Model-free price bounds under dynamic option trading
aa r X i v : . [ q -f i n . M F ] J a n MODEL-FREE PRICE BOUNDS UNDER DYNAMIC OPTION TRADING
ARIEL NEUFELD AND JULIAN SESTER
Abstract.
In this paper we extend discrete time semi-static trading strategies by also allowingfor dynamic trading in a finite amount of options, and we study the consequences for the model-independent super-replication prices of exotic derivatives. These include duality results as well as aprecise characterization of pricing rules for the dynamically tradable options triggering an improve-ment of the price bounds for exotic derivatives in comparison with the conventional price boundsobtained through the martingale optimal transport approach.
Keywords:
Martingale optimal transport, European options, Price bounds, Sensitivity Introduction
In practice it is a well-known and often faced problem that given a specific market model, super-hedging strategies for financial derivatives are very expensive to implement (see for example [14],[21], [29, Example 7.21], [36], or [39]). If an investor is interested in considering hedging strategiesthat super-replicate the payoff of a derivative under parameter-uncertainty of a specific modelclass or even completely model-independent, then prices for super-hedges become even higher thanmodel-specific hedges, compare for this the model-independent approaches from [11], [24] as wellas the robust approach from [15]. It has therefore become an agile recent research topic to reducemodel-independent super-hedging prices through the introduction of different properties of modelsinferred from financial markets or via the inclusion of additional information (compare e.g. [4], [5],[28], [37], and [41]).We contribute to the literature on model-free pricing by studying a novel approach to reduceprices of model-independent super-replication strategies. The approach relies on extending theusually considered class of semi-static trading strategies by additionally allowing dynamic tradingin a finite amount of liquid European options. As we will show, this larger class of trading strategiesmay then allow to realize more flexible payoffs and thus to reduce super-hedging prices in manysituations.In an n -period financial market model, the assumption of being able to trade in European calloptions with expiration date t j (and other liquid kind of options) not only at initial time t , but alsoat intermediate times ( t i ) i =1 ,...,j − can be motivated by an observation that can be made on manyreal financial markets. Usually, expiration dates of European call options are bound to specificdates , i.e., if options with maturity t j are assumed to be liquidly traded at time t , then thereare also quotes available for the same maturity (and a shorter time-to-maturity) at time t i > t with t i < t j , and the options can also assumed to be tradeable at the later date. Thus, from apractical point of view, it seems natural to consider strategies incorporating dynamic trading in theunderlying security as well as in European options.It is one of the fundamental ideas in mathematical finance that minimizing prices over specificclasses of super-replication strategies yields in many situations the same value as the maximal(risk-neutral) expectation of the payoff that is super-replicated, where the expectation is takenw.r.t. (martingale) measures from a specific model-class which is strongly related to the class ofadmissible super-hedges. The precise mathematical formulation of this result is known as super-hedging duality and it can be derived in different settings, compare [1], [2], [10], [11], [12], [15],[17], [19], [20], and [25], to name but a few. In model-independent approaches, the dual model-classconsists of all martingale models (with undefined dynamics of the underlying stochastic process)that are consistent with prices of call options, whereas the trading strategies that super-replicate apayoff pointwise are semi-static.In this sense, the model-free duality result from [11] reveals that the dynamic trading positionin the underlying security can be considered as a natural counterpart of the martingale property of For many call options this is the last trading day before the 20th of a month. measures, whereas the static positions in European option corresponds to information on marginaldistributions. We will describe the dual counterpart of a dynamic trading position in Europeanoptions through a martingale property for the prices of these options, i.e., the model-class that isconsidered for the maximization of the payoff consists of call option-calibrated martingale measuresunder which the prices of the traded European options are also martingales.We study extensively the consequences of the modified model-independent setting for upperbounds of prices for exotic derivatives that emerge as minimal prices among super-replication strate-gies involving European options and simultaneously as maximal prices over the above described classof financial models.The remainder of the paper is as follows. Section 2 introduces the setting and provides the mainresults. Section 3 provides several numerical examples. Section 4 contains all the mathematicalproofs. Moreover, in Appendix A we provide extensions of the presented approach to frictions,multiple securities and other dynamically traded options.2.
Setup and main Results
Setup.
We consider at t = 0 a frictionless discrete-time financial market with a fixed amountof n ∈ N times t , . . . , t n and one underlying asset S = ( S t i ) i =0 , ,...,n . Extensions of this setup arediscussed in Appendix A.In the classical setup for model-independent pricing, which is referred to as the martingale optimaltransport (MOT) case (introduced in [11]), prices for call options with all strikes and all maturities t j , j = 1 , . . . , n , are observable at initial time t and available for static trading. This means that oneis able to initiate a buy-and-hold strategy into these call options. Since in this situation every twice-differentiable European payoff u i ( S t i ) with u i ∈ C ( R ) can be replicated using call options withdifferent strikes (compare [18]), it is nearby to also allow initiating static investments in Europeanoptions.Moreover, one allows to initiate a trading strategy into the underlying security that is dynamicallyadjusted over time. Dynamic trading of European options is however not considered in the MOTsetting. In contrast, in this paper, we consider dynamic trading in a finite amount of Europeanoptions. We assume that for each maturity t j the market offers N ∈ N European options possessingthis expiration date available for trading at all times t i < t j . We denote the set of options availablefor dynamic trading by V , with V = n · N . The reduction to the finite subset V accounts for apossible lack of liquidity in European options over time, see also [40]. Additionally, we discuss indetail the case with infinitely many traded options in Section 2.4.We denote by P t i ( v j,k ) the price at time t i for a European option v j,k ∈ V with a non-negativeBorel-measurable payoff function v j,k : R + → R + , where the index j refers to the maturity t j and k ∈ { , . . . , N } labels the options.As underlying sample space we consider Ω := R n + × R n · N + . We use for ω ∈ Ω the representation ω = ( s , . . . , s n , p , , , . . . , p n,n,N ) = ( s, p ) with s ∈ R n + and p ∈ R n · N + . Then ( S t i ) i =1 ,...,n is assumedto be the canonical process on the first n components, i.e., for i = 1 , . . . , n and all ( s, p ) ∈ Ω wehave S t i ( s, p ) = s i , and where we set S = s for some fixed s ∈ R + . Moreover, we set for i, j ∈ { , . . . , n } , k ∈{ , . . . , N } P t i ( v j,k ) ( s, p ) = p i,j,k . Furthermore, we denote by P (Ω) the set of all probability measures on Ω and we define the filtration F = ( F t s ) s =0 , ,...,n through F t s := σ ( { S t i for 0 ≤ i ≤ s } )with F t being trivial. Next, we allow the financial agent to restrict to price paths she considersadmissible. Thus, we introduce, similar as in [8], a subset Ξ ⊂ Ω of admissible price paths. Below,in Remark 2.1, we discuss the role of Ξ in our setting in different examples. In practice there may be a different amount of options available for each maturity t j . We then would, purely fortechnically reasons, additionally consider options with constant payoff 0 to be able to consider an equal amount oftradable options among maturities. ODEL-FREE PRICE BOUNDS UNDER DYNAMIC OPTION TRADING 3
Remark 2.1 (Choices of Ξ) . (a) It is of course possible to set
Ξ := Ω and hence not to impose any restrictions on the set offuture paths. (b)
According to [23] and [39] , the maximal super-replication price at time t for Europeanoptions in fully incomplete markets coincides with the buy-and-hold-super-replication priceand is given by todays’ value of its concave envelope, i.e., the smallest concave functionlarger or equal than the payoff function f itself, here denoted by f conc . For the sub-replicationprice one correspondingly considers the convex envelope, f conv , which is the greatest convexfunction smaller than the payoff. This means, we obtain a pricing rule of the form P t i ( v j,k ) ∈ (cid:2) v conv j,k ( S t i ) , v conc j,k ( S t i ) (cid:3) , i.e., Ξ = (cid:8) ( s, p ) ∈ Ω (cid:12)(cid:12) p i,j,k ∈ (cid:2) v conv j,k ( s i ) , v conc j,k ( s i ) (cid:3) for all i, j, k (cid:9) . Using this approach, we also rediscover the standard no-arbitrage bounds for call options(see also [21] , [26] , and [30] ) P t i (( S t j − K ) + ) ∈ (cid:2) ( S t i − K ) + , S t i (cid:3) . (c) Suppose an investor believes that it is accurate to price call options under the risk-neutralmeasure of a Black–Scholes model with volatility b σ . To incorporate uncertainty w.r.t. thechoice of the volatility σ one allows for σ ∈ [ b σ − ε, b σ + ε ] for some ε > such that b σ − ε > .Due to the positive vega, pricing of call options in a Black–Scholes model is monotone w.r.t.the choice of σ and we obtain a pricing rule of the form (2.1) P t i (cid:0) ( S t j − K ) + (cid:1) ∈ (cid:20) S t i N (cid:0) d , b σ − ε ( S t i , K ) (cid:1) − K N (cid:0) d , b σ − ε ( S t i , K ) (cid:1) ,S t i N (cid:0) d , b σ + ε ( S t i , K ) (cid:1) − K N (cid:0) d , b σ + ε ( S t i , K ) (cid:1) (cid:21) for N ( · ) describing the cumulative distribution function of the standard normal distributionand with (2.2) d ,σ ( x, K ) = ln ( x/K ) + σ ( t j − t i ) σ √ t j − t i , d ,σ ( x, K ) = d ,σ ( x, K ) − σ p t j − t i . This approach can, in principal, be extended to any kind of parametric model. In particular,as shown above, when prices of convex payoffs are increasing w.r.t. the input parameter,then the price bounds are attained by the bounds of the interval. (d)
The pricing rule can also be robust in the sense that it reflects general properties of the marketor an admissible underlying process. For example, a Markov property for the valuation ofoptions (similar as in [41] ) can be incorporated through
Ξ = (cid:8) ( s, p ) ∈ Ω | For all i, j, k there exists some Borel-measurable function f i,j,k : R → R such that p i,j,k = f i,j,k ( s i ) (cid:9) . Or if the difference t i +1 − t i is constant for all i = 1 , . . . , n − , then a homogeneity assumptionsimilar as in [28] can be modeled through Ξ = (cid:8) ( s, p ) ∈ Ω | p i,j,k = p i + l,j, + l,k for all i, j, k, l s.t. ≤ i + l, j + l ≤ n, k = 1 , . . . , N (cid:9) . (e) Given some Ξ ⊂ Ω we define for m ∈ N s i , . . . , s mi , p i,j,k , . . . , p mi,j,k ∈ R + Ξ m grid := (cid:8) ( s, p ) ∈ Ξ (cid:12)(cid:12) s i ∈ { s i , . . . , s mi } , p i,j,k ∈ { p i,j,k , . . . , p mi,j,k } for all i, j, k (cid:9) . This allows to consider the valuation problem on a discrete grid, which is particularly use-ful for implementing the approach numerically, i.e., via linear programming, compare alsoAlgorithm 1.
ARIEL NEUFELD AND JULIAN SESTER
Valuation of Derivatives.
We are interested in finding model-free price bounds for someexotic financial derivative Φ( S t , . . . , S t n ), where Φ : R n + → R + is Borel-measurable. For notationalsimplicity, we focus on finding the upper bound .2.2.1. The primal approach.
