its components following Madan (2015). Definition 1. Let F={{∅,Ω}=F0⊂ F1⊂ · · · ⊂ FT=F} be an information filtration on the probability space (Ω,F,P) ,where Ω={ω1,ω2, . . . , ωN} is a finite sample space, representing the economic states. We consider a T -period financial model where T≥ 1and for any 0 ≤t≤T , cash flows (in \ out) are being traded at time t. Definition 2. Let Xdenote the space of all F−adapted cash flows x = (xt)T t=0with the inner product <x,y>=E"T ∑ t=o xtyt# Then Xis a finite dimensional Hilbert space.
Risks 2020,8, 66 4 of 34 The market consists of M risky assets Sm∈ X , m= 1, 2, . . . , M and T risk-free zero coupon bonds 1 u , u= 1, 2, . . . , T where 1 u u= 1 and 1 u t= 0 for t6=u . At time t , there is a bid and ask price pair bi t≤ai t . Paying ai t , one will receive the income flow Si tT s=t+1 and receiving bi t one will get the income flow −Si tT s=t+1 . For risk-less bonds, the bid and ask prices of 1 u , paid/received at time t<u is denoted by gu t≤hu t≤1. There are two zero cost cash flows associated with each pair of bid and ask prices. Sit s= 0s<t −ai ts=t Si ss>t and ˜ Sit s= 0s<t bi ts=t −Si ss>t Similarly the bond maturing at time u, creates two zero cost cash flows as below: 1ut s= 0s6=u,t −hu ts=t 1s=u and ˜ 1ut s= 0s6=u,t gu ts=t −1s=u 2.2. Set of Zero-Cost Cash Flows Assuming that one can trade any non-negative multiple of above zero-cost cash flows, and suppose αi t , ˜ αi t , βu t , ˜ βu t , for i= 1, . . . , M , u= 1, . . . , T are non-negative F -adapted random variables, then Zis defined to be the set of all zero cost cash flows of the form z= T ∑ t=0"M ∑ i=1αi tSit +˜ αi t˜ Sit+ T ∑ u=1βu t1ut s+˜ βu t˜ 1ut s#(1) It can be seen from the definition that Z is a cone. Define C to be the set of all cash flows c∈ X such that there is z∈ Z with z≥c . Then C is the set of adapted processes, super-replicable at zero cost. It is clear that Ca closed convex cone and Z ⊂ C. Figure 1is a geometrical demonstration (in R2 ) of a cone Z and the convex cone C , set of all vectors dominated by some elements of Z(z≥c). Z C C Figure 1. Set of zero-cost cash flows Z, and the cone C.
Risks 2020,8, 66 5 of 34 2.3. The Characterization of No-Arbitrage We now define arbitrage and will be characterizing no-arbitrage using a utility maximization problem. Definition 3. We say that a cash flow c = (ct)T t=0∈ C\{0}is an arbitrage if ct≥0for all t =0, 1, . . . , T. Definition 4. (Utility Function) An extended real valued function u:R→R∪ {+∞} is called a utility function if it satisfies the following three characteristics: 1. (Risk Aversion) u is strictly concave, 2. (Profit Seeking) u is strictly increasing and lim t→+∞u(t) = +∞, 3. (Bankruptcy Forbidden) for any t <0, u(t) = −∞. Let X+ denote the non-negative cone in X , also w0∈ X + be the initial endowment and (ct)T t=0= (c0,c1, ..., cT)∈ C be a cash flow. Consider the following optimal trading problem p=max (E"T ∑ t=0 u(ct)#;c= (ct)T t=0∈w0+C) where u is a utility function. We are able to characterize the no-arbitrage principle in terms of this utility maximization problem. Theorem 1 (Characterization of Utility Maximization) . The financial market has no-arbitrage opportunity if and only if the optimal trading problem above has a finite value p <∞. Proof. In Appendix A. 3. FTAP for Multi-Period Model In this section, we establish FTAP for conic finance model, characterizing no-arbitrage in terms of the extension of pricing factors. A first version of this theorem was proven by Harrison and Kreps (1979) . More general versions of the theorem were proven in 1981 by Harrison and Pliska (1981) and in 1994 by Delbaen and Schachermayer (1994). Theorem 2 (The First Fundamental Theorem of Asset Pricing) . A financial market with time horizon T and price processes of the risky asset and risk-less bond given by S1 , ..., ST and S0 1 , ..., S0 T , respectively, is arbitrage-free under the probability P if and only if there exists another probability measure Q such that i. For any event A , P(A) = 0if and only if Q(A) = 0(We say in this case that P and P are equivalent probability measures). ii. The discounted price process, X1:=S1 S0 1 , ..., XT :=ST S0 T is a martingale under Q . A measure Q that satisfies (i) and (ii) is known as a risk neutral measure. We now start by considering dual of the utility maximization problem in Section 2. First, we rephrase the utility maximization problem in Section 2as an abstract convex programming problem.
