Arbitrage Bounds for Weighted Variance Swap Prices

Transcription

1 Arbitrage Bounds for Weighted Variance Swap Prices Mark Davis Imperial College London Jan Ob lój University of Oxford and Vimal Raval Imperial College London January 13, 21 Abstract Consider a frictionless market trading a finite number of co-maturing European call and put options written on a risky asset plus an instrument with path-dependent payoff known as a weighted variance swap, e.g. a vanilla variance swap or a corridor variance swap. he question we ask is: Do the traded prices admit an arbitrage opportunity? We determine necessary and sufficient model-free conditions for the price of a continuously monitored weighted variance swap to be consistent with absence of arbitrage. We discuss in detail the types of arbitrage that may arise when the determined conditions are not satisfied. In particular we find that prices of European call/puts are