A first approach to determine the value of Φ is to compute the expec-tation of Φ under a risk-neutral pricing measure associated to a potential model of an underlyingfinancial market, i.e., among all measures restricted to paths Ξ that are arbitrage-free. The maximalmodel-price determines the upper price bound. We call this approach the primal approach.To determine the set of such models, we observe that under any admissible pricing measure Q ,the price process (P t i ( v j,k )) i =1 ,...,n is required to be a martingale, from which we obtain the requiredrepresentation that(2.3) P t i ( v j,k ) = E Q (cid:2) P t j ( v j,k ) (cid:12)(cid:12) F t i (cid:3) = E Q (cid:2) v j,k ( S t j ) (cid:12)(cid:12) F t i (cid:3) Q -a.s.for all 1 ≤ i ≤ j ≤ n , k = 1 , . . . , N . We further, extend the validity of (2.3) to all 1 ≤ i, j ≤ n ,since this definition implies P t i ( v j,k ) = v j,k Q -a.s. if j ≤ i , i.e., the price processes are assumed tobe constant after expiration date (for a fixed price path).Additionally, we only consider such models consistent with market prices of call options . Accord-ing to [16], consistency of pricing measures w.r.t. call option prices for all maturities t , . . . , t n andfor all strikes determines the one-dimensional marginal distributions of S t i for all i = 1 , . . . , n , whichwe from now on denote by µ i , i.e., S t i ∼ µ i . To ensure absence of model-independent arbitragethrough static option trading, we assume µ (cid:22) µ (cid:22) · · · (cid:22) µ n , where (cid:22) denotes the usual convexorder for measures with finite first moments, compare also [1], [35] and [42]. Thus, we include thiscondition in the following standing assumption. Assumption 2.2 (Standing Assumption) . (a) The marginals µ , . . . , µ n have finite first moments and it holds µ (cid:22) µ (cid:22) · · · (cid:22) µ n . (b) The set Ξ ⊂ Ω is Borel-measurable. In any potential arbitrage-free model of a financial market, S and P are martingales and themarginals of S are fixed by µ , . . . , µ n through the required consistency with the observed vanillaoption prices. A risk-neutral measure of an admissible model is consequently an element of M V (Ξ , µ , . . . , µ n ) := (cid:26) Q ∈ P (Ω) (cid:12)(cid:12)(cid:12)(cid:12) Q (Ξ) = 1; E Q [ S t i |F t i − ] = S t i − Q -a.s. for all i = 1 , . . . , n ; Q ◦ S − t i = µ i for all i = 1 , . . . , n ; E Q [ v j,k ( S t j ) |F t i ] = P t i ( v j,k ) Q -a.s. for all i = 1 , . . . , n, j = 1 , . . . , n, k = 1 , . . . , N (cid:27) . Remark 2.3.
For all Q ∈ M V (Ξ , µ , . . . , µ n ) and for all i, j = 1 , . . . , n the random variable P t i ( v j,k ) is F Q t i -measurable by (2.3) , where F Q t i denotes the Q - F -completion of F t i . In particu-lar, (P t i ( v j,k )) i =1 ,...,n is adapted to F Q = (cid:16) F Q t i (cid:17) i =1 ,...,n . Moreover, note that the martingale propertyin the definition of M V (Ξ , µ , . . . , µ n ) does not change if we define it w.r.t. F Q instead of F . The upper price bound of Φ using the primal approach is then given by the maximal expectationof Φ w.r.t. measures from M V (Ξ , µ , . . . , µ n ), namely(2.4) P Ξ (Φ) := sup Q ∈M V (Ξ ,µ ,...,µ n ) Z Ω Φ( s ) d Q ( s, p ) . This is no restriction, since with the same approach one can easily obtain the lower bound through the relationinf x f ( x ) = − sup x − f ( x ). A model with associated pricing measure Q is said to be consistent with the market price π ( v ) of the option v if E Q [ v ] = π ( v ). From now on, we always assume that Assumption 2.2 holds and we do not repeat it in the statements of ourresults.
ODEL-FREE PRICE BOUNDS UNDER DYNAMIC OPTION TRADING 5
The four properties for a measure Q to be in M V (Ξ , µ , . . . , µ n ) – i.e., the paths of S are restrictedto Ξ, S is a Q -martingale, S possesses correct fixed marginals, and that P t i ( v j,k ) can be regardedas a conditional expectation of v j,k – can also be characterized by integral equations. Thus, the set M V (Ξ , µ , . . . , µ n ) writes equivalently as(2.5) M V (Ξ , µ , . . . , µ n ) = (cid:26) Q ∈ P (Ω) (cid:12)(cid:12)(cid:12)(cid:12) Z Ω Ξ ( s, p ) d Q ( s, p ) = 1; Z Ω H ( s , . . . , s i )( s i +1 − s i ) d Q ( s, p ) = 0for all H ∈ C b ( R i + ) , i = 1 , . . . , n ; Z R + u i ( s i ) d µ i ( s i ) = Z Ω u i ( s i ) d Q ( s, p )for all u i ∈ C lin ( R + , R + ) , i = 1 , . . . , n ; Z Ω H ( s , . . . , s i )( v j,k ( s j ) − p i,j,k ) d Q ( s, p ) = 0for all H ∈ C b ( R i + ) , i, j = 1 , . . . , n, k = 1 , . . . , N (cid:27) , where for any k ∈ N C lin (cid:16) R k + , R + (cid:17) := ( f ∈ C (cid:16) R k + , R + (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup ( x ,...,x k ) ∈ R k + f ( x , . . . , x k )1 + P ki =1 x i < ∞ ) denotes the class of positive continuous functions with linear growth on R k + . In the MOT caseAssumption 2.2 (a) ensures non-emptiness of the set M ( µ , . . . , µ n ) := (cid:26) Q ∈ P ( R n + ) (cid:12)(cid:12)(cid:12)(cid:12) E Q [ S t i +1 |F t i ] = S t i Q -a.s. , Q ◦ S − t i = µ i for all i = 1 , . . . , n (cid:27) , see for example [35]. We ensure in our setting the non-emptiness of M V (Ξ , µ , . . . , µ n ) by anadditional condition, see Theorem 2.4 (a).2.2.2. The dual approach.
A second valuation approach relies on the determination of the smallestprice among model-independent super-replication strategies of Φ on Ξ. We refer also to [8], [9], [34],and [38]. This means we consider strategies of the formΨ V ( H i ) , ( H i,j,k ) , ( u i ) ( s, p ) := n X i =1 u i ( s i ) + n − X i =1 H i ( s , . . . , s i )( s i +1 − s i )+ n − X i =1 n X j = i +1 N X k =1 H i,j,k ( s , . . . , s i ) ( v j,k ( s j ) − p i,j,k ) . for functions u i : R + → R , H i , H i,j,k : R i + → R that can be interpreted as trading positions. Inaddition to the semi-static trading from the martingale optimal transport approach we encounter P n − i =1 P nj = i +1 P Nk =1 H i,j,k ( s , . . . , s i ) ( v j,k ( s j ) − p i,j,k ), which is the profit of a self-financing dynam-ically adjusted trading position in European options. We call this approach the dual approach.Given the marginal distributions µ i for i = 1 , . . . , n , the fair price of a strategy Ψ V ( H i ) , ( H i,j,k ) , ( u i ) ( s, p )calculates as P ni =1 R R u i ( s i ) d µ i ( s i ), since the terms n − X i =1 H i ( s , . . . , s i )( s i +1 − s i ) + n − X i =1 n X j = i +1 N X k =1 H i,j,k ( s , . . . , s i )( v j,k ( s j ) − p i,j,k ) ARIEL NEUFELD AND JULIAN SESTER are profits of self-financing strategies and therefore are considered to be costless. The upper pricebound for Φ using the dual approach is thus given through the super-replication functional:(2.6) D Ξ (Φ) := inf u i ∈ C lin ( R + , R + ) H i ,H i,j,k ∈ C b ( R i + ) (cid:26) n X i =1 Z R + u i ( s i ) d µ i ( s i ) (cid:12)(cid:12)(cid:12)(cid:12) Ψ V ( H i ) , ( H i,j,k ) , ( u i ) ( s, p ) ≥ Φ( s )for all ( s, p ) ∈ Ξ (cid:27) . Main Results.
Our first main result imposes that - under mild conditions - the two presentedvaluation approaches yield the same value. Furthermore, we state criteria for the non-emptiness andcompactness of M V (Ξ , µ , . . . , µ n ), guaranteeing the existence of an optimal pricing measure. Werefer to Section 4 for the corresponding proofs of the main results stated in the following theoremsand remarks. Theorem 2.4.
Let v j,k ∈ C lin ( R + , R + ) for all j = 1 , . . . , n , k = 1 , . . . , N . Then, the followingholds. (a) The set M V (Ξ , µ , . . . , µ n ) ⊂ P (Ω) is non-empty if and only if there exists some Q ∈M ( µ , . . . , µ n ) ⊂ P ( R n + ) such that it holds (2.7) (cid:18) S t i , (cid:0) E Q [ v j,k ( S t j ) | F t i ] (cid:1) i,j =1 ,...,n,k =1 ,...,N (cid:19) ∈ Ξ Q -a.s. (b) Assume that M V (Ξ , µ , . . . , µ n ) = ∅ , that Ξ is closed, and that (2.8) sup i,j ∈{ ,...,n } ,k ∈{ ,...,N } ( s,p ) ∈ Ξ p i,j,k P iℓ =1 s ℓ < ∞ . Then, the set M V (Ξ , µ , . . . , µ n ) is compact in the weak topology. (c) Let Φ ∈ C lin (cid:0) R n + , R + (cid:1) . Further assume that M V (Ξ , µ , . . . , µ n ) = ∅ . Then it holds P Ξ (Φ) = D Ξ (Φ) . Moreover, under the assumptions of (b), the primal value in (2.4) is attained.
Remark 2.5. (a)
As a consequence of Theorem 2.4 (a) it holds P Ξ (Φ) = inf Q ∈M ( µ ,...,µ n ):(2.7) holds Z Φ( s ) d Q ( s ) , i.e., P Ξ (Φ) can be considered as a constrained martingale optimal transport problem. (b) If Ξ = Ω , then, according to Theorem 2.4, the non-emptiness of M V (Ξ , µ , . . . , µ n ) is equiv-alent to the non-emptiness of M ( µ , . . . , µ n ) , independent of V . (c) Assume that
Ξ = { ( s, p ) ∈ Ω | For all i, j, k there exists f i,j,k ∈ F i,j,k s.t. p i,j,k = f i,j,k ( s , . . . , s i ) } for some classes F i,j,k ⊂ C lin (cid:0) R i + , R + (cid:1) whose restrictions onto any compact set K ⊂ R n + arecompact in the uniform topology on C ( K ) and which fulfill (2.9) sup f i,j,k ∈ F i,j,k sup ( s ,...,s i ) ∈ R i f i,j,k ( s , . . . , s i )1 + P iℓ =1 s ℓ < ∞ . Note that, as a consequence of the compactness of F i,j,k , the set Ξ is closed. Under theseconditions, we define Q ( F i,j,k ) := (cid:26) Q ∈ M ( µ , . . . , µ n ) (cid:12)(cid:12)(cid:12)(cid:12) For all i, j, k there exists f i,j,k ∈ F i,j,k s.t. E Q [ v j,k ( S t j ) | F t i ] = f i,j,k Q -a.s. (cid:27) By abuse of notation ( S t i ) denotes in (2.7) the canonical process on R n + . ODEL-FREE PRICE BOUNDS UNDER DYNAMIC OPTION TRADING 7 and D ( F i,j,k ) (Φ) := inf u i ∈ C lin ( R + , R + ) H i ,H i,j,k ∈ C b ( R i + ) (cid:26) n X i =1 Z R + u i d µ i (cid:12)(cid:12)(cid:12)(cid:12) Ψ V ( H i ) , ( H i,j,k ) , ( u i ) ( s, ( f i,j,k ( s , . . . , s i )) i,j,k ) ≥ Φ( s ) for all s ∈ R n + , f i,j,k ∈ F i,j,k (cid:27) . Then, under the assumptions of Theorem 2.4 (c), we obtain that P Ξ (Φ) = sup Q ∈Q ( F i,j,k ) Z R n + Φ( s ) d Q ( s ) = D ( F i,j,k ) (Φ) . This particular structure of Ξ is fulfilled for example in the setting discussed in Remark 2.1 (b)and Remark 2.1 (c). Indeed, in Remark 2.1 (c) we obtain for the case of call options withstrikes K k ∈ R + that F i,j,k = { ( s , . . . , s i ) ( s i − y ) + for y ∈ [0 , K k ] } . Moreover, in the setting of Remark 2.1 (c) we get F i,j,k = { ( s , . . . , s i ) s i N ( d ,σ ( s i , K k )) − K k N ( d ,σ ( s i , K k )) for σ ∈ [ b σ − ε, b σ + ε ] } , which is compact due to the Arzel`a–Ascoli theorem. Further, we have s i N ( d ,σ ( s i , K k )) − K k N ( d ,σ ( s i , K k )) ≤ s i + K k , i.e., the set is indeed contained in C lin (cid:0) R i + , R + (cid:1) . The following remark shows that without restricting the set of possible prices of the dynamicallytraded options, i.e., when setting Ξ := Ω, we do not obtain any improved price bounds in comparisonwith the classical MOT formulation. This motivates to define a set of restricted pricing rules forEuropean options in order to obtain improved price bounds for Φ.