Risks 2020,8, 66 6 of 34 For x= (xt)T t=0∈ X , define g(x) = ιw0+C(x) and f(x) = E"T ∑ t=0 (−u)(xt)# . Then we can rewrite the optimal trading problem in Equation (1) as p=−min (E"T ∑ t=0 (−u)(ct)#|c= (c)T t=0∈w0+C) =−min x∈X (E"T ∑ t=0 (−u)(xt)#+ιw0+C(x)) =−min x∈X {f(x) + g(x)} The dual representation of the problem above gives us an extension of the Fundamental Theorem of Asset Pricing (FTAP) in conic finance. We approach FTAP via a utility optimization problem. Definition 5 (Pricing Factor) . For t<u≤T , we say fu t is a pricing factor from time t to time u and compatible with financial market, if the two following conditions are satisfied gu t≤Et"T ∑ s=t+1 fu t1ut s#≤hu t, (2) and also for 1≤i≤M bi t≤Et"T ∑ s=t+1 fu tSi t#≤ai t. Theorem 3. (FTAP in conic finance) Let u(t) be a utility function and consider a two-price financial market consisting of price processes of the risky assets and risk-free bonds. Then the market is arbitrage-free under the probability P if and only if there exists a pricing factor. Proof. In Appendix A. Remark 1. Through the proof of Theorem 3, there are a few things we would like to point out here: 1. Pricing factor is not unique. This is clear since the existence of a pricing factor in FTAP comes from the existence of a solution to the dual problem, and we know the dual problem does not necessary have a unique solution. 2. The pricing factor is related to the utility function u via the duality. In fact we saw specifically that ¯ z∈ −∂(−u)( ¯ x). 3. The pricing factors can be used to generate prices for cash flows c∈ C with no-arbitrage existing. We will explain this more in detail on the coming sub-section. 4. Price Bounds and Their Estimates A distinct feature of conic finance is that prices of assets are not unique. In fact, we see that every price factor will provide a non-arbitrage price. We will see that the set of price factors will provide us price bounds, outside of which arbitrage arises. 4.1. Definition of the Bounds Define PF to be the set of all price factors fs t that are compatible with the financial market which is PF =(fs t|gu t≤Et"T ∑ s=t+1 fu t1ut s#≤hu t,bi t≤Et"T ∑ s=t+1 fs tSi s#≤ai t)(3)
Risks 2020,8, 66 7 of 34 Let c= (ct)∈ C. We define ut(c) = sup (Et"T ∑ s=t+1 fs tcs#|fs t∈ PF)(4) and lt(c) = inf (Et"T ∑ s=t+1 fs tcs#|fs t∈ PF)(5) Remark 2. The values of ut(c) and lt(c) are the lower and upper bounds for bid and ask prices of the cash flow c= (ct)∈ C , equivalent to existing no-arbitrage. Equivalently, [lt(c) , ut(c)] is the no-arbitrage region for the price of a given cash flow c = (ct)∈ C. Remark 3. As we know, any of fu t∈ PF has a one to one correspondence relation to a solution of the dual problem say ¯ z∈ Co so that ¯ zt=ΓtMt=ΓtQ P , where Q is a P -equivalent martingale measure defined as Q=MtP. Since f u tare in direct relation to the solution of dual problem, this leads to Et"T ∑ s=t+1 ¯ zscs#=Et"¯ zt T ∑ s=t+1 ¯ zs ¯ zt cs#=¯ ztEt"T ∑ s=t+1 fs tcs# which is a linear combination in terms of the pricing factor we defined above, so that the corresponding maximization and minimization above (and later in our text) becomes a linear programming problem. A question that one can ask is, if the market (bid and ask) prices of a cash flow c= (ct)∈ C , are outside of the no-arbitrage region [lt(c) , ut(c)] , will any arbitrage opportunity become possible? We will answer this question in the next section. 4.2. Computations in One-Period To illustrate the general pattern, we start with a simple example. Example 1. Consider a one-period model ( T= 1) with a sample space Ω={ω1,ω2} containing only two elements and let M =1(only one risky asset, S1=S). Additionally, consider a bond 11,0 = (−h1 0, 1)˜ 11,0 = (g1 0,−1) where the ask and prices are h1 0= 0.9 and g1 0= 0.8, respectively. Assume the asset St over times t= 0, 1 has standard initial price 1 and payoff as following S0= 1 S1(ω1) = 2 S1(ω2) = 0.5 Now consider the following primal maximization problem p1:= u0=max Ehf1 0S1i g1 0≤Ehf1 0i≤h1 0 =(u0=max hf1 0(ω1)P(ω1)S1(ω1) + f1 0(ω2)P(ω2)S1(ω2)i g1 0≤f1 0(ω1)P(ω1) + f1 0(ω2)P(ω2)≤h1 0 (6) For simplicity we let r1:=f1 0(ω1)P(ω1) and r2:=f1 0(ω2)P(ω2) so that the linear programming problem in Equation (6) becomes
