A note on the Fundamental Theorem of Asset Pricing under model uncertainty
aa r X i v : . [ q -f i n . P R ] S e p A NOTE ON THE FUNDAMENTAL THEOREM OF ASSET PRICING UNDERMODEL UNCERTAINTY
ERHAN BAYRAKTAR, YUCHONG ZHANG, AND ZHOU ZHOU
Abstract.
We show that the results of [3] on the Fundamental Theorem of Asset Pricing andthe super-hedging theorem can be extended to the case in which the options available for statichedging ( hedging options ) are quoted with bid-ask spreads. In this set-up, we need to work with thenotion of robust no-arbitrage which turns out to be equivalent to no-arbitrage under the additionalassumption that hedging options with non-zero spread are non-redundant . A key result is theclosedness of the set of attainable claims, which requires a new proof in our setting. introduction We consider a discrete time financial market in which stocks are traded dynamically and optionsare available for static hedging. We assume that the dynamically traded asset is liquid and trading inthem does not incur transaction costs, but that the options are less liquid and their prices are quotedwith a bid-ask spread. (The more difficult problem with transaction costs on a dynamically tradedasset is analyzed in [2] and [7].) As in [3] we do not assume that there is a single model describingthe asset price behavior but rather a collection of models described by the convex collection P ofprobability measures, which does not necessarily admit a dominating measure. One should thinkof P as being obtained from calibration to the market data. We have a collection rather than asingle model because generally we do not have point estimates but a confidence intervals for theparameters of our models. Our first goal is to obtain a criteria for deciding whether the collectionof models represented by P is viable or not. Given that P is viable we would like to obtain therange of prices for other options written on the dynamically traded assets. The dual elements inthese result are martingale measures that price the hedging options correctly (i.e. consistent withthe quoted prices). As in classical transaction costs literature, we need to replace the no-arbitragecondition by the stronger robust no-arbitrage condition , as we shall see in Section 2. In Section 3we will make the additional assumption that the hedging options with non-zero spread are non-redundant (see Definition 3.1). We will see that under this assumption no-arbitrage and robustno-arbitrage are equivalent. Our main results are Theorems 2.1 and 3.1. Key words and phrases.
Model uncertainty, bid-ask prices for options, semi-static hedging, non-dominated col-lection of probability measures, Fundamental Theorem of Asset Pricing, super-hedging, robust no-arbitrage, non-redundant options.This research is supported by the National Science Foundation under grant DMS-0955463. Fundamental Theorem with Robust No Arbitrage
Let S t = ( S t , . . . , S dt ) be the prices of d traded stocks at time t ∈ { , , . . . , T } and H be the set ofall predictable R d -valued processes, which will serve as our trading strategies. Let g = ( g , . . . , g e )be the payoff of e options that can be traded only at time zero with bid price g and ask price g ,with g ≥ g (the inequality holds component-wise). We assume S t and g are Borel measurable, andthere are no transaction costs in the trading of stocks. Definition 2.1 (No-arbitrage and robust no-arbitrage) . We say that condition NA( P ) holds if forall ( H, h ) ∈ H × R e , H • S T + h + ( g − g ) − h − ( g − g ) ≥ P − quasi-surely ( -q.s. ) implies H • S T + h + ( g − g ) − h − ( g − g ) = 0 P -q.s.,where h ± are defined component-wise and are the usual positive/negative part of h . We say that condition
N A r ( P ) holds if there exists g ′ , g ′ such that [ g ′ , g ′ ] ⊆ ri [ g, g ] and NA( P )holds if g has bid-ask prices g ′ , g ′ . Definition 2.2 (Super-hedging price) . For a given a random variable f , its super-hedging price isdefined as π ( f ) := inf { x ∈ R : ∃ ( H, h ) ∈ H × R e such that x + H • S T + h + ( g − g ) − h − ( g − g ) ≥ f P -q.s. } . Any pair ( H, h ) ∈ H × R e in the above definition is called a semi-static hedging strategy. Remark 2.1. [1]