Remark 2.6.
Let Φ ∈ C lin (cid:0) R n + , R + (cid:1) . If it holds for all s ∈ R n + that (2.10) ( s, e p ) ∈ Ξ for e p ∈ R n · N + with e p i,j,k = v j,k ( s j ) for all i, j = 1 , . . . , n , k = 1 , . . . , N , then it holds (2.11) D Ξ (Φ) = sup Q ∈M ( µ ,...,µ n ) Z R n + Φ( s ) d Q ( s ) . Note that condition (2.10) holds particularly if
Ξ := Ω . Further, note that if (2.10) holds, thenequality (2.11) holds independent of the amount of traded options N ∈ N , i.e., gradually increasingthe number of dynamically traded options only comes with improved price bounds if we introducefurther restrictive pricing rules, as done in the following Theorem 2.8. Next, we study the influence of a change in the pricing rule on the upper price bound of theexotic derivative Φ.For the following studies we impose an additional assumption on the shape of the pricing rules.
Assumption 2.7.
Let Ξ ⊂ Ω be of the form Ξ ≡ Ξ ( p i,j,k ,p i,j,k ) := n ( s, p ) ∈ Ω (cid:12)(cid:12)(cid:12) p i,j,k ∈ h p i,j,k ( s , . . . , s i ) , p i,j,k ( s , . . . , s i ) i for all i, j, k o for some Borel-measurable functions p i,j,k , p i,j,k : R i + → R + for i, j, = 1 , . . . , n , k = 1 , . . . , N . We investigate how the boundaries [ p i,j,k ( · ) , p i,j,k ( · )] need to be chosen to directly imply improvedprice bounds of Φ in comparison with the MOT formulation, i.e., to obtain P Ξ (Φ) < P Ω (Φ). Thefollowing theorem asserts precisely how the pricing rules p i,j,k , p i,j,k have to be defined to obtainimproved price bounds for Φ. For this, we define for a fixed financial derivative Φ ∈ C lin (cid:0) R n + , R + (cid:1) the set of optimizers of the primal problem c M V (Ξ , µ , . . . , µ n ) := (cid:26) Q ∈ M V (Ξ , µ , . . . , µ n ) s.t. Z Ξ Φ d Q = P Ξ (Φ) (cid:27) , which is non-empty under the assumptions of Theorem 2.4 (b). ARIEL NEUFELD AND JULIAN SESTER
Theorem 2.8.
Let the assumptions of Theorem 2.4 (b) hold and let Ξ be of the form as describedin Assumption 2.7. Then the following holds. (a) It holds P Ξ ( pi,j,k,pi,j,k ) (Φ) < P Ω (Φ) if and only if for all Q ∈ c M V (Ω , µ , . . . , µ n ) there exist i, j ∈ { , . . . , n } , k ∈ { , . . . , N } such that (2.12) p i,j,k > E Q (cid:2) v j,k ( S t j ) (cid:12)(cid:12) F t i (cid:3) or (2.13) p i,j,k < E Q (cid:2) v j,k ( S t j ) (cid:12)(cid:12) F t i (cid:3) on some Borel-measurable set A ⊂ Ω with Q ( A ) > . (b) For all ε > such that M V (cid:16) Ξ ( p i,j,k + ε,p i,j,k ) , µ , . . . , µ n (cid:17) = ∅ it holds that P Ξ ( pi,j,k + ε,pi,j,k ) (Φ) < P Ξ ( pi,j,k,pi,j,k ) (Φ) if and only if for all Q ∈ c M V (cid:16) Ξ ( p i,j,k ,p i,j,k ) , µ , . . . , µ n (cid:17) there exist i, j ∈ { , . . . , n } , k ∈{ , . . . , N } such that (2.14) p i,j,k + ε > E Q (cid:2) v j,k ( S t j ) (cid:12)(cid:12) F t i (cid:3) on some Borel-measurable set A ⊂ Ω with Q ( A ) > . (c) For ε > such that M V (cid:16) Ξ ( p i,j,k ,p i,j,k − ε ) , µ , . . . , µ n (cid:17) = ∅ it holds that P ( p i,j,k ) , ( p i,j,k − ε ) (Φ) < P ( p i,j,k ) , ( p i,j,k ) (Φ) if and only if for all Q ∈ c M V (cid:16) Ξ ( p i,j,k ,p i,j,k ) , µ , . . . , µ n (cid:17) there exist i, j ∈ { , . . . , n } , k ∈{ , . . . , N } such that (2.15) p i,j,k − ε < E Q (cid:2) v j,k ( S t j ) (cid:12)(cid:12) F t i (cid:3) on some Borel-measurable set A ⊂ Ω with Q ( A ) > . Infinitely many European call options.
If we do not restrict the set of options availablefor dynamic trading to a fixed finite amount of European options, but a priori consider infinitelymany call options with a continuous range of strikes reaching from 0 to + ∞ , then, by following therationale from [18], each positive European payoff can be replicated by call options and thus everyEuropean payoff which only depends on a single value of the underlying security can be consideredas being available for dynamic trading. Hence, from now on, we allow for dynamic trading in alloptions with payoff v j ( S t j ) for v j ∈ C lin ( R + , R + ) and j ∈ { , . . . , n } .Similar to Remark 2.5 we consider the following formulation of a super-hedging problem. Given aBorel-measurable set e Ξ ⊂ R n + and some sets F i,j ( v j ) ⊂ C lin (cid:0) R i + , R + (cid:1) for functions v j ∈ C lin ( R + , R + ), i, j = 1 , . . . , n , we define(2.16) e D ( F i,j ) (Φ) := inf ui,vj ∈ C lin( R + , R +) Hi,Hi,j ∈ Cb ( R i +) (cid:26) n X i =1 Z R + u i ( s i ) d µ i ( s i ) (cid:12)(cid:12)(cid:12)(cid:12) n X i =1 u i ( s i ) + n − X i =1 H i ( s , . . . , s i )( s i +1 − s i )+ n − X i =1 n X j = i +1 H i,j ( s , . . . , s i ) ( v j ( s j ) − f i,j ( s , . . . , s i )) ≥ Φ( s )for all s ∈ e Ξ , f i,j ∈ F i,j ( v j ) (cid:27) This means e D ( F i,j ) corresponds to the minimal super-replication price among strategies where dy-namic trading in all options v j ∈ C lin ( R + , R + ) is allowed at time t i . The time t i -price of this optionis associated to some f i,j ∈ F i,j ( v j ) which is unknown for the financial agent. Thus, the consideredstrategies super-replicate Φ pointwise on e Ξ and among all potential prices f i,j in F i,j ( v j ). ODEL-FREE PRICE BOUNDS UNDER DYNAMIC OPTION TRADING 9
We obtain the following duality result that allows an interpretation of the super-hedging problemas a maximization problem of expected values of Φ w.r.t. martingale measures Q s.t. the F t i -conditional expectations of v j ∈ C lin ( R + , R + ) can be written in terms of some function f i,j from F i,j ( v j ). Theorem 2.9.
Let Φ ∈ C lin (cid:0) R n + , R + (cid:1) , let each ( F i,j ( v j )) vj ∈ C lin ( R i + , R + ) ,i,j ∈{ ,...,n } ⊂ C lin ( R + , R + ) satisfy forall compact K ⊂ R n + that F i,j ( v j ) | K is compact in the uniform topology on C ( K ) and such that forall v j ∈ C lin ( R + , R + )(2.17) sup fi,j ∈ F i,j ( vj )( s ,...,si ) ∈ R i + f i,j ( s , . . . , s i )1 + P iℓ =1 s ℓ < ∞ . If the set e Q ( F i,j ) = (cid:26) Q ∈ M ( µ , . . . , µ n ) : Q (cid:0)e Ξ (cid:1) = 1 , for all i, j and all v j ∈ C lin ( R + , R + ) there exists f i,j ∈ F i,j ( v j ) s.t. E Q [ v j ( S t j ) | F t i ] = f i,j Q -a.s. (cid:27) is non-empty, then it holds e D ( F i,j ) (Φ) = sup Q ∈ e Q ( F i,j ) Z R n + Φ( s ) d Q ( s ) . Remark 2.10.
An example of sets F i,j ( v j ) fulfilling the assumptions of Theorem 2.9 includes forgiven i, j, ∈ { , . . . , n } , v j ∈ C lin ( R + , R + ) the sets F i,j ( v j ) = (cid:26) ( s , . . . , s i ) g ( s i ) (cid:12)(cid:12)(cid:12)(cid:12) g being -Lipschitz with g (0) = 0 , and g ( S t i ) = E Q [ v j ( S t j ) | S t i ] Q -a.s. for some Q ∈ M ( µ , . . . , µ n ) (cid:27) of prices following a Markovian pricing rule. Remark 2.11.
The assertion of Theorem 2.9 can easily be modified by only allowing for trading inoptions v j that are contained in some (possibly infinitely large) subset e V ⊂ C lin (cid:0) R i + , R + (cid:1) . This canbe done to account for a possible lack in liquidity. Therefore, one possible choice for e V includes allpayoffs of call and put options for a predefined range of strikes. If e V only contains a finite amountof payoffs, then we rediscover the result discussed in Remark 2.5 (c). Examples and Numerics
Examples.
In this section we provide several examples. In particular, we compare our ap-proach with the conventional martingale transport approach where semi-static hedging withoutdynamic trading in options is involved. We start with an empirical study indicating how to choosepricing rules for European call options.
Example 3.1 (Market Implied Marginals from real financial data) . We consider the marginaldistributions µ and µ derived from call and put options on the stock of Apple Inc.