Risks 2020,8, 66 8 of 34 p1=(u0=max [r1S1(ω1) + r2S1(ω2)] g1 0≤r1+r2≤h1 0 = u0=max [r1S1(ω1) + r2S1(ω2)] r1+r2≤h1 0 −r1−r2≤ −g1 0 (7) A systematical way of deriving the arbitrage strategy is to use linear programming duality. By using the linear programming duality, we can write the dual problem to Equation (7) as d1:= u0=min −g1 0t1+h1 0t2 t2−t1≥S1(ω1) t2−t1≥S1(ω2) t1,t2≥0 = u0=min −g1 0t1+h1 0t2 t2≥t1+max ω∈ΩS1(ω) t1,t2≥0 (8) If we use the constraint condition in Equation (8) and substitute in the objective function, we have u0=min t1≥0−g1 0t1+h1 0t1+max ω∈ΩS1(ω)=min t1≥0t1h1 0−g1 0+h1 0max ω∈ΩS1(ω) but since h1 0−g1 0≥0, minimum happens when t1=0so that u0=h1 0max ω∈ΩS1(ω)=0.9 (2) = 1.8 (9) Therefore 1.8 is the highest ask price for S1 offered at time t= 0under no-arbitrage condition. That means for any offered price higher than 1.8 one should be able to build a portfolio creating an arbitrage. In fact for a bid price b0> 1.8, since the solution to Equation (8) is t1= 0, t2= 2, one can buy 2 unites of bond 1 1,0 = (− 0.9, 1 ) at time t= 0for the price of 1.8 and receive 2 at time t= 1so that S1 can be delivered and still making a profit of b0−1.8 0.9 >0. Similarly we can find a lowest bid price l0 for which one can construct a portfolio providing arbitrage if any ask price lower than l0is available. The approach illustrated above can also be used in financial market that involves both bonds and stocks. A general result is summarized in the following theorem. We will use the set of all possible pricing factors corresponding to all solutions of the dual problem to provide a no-arbitrage region for a given cash flow. We state the result for super-hedging, i.e., upper bound for no-arbitrage. Sub-hedging can be discussed similarly. Theorem 4 (Super-Hedging).Suppose that (c0,c1)∈ C is an acceptable cash flow, then u0=sup f1 0∈PF nEhf1 0c1io =min (Λ,γ)0∈R2M+2 +(−p0((Λ,γ)0) + sup ω∈Ωhg1 0(c1(ω)−p1((Λ,γ)0)(ω))i)
Risks 2020,8, 66 15 of 34 S(B1,1) S(B2,2) S(B2,1) that is p1:= u2 1(B1,1) = max [r1(B1,1)c2(B2,1) + r2(B1,1)c2(B2,2)](B1,1) g2 1(B1,1)≤r1(B1,1) + r2(B1,1)≤h2 1(B1,1), b2 1(B1,1)≤r1(B1,1)S2(B2,1) + r2(B1,1)S2(B2,2)≤a2 1(B1,1). (23) for which the constrains in all one-sided inequalities can be written as p1:= u2 1(B1,1) = max [r1(B1,1)c2(B2,1) + r2(B1,1)c2(B2,2)](B1,1) r1(B1,1) + r2(B1,1)≤h2 1(B1,1), −r1(B1,1)−r2(B1,1)≤ −g2 1(B1,1), r1(B1,1)S2(B2,1) + r2(B1,1)S2(B2,2)≤a2 1(B1,1), −r1(B1,1)S2(B2,1)−r2(B1,1)S2(B2,2)≤ −b2 1(B1,1), r1,r2≥0. (24) and having the objective function and constrains in matrix form we have p1:= u2 1(B1,1) = max hc2(B2,1)c2(B2,2)i"r1(B1,1) r2(B1,1)#! 1 1 −1−1 S2(B2,1)S2(B2,2) −S2(B2,1)−S2(B2,2) "r1(B1,1) r2(B1,1)#≤ h2 1(B1,1) −g2 1(B1,1) a2 1(B1,1) −b2 1(B1,1) , r1,r2≥0. (25) Now we write the dual to this linear programming problem which is d1:= u2 1(B1,1) = min hh2 1(B1,1)−g2 1(B1,1)a2 1(B1,1)−b2 1(B1,1)i t1 t2 t3 t4 "1−1S2(B2,1)−S2(B2,1) 1−1S2(B2,2)−S2(B2,2)# t1 t2 t3 t4 ≥"c2(B2,1) c2(B2,2)#, t1,t2,t3,t4≥0. (26)
Risks 2020,8, 66 16 of 34 or equivalently d1:= u2 1(B1,1) = min hh2 1(B1,1)−g2 1(B1,1)a2 1(B1,1)−b2 1(B1,1)i t1 t2 t3 t4 "−1 1 −S2(B2,1)S2(B2,1) −1 1 −S2(B2,2)S2(B2,2)# t1 t2 t3 t4 ≤"−c2(B2,1) −c2(B2,2)#, t1,t2,t3,t4≥0. (27) If we let, as an example, a2 1(B1,1) = 3, b2 1(B1,1) = 2, c2(B2,1) = 3 and c2(B2,2) = 2, then by solving Equation (27) using linear programming with complementary slackness conditions (we used MATLAB for our purpose) we have t1=5 3 , t3=1 3 , t2=t4= 0 (which is the solution to the dual problem d1 and indicates the portfolio strategy one needs to choose in the case of existence of any arbitrage opportunity) and u2 1(B1,1) = 2.5. Similarly by solving the dual problem for the other part of the diagram S(B1,2) S(B2,4) S(B2,3) that is d1:= u2 1(B1,2) = min hh2 1(B1,2)−g2 1(B1,2)a2 1(B1,2)−b2 1(B1,2)i t1 t2 t3 t4 "−1 1 −S2(B2,3)S2(B2,3) −1 1 −S2(B2,4)S2(B2,4)# t1 t2 t3 t4 ≤"−c2(B2,3) −c2(B2,4)#. t1,t2,t3,t4≥0. (28) and if we let a2 1(B1,2) = 0.7 and b2 1(B1,2) = 0.4 also let c2(B2,3) = 1 and c2(B2,2) = 0.5, then the solution is t1=1 3,t2=t4=0, t3=2 3and u2 1(B1,2) = 0.7833. Now for the last part consider the following picture and its corresponding linear programming in dual form