Let ˆ π ( g i ) and ˆ π ( − g i ) be the super-hedging prices of g i and − g i , where the hedgingis done using stocks and options excluding g i . NA r ( P ) implies either − ˆ π ( − g i ) ≤ g i = g i ≤ ˆ π ( g i ) or − ˆ π ( − g i ) ≤ ( g ′ ) i < g i and g i < ( g ′ ) i ≤ ˆ π ( g i ) (2.1) where g ′ , g ′ are the more favorable bid-ask prices in the definition of robust no-arbitrage. Thereason for working with robust no-arbitrage is to be able to have the strictly inequalities in (2.1) foroptions with non-zero spread, which turns out to be crucial in the proof of the closedness of the setof hedgeable claims in (2.3) (hence the existence of an optimal hedging strategy), as well as in theconstruction of a dual element (see (2.6) ). [2] Clearly NA r ( P ) implies NA ( P ) , but the converse is not true. For example, assume in themarket there is no stock, and there are only two options: g ( ω ) = g ( ω ) = ω, ω ∈ Ω := [0 , . Let P be the set of probability measures on Ω , g = g = 1 / , g = 1 / and g = 1 / . Then NA ( P ) holds while NA r ( P ) fails. A set is P -polar if it is P -null for all P ∈ P . A property is said to hold P -q.s. if it holds outside a P -polar set. When we multiply two vectors, we mean their inner product. “ri” stands for relative interior. [ g ′ , g ′ ] ⊆ ri [ g, g ] means component-wise inclusion. NOTE ON FTAP 3
For b, a ∈ R e , let Q [ b,a ] := { Q ≪ P : Q is a martingale measure and E Q [ g ] ∈ [ b, a ] } where Q ≪ P means ∃ P ∈ P such that Q ≪ P . Let Q [ b,a ] ϕ := { Q ∈ Q : E Q [ ϕ ] < ∞} . When[ b, a ] = [ g, g ], we drop the superscript and simply write Q , Q ϕ . Also define Q s := { Q ≪ P : Q is a martingale measure and E Q [ g ] ∈ ri [ g, g ] } and Q sϕ := { Q ∈ Q s : E Q [ ϕ ] < ∞} . Theorem 2.1.
Let ϕ ≥ be a random variable such that | g i | ≤ ϕ ∀ i = 1 , . . . , e . The followingstatements hold: (a) (Fundamental Theorem of Asset Pricing): The following statements are equivalent (i) NA r ( P ) holds. (ii) There exists [ g ′ , g ′ ] ⊆ ri [ g, g ] such that ∀ P ∈ P , ∃ Q ∈ Q [ g ′ ,g ′ ] ϕ such that P ≪ Q . (b) (Super-hedging) Suppose NA r ( P ) holds. Let f : Ω → R be Borel measurable such that | f | ≤ ϕ . The super-hedging price is given by π ( f ) = sup Q ∈Q sϕ E Q [ f ] = sup Q ∈Q ϕ E Q [ f ] ∈ ( −∞ , ∞ ] , (2.2) and there exists ( H, h ) ∈ H × R e such that π ( f ) + H • S T + h + ( g − g ) − h − ( g − g ) ≥ f P -q.s..Proof. It is easy to show ( ii ) in (a) implies that NA( P ) holds for the market with bid-ask prices g ′ , g ′ , Hence NA r ( P ) holds for the original market. The rest of our proof consists two parts asfollows. Part 1: π ( f ) > −∞ and the existence of an optimal hedging strategy in (b). Once weshow that the set C g := { H • S T + h + ( g − g ) − h − ( g − g ) : ( H, h ) ∈ H × R e } − L (2.3)is P − q.s. closed (i.e., if ( W n ) ∞ n =1 ⊂ C g and W n → W P − q.s. , then W ∈ C g ), the argument usedin the proof of [3, Theorem 2.3] would conclude the results in part 1. We will demonstrate theclosedness of C g in the rest of this part.Write g = ( u, v ), where u = ( g , . . . , g r ) consists of the hedging options without bid-ask spread,i.e, g i = g i for i = 1 , . . . , r , and v = ( g r +1 , . . . , g e ) consists of those with spread, i.e., g i < g i for i = r + 1 , . . . , e , for some r ∈ { , . . . , e } . Denote u := ( g , . . . , g r ) and similarly for v and v . Define C := { H • S T + α ( u − u ) : ( H, α ) ∈ H × R r } − L . Then C is P − q.s. closed by [3, Theorem 2.2].Let W n → W P − q.s. with W n = H n • S T + α n ( u − u ) + ( β n ) + ( v − v ) − ( β n ) − ( v − v ) − U n ∈ C g , (2.4) E Q [ g ] ∈ [ b, a ] means E Q [ g i ] ∈ [ b i , a i ] for all i = 1 , . . . , e . ERHAN BAYRAKTAR, YUCHONG ZHANG, AND ZHOU ZHOU where ( H n , α n , β n ) ∈ H × R r × R e − r and U n ∈ L . If ( β n ) n is not bounded, then by passing tosubsequence if necessary, we may assume that 0 < || β n || → ∞ and rewrite (2.4) as H n β n • S T + α n || β n || ( u − u ) ≥ W n || β n || − (cid:18) β n || β n || (cid:19) + ( v − v ) + (cid:18) β n || β n || (cid:19) − ( v − v ) ∈ C , where || · || represents the sup-norm. Since C is P − q.s. closed, the limit of the right hand sideabove is also in C , i.e., there exists some ( H, α ) ∈ H × R r , such that H • S T + α ( u − u ) ≥ − β + ( v − v ) + β − ( v − v ) , P − a.s., where β is the limit of ( β n ) n along some subsequence with || β || = 1. NA( P ) implies that H • S T + α ( u − u ) + β + ( v − v ) − β − ( v − v ) = 0 , P − a.s.. (2.5)As β =: ( β r +1 , . . . , β e ) = 0, we assume without loss of generality (w.l.o.g.) that β e = 0. If β e < g e + Hβ − e • S T + αβ − e ( u − u ) + e − X i = r +1 (cid:20) β + i β − e ( g i − g i ) − β − i β − e ( g i − g i ) (cid:21) = g e , P − a.s..