The data wasobserved at t = 24 July for S t = 389 . . The considered time to maturities are t − t = 84 days and t − t = 175 days respectively. Due to the short maturities we neglect dividend yields aswell as interest rates and discretize the resultant marginal distributions on a discrete grid with supporting values, where the discretization is performed according to the method proposed in [6] and [31] to be able to apply the linear programming approach that is described in Algorithm 1. We allowfor dynamic trading in call options with maturity t and strikes K k , i.e., v ,k ( S t ) = ( S t − K k ) + ,where K = 360 , K = 340 , K = 320 . We set the standard price bounds p , ,k ( S t ) = ( S t − K k ) + and p , ,k ( S t ) = S t for k = 1 , , , see also Remark 2.1 (b). Now, we compute numerically the All the codes are available under https://github.com/juliansester/dynamic option trading We only consider dynamic trading in options with maturity t as trading in an option with maturity t would notinduce a proper dynamic trading position, since such positions are implicitly subsumed in the static component u . quantities P Ξ ( pi,j,k + ε,pi,j,k ) (Φ) and P Ξ ( pi,j,k,pi,j,k − ε ) (Φ) for different values of ε and for different payofffunctions Φ . Further we illustrate the differences between considering V = { v , } , V = { v , , v , } and V = { v , , v , , v , } respectively, i.e. we study the effect of including more options for dynamictrading. The results, using Algorithm 1, are depicted in Figure 1. No dynamically traded options1 dynamically traded option2 dynamically traded options3 dynamically traded options ε P Ξ ( p i , j , k + ε , p i , j , k ) ( Φ )
200 300 ε . . . P Ξ ( p i , j , k , p i , j , k − ε ) ( Φ ) Φ( S t , S t ) = | S t − S t | ε P Ξ ( p i , j , k + ε , p i , j , k ) ( Φ )
280 300 320 340 ε P Ξ ( p i , j , k , p i , j , k − ε ) ( Φ ) Φ( S t , S t ) = (max( S t , S t ) − + ε . . . P Ξ ( p i , j , k + ε , p i , j , k ) ( Φ )
200 250 300 350 ε . . . P Ξ ( p i , j , k , p i , j , k − ε ) ( Φ ) Φ( S t , S t ) = (0 . S t + S t ) − + ε P Ξ ( p i , j , k + ε , p i , j , k ) ( Φ )
200 250 300 350 ε . . . . P Ξ ( p i , j , k , p i , j , k − ε ) ( Φ ) Φ( S t , S t ) = | S t − . · S t | Figure 1.
The upper price bound for different payoff functions in dependence of achange in the bounds of the pricing rule and in dependence of a different number ofconsidered options for dynamic trading. The price bounds without dynamic optiontrading (but still with semi-static trading) are indicated by a black dashed line.
ODEL-FREE PRICE BOUNDS UNDER DYNAMIC OPTION TRADING 11
Example 3.2 (Model-Implied Pricing Rules) . We consider the same market-implied marginals andthe same sets V as in Example 3.1. Then we consider for dynamically traded vanilla options aBlack-Scholes type pricing rule of the form (2.1) and denote for ε, b σ > by Ξ b σ − ε, b σ + ε := { ( s, p ) ∈ Ω | (2.1) holds for σ ∈ [ b σ − ε, b σ + ε ] } the set of admissible paths under a Black-Scholes model with uncertainty in the volatility parameter.We set b σ = 0 . and depict in Figure 2 how the robust upper price bounds for several payoff functions Φ behave under dynamic option trading for a varying number of call options in dependence of ε ,using Algorithm 1. No dynamically traded options1 dynamically traded option2 dynamically traded options3 dynamically traded options . . . ε P Ξ . − ε , . + ε ( Φ ) Φ( S t , S t ) = | S t − S t | . . . ε . . . P Ξ . − ε , . + ε ( Φ ) Φ( S t , S t ) = (max( S t , S t ) − + . . . ε . . . . P Ξ . − ε , . + ε ( Φ ) Φ( S t , S t ) = (0 . S t + S t ) − + . . . ε P Ξ . − ε , . + ε ( Φ ) Φ( S t , S t ) = | S t − . · S t | Figure 2.
The figure shows how the upper robust price bounds behave under adifferent number of traded options which are priced according to a robust Black-Scholes pricing rule as in (2.1) in dependence of ε and for b σ = 0 . Example 3.3 (Three Times, Continuous Marginals) . We consider log-normally distributed marginals S t i ∼ S t exp (cid:18) σ √ t i N i − σ t i (cid:19) for i = 1 , , , with S t = 100 , σ = 10 , t i = i and N i ∼ N (0 , i.i.d. for i = 1 , , . The payoff function isan Asian call option of the form Φ( S ) = (cid:16) P i =1 S t i − (cid:17) + . As dynamically traded options wetake into account European call options v , = ( S t − + and v , = ( S t − + . We consideras price bounds for the European options p ,l, ( S t l ) = ( S t l − + and p ,l, ( S t l ) = S t l for l = 2 , respectively. Then we study, using the neural networks approach which is explained in Section 3.2.2,how the price bounds P Ξ ( pi,j,k + ε ,pi,j,k − ε (Φ) behave for increasing ε , ε . The results are illustratedin Figure 3, where we can observe that increasing ε , ε simultaneously may lead to an even strongerimprovement of the price bounds of Φ in comparison with only increasing either ε or ε . ε ε
65 67.5 70 72.5 75 77.7 80 82.5 85 P Ξ ( p i , j , k + ε , p i , j , k − ε ) ( Φ ) Figure 3.
The price bounds P Ξ ( pi,j,k + ε ,pi,j,k − ε (Φ) of an Asian option with log-normally distributed marginal distributions and simultaneously increased pricebounds of dynamically traded European options. Example 3.4 (No arbitrage bounds for call-option prices using S & P
500 data) . We study prices forcall options written on constituents of the S & P index at June . In total we investigate options. We study to which degree ask and bid prices deviate from the standard no-arbitragebounds ( S t − K ) + and S t respectively, see also Remark 2.1 (b). As we do not consider interestrates, we only take into account those options with a short time-to-maturity. Here, we consideronly options with time-to-maturity less than days. The deviation of the average of all normalizedprices (in percentage) and of the and -quantile of all normalized prices from the no-arbitragebounds is illustrated in Figure 4. We observe a certain amount of options with prices lower thanthe lower no-arbitrage bound, which can be explained through interest rates and dividend yields. Strike (% of spot price) P r i c e ( % o f s p o t p r i c e ) Average Call Option Price5 % Quantile of Call Option Prices95 % Quantile of Call Option PricesNo Arbitrage Price Bounds
Figure 4.
The plot shows how prices of call options written on the S & P
500 deviatefrom the lower no-arbitrage bound ( S t − K ) + and the upper no-arbitrage bound S t ,respectively. In particular, we realize that the deviation from the upper no-arbitrage bound is much larger thanfrom the lower bound. This is because the payoff functions of call options are convex functions andthus the upper price bound S t , which is the concave envelope of ( S t j − K ) + , is relatively distant fromthe payoff itself, whereas the convex envelope ( S t − K ) + is closer to the convex payoff function. Forconcave payoff functions the situation turns out to be exactly opposite, i.e., the concave envelope iscloser to the payoff function than the convex envelope (that appears as a lower bound). ODEL-FREE PRICE BOUNDS UNDER DYNAMIC OPTION TRADING 13
Numerics.
In this section we sketch and discuss two algorithms to solve the problem of thecomputation of D Ξ (Φ) and P Ξ (Φ) numerically.3.2.1. Linear Programming.
Given that for all i = 1 , . . . , n, j = 1 , . . . , n , k = 1 , . . . , N we have S t i ∈ { s , . . . , s n i i } and P t i ( v j,k ) ∈ { p i,j,k , . . . , p n i,j,k i,j,k } - which can always be achieved through acareful discretization of the underlying space - we can formulate a linear program to solve theprimal problem P Ξ (Φ) as well as the dual problem D Ξ (Φ). Here we need to remark that this linearprogramming approach, however, scales badly with dimensions, but on the contrary yields preciseand fast results in low dimensions. Algorithm 1:
Computation of D Ξ (Φ) via linear programming Input :
Marginals µ , . . . , µ n ; Payoff function Φ; Set of dynamically tradable options V = { v j,k } ; Grid Ξ m grid as in Remark 2.1 (e); Output:
Minimal P ni =1 R u i d µ i such that (3.1) holds;Minimal u i ( s ) , H i ( s , . . . , s i ) , H i,j,k ( s , . . . , s i ) such that (3.1) holds.Discretize marginals such that supp( µ i ) ⊂ Ξ m grid , e.g. by the methods from [6] and [31]; for ( s, p ) ∈ Ξ m grid do Add inequality constraints of the form(3.1) n X i =1 u i ( s ) + n − X i =1 H i ( s , . . . , s i )( s i +1 − s i )+ n − X i =1 n X j = i +1 N X k =1 H i,j,k ( s , . . . , s i ) ( v j,k ( s j ) − p i,j,k ) ≥ Φ( s ) end Minimize n X i =1 Z u i d µ i = n X i =1 X ( s,p ) ∈ Ξ m grid u i ( s ) µ i ( { s } )w.r.t. u i ( s ) , H i ( s , . . . , s i ) , H i,j,k ( s , . . . , s i ) such that the imposed inequality constraints(3.1) are fulfilled. This is possible e.g. via the simplex algorithm, compare [22].For the computation of P Ξ (Φ), in addition to the the linear programming approach for martingaleoptimal transport, we obtain supplementary constraints associated to the property P t i ( v j,k ) = E Q [ v j,k ( S t j ) |F t i ] for all i = 1 , . . . , n For the computation of D Ξ (Φ) one obtains for the hedging strategies additional terms of the form H i,j,k ( s , . . . , s i )( v j,k ( s j ) − p i,j,k ) that will be considered on the grid induced by the discrete valuesfor S t i and P t i ( v j,k ).For further details of the approach in the martingale optimal transport setting we refer to theAlgorithm 1 and to [31], [32].3.2.2. Neural networks and penalization.
We explain how to adjust the approach from [27] to com-pute the price bounds involving dynamic option trading. The adapted algorithm from [27] isstated in Algorithm 2 for the case p i,j,k ∈ [ p i,j,k , p i,j,k ] in which one only needs to consider values p i,j,k ∈ { p i,j,k , p i,j,k } since these values lead to the extremal values of the super-hedging strategies.In contrast to the linear programming approach, this algorithm scales very well with dimensions,i.e., with an increasing number of marginals and of considered underlying securities. However, thechoice of the involved hyper-parameters turns out to be a rather complicated task, as it was alreadyobserved in [33]. Within our numerical examples we decided to mainly stick to the parameters usedin [27] and [5] by choosing γ = 10000, neural networks with 5 hidden layers, 64 · n neurons and ReLu activation functions. The batch size was 2 n and the optimization was performed by anAdam optimizer with standard parameters for N = 50000 iterations. To reduce the variance of theresults we finally average over 30 independent simulations. We remark first that the discretization needs to be performed such that the discrete marginals keep increasing inconvex order, compare [3], and second, even if the marginals are supported on a discrete grid and do not require adiscretization, the price process always needs to be discretized.