Risks 2020,8, 66 17 of 34 S(B0,1) S(B1,2) S(B1,1) with d1:= u2 0(B0,1) = min hh1 0(B0,1)−g1 0(B0,1)a1 0(B0,1)−b1 0(B0,1)i t1 t2 t3 t4 "−1 1 −S1(B1,1)S1(B1,1) −1 1 −S1(B1,2)S1(B1,2)# t1 t2 t3 t4 ≤"−u2 1(B1,1) −u2 1(B1,2)#, t1,t2,t3,t4≥0. (29) for which when a1 0(B0,1) = 1.6, b1 0(B0,1) = 1.5 (here we simply assumed that c1= 0), the corresponding solution becomes t1=19 90 , t3=103 90 , t2=t4= 0 and u2 0(B0,1) = 2.0317. For any bid price offered at time t=0 larger than 2.0317, there is an arbitrage and the trading strategy is as following: 19 90 1 10 +103 90 S01 1 3 1 21 +2 3S12 5 3 1 21 +1 3S12 buy 19 90 units of 1 10 and 103 90 units of S01 and at time t= 1 in either case we are able cover the asked price of c2at time t=2. In fact in the case of B1,1 possibility, we buy 5 3units of 121 and 1 3units of S12, and in the case of B1,2 possibility, we buy 1 3 units of 1 21 and 2 3 units of S12 , and in either case c2 will be covered to deliver and the difference will be an arbitrage. As we noted in the last diagram, at least 1 3 of the risk-free bond 1 21 is needed to be bought at time t= 1. Now we consider a trading strategy which purchases 1 3 of the bond 1 20 bought at time t= 0. Therefore the trading strategy can be replaced by
Risks 2020,8, 66 18 of 34 1 3 1 20 +t1 1 10 +t2˜ 110 +t3S01 +t4˜ S01 2 3S12 4 3 1 21 +1 3S12 where t1,t2,t3,t4are the solutions to the new linear programming problem below min hh1 0(B0,1)−g1 0(B0,1)a1 0(B0,1)−b1 0(B0,1)i t1 t2 t3 t4 "−1 1 −S1(B1,1)S1(B1,1) −1 1 −S1(B1,2)S1(B1,2)# t1 t2 t3 t4 ≤"−˜ u2 1(B1,1) −˜ u2 1(B1,2)#, t1,t2,t3,t4≥0. (30) and ˜ u2 1(B1,1) = 4 3h2 1(B1,1) + 1 3a2 1(B1,1) = 11 5 ˜ u2 1(B1,2) = 2 3a2 1(B1,2) = 7 15 (31) Solution to the linear programming above is t2=t4= 0, t1=1 9 , t3=73 72 so that the largest ask price for c2in this case becomes 1 3h2 0+1 9h1 0+73 72a1 0=2.02778 and as we see, this upper bound is smaller and much more accurate than the upper bound u2 0= 2.0317 which we found above without using the 120 bond. Now consider the cash flow ( ., c1 , c2) and let α to be the portion of 1 20 = (−h2 0 , 0, 1 ) that we carry in the trading strategy at time t= 0. We would like to find the highest ask-price at time t= 0 for this cash flow. Therefore we consider the following maximization problem and unlike the example above, we try to solve this problem in one step u2 0=max E0hf1 0c1+f2 0c2i=max E0hf2 0α+f1 0c1+E1hf2 0(c2−α)ii g1 0≤E0hf1 0i≤h1 0, g2 0≤E0hf2 0i≤h2 0, b2 0≤E0hf1 0S1+f2 0S2i≤a2 0, (32) and by the definition of expectation and letting rij =fi 0(Bi,j)Q(Bi,j)we have u2 0(α) = max{α[s21 +s22 +s23 +s24]+r11 [c1(B11) + r21c2(B21) + r22c2(B22)] +r12 [c1(B12) + r23c2(B23) + r24c2(B24)]−α[r11(r21 +r22) + r12(r23 +r24)]}
Risks 2020,8, 66 19 of 34 where r11 +r12 ≤h1 0, −r11 −r12 ≤ −g1 0, s21 +s22 +s23 +s24 ≤h2 0, −s21 −s22 −s23 −s24 ≤ −g2 0, r11 [S1(B11) + r21S2(B21) + r22S2(B22)]+r12 [S1(B12) + r23S2(B23) + r24S2(B24)]≤a2 0, −r11 [S1(B11) + r21S2(B21) + r22S2(B22)]−r12 [S1(B12) + r23S2(B23) + r24S2(B24)]≤b2 0. (33) As we can see, the above linear programming is a ten variable linear programming with one parameter α and this is only for a binary priced market where at each time there are only two possibilities for the stock and cash flow price. We are only considering one risky asset. 4.4. Complexity of Multi-Period Model Consider a 2-period model and let v2 be the number of variables in the linear programming associated with this model and p2 be the number of parameters associated with 2-period bonds in this model. We already know that v1= 2 and p1= 0. We can see that adding a single 1-period bond into the model, adds 2 variables and one parameter to the linear programming problem. On the other hand a 2-period model is equivalent to a composition of a 1-period model followed by two 1-period models as illustrated in following diagram 1-period 1-period 1-period Since a 2-period model is the composition of a 1-period followed by two 1-period models, then v2 p2! = v1 p1! + 2 v1 p1! + 2-period bond = 2 0! +2 2 0! + 4 1! = 10 1! which coincides with what we found before. This is the case for a over simplified case, binary model with a two states at each time. Now consider a T -period model and let vT be the number of variables in the linear programming associated with this model and pT be the number of parameters associated with multi-period bonds in this model. A T -period model is equivalent to a composition of a 1-period model followed by two (T−1)-period models as illustrated in following diagram 1-period (T−1)-period (T−1)-period For example, a 3-period model is the composition of a 1-period followed by two 2-period models, so that