Therefore ˆ π ( g e ) ≤ g e , which contradicts the robust no-arbitrage property (see (2.1)) of g e . Hereˆ π ( g e ) is the super-hedging price of g e using S and g excluding g e . Similarly we get a contradictionif β e > β n ) n is bounded, and has a limit β ∈ R e − r along some subsequence ( n k ) k . Since by (2.4) H n • S T + α n ( u − u ) ≥ W n − ( β n ) + ( v − v ) + ( β n ) − ( v − v ) ∈ C , the limit of the right hand side above along ( n k ) k , W − β + ( v − v ) + β − ( v − v ), is also in C by itsclosedness, which implies W ∈ C g . Part 2: ( i ) ⇒ ( ii ) in part (a) and (3.3) in part (b). We will prove the results by an inductionon the number of hedging options, as in [3, Theorem 5.1]. Suppose the results hold for the marketwith options g , . . . , g e . We now introduce an additional option f ≡ g e +1 with | f | ≤ ϕ , availableat bid-ask prices f < f at time zero. (When the bid and ask prices are the same for f , then theproof is identical to [3].)( i ) = ⇒ ( ii ) in (a): Let π ( f ) be the super-hedging price when stocks and g , . . . , g e are availablefor trading. By NA r ( P ) and (3.3) in part (b) of the induction hypothesis, we have f > f ′ ≥ − π ( − f ) = inf Q ∈Q sϕ E Q [ f ] and f < f ′ ≤ π ( f ) = sup Q ∈Q sϕ E Q [ f ] (2.6)where [ f ′ , f ′ ] ⊆ ( f , f ) comes from the definition of robust no-arbitrage. This implies that thereexists Q + , Q − ∈ Q sϕ such that E Q + [ f ] > f ′′ and E Q − [ f ] < f ′′ where f ′′ = ( f + f ′ ), f ′′ = ( f + f ′ ).By (a) of induction hypothesis, there exists [ b, a ] ⊆ ri [ g, g ] such that for any P ∈ P , we can find Q ∈ Q [ b,a ] ϕ satisfying P ≪ Q ≪ P . Define g ′ = min( b, E Q + [ g ] , E Q − [ g ]) , and g ′ = max( a, E Q + [ g ] , E Q − [ g ]) NOTE ON FTAP 5 where the minimum and maximum are taken component-wise. We have [ b, a ] ⊆ [ g ′ , g ′ ] ⊆ ri [ g, g ]and Q + , Q − ∈ Q [ g ′ ,g ′ ] ϕ .Now, let P ∈ P . (a) of induction hypothesis implies the existence of a Q ∈ Q [ b,a ] ϕ ⊆ Q [ g ′ ,g ′ ] ϕ satisfying P ≪ Q ≪ P . Define Q := λ − Q − + λ Q + λ + Q + . Then Q ∈ Q [ g ′ ,g ′ ] ϕ and P ≪ Q . By choosing suitable weights λ − , λ , λ + ∈ (0 , , λ − + λ + λ + = 1,we can make E Q [ f ] ∈ [ f ′′ , f ′′ ] ⊆ ri [ f , f ].(3.3) in (b): Let ξ be a Borel measurable function such that | ξ | ≤ ϕ . Write π ′ ( ξ ) for its super-hedging price when stocks and g , . . . , g e , f ≡ g e +1 are traded, Q ′ ϕ := { Q ∈ Q ϕ : E Q [ f ] ∈ [ f , f ] } and Q ′ sϕ := { Q ∈ Q sϕ : E Q [ f ] ∈ ( f , f ) } . We want to show π ′ ( ξ ) = sup Q ∈Q ′ sϕ E Q [ ξ ] = sup Q ∈Q ′ ϕ E Q [ ξ ] . (2.7)It is easy to see that π ′ ( ξ ) ≥ sup Q ∈Q ′ ϕ E Q [ ξ ] ≥ sup Q ∈Q ′ sϕ E Q [ ξ ] (2.8)and we shall focus on the reverse inequalities. Let us assume first that ξ is bounded from above,and thus π ′ ( ξ ) < ∞ . By a translation we may assume π ′ ( ξ ) = 0.First, we show π ′ ( ξ ) ≤ sup Q ∈Q ′ ϕ E Q [ ξ ]. It suffices to show the existence of a sequence { Q n } ⊆ Q ϕ such that lim n E Q n [ f ] ∈ [ f , f ] and lim n E Q n [ ξ ] = π ′ ( ξ ) = 0. (See page 30 of [3] for why this issufficient.) In other words, we want to show that { E Q [( f, ξ )] : Q ∈ Q ϕ } ∩ (cid:0) [ f , f ] × { } (cid:1) = ∅ . Suppose the above intersection is empty. Then there exists a vector ( y, z ) ∈ R with | ( y, z ) | = 1that strictly separates the two closed, convex sets, i.e. there exists b ∈ R s.t.sup Q ∈Q ϕ E Q [ yf + zξ ] < b and inf a ∈ [ f,f ] ya > b. (2.9)It follows that y + f − y − f + π ′ ( zξ ) ≤ π ′ ( yf + zξ ) ≤ π ( yf + zξ ) = sup Q ∈Q ϕ E Q [ yf + zξ ] < b < y + f − y − f , (2.10)where the first inequality is because one can super-replicate zξ = ( yf + zξ ) + ( − yf ) from initialcapital π ′ ( yf + zξ ) − y + f + y − f , the second inequality is due to the fact that having more optionsto hedge reduces hedging cost, and the middle equality is by (b) of induction hypothesis. The lasttwo inequalities are due to (2.9).It follows from (2.10) that π ′ ( zξ ) <
0. Therefore, we must have that z <
0, otherwise π ′ ( zξ ) = zπ ′ ( ξ ) = 0 (since the super-hedging price is positively homogenous). Recall that we have provedin part (a) that Q ′ ϕ = ∅ . Let Q ′ ∈ Q ′ ϕ ⊆ Q ϕ . The part of (2.10) after the equality implies that yE Q ′ [ f ] + zE Q ′ [ ξ ] < y + f − y − f . Since E Q ′ [ f ] ∈ [ f , f ], we get zE Q ′ [ ξ ] < y + ( f − E Q ′ [ f ]) − y − ( f − E Q ′ [ f ]) ≤
0. Since z < E Q ′ [ ξ ] >
0. But by (2.8), E Q ′ [ ξ ] ≤ π ′ ( ξ ) = 0, which is a contradiction. ERHAN BAYRAKTAR, YUCHONG ZHANG, AND ZHOU ZHOU
Next, we show sup Q ∈Q ′ ϕ E Q [ ξ ] ≤ sup Q ∈Q ′ sϕ E Q [ ξ ]. It suffices to show for any ε > Q ∈ Q ′ ϕ , we can find Q s ∈ Q ′ sϕ such that E Q s [ ξ ] > E Q [ ξ ] − ε . To this end, let Q ′ ∈ Q ′ sϕ which isnonempty by part (a). Define Q s := (1 − λ ) Q + λQ ′ . We have Q s ≪ P by the convexity of P , and Q s ∈ Q ′ sϕ if λ ∈ (0 , E Q s [ ξ ] = (1 − λ ) E Q [ ξ ] + λE Q ′ [ ξ ] → E Q [ ξ ] as λ → . So for λ > Q s constructed above satisfies E Q s [ ξ ] > E Q [ ξ ] − ε . Hencewe have shown that the supremum over Q ′ ϕ and Q ′ sϕ are equal. This finishes the proof for upperbounded ξ .Finally when ξ is not bounded from above, we can apply the previous result to ξ ∧ n , and thenlet n → ∞ and use the closedness of C g in (2.3) to show that (3.3) holds. The argument would bethe same as the last paragraph in the proof of [3, Thoerem 3.4] and we omit it here. (cid:3) A Sharper Fundamental Theorem with the non-redundancy assumption
We now introduce the concept of non-redundancy. With this additional assumption we willsharpen our result.