Algorithm 2:
Computation of D Ξ (Φ) via penalization Input :
Marginals µ , . . . , µ n ; Batch size B ; Payoff function Φ; Set of dynamically tradableoptions V = { v j,k } ; Penalization parameter γ ; Price bound functions p i,j,k , p i,j,k forpricing rules of European options; Number of iterations N ; Architecture of NeuralNetworks; Parameters for Adam optimizer; Output:
AverageLoss.Initialize neural networks H i , H i,j,k , u i , with random weights;iter ← while iter < N dofor b = 1 : B dofor i = 1 : n do Sample x bi ∼ µ i ; for j = 1 : n dofor k = 1 : N do Sample p bi,j,k ∼ U ( { p i,j,k ( x b , . . . , x bi ) , p i,j,k (( x b , . . . , x bi ) } ); endendendend Loss[iter] ← B B X p =1 n X j =1 u i ( x bi ) + 12 γ B max (cid:26) B X p =1 (cid:18) Φ( x b , · · · , x bn ) − n X i =1 u i ( x bi ) − n − X i =1 H i ( x b , . . . , x bi )( x bi +1 − x bi ) − n X i =1 n X j = i +1 N X k =1 H i,j,k ( x b , . . . , x bi )( v j,k ( x bj ) − p bi,j,k ) (cid:19) , (cid:27) ;Use Adam optimizer to minimize the weights of H i , H i,j,k , u i w.r.t. Loss[iter];iter ← iter +1; end AverageLoss ← Loss[0 . N : N ]; // Average loss over the last of Iterations Proofs
In this section we provide all proofs of the mathematical statements from the previous sections.
Proof of Theorem 2.4 (a).
First, let M V (Ξ , µ , . . . , µ n ) be non-empty. Then pick some measure Q ∈ M V (Ξ , µ , . . . , µ n ) ⊂ P (Ω). We define a measure Q ∈ P ( R n + ) through Q := Q ◦ S − with S : Ω → R n + , S ( s, p ) = s . The measure Q is contained in M ( µ , . . . , µ n ) as martingaleand marginal properties of Q are inherited from Q . Moreover, it holds for all i, j = 1 , . . . , n , ODEL-FREE PRICE BOUNDS UNDER DYNAMIC OPTION TRADING 15 k = 1 , . . . , N and all Borel-measurable sets A ⊂ R i + that Z Ω A ( s , . . . , s i ) E Q (cid:2) v j,k ( S t j ) (cid:12)(cid:12) F t i (cid:3) ( s, p ) d Q ( s, p )= Z Ω A ( s , . . . , s i ) v j,k ( s j ) d Q ( s, p )= Z R n + A ( s , . . . , s i ) v j,k ( s j ) d Q ( s )= Z R n + A ( s , . . . , s i ) E Q (cid:2) v j,k ( S t j ) (cid:12)(cid:12) F t i (cid:3) ( s ) d Q ( s )= Z Ω A ( s , . . . , s i ) E Q (cid:2) v j,k ( S t j ) (cid:12)(cid:12) F t i (cid:3) ◦ S ( s, p ) d Q ( s, p ) . Thus, it holds Q -almost surely that E Q (cid:2) v j,k ( S t j ) (cid:12)(cid:12) F t i (cid:3) ◦ S = E Q (cid:2) v j,k ( S t j ) (cid:12)(cid:12) F t i (cid:3) = P t i ( v j,k ) . This implies, by using the definition of Q , that Q (cid:16)n s ∈ R n + (cid:12)(cid:12)(cid:12) (cid:16) s, E Q (cid:2) v j,k ( S t j ) (cid:12)(cid:12) F t i (cid:3) ( s ) i,j =1 ,...,n,k =1 ,...,N (cid:17) ∈ Ξ o(cid:17) = Q (cid:16)n ( s, p ) ∈ Ω (cid:12)(cid:12)(cid:12) (cid:16) S ( s, p ) , E Q (cid:2) v j,k ( S t j ) (cid:12)(cid:12) F t i (cid:3) ◦ S ( s, p ) i,j =1 ,...,n,k =1 ,...,N (cid:17) ∈ Ξ o(cid:17) = Q (cid:16)n ( s, p ) ∈ Ω (cid:12)(cid:12)(cid:12) (cid:16) S ( s, p ) , P t i ( v j,k )( s, p ) i,j =1 ,...,n,k =1 ,...,N (cid:17) ∈ Ξ o(cid:17) = Q (Ξ) = 1 , and thus (2.7) is fulfilled. Conversely, let (2.7) be valid for some Q ∈ M ( µ , . . . , µ n ) ⊂ P ( R n + ). Wedefine a measure Q ∈ P (Ω) through(4.1) Q := Q ◦ g − for g : s (cid:18) s, (cid:0) E Q [ v j,k ( S t j ) | F t i ]( s ) (cid:1) i,j, =1 ,...,nk =1 ,...,N (cid:19) Then Q (Ξ) = Q (cid:0)(cid:8) s ∈ R n + (cid:12)(cid:12) g ( s ) ∈ Ξ (cid:9)(cid:1) = 1 is ensured through (2.7), and it further holds for all i, j = 1 , . . . , n , k = 1 , . . . , N and H ∈ C b ( R i + ) Z Ω H ( s , . . . , s i )( v j,k ( s j ) − p i,j,k ) d Q ( s, p )= Z R n + H ( s , . . . , s i )( v j,k ( s j ) − E Q [ v j,k ( S t j ) | F t i ]( s )) d Q ( s ) = 0 . Hence Q ∈ M V (Ξ , µ , . . . , µ n ), since the martingale and marginal constraints are inherited from Q . (cid:3) Proof of Theorem 2.4 (b).
We define for i ∈ { , . . . , n } , H ∈ C b ( R i ) , v j,k ∈ V the map φ i, ( v j,k ) ,H : P (Ω) → RQ Z Ω H ( s , . . . , s i )( v j,k ( s j ) − p i,j,k ) d Q ( s, p )and consider S : Ω → R n + , S ( s, p ) = s . With these definitions, we obtain the representation(4.2) M V (Ξ , µ , . . . , µ n ) = (cid:8) Q ∈ P (Ω) (cid:12)(cid:12) Q ◦ S − ∈ M ( µ , . . . , µ n ) (cid:9) ∩ { Q ∈ P (Ω) | Q (Ξ) = 1 } \ i ∈{ ,...,n } ,v j,k ∈ V,H ∈ C b ( R i ) n φ − i, ( v j,k ) ,H ( { } ) o . To show that M V (Ξ , µ , . . . , µ n ) is compact in the weak topology we first show that the set (cid:8) Q ∈ P (Ω) (cid:12)(cid:12) Q ◦ S − ∈ M ( µ , . . . , µ n ) (cid:9) is compact and then that { Q ∈ P (Ω) | Q (Ξ) = 1 } as well asthe sets n φ − i, ( v j,k ) ,H ( { } ) o ∩ (cid:8) Q ∈ P (Ω) (cid:12)(cid:12) Q ◦ S − ∈ M ( µ , . . . , µ n ) (cid:9) are closed for all i ∈ { , . . . , n } , v j,k ∈ V , H ∈ C b ( R i ). Then, due to the representation (4.2), M V (Ξ , µ , . . . , µ n ) is indeed compactas a closed subset of a compact set.First, let (cid:0) Q ( m ) (cid:1) m ∈ N be a sequence of measures in P (Ω) with Q ( m ) ◦ S − ∈ M ( µ , . . . , µ n ) for all m ∈ N . Then by the weak-compactness of M ( µ , . . . , µ n ) (compare [11, Proposition 2.4.]), thereexists a subsequence (labelled identically) and some P ∈ M ( µ , . . . , µ n ) such that Q ( m ) ◦ S − → P weakly for m → ∞ . With an analogue construction as in (4.1), we conclude the existence of some Q ∈ P (Ω) with Q ◦ S − = P . Consequently it holds that the set (cid:8) Q ∈ P (Ω) (cid:12)(cid:12) Q ◦ S − ∈ M ( µ , . . . , µ n ) (cid:9) is weakly compact.Next, pick a sequence (cid:0) Q ( m ) (cid:1) m ∈ N in P (Ω) with Q ( m ) (Ξ) = 1 for all m ∈ N , which convergesweakly to some Q for m → ∞ . Then, by the Portmanteau theorem and by the assumption that Ξis closed it holds 1 = lim sup m →∞ Q ( m ) (Ξ) ≤ Q (Ξ)and thus Q (Ξ) = 1. Hence, { Q ∈ P (Ω) | Q (Ξ) = 1 } is closed.Next, we pick i ∈ { , . . . , n } , H ∈ C b ( R i ) , v j,k ∈ V and claim that the function φ i, ( v j,k ) ,H iscontinuous on (cid:8) Q ∈ P (Ω) (cid:12)(cid:12) Q ◦ S − ∈ M ( µ , . . . , µ n ) (cid:9) . Indeed, pick a sequence (cid:0) Q ( m ) (cid:1) m ∈ N fromthis set which converges weakly to some measure Q , which is by the above shown closedness againcontained in the set. Then for all a ≥ m ∈ N we have φ i, ( v j,k ) ,H ( Q ( m ) ) = Z [ − a,a ] n × R n N + H ( s , . . . , s i )( v j,k ( s j ) − p i,j,k ) d Q ( m ) ( s, p )+ Z ( R n \ [ − a,a ] n ) × R n N + H ( s , . . . , s i )( v j,k ( s j ) − p i,j,k ) d Q ( m ) ( s, p ) . Since v j,k ∈ C lin ( R + , R + ) and by validity of (2.8) there exist constants c , c ∈ R such that thesecond integral is smaller or equal thansup m ∈ N (Z R n + \ [ − a,a ] n " | H ( s , . . . , s i ) | c (1 + s j ) + c i X k =1 s k !! d( Q ( m ) ◦ S − )( s ) ) , which converges to 0 for a → ∞ as Q ( m ) ◦ S − ∈ M ( µ , . . . , µ n ) possesses fixed marginals µ , . . . , µ n for all m ∈ N . This, on the other hand implies, as in the proof of [11, Lemma 2.2.] the conver-gence φ i, ( v j,k ) ,H ( Q ( m ) ) → φ i, ( v j,k ) ,H ( Q ) for m → ∞ . The continuity yields then the closedness of n φ − i, ( v j,k ) ,H ( { } ) o ∩ (cid:8) Q ∈ P (Ω) (cid:12)(cid:12) Q ◦ S − ∈ M ( µ , . . . , µ n ) (cid:9) .Thus, in total we have shown that M V (Ξ , µ , . . . , µ n ) is compact as a closed subset of a compactset. (cid:3) Proof of Theorem 2.4 (c).