Risks 2020,8, 66 20 of 34 v3 p3!= v1 p1!+ 2 v2 p2!+ 2-period bond + 3-period bond = 2 0!+2 10 1!+ 4 1!+ 8 1!= 34 4! Now for a T-period model we have vT pT! = v1 p1! + 2 vT−1 pT−1! + 2-period bond + 3-period bond + . . . + T -period bond = 2 0! +2 vT−1 pT−1!+ 4 1!+ 8 1!+ . . . + 2T 1!= 2vT−1+2+4+8+· · · +2T 2pT−1+T−1! or vT pT!= 2vT−1+2T+1−2 2pT−1+T−1! As it can be seen, when the periods of the model increases, the number of variables and parameters increase at an exponential rate. Considering the fact that this calculation was only done for the simplest case of a model (binary prices and one risky asset) we notice that for a T -period model, the parametrized linear programming with pT parameters and vT variables make the problem very complicated to solve. Lemma 1. For a T -period model with vT as the number of variables in the linear programming associated with the model and pTas the number of parameters associated with multi-period bonds, we have v2 p2!= 2 0!and vT pT!= 2vT−1+2T+1−2 2pT−1+T−1!(34) As a result of the reason explained above, we suggest solving the problem one step at a time, which means (similar to the 2-period linear programming problem that we solved) starting form the end and solving multiple 1-period problems on each step and continuing this process backward until we reach the initial time on the problem. 4.5. Estimate of Multi-Period Bounds (Breaking into One Periods) Consider the following portfolio of zero-cost cash flows (p0,p1,p2) = t1110 +t2120 +t3S10 +t4S20 +t5S21 +t6121 =t1−h1 0, 1, 0+t2−h2 0, 0, 1+t3−a1 0,S1, 0+t4(−a2 0, 0, S2) +t50, −a2 1, 1+t60, −h2 1, 1(35) where p0=−t1h1 0−t2h2 0−t3a1 0−t4a2 0 p1=t1+t3S1−t5a2 1−t6h2 1 p2=t2+t4S2+t5S2+t6(36) also t1 , t2 , t3 , t4∈ F0 and t5 , t6∈ F1 . In fact we have the following trading diagram corresponding to the above portfolio at time t=0 and t=1
Risks 2020,8, 66 21 of 34 t1 1 10 +t2 1 20 +t3S10 +t4S20 t5(B12)S21(B12) + t6(B12) 1 21(B12) t5(B11)S21(B11) + t6(B11) 1 21(B11) Since we are assuming to deliver c2∈ F2 at t= 2, indeed we have to solve the linear programing problem min(−p0) such that; p1≥0 p2−c2≥0 t1, . . . , t6≥0 (37) We can define the corresponding Lagrangian as L(t1, . . . , t6,λ1,λ2) = (−p0)−λ1p1−λ2(p2−c2)(38) where λ1∈ F1 and λ2∈ F2 are the random variable Lagrange multipliers. Therefore our minimization problem can be stated as inf t≥0sup λ≥0 L(t1, . . . , t6,λ1,λ2)(39) where L(t1, . . . , t6,λ1(B11),λ1(B12),λ2(B21),λ2(B22),λ2(B24),λ2(B24)) = t1h1 0+t2h2 0+t3a1 0+t4a2 0 −λ1(B11)ht1+t3S1(B11)−t5(B11)a2 1(B11)−t6(B11)h2 1(B11)i −λ1(B12)ht1+t3S1(B12)−t5(B12)a2 1(B12)−t6(B12)h2 1(B12)i −λ2(B21)[t2+t4S2(B21) + t5(B11)S2(B21) + t6(B11)−c2(B21)] −λ2(B22)[t2+t4S2(B22) + t5(B11)S2(B22) + t6(B11)−c2(B22)] −λ2(B23)[t2+t4S2(B23) + t5(B12)S2(B23) + t6(B12)−c2(B23)] −λ2(B24)[t2+t4S2(B24) + t5(B12)S2(B24) + t6(B12)−c2(B24)] (40) Now by Linearity Constraint Qualification the problem is feasible so that by Lagrangian strong duality we have inf t≥0sup λ≥0 L(t1, . . . , t6,λ1,λ2) = sup λ≥0 inf t≥0L(t1, . . . , t6,λ1,λ2)(41)
Risks 2020,8, 66 22 of 34 hence L(t1, . . . , t6,λ1,λ2)can be written as L(t1, . . . , t6,λ1(B11),λ1(B12),λ2(B21),λ2(B22),λ2(B24),λ2(B24)) =t1"h1 0− 2 ∑ j=1 λ1(B1j)#+t2"h2 0− 4 ∑ k=1 λ2(B2k)# +t3"a1 0− 2 ∑ j=1 λ1(B1j)S1(B1j)#+t4"a2 0− 4 ∑ k=1 λ2(B2k)S2(B2k)# +t5(B11)hλ1(B11)a2 1(B11)−λ2(B21)S2(B21)−λ2(B22)S2(B22)i +t5(B12)hλ1(B12)a2 1(B12)−λ2(B23)S2(B23)−λ2(B24)S2(B24)i +t6(B11)hλ1(B11)h2 1(B11)−λ2(B21)−λ2(B22)i +t6(B12)hλ1(B12)h2 1(B12)−λ2(B23)−λ2(B24)i + 4 ∑ k=1 λ2(B2k)c2(B2k) (42) We can see that the corresponding maximization problem to the Lagrangian above is max 4 ∑ k=1 λ2(B2k)c2(B2k)! such that; λ1(B11) + λ1(B12)≤h1 0 λ2(B21) + λ2(B22) + λ2(B23) + λ2(B24)≤h2 0 λ1(B11)S1(B11) + λ1(B12)S1(B12)≤a1 0 λ2(B21)S2(B21) + λ2(B22)S2(B22) + λ2(B23)S2(B23) + λ2(B24)S2(B24)≤a2 0 λ2(B21) λ1(B11)S2(B21) + λ2(B22) λ1(B11)S2(B22)≤a2 1(B11) λ2(B23) λ1(B12)S2(B23) + λ2(B24) λ1(B12)S2(B24)≤a2 1(B12) λ2(B21) λ1(B11)+λ2(B22) λ1(B11)≤h2 1(B11) λ2(B23) λ1(B12)+λ2(B24) λ1(B12)≤h2 1(B12) λ1,λ2≥0 (43) Unsurprisingly, we observe that λ1 , λ2 are the pricing factors in the expectation form. In fact, as we defined before, if we let λ1(B1j) = f1 0(B1j) = r1jj=1, 2 λ2(B1k) = f2 0(B1k) = ukk=1, 2, 3, 4 λ2(B1k) = f2 1(B2k) = r2kk=1, 2, 3, 4 (44)