Definition 3.1 (Non-redundancy) . A hedging option g i is said to be non-redundant if it is notperfectly replicable by stocks and other hedging options, i.e. there does not exist x ∈ R and asemi-static hedging strategy ( H, h ) ∈ H × R e such that x + H • S T + X j = i h j g j = g i P -q.s. . Remark 3.1. NA r ( P ) does not imply non-redundancy. For Instance, having only two identicaloptions in the market whose payoffs are in [ c, d ] , with identical bid-ask prices b and a satisfying b < c and a > d , would give a trivial counter example where NA r ( P ) holds yet we have redundancy. Lemma 3.1.
Suppose all hedging options with non-zero spread are non-redundant. Then NA ( P ) implies NA r ( P ) .Proof. Let g = ( g , . . . , g r + s ), where u := ( g , . . . , g r ) consists of the hedging options withoutbid-ask spread, i.e, g i = g i for i = 1 , . . . , r , and ( g r +1 , . . . , g r + s ) consists of those with bid-askspread, i.e., g i < g i for i = r + 1 , . . . , r + s . We shall prove the result by induction on s . Obviouslythe result holds when s = 0. Suppose the result holds for s = k ≥
0. Then for s = k + 1, denote v := ( g r +1 . . . , g r + k ), v := ( g r +1 , . . . , g r + k ) and v := ( g r +1 , . . . , g r + k ). Denote f := g r + k +1 .By the induction hypothesis, there exists [ v ′ , v ′ ] ⊂ ( v, v ) be such that NA( P ) holds in the marketwith stocks, options u and options v with any bid-ask prices b and a satisfying [ v ′ , v ′ ] ⊂ [ b, a ] ⊂ ( v, v ).Let v n ∈ ( v, v ′ ), v n ∈ ( v ′ , v ), f n > f and f n < f , such that v n ց v , v n ր v , f n ց f and f n ր f .We shall show that for some n , NA( P ) holds with stocks, options u , options v with bid-ask prices v n and v n , option f with bid-ask prices f n and f n . We will show it by contradiction. NOTE ON FTAP 7
If not, then for each n , there exists ( H n , h nu , h nv , h nf ) ∈ H × R r × R k × R such that H n • S T + h nu ( u − u )+( h nv ) + ( v − v n ) − ( h nv ) − ( v − v n )+( h nf ) + ( f − f n ) − ( h nf ) − ( f − f n ) ≥ , P − q.s., (3.1)and the strict inequality for the above holds with positive probability under some P n ∈ P . Hence h nf = 0. By a normalization, we can assume that | h nf | = 1. By extracting a subsequence, we canw.l.o.g. assume that h nf = − h nf = 1 is similar). If ( h nu , h nv ) n is notbounded, then w.l.o.g. we assume that 0 < c n := || ( h nu , h nv ) || → ∞ . By (3.1) we have that H n c n • S T + h nu c n ( u − u ) + ( h nv ) + c n ( v − v n ) − ( h nv ) − c n ( v − v n ) − c n ( f − f n ) ≥ , P − q.s..
By [3, Theorem 2.2], there exists H ∈ H , such that H • S T + h u ( u − u ) + h + v ( v − v ) − h − v ( v − v ) ≥ , P − q.s., where ( h u , h v ) is the limit of ( h nu /c n , h nu /c n ) along some subsequence with || ( h u , h v ) || = 1. NA( P )implies that H • S T + h u ( u − u ) + h + v ( v − v ) − h − v ( v − v ) = 0 , P − q.s.. (3.2)Since ( h u , h v ) = 0, (3.2) contradicts the non-redundancy assumption of ( u, v ).Therefore, ( h nu , h nv ) n is bounded, and w.l.o.g. assume it has the limit (ˆ h u , ˆ h v ). Then applying [3,Theorem 2.2] in (3.1), there exists ˆ H ∈ H such thatˆ H • S T + ˆ h u ( u − u ) + ˆ h + v ( v − v ) − ˆ h − v ( v − v ) − ( f − f ) ≥ , P − q.s..