We aim at applying the biconjugate duality theorem [7, Theorem 2.2.]to D Ξ . A similar proof of a martingale transport duality under additional constraints can be foundin [28] and [4]. First, we extend the domain of the super-replication functional D Ξ ( · ) from functionsdefined only on R n + to payoffs defined on Ω and consider D Ξ : C Ω → R with C Ω := ( f : Ω → R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup ( s,p ) ∈ Ω | f ( s, p ) | (1 + P ni =1 s i ) < ∞ ) . Observe that D Ξ ( · ) is convex and increasing on C Ω as well as, by abuse of notation, that we have C lin (cid:0) R n + , R + (cid:1) ⊂ C Ω . The fulfilment of condition (R1) from [7, Theorem 2.2.] follows analogue as inthe proof of [28, Theorem 3.3.].We define P + := ( Q ∈ P (Ω) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z Ω n X i =1 s i d Q ( s, p ) < ∞ ) . An application of [7, Theorem 2.2.] yields(4.3) D Ξ (Φ) = sup Q ∈P + (cid:18)Z Ω Φ( s ) d Q ( s, p ) − D ∗ Ξ ( Q ) (cid:19) , ODEL-FREE PRICE BOUNDS UNDER DYNAMIC OPTION TRADING 17 where the convex conjugate D ∗ Ξ of D Ξ is defined throughD ∗ Ξ ( Q ) = sup f ∈ C Ω (cid:26)Z Ω f ( s, p ) d Q ( s, p ) − D Ξ ( f ) (cid:27) . We want to show that D ∗ Ξ ( Q ) = ( Q ∈ M V (Ξ , µ , . . . , µ n ) , ∞ else.W.l.o.g. assume Ξ = Ω, else Q (Ξ) = 1 is trivially satisfied. By Urysohn’s Lemma, there existfunctions ( f m ) m ∈ N ⊂ C b (Ω) ⊂ C Ω which are 0 on Ξ and converge pointwise and monotonically to ∞ · Ξ c for m → ∞ . Thus D Ξ ( f m ) ≤ m ∈ N and we obtain(4.4) D ∗ Ξ ( Q ) ≥ sup m (cid:26)Z Ω f m ( s, p ) d Q ( s, p ) − D Ξ ( f m ) (cid:27) ≥ ∞ · Q (Ξ c ) . Therefore D ∗ Ξ ( Q ) = ∞ if Q (Ξ c ) >
0. Assume from now on that Q (Ξ) = 1. Next, we computeD ∗ Ξ ( Q ). We first use the relation − inf − f = sup f and obtainD ∗ Ξ ( Q ) = sup f ∈ C Ω sup u i ∈ C lin ( R + , R + ) H i ,H i,j,k ∈ C b ( R i ):Ψ V ( Hi ) , ( Hi,j,k ) , ( ui ) ≥ f on Ξ (Z Ω f ( s, p ) d Q ( s, p ) − n X i =1 Z R + u i ( s ) d µ i ( s ) ) . We observe that Ψ V ( H i ) , ( H i,j,k ) , ( u i ) ∈ C Ω . Thus we may plug in for f the strategy Ψ V ( H i ) , ( H i,j,k ) , ( u i ) toget D ∗ Ξ ( Q ) = sup u i ∈ C lin ( R + , R + ) n X i =1 (cid:18)Z Ξ u i ( s i ) d Q ( s, p ) − Z R + u i ( s i ) d µ i ( s i ) (cid:19) + sup H i ∈ C b ( R i + ) n − X i =1 (cid:18)Z Ξ H i ( s , . . . , s i )( s i +1 − s i ) d Q ( s, p ) (cid:19) + sup H i,j,k ∈ C b ( R i + ) n − X i =1 n X j = i +1 N X k =1 (cid:18)Z Ξ H i,j,k ( s , . . . , s i )( v j,k ( s j ) − p i,j,k ) d Q ( s, p ) (cid:19) . Then, by the characterization of M V (Ξ , µ , . . . , µ n ) through integrals in (2.5), and by (4.4), we seethat D ∗ Ξ ( Q ) = ( Q ∈ M V (Ξ , µ , . . . , µ n ) , ∞ else . Hence, we conclude from (4.3) thatD Ξ (Φ) = sup Q ∈P + (cid:18)Z Ω Φ( s ) d Q ( s, p ) − D ∗ Ξ ( Q ) (cid:19) = sup Q ∈M V (Ξ ,µ ,...,µ n ) Z Ω Φ( s ) d Q ( s, p )= P Ξ (Φ) . Finally, attainment of the primal value follows under the assumptions of Theorem 2.4 (b), since theset M V (Ξ , µ , . . . , µ n ) is compact and Q R Φ d Q is continuous on M V (Ξ , µ , . . . , µ n ), (compare[11, Lemma 2.2.]). (cid:3) Proof of Remark 2.5 (a).
W.l.o.g. assume that M V (Ξ , µ , . . . , µ n ) = ∅ which by Theorem 2.4 (a)is equivalent to the non-emptiness of { Q ∈ M ( µ , . . . , µ n ) : (2.7) holds } , else the assertion ofRemark 2.5 (a) holds trivially. Let Q ∈ M V (Ξ , µ , . . . , µ n ). Then, according to the proof ofTheorem 2.4 (a), there exists some Q ∈ M ( µ , . . . , µ n ) such that (2.7) holds and such that wefurther have R Ω Φ d Q = R R n + Φ d Q . Analogously, for each Q ∈ M ( µ , . . . , µ n ) such that (2.7)holds we can find some Q ∈ M V (Ξ , µ , . . . , µ n ) with R Ω Φ d Q = R R n + Φ d Q . (cid:3) Proof of Remark 2.5 (c).
We first note that the validity of (2.7) for Q ∈ M ( µ , . . . , µ n ) is equivalentto the fact that for all i, j = 1 , . . . , n , k = 1 , . . . , N there exists some f i,j,k ∈ F i,j,k such that E Q [ v j,k ( S t j ) | S t i , . . . , S t ] = f i,j,k Q -a.s. According to Remark 2.5 (a), this explains P Ξ (Φ) =sup Q ∈Q ( F i,j,k ) R R n + Φ( s ) d Q ( s ).Next, we see that(4.5) Ψ V ( H i ) , ( H i,j,k ) , ( u i ) ( s, ( f i,j,k ( s , . . . , s i )) i,j,k ) ≥ Φ( s ) for all s ∈ R n + , f i,j,k ∈ F i,j,k if and only if inf f i,j,k ∈ F i,j,k Ψ V ( H i ) , ( H i,j,k ) , ( u i ) ( s, ( f i,j,k ( s , . . . , s i )) i,j,k ) ≥ Φ( s ) for all s ∈ R n + . (4.6)As in the proof of Theorem 2.4 (c), in equation (4.3), we compute the biconjugate representation ofthe super-replication functional D ( F i,j,k ) . To that end, we first obtain for every measure Q ∈ P ( R n + )with finite first moments that its convex conjugate satisfiesD ∗ ( F i,j,k ) ( Q ) = sup f ∈ C lin ( R n + , R + ) sup u i ∈ C lin ( R + , R + ) H i ,H i,j,k ∈ C b ( R i ):(4.6) holds (Z R n + f ( s ) d Q ( s ) − n X i =1 Z R + u i ( s ) d µ i ( s ) ) . (4.7)Analogue to the proof of Theorem 2.4 (c) we aim at showing that(4.8) D ∗ ( F i,j,k ) ( Q ) = ( Q ∈ Q ( F i,j,k ) , ∞ if Q
6∈ Q ( F i,j,k ) . To this end, we first want to show that for all i, j, k we have(4.9) inf f i,j,k ∈ F i,j,k Ψ V ( H i ) , ( H i,j,k ) , ( u i ) ( s, ( f i,j,k ( s , . . . , s i )) i,j,k ) ∈ C lin (cid:0) R n + , R + (cid:1) . For every H i,j,k ∈ C b ( R i + ) and every i, j, k define g : F i,j,k × R i + → R ( f i,j,k , ( s , . . . , s i )) H i,j,k ( s , . . . , s i ) ( v j,k ( s j ) − f i,j,k ( s , . . . , s i )) , and for any compact set K ⊂ R i + let g | K : F i,j,k × K → R be the restriction of g onto F i,j,k × K .Note that g is continuous, as for any (cid:16) f ( N ) i,j,k (cid:17) N ∈ N ⊂ F i,j,k converging uniformly on K to some f i,j,k and (cid:16) s ( N )1 , . . . , s ( N ) i (cid:17) N ∈ N ⊂ K converging to some ( s , . . . , s i ) for N → ∞ , we have that f ( N ) (cid:16) s ( N )1 , . . . , s ( N ) i (cid:17) → f ( s , . . . , s n ) for N → ∞ . Moreover, since by assumption F i,j,k is compactwhen restricted to K , we can apply Berge’s maximum theorem (compare e.g. [13, Chapter VI,Section 3]) to g | K implying the continuity of( s , . . . , s i ) inf f i,j,k ∈ F i,j,k H i,j,k ( s , . . . , s i ) ( v j,k ( s j ) − f i,j,k ( s , . . . , s i )) on K. Since K was chosen arbitrarily the map g is continuous on the whole space R i . Further, due to(2.9), we obtain sup ( s ,...,s i ) ∈ R i + inf f i,j,k ∈ F i,j,k Ψ V ( H i ) , ( H i,j,k ) , ( u i ) ( s, ( f i,j,k ( s , . . . , s i )) i,j,k )1 + P iℓ =1 s ℓ < ∞ . Hence, using the definition of Ψ V ( H i ) , ( H i,j,k ) , ( u i ) , we conclude (4.9). Due to the validity of (4.9), whencomputing (4.7) we get for every Q ∈ P ( R n + ) with finite first moments thatD ∗ ( F i,j,k ) ( Q ) = sup u i ∈ C lin ( R + , R + ) n X i =1 Z R n + u i ( s i ) d Q ( s ) − Z R + u i ( s i ) d µ i ( s i ) ! + sup H i ∈ C b ( R i + ) n − X i =1 Z R n + H i ( s , . . . , s i )( s i +1 − s i ) d Q ( s ) ! + sup H i,j,k ∈ C b ( R i + ) n − X i =1 n X j = i +1 N X k =1 Z R n + inf f i,j,k ∈ F i,j,k H i,j,k ( s , . . . , s i )( v j,k ( s j ) − f i,j,k ( s , . . . , s i )) d Q ( s ) ! . ODEL-FREE PRICE BOUNDS UNDER DYNAMIC OPTION TRADING 19
As in the proof of Theorem 2.4 (c) the first two summands vanish if and only if Q fulfils theassociated marginal and martingale constraints. Moreover, the last summand is greater or equal 0which can be seen through setting H i,j,k ≡
0. On the other handsup H i,j,k ∈ C b ( R i + ) n − X i =1 n X j = i +1 N X k =1 Z R n + inf f i,j,k ∈ F i,j,k H i,j,k ( s , . . . , s i )( v j,k ( s j ) − f i,j,k ( s , . . . , s i )) d Q ( s ) ! ≤ sup H i,j,k ∈ C b ( R i + ) inf f i,j,k ∈ F i,j,k n − X i =1 n X j = i +1 N X k =1 Z R n + H i,j,k ( s , . . . , s i )( v j,k ( s j ) − f i,j,k ( s , . . . , s i )) d Q ( s ) ! which vanishes if and only if for all i, j, k there exists some f i,j,k ∈ F i,j,k such that it holds E Q [ v j,k ( S t j ) | S t i , . . . , S t ] = f i,j,k Q -a.s., and can be scaled infinitely large otherwise. This provesthat the conjugate D ∗ ( F i,j,k ) satisfies (4.8). (cid:3) Proof of Remark 2.6.
Consider some super-replication strategy Ψ V ( H i ) , ( H i,j,k ) , ( u i ) such thatΨ V ( H i ) , ( H i,j,k ) , ( u i ) ( s, p ) ≥ Φ( s ) for all ( s, p ) ∈ Ξ . Then by (2.10) assumed on Ξ it holds directly Ψ V ( H i ) , ( H i,j,k ) , ( u i ) ( s, e p ) ≥ Φ( s ) for all s ∈ R n + and forthe particular choice of e p ∈ R nN with e p i,j,k = v j,k ( s j ). Moreover, since n − X i =1 n X j = i +1 N X k =1 H i,j,k ( s , . . . , s i ) ( v j,k ( s j ) − e p i,j,k ) = 0 , this implies thatΨ ( H i ) , ( u i ) ( s ) := n X i =1 u i ( s i ) + n − X i =1 H i ( s , . . . , s i )( s i +1 − s i ) ≥ Φ( s ) for all s ∈ R n + . Hence, we obtain D Ξ (Φ) = inf u i ∈ C lin ( R + , R + ) H i ,H i,j,k ∈ C b ( R i ):Ψ V ( Hi ) , ( Hi,j,k ) , ( ui ) ≥ Φ n X i =1 Z u i ( s i ) d µ i ( s i ) ≥ inf u i ∈ C lin ( R + , R + ) H i ∈ C b ( R i ):Ψ ( Hi ) , ( ui ) ≥ Φ n X i =1 Z u i ( s i ) d µ i ( s i )= sup Q ∈M ( µ ,...,µ n ) Z R n + Φ( s ) d Q ( s )where the last equality is the martingale optimal transport duality from [11, Corollary 1.2.]. Thereverse inequality follows immediately by definition. (cid:3) Proof of Theorem 2.8.