Risks 2020,8, 66 23 of 34 then the linear programming in Equation (43) becomes max (u1c2(B21) + u2c2(B22) + u3c2(B23) + u4c2(B24)) such that; r11 +r12 ≤h1 0 u1+u2+u3+u4≤h2 0 r11S1(B11) + r12S1(B12)≤a1 0 u1S2(B21) + u2S2(B22) + u3S2(B23) + u4S2(B24)≤a2 0 r21S2(B21) + r22S2(B22)≤a2 1(B11) r23S2(B23) + r24S2(B24)≤a2 1(B12) r21 +r22 ≤h2 1(B11) r23 +r24 ≤h2 1(B12) r11,r12,r21,r22,r23,r24,u1,u2,u3,u4≥0 (45) which, as we can see, is the primal maximization problem (the dual of the dual in this case) corresponding to our initial portfolio. We can also observe that the problem in Equation (45) is indeed the expectation form of a super-hedging problem as max E0hf2 0c2i such that; E0hf1 0i≤h1 0 E0hf2 0i≤h2 0 E0hf1 0S1i≤a1 0 E0hf2 0S2i≤a2 0 E1hf2 1S2i≤a2 1 E1hf2 1i≤h2 1 (46) Solving the linear programming problem in Equation (45) we have the following solutions t11.3327 t5(B11)0.1371 t20.5256 t5(B12)0.4847 t30.2725 t6(B11)1.5377 t40.0969 t6(B12)0.6964 with a super-hedging value of u2 0=2.5649 As we notice, there is slightly small decrease on the value of upper bound for the asked price comparing with the case where we calculated this upper bound one step at a time. This is completely justified because the minimization above has more constraints since we are using the two-period bond and asset to formulate our problem. As we stated before, for multi-period cases (that requires us to solve linear models with multiple variables and parameters with a very small adjustments) we will find the upper-bound and lower-bound by solving linear programming problems one at a time.
Risks 2020,8, 66 24 of 34 4.6. The General 2-Period Model In this section we use the results from the examples in this section to summarize the two different cases on a 2-period model. First case is when one can use all possibilities of bonds and assets (both 1-period and 2-periods). Theorem 5. Let T= 2, t= 0, 1, c= (c1 , c2)∈ C an acceptable cash flow and consider the following linear programing problem u2 0(II) = sup E0hf1 0c1+f2 0c2i subject to E0hf1 0i≤h1 0,E0hf1 0(−1)i≤ −g1 0, E0hf2 0i≤h2 0,E0hf2 0(−1)i≤ −g2 0, E1hf2 1i≤h2 1,E1hf2 1(−1)i≤ −g2 1, E1hf2 1Si 2i≤ai 1,E1hf2 1−Si 2i≤ −bi 1, E0hf1 0Si 1+f2 0Si 2i≤ai 0, E0hf1 0−Si 1+f2 0−Si 2i≤ −bi 0. (47) Then u2 0(II)is a super-hedging bound and the solution to dual problem is a portfolio by which one is able to construct an arbitrage opportunity if the market bid price of c exceeds u2 0(II). Proof. In Appendix A. In this version of the problem one is able to take advantage of the both 2-period bond and assets, however as we saw before there is some complexity in solving and finding the right strategy to construct the portfolio, where as the number of assets and their price possibilities increase, the complexity of finding the right portfolio strategy increases exponentially. On the other hand, we can find a super-hedge value for a 2-period linear programming problem by solving two 1-period problems successively, in which case we have the following theorem. Theorem 6. Let c = (c1,c2)∈ C be an acceptable cash flow and suppose u2 1=sup E1hf2 1c2i subject to E1hf2 1i≤h2 1,E1hf2 1(−1)i≤ −g2 1, E1hf2 1Si 2i≤ai 1,E1hf2 1−Si 2i≤ −bi 1. (48) is a super-hedging bound for c2 paid at time t= 1(the largest ask price for c2 under no-arbitrage assumption). Now consider the following linear programming problem u2 0(I) = sup E0hf1 0c1+u2 1i subject to E0hf1 0i≤h1 0,E0hf1 0(−1)i≤ −g1 0, E0hf1 0Si 1i≤ai 0, E0hf1 0−Si 1i≤ −bi 0. (49) then the solution to linear programming problems in Equations (48) and (49) and their duals gives us a trading strategy by which we can take advantage if any arbitrage opportunity is available. Proof. In Appendix A.