NA( P ) implies thatˆ H • S T + ˆ h u ( u − u ) + ˆ h + v ( v − v ) − ˆ h − v ( v − v ) − ( f − f ) = 0 , P − q.s., which contradicts the non-redundancy assumption of f . (cid:3) We have the following FTAP and super-hedging result in terms of NA( P ) instead of NA r ( P ),when we additionally assume the non-redundancy of g . Theorem 3.1.
Suppose all hedging options with non-zero spread are non-redundant. Let ϕ ≥ bea random variable such that | g i | ≤ ϕ ∀ i = 1 , . . . , e . The following statements hold: (a’) (Fundamental Theorem of Asset Pricing): The following statements are equivalent (i) NA ( P ) holds. (ii) ∀ P ∈ P , ∃ Q ∈ Q ϕ such that P ≪ Q . (b’) (Super-hedging) Suppose NA ( P ) holds. Let f : Ω → R be Borel measurable such that | f | ≤ ϕ . The super-hedging price is given by π ( f ) = sup Q ∈Q ϕ E Q [ f ] ∈ ( −∞ , ∞ ] , (3.3) and there exists ( H, h ) ∈ H × R e such that π ( f ) + H • S T + h + ( g − g ) − h − ( g − g ) ≥ f P -q.s.. ERHAN BAYRAKTAR, YUCHONG ZHANG, AND ZHOU ZHOU
Proof. (a’)(ii) = ⇒ (a’)(i) is trivial. Now if (a’)(i) holds, then by Lemma 3.1, (a)(i) in Theorem 2.1holds, which implies (a)(ii) holds, and thus (a’)(ii) holds. Finally, (b’) is implied by Lemma 3.1and Theorem 2.1(b). (cid:3) Remark 3.2.
Theorem 3.1 generalizes the results of [3] to the case when the option prices arequoted with bid-ask spreads. When P is the set of all probability measures and the given optionsare all call options written on the dynamically traded assets, a result with option bid-ask spreadssimilar to Theorem 3.1-(a) had been obtained by [4] ; see Proposition 4.1 therein, although the non-redundancy condition did not actually appear. (The objective of [4] was to obtain relationshipsbetween the option prices which are necessary and sufficient to rule out semi-static arbitrage andthe proof relies on determining the correct set of relationships and then identifying a martingalemeasure.)However, the no arbitrage concept used in [4] is different: the author there assumes that there isno weak arbitrage in the sense of [6] ; see also [5] and [1] . (Recall that a market is said to haveweak arbitrage if for any given model (probability measure) there is an arbitrage strategy which isan arbitrage in the classical sense.) The arbitrage concept used here and in [3] is weaker, in thatwe say that a non-negative wealth ( P -q.s.) is an arbitrage even if there is a single P under whichthe wealth process is a classical arbitrage. Hence our no-arbitrage condition is stronger than theone used in [4] . But what we get out from a stronger assumption is the existence of a martingalemeasure Q ∈ Q ϕ for each P ∈ P . Whereas [4] only guarantees the existence of only one martingalemeasure which prices the hedging options correctly. References [1] B. Acciaio, M. Beiglb¨ock, F. Penkner, and W. Schachermayer. A model-free version of the fundamental theoremof asset pricing and the super-replication theorem.
To appear in Mathematical Finance . Also available on ArXivas 1301.5568.[2] E. Bayraktar and Y. Zhang. Fundamental Theorem of Asset Pricing under Transaction costs and Model uncer-tainty.
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Math. Finance , 17(1):1–14, 2007.[7] Yan Dolinsky and H. Mete Soner. Robust hedging with proportional transaction costs.
Finance Stoch. , 18(2):327–347, 2014.[8] Y. Kabanov and M. Safarian.
Markets with Transaction Costs, Mathematical Theory . Springer-Verlag BerlinHeidelberg, 2009. The no-arbitrage assumption in [1] is the model independent arbitrage of [6]. However that paper rules out themodel dependent arbitrage by assuming that a superlinearly growing option can be bought for static hedging.
NOTE ON FTAP 9 (Erhan Bayraktar)
Department of Mathematics, University of Michigan, 530 Church Street, AnnArbor, MI 48104, USA
E-mail address : [email protected] (Yuchong Zhang) Department of Mathematics, University of Michigan, 530 Church Street, Ann Ar-bor, MI 48104, USA
E-mail address : [email protected] (Zhou Zhou) Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor,MI 48104, USA
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