We first prove the assertion from (a). W.l.o.g. assume (2.12) holds true, asthe case (2.13) can be argued analogously. First we claim that c M V (Ω , µ , . . . , µ n ) ∩ c M V (Ξ ( p i,j,k ,p i,j,k ) , µ , . . . , µ n ) = ∅ . Assume by contradiction that there exists Q ∈ c M V (Ω , µ , . . . , µ n ) ∩ c M V (Ξ ( p i,j,k ,p i,j,k ) , µ , . . . , µ n ).In particular, it holds for this Q that Q (cid:16) P t i ( v j,k ) ∈ [ p i,j,k , p i,j,k ] (cid:17) = 1 for all i, j ∈ { , . . . , n } , k ∈{ , . . . , N } . Set e A = A ∩ { ( s, p ) ∈ Ω | p i,j,k ∈ [ p i,j,k , p i,j,k ] } where i, j, k are the indices and A is theset corresponding to (2.12). Then, by validity of (2.12), we obtain the following inequality Z e A ( v j,k ( s j ) − p i,j,k ) d Q ( s, p ) ≤ Z e A ( v j,k ( s j ) − p i,j,k ( s , . . . , s i )) d Q ( s, p ) < which contradicts the definition of P t i ( v j,k ) which coincides Q -a.s. with the F t i -conditional expec-tation of v j,k ( S t j ). Thus c M V (Ω , µ , . . . , µ n ) ∩ c M V (Ξ ( p i,j,k ,p i,j,k ) , µ , . . . , µ n ) = ∅ . Moreover, by Theorem 2.4 (c), there exists some Q ∈ c M V (Ξ ( p i,j,k ,p i,j,k ) , µ , . . . , µ n ). It thereforeholds for all Q ∈ c M V (Ω , µ , . . . , µ n ), as M V (Ξ ( p i,j,k ,p i,j,k ) , µ , . . . , µ n ) ⊂ M V (Ω , µ , . . . , µ n ), thatP Ξ ( pi,j,k,pi,j,k ) (Φ) = Z Φ d Q < Z Φ d Q = P Ω (Φ) . On the other hand, if neither (2.12) nor (2.13) hold true, then there exists some measure Q ∈ c M V (Ω , µ , . . . , µ n ) such that p i,j,k ≤ E Q (cid:2) v j,k ( S t j ) (cid:12)(cid:12) F t i (cid:3) ≤ p i,j,k Q -a.s. for all i, j, k. Hence Q ∈ M V (Ξ ( p i,j,k ,p i,j,k ) , µ , . . . , µ n ) and consequentlyP Ξ ( pi,j,k,pi,j,k ) (Φ) ≥ Z Φ d Q = P Ω (Φ) , which in turn implies equality.For the assertion from (b) one can show analogously that if (2.14) holds, then c M V (Ξ ( p i,j,k ,p i,j,k ) , µ , . . . , µ n ) ∩ c M V (Ξ ( p i,j,k + ε,p i,j,k ) , µ , . . . , µ n ) = ∅ and conclude that for all Q ∈ c M V (Ξ ( p i,j,k ,p i,j,k ) , µ , . . . , µ n )P Ξ ( pi,j,k + ε,pi,j,k ) (Φ) < Z Φ d Q ( s, p ) = P Ξ ( pi,j,k,pi,j,k ) (Φ) . For the reverse direction we remark that if (2.14) does not hold, then there exists some Q ∈ c M V (Ξ ( p i,j,k ,p i,j,k ) , µ , . . . , µ n ) such that(4.10) p i,j,k + ε ≤ E Q (cid:2) v j,k ( S t j ) (cid:12)(cid:12) F t i (cid:3) ≤ p i,j,k Q -a.s. for all i, j, k. Hence P Ξ ( pi,j,k + ε,pi,j,k ) (Φ) ≥ Z Φ d Q = P Ω (Φ) , which in turn implies equality. The assertion from (c) follows in the same way as in the proof of(b). (cid:3) Proof of Theorem 2.9.
The proof is analogue to the proof of Remark 2.5 (c). Analogue to equation(4.4) we see that(4.11) e D ∗ ( F i,j ) ( Q ) = ∞ if Q ( e Ξ c ) > . Moreover, when computing the convex conjugate of e D ( F i,j ) (Φ) we obtain for every Q ∈ P ( R n + ) withfinite first moments satisfying Q (Ξ) = 1 that e D ∗ ( F i,j ) ( Q ) = sup u i ∈ C lin ( R + , R + ) n X i =1 (cid:18)Z e Ξ u i ( s i ) d Q ( s ) − Z R + u i ( s i ) d µ i ( s i ) (cid:19) + sup H i ∈ C b ( R i + ) n − X i =1 (cid:18)Z e Ξ H i ( s , . . . , s i )( s i +1 − s i ) d Q ( s ) (cid:19) + sup Hi,j ∈ Cb ( R i +) vj ∈ C lin( R + , R +) n − X i =1 n X j = i +1 (cid:18)Z e Ξ inf f i,j ∈ F i,j ( v j ) H i,j ( s , . . . , s i )( v j ( s j ) − f i,j ( s , . . . , s i )) d Q ( s ) (cid:19) . The first two suprema vanish if and only if Q fulfils the corresponding martingale and marginalproperty, whereas the last supremum vanishes if and only if for all i, j = 1 , . . . , n and all v j ∈ C lin ( R + , R + ) there exist f i,j ∈ F i,j ( v j ) such that E Q [ v j ( S t j ) | S t i , . . . , S t ] = f i,j Q -a.s. ODEL-FREE PRICE BOUNDS UNDER DYNAMIC OPTION TRADING 21
This means, together with (4.11) that e D ∗ ( F i,j ) ( Q ) = 0 if and only if Q ∈ e Q ( F i,j ) , and otherwise e D ∗ ( F i,j ) ( Q ) becomes infinitely large. Hence we conclude the results by the biconjuagte representationfor e D ( F i,j ) similar to (4.3). (cid:3) Proof of Remark 2.10.
Condition (2.17) is fulfilled since for all f ∈ F i,j ( v j ) and all ( s , . . . , s i ) ∈ R i + the Lipschitz property ensures that(4.12) | f ( s , . . . , s i ) | ≤ | g ( s i ) − g (0) | + | g (0) | ≤ s i . To see that for every compact set K ⊂ R n + the set F i,j ( v j ) is compact when restricted onto K , picka sequence ( f ( N ) i,j ) N ∈ N with f ( N ) i,j ∈ F i,j ( v j ) for all N ∈ N . Then we obtain for all N ∈ N a represen-tation f ( N ) i,j ( s , . . . , s i ) = g ( N ) ( s i ) for some 1-Lipschitz function g ( N ) . By the 1-Lipschitz propertyof g ( N ) the sequence ( g ( N ) ) N ∈ N is uniformly equicontinuous and pointwise bounded according to(4.12).Thus, the Arzel`a–Ascoli theorem implies the existence of a uniformly convergent subsequence(labelled identically) with g ( N ) → g for N → ∞ for some function g . Then g is 1-Lipschitz as itholds for all x, y ∈ R + that | g ( x ) − g ( y ) | = lim N →∞ | g ( N ) ( x ) − g ( N ) ( y ) | ≤ | x − y | . Further, we have g (0) = lim N →∞ g ( N ) (0) = 0. It remains to show that g admits a representation of the form E Q [ v j ( S t j ) | S t i ] = g Q -a.s. for some Q ∈ M ( µ , . . . , µ n ) . By definition of F i,j ( v j ), we have for all N ∈ N the representation(4.13) g ( N ) = E Q ( N ) [ v j ( S t j ) | S t i ] Q ( N ) -a.s. for some Q ( N ) ∈ M ( µ , . . . , µ n ) . Then by the weak compactness of M ( µ , . . . , µ n ) there exists some subsequence of Q ( N ) (denotedidentically) converging weakly to some Q ∈ M ( µ , . . . , µ n ). Similar to [41, Lemma 3.3.] we obtainfor all ∆ ∈ C b ( R + ) and all i, j = 1 , . . . , n that Z R n + ∆( s i ) lim N →∞ g ( N ) ( s i ) d Q ( s , . . . , s i )(4.14) = lim N →∞ Z R n + ∆( s i ) g ( N ) ( s i ) d Q ( N ) ( s , . . . , s i )(4.15) = lim N →∞ Z R n + ∆( s i ) v j ( s j ) d Q ( N ) ( s , . . . , s i )(4.16) = Z R n + ∆( s i ) v j ( s j ) d Q ( s , . . . , s i ) , (4.17)where the equality between (4.14) and (4.15) follows due to dominated convergence (which can beseen through the validity of (2.17) and by Q ∈ M ( µ , . . . , µ n )) and since Q ( N ) ◦ S − t i = Q ◦ S − t i , theequality between (4.15) and (4.16) is a consequence of (4.13), and (4.17) follows from [11, Lemma2.2.]. Hence, we conclude that lim N →∞ g ( N ) = E Q [ v j ( S t j ) | S t i ] Q -a.s.and thus g = E Q [ v j ( S t j ) | S t i ] Q -a.s. (cid:3) Acknowledgements
Financial support of the NAP Grant
Machine Learning based Algorithms in Finance and Insur-ance is gratefully acknowledged. Further we acknowledge the Singapore National SupercomputingCentre (NSCC) which provided computing power to conduct the numerical examples for the re-search.
Appendix A. Extensions
In this section we discuss various extensions of the presented results in Section 2. We extend ourconsiderations to multiple securities, market frictions such as transaction costs and the inclusion ofdifferent kind of (path-dependent) options for dynamic trading.A.1.
Transaction costs.