Risks 2020,8, 66 31 of 34 (b) If we assume that there is only one asset with two options (a 2-period S2 06= 0 or two 1-period S1 0andS2 1) then the difference in the hedging-price values is a1 0max ω∈Ωa2 1(ω)−a2 0max ω∈Ωc2 S2 (ω)≤a1 0max ω∈Ωa2 1(ω)−a2 0maxω∈Ωc2(ω) minω∈ΩS2(ω)(A17) (c) Now if we use both bond and asset then we have an upper bound for the difference as "h1 0max ω∈Ωh2 1(ω)−h2 0+ a1 0maxω∈Ωa2 1(ω)−a2 0 minω∈ΩS2(ω)!#max ω∈Ωc2(ω)(A18) (d) Therefore for a super-hedging with both bonds and finite number (M) of assets we have u2 0(I)−u2 0(II)≤"h1 0max ω∈Ωh2 1(ω)−h2 0+ M ∑ i=1 ai1 0maxω∈Ωai2 1(ω)−ai2 0 minω∈ΩSi 2(ω)!#max ω∈Ωc2(ω)(A19) Proof of Theorem 8. As a result of what was discussed in Lemma 1(the complexity in solving the dual linear programming problem) we solve and find the trading strategy starting at T− 1 going backward and one step at a time. So first let uT T−1=sup ET−1hfT T−1cTi subject to ET−1hfT T−1i≤hT T−1,ET−1hfT T−1(−1)i≤ −gT T−1 ET−1hfT T−1Si T−1i≤ai T−1,ET−1hfT T−1−Si T−1i≤ −bi T−1,i=1, . . . , M (A20) We formulate the dual problem by using the Lagrangian. First let (Λ,γ)T−1= (λ1 T−1, . . . , λM T−1,˜ λ1 T−1, . . . , ˜ λM T−1,γT T−1,˜ γT T−1)∈RV R2M+2 +,FT−1(A21) be the Lagrange multiplier of our linear programing problem. So we can write the Lagrangian as L(f,(Λ,γ)T−1) = ET−1fT T−1cT+∑M i=1λi T−1ai T−1−ET−1fT T−1Si T−1+ ∑M i=1˜ λi T−1ET−1fT T−1Si T−1−bi T−1+γT T−1hT T−1−ET−1fT T−1+ ˜ γT T−1ET−1fT T−1−gT T−1(A22) Now we can observe that inf (Λ,γ)T−1∈RV(R2M+2 +,FT−1)L(f,(Λ,γ)T−1) = (ET−1hfT T−1cTif∈ PD −∞otherwise (A23) So that, by strong linear programming duality we have uT−1=supf∈PD inf(Λ,γ)T−1∈RV(R2M+2 +,FT−1)L(f,(Λ,γ)T−1) =inf(Λ,γ)T−1∈RV(R2M+2 +,FT−1)supf∈PD L(f,(Λ,γ)T−1)(A24) On the other hand, the Lagrangian can be rewritten as L(f,(Λ,γ)T−1) = ET−1nfT T−1cT−h∑M i=1λT T−1−˜ λi T−1Si T+γT T−1−˜ γT T−1io −∑M i=1−λi T−1ai T−1+˜ λi T−1bi T−1+−γT T−1hT T−1+˜ γT T−1gT T−1(A25)
Risks 2020,8, 66 32 of 34 Now for (Λ,γ)T−1∈RV R2M+2 +,FT−1consider the zero-cost portfolio of cash flow pT−1(Λ,γ)T−1,pT(Λ,γ)T−1 (A26) =γT T−11T,T−1+˜ γT T−1˜ 1T,T−1+ M ∑ i=1λT T−1Si,T−1+˜ λi T−1˜ Si,T−1(A27) so that pT−1(Λ,γ)T−1= M ∑ i=1−λi T−1ai T−1+˜ λi T−1bi T−1+−γT T−1hT T−1+˜ γT T−1gT T−1(A28) and pT(Λ,γ)T−1= M ∑ i=1λT T−1−˜ λi T−1Si T+γT T−1−˜ γT T−1(A29) Since the term pT−1(Λ,γ)T−1 is an F measurable random variable and independent of f∈ PD , then we see that Equation (A25) can be written as L(f,(Λ,γ)T−1) = ET−1nfT T−1cT−pT(Λ,γ)T−1o−pT−1(Λ,γ)T−1(A30) Now let ¯ ω∈arg max cT−pT(Λ,γ)T−1 (A31) Let fT T−1:=gT T−1for all ω∈Ω. Then Equation (A24) becomes uT T−1=inf(Λ,γ)T−1∈RV(R2M+2 +,FT−1)gT T−1supω∈ΩcT(ω)−pT(Λ,γ)T−1(ω)−pT−1(Λ,γ)T−1 (A32) If (¯ Λ,¯ γ)T−1 is a solution to the minimization problem in Equation (A32), then it determines the trading strategy from time T− 1 to time T while uT T−1 is the highest price for cT (with no-arbitrage) paid at time T− 1. Therefore we set up a linear programming problem that determines a super-hedging value of both uT T−1and cT−1paid at time T−2, as following uT−1 T−2=sup ET−2hfT T−1cT−1+uT T−1i subject to ET−2hfT−1 T−2i≤hT−1 T−2,ET−2hfT−1 T−2(−1)i≤ −gT−1 T−2 ET−2hfT−1 T−2Si T−2i≤ai T−2,ET−2hfT−1 T−2−Si T−2i≤ −bi T−2,i=1, . . . , M (A33) Similar to what we did above on establishing and solving the dual problem, suppose (¯ Λ,¯ γ)T−2 is the solution to dual linear programming problem of Equation (A33), then it determines the trading strategy form time T− 2 to T− 1. Continuing this backward process and one step at a time, we will have a sequence of pairs, super-hedges and non-negative random variable coefficients ut+2 t+1,(¯ Λ,¯ γ)t+1,· · · ,uT−2 T−3,(¯ Λ,¯ γ)T−3,uT−1 T−2,(¯ Λ,¯ γ)T−2,uT T−1,(¯ Λ,¯ γ)T−1(A34) where ur r−1=sup f∈PD Er−1hfr r−1cr+ur+1 ri (A35)
Risks 2020,8, 66 33 of 34 and the first term in Equation (A34) is the super-hedging value and the corresponding trading strategy coefficients determined at time t+ 1. Now we set-up our last linear programming problem where we find the super-hedging bound for c= (0, · · · , 0, ct+1,ct+2,· · · ,cT). Let uT t=sup Ethft+1 tct+1+ut+2 t+1i subject to Ethft+1 ti≤ht+1 t,Ethft+1 t(−1)i≤ −gt+1 t Ethft+1 tSi ti≤ai t,Ethft+1 t−Si ti≤ −bi t,i=1, . . . , M (A36) with the corresponding Lagrangian L(f,(Λ,γ)t) = Etnft+1 tct+ut+2 t+1−pt+1((Λ,γ)t)o−pt((Λ,γ)t)(A37) so that uT t=inf (Λ,γ)t∈RV(R2M+2 +,FT−1)sup f∈PD L(f,(Λ,γ)t)(A38) =inf (Λ,γ)t∈RV(R2M+2 +,FT−1)(gt+1 tsup ω∈Ωhct+1(ω) + ut+2 t+1(ω)−pt+1((Λ,γ)t)(ω)i−pt((Λ,γ)t))(A39) If (¯ Λ,¯ γ)t is a solution to the minimization problem in Equation (A39) then, in fact, it determines the trading strategy from time t to time t+ 1 and along with the other non-negative random variable coefficients (¯ Λ,¯ γ)r,r=t+1, · · · ,T−1 we have created a trading strategy corresponding to (¯ Λ,¯ γ)t,(¯ Λ,¯ γ)t+1,· · · ,(¯ Λ,¯ γ)T−1(A40) with the super-hedging bound uT t by which one can create an arbitrage opportunity if any higher price in the market is available. References Artzner, Philippe, Freddy Delbaen, Jean-Marc Eber, and David Heath. 1999. Coherent measures of risk. Mathematical Finance 9: 203–28. [CrossRef] Bion-Nadal, Jocelyne. 2009. Bid-ask dynamic pricing in financial markets with transaction costs and liquidity risk. Journal of Mathematical Economics 45: 738–50. [CrossRef] Carr, Peter, Helyette Geman, and Dilip B. Madan. 2001. Pricing and Hedging in Incomplete Markets. Journal of Financial Economics 62: 131–67. [CrossRef] Carr, Peter, and Qiji Jim Zhu. 2018. Convex Duality and Financial Mathematics. Berlin and Heidelberg: Springer. Cherny, Alexander, and Dilip B. Madan. 2009. New Measures for Performance Evaluation. Review of Financial Studies 22: 2571–606. [CrossRef] Cherny, Alexander, and Dilip B. Madan. 2010. Markets as a Counterparty: An Introduction to Conic Finance. International Journal of Theoretical and Applied Finance 13: 1149–77. Constantinides, George M. 1986. Capital Market Equilibrium with Transaction Costs. Journal of Political Economy 94: 842–62. [CrossRef] Copeland, Thomas E., and Dan Galai. 1983. Information Effects on the Bid-Ask Spread. Journal of Finance 38: 1457–69. [CrossRef] Delbaen, Freddy, and Walter Schachermayer. 1994. A General Version of the Fundamental Theorem of Asset Pricing. In The Mathematics of Arbitrage. Berlin and Heidelberg: Springer, pp. 149–205. Eberlein, Ernst, Thomas Gehrig, and Dilip B. Madan. 2009. Accounting to Acceptability: With Applications to the Pricing of One’s Own Credit Risk. Robert H. Smith School Research Paper No. RHS 06-113. SSRN Electronic Journal. [CrossRef]
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