Since the considered strategies include an additional dynamic tradingcomponent which may cause a significant additional amount of transaction costs, it is important todiscuss how to incorporate transactions costs of the form considered for example in [20, Section 3.1.].For simplicity, we stick to the setting described in [20] where one considers only a finite amountof staticly traded call options instead of general European options. When considering transactioncosts, the profits of the considered strategies (without pricing rules) will be reduced and change for( s, p ) ∈ Ξ to:(A.1) n X i =1 M i X j =1 θ i,j ( s i − K i,j ) + − h i,j ( θ i,j )+ n − X i =1 H i ( s , . . . , s i )( s i +1 − s i ) − g stock i (∆ H i +1 · s i )+ n − X i =1 n X j = i +1 N X k =1 (cid:18) H i,j,k ( s , . . . , s i ) ( v j,k ( s j ) − p i,j,k ) − g option i (∆ H i +1 ,j,k · p i,j,k ) (cid:19) with θ i,j , K i,j ∈ R + , M i ∈ N , ∆ H i +1 = H i +1 − H i , ∆ H i +1 ,j,k = H i +1 ,j,k − H i,j,k , and h i,j , g stock i , g option i real-valued functions associated to the respective trading positions.In the case of proportional transaction costs one has h i,j ( θ i,j ) = θ + i,j h + i,j − θ − i,j h − i,j where h + i,j , h − i,j denote ask and bid prices of the considered call options. Moreover, we have g stock i ( x ) = ε stock i | x | and g option i ( x ) = ε option i | x | for some ε stock i , ε option i ≥
0. In this case combining our duality argumentwith the argumentation of [20], we obtain that minimizing prices of (A.1)-type super-replicationstrategies of Φ( S ) for Φ ∈ C lin (cid:0) R n + , R + (cid:1) is equivalent tosup Q ∈M prop Z Ω Φ( s ) d Q ( s, p ) , where M prop is the set of all probability measures Q on Ω with(i) (1 − ε stock i ) S t i ≤ E Q [ S t i +1 |F t i ] ≤ (1 + ε stock i ) S t i Q -a.s. for all i = 1 , . . . , n ,(ii) (1 − ε option i ) P t i ( v j,k ) ≤ E Q [ v j,k ( S t j ) |F t i ] ≤ (1 + ε option i ) P t i ( v j,k ) Q -a.s. for all i, j = 1 , . . . , n , k = 1 , . . . .N ,(iii) h − i,j ≤ E Q [( S t i − K i,j ) + ] ≤ h + i,j Q -a.s. for all i = 1 , . . . , n, j = 1 , . . . , M i ,(iv) Q (Ξ) = 1.This means that on the primal side we obtain an optimization problem over a set of measures withrelaxed inequality constraints which will eventually lead to higher maximal prices compared withthe formulation without transaction costs.A.2. Multiple securities.
The considerations from Section 2 can be extended straightforward toa high-dimensional market in which we consider d ≥ n ∈ N future times. Inthis case one considers trading strategies of the form d X l =1 (cid:18) n X i =1 u li ( s li ) + n − X i =1 H li ( s , . . . , s i )( s li +1 − s li )+ n − X i =1 n X j = i +1 N X k =1 H li,j,k ( s , . . . , s i ) (cid:16) v lj,k ( s j ) − p li,j,k (cid:17) (cid:19) . for ( s , . . . , s n ) = ( s , . . . , s dn ) ∈ R nd and p = ( p , , , . . . , p dn,n,N ) ∈ R nNd . We stress that thestrategies H li and H li,j,k are for all l = 1 , . . . , d allowed to depend on the price paths of all of the ODEL-FREE PRICE BOUNDS UNDER DYNAMIC OPTION TRADING 23 other securities under considerations, i.e., all available information is taken into account for trading.On the primal side this corresponds to joint martingale properties of the form E Q h S lt i +1 (cid:12)(cid:12)(cid:12) S t i , . . . , S dt i , . . . , S dt i = S lt i Q -a.s. , E Q h v lj,k ( S lt j ) (cid:12)(cid:12)(cid:12) S t i , . . . , S dt i , . . . , S dt i = P t i ( v lj,k ) Q -a.s.for all i = 1 , . . . , n, j = i + 1 , . . . , n, l = 1 , . . . , d. A.3.
Path-dependent traded options.
From a mathematical point of view there is no need torestrict the considerations to the case of dynamic trading in European options, i.e., to options wherethe associated payoff function only depends on a sole value of an underlying security. However, theassumption to allow dynamic trading over time requires from a practical point of view that theinvolved option is traded in a sufficiently liquid amount over time. This is very often only fulfilledfor specific European options such as call and put options. However, if the liquidity of options isensured, it is also thinkable to allow for trading in other kind of options that are possibly dependingon the whole path of an underlying security. If v j,k depends on the whole path until time t j , thenwe substitute (2.3) by(A.2) P t i ( v j,k ) = E Q (cid:2) v j,k ( S t , . . . , S t j ) (cid:12)(cid:12) F t i (cid:3) Q -a.s.and accordingly on the dual side the term expressing the dynamic position in the traded optionschanges to(A.3) n − X i =1 n X j = i +1 N X k =1 H i,j,k ( s , . . . , s i ) ( v j,k ( s , . . . , s j ) − p i,j,k ) . Similarly one can include dynamically traded basket options, i.e., (possibly path-dependent) optionsthat depend on a multitude of underlying securities.
References [1] Beatrice Acciaio, Mathias Beiglb¨ock, Friedrich Penkner, and Walter Schachermayer. A model-free version of thefundamental theorem of asset pricing and the super-replication theorem.
Mathematical Finance , 26(2):233–251,2016.[2] Anna Aksamit, Shuoqing Deng, Jan Ob l´oj, and Xiaolu Tan. The robust pricing–hedging duality for Americanoptions in discrete time financial markets.
Mathematical Finance , 29(3):861–897, 2019.[3] Aur´elien Alfonsi, Jacopo Corbetta, and Benjamin Jourdain. Sampling of one-dimensional probability measuresin the convex order and computation of robust option price bounds.
International Journal of Theoretical andApplied Finance , 22(03):1950002, 2019.[4] Jonathan Ansari, Eva L¨utkebohmert, Ariel Neufeld, and Julian Sester. Improved robust price bounds for multi-asset derivatives under market-implied dependence information.
Preprint , 2021.[5] Luca De Gennaro Aquino and Carole Bernard. Bounds on multi-asset derivatives via neural networks. arXivpreprint arXiv:1911.05523 , 2019.[6] David Baker. Martingales with specified marginals.
Theses, Universit´e Pierre et Marie Curie-Paris VI , 2012.[7] Daniel Bartl, Patrick Cheridito, and Michael Kupper. Robust expected utility maximization with medial limits.
Journal of Mathematical Analysis and Applications , 471(1-2):752–775, 2019.[8] Daniel Bartl, Michael Kupper, and Ariel Neufeld. Pathwise superhedging on prediction sets.
Finance and Stochas-tics , 24(1):215–248, 2020.[9] Daniel Bartl, Michael Kupper, David J Pr¨omel, and Ludovic Tangpi. Duality for pathwise superhedging incontinuous time.
Finance and Stochastics , 23(3):697–728, 2019.[10] Erhan Bayraktar and Zhou Zhou. On arbitrage and duality under model uncertainty and portfolio constraints.
Mathematical Finance , 27(4):988–1012, 2017.[11] Mathias Beiglb¨ock, Pierre Henry-Labord`ere, and Friedrich Penkner. Model-independent bounds for optionprices—a mass transport approach.
Finance and Stochastics , 17(3):477–501, 2013.[12] Mathias Beiglb¨ock, Marcel Nutz, and Nizar Touzi. Complete duality for martingale optimal transport on the line.
The Annals of Probability , 45(5):3038–3074, 2017.[13] Claude Berge.
Topological Spaces: including a treatment of multi-valued functions, vector spaces, and convexity .Courier Corporation, 1997.[14] Sara Biagini, Marco Frittelli, et al. On the super replication price of unbounded claims.
The Annals of AppliedProbability , 14(4):1970–1991, 2004.[15] Bruno Bouchard and Marcel Nutz. Arbitrage and duality in nondominated discrete-time models.
The Annals ofApplied Probability , 25(2):823–859, 2015.[16] Douglas T Breeden and Robert H Litzenberger. Prices of state-contingent claims implicit in option prices.
Journalof business , pages 621–651, 1978. [17] Matteo Burzoni, Marco Frittelli, and Marco Maggis. Model-free superhedging duality.
Ann. Appl. Probab. ,27(3):1452–1477, 2017.[18] Peter Carr and Dilip Madan. Towards a theory of volatility trading.
Option Pricing, Interest Rates and RiskManagement, Handbooks in Mathematical Finance , pages 458–476, 2001.[19] Patrick Cheridito, Matti Kiiski, David J Pr¨omel, and H Mete Soner. Martingale optimal transport duality.
Mathematische Annalen , pages 1–28, 2020.[20] Patrick Cheridito, Michael Kupper, and Ludovic Tangpi. Duality formulas for robust pricing and hedging indiscrete time.
SIAM Journal on Financial Mathematics , 8(1):738–765, 2017.[21] Jakˇsa Cvitani´c, Huyen Pham, and Nizar Touzi. A closed-form solution to the problem of super-replication undertransaction costs.
Finance and stochastics , 3(1):35–54, 1999.[22] George Bernard Dantzig.
Linear programming and extensions , volume 48. Princeton university press, 1998.[23] Yan Dolinsky and Ariel Neufeld. Super-replication in fully incomplete markets.
Mathematical Finance , 28(2):483–515, 2018.[24] Yan Dolinsky and H Mete Soner. Martingale optimal transport and robust hedging in continuous time.
ProbabilityTheory and Related Fields , 160(1-2):391–427, 2014.[25] Yan Dolinsky and H. Mete Soner. Martingale optimal transport and robust hedging in continuous time.
Probab.Theory Related Fields , 160(1-2):391–427, 2014.[26] Ernst Eberlein and Jean Jacod. On the range of options prices.
Finance and Stochastics , 1(2):131–140, 1997.[27] Stephan Eckstein and Michael Kupper. Computation of optimal transport and related hedging problems viapenalization and neural networks.
Applied Mathematics & Optimization , pages 1–29, 2019.[28] Stephan Eckstein and Michael Kupper. Martingale transport with homogeneous stock movements.
QuantitativeFinance , pages 1–10, 2020.[29] Hans F¨ollmer and Alexander Schied.
Stochastic finance: an introduction in discrete time . Walter de Gruyter,2016.[30] R¨udiger Frey and Carlos A Sin. Bounds on European option prices under stochastic volatility.
MathematicalFinance , 9(2):97–116, 1999.[31] Gaoyue Guo, Jan Ob l´oj, et al. Computational methods for martingale optimal transport problems.
The Annalsof Applied Probability , 29(6):3311–3347, 2019.[32] Pierre Henry-Labord`ere. Automated option pricing: Numerical methods.
International Journal of Theoreticaland Applied Finance , 16(08):1350042, 2013.[33] Pierre Henry-Labordere. (martingale) optimal transport and anomaly detection with neural networks: A primal-dual algorithm.
Available at SSRN 3370910 , 2019.[34] Zhaoxu Hou and Jan Ob l´oj. Robust pricing–hedging dualities in continuous time.
Finance and Stochastics ,22(3):511–567, 2018.[35] Hans G Kellerer. Markov-Komposition und eine Anwendung auf Martingale.
Mathematische Annalen , 198(3):99–122, 1972.[36] Shlomo Levental and Anatolii V Skorohod. On the possibility of hedging options in the presence of transactioncosts.
The Annals of Applied Probability , 7(2):410–443, 1997.[37] Eva L¨utkebohmert and Julian Sester. Tightening robust price bounds for exotic derivatives.
Quantitative Finance ,19(11):1797–1815, 2019.[38] Per Aslak Mykland. Financial options and statistical prediction intervals.
The Annals of Statistics , 31(5):1413–1438, 2003.[39] Ariel Neufeld. Buy-and-hold property for fully incomplete markets when super-replicating markovian claims.
International Journal of Theoretical and Applied Finance , 21(08):1850051, 2018.[40] Ariel Neufeld, Antonis Papapantoleon, and Qikun Xiang. Model-free bounds for multi-asset options using option-implied information and their exact computation. arXiv preprint arXiv:2006.14288 , 2020.[41] Julian Sester. Robust price bounds for derivative prices in markovian models.
International Journal of Theoreticaland Applied Finance , 23(3):2050015, 2020.[42] Volker Strassen. The existence of probability measures with given marginals.
The Annals of Mathematical Sta-tistics , 36(2):423–439, 1965., 36(2):423–439, 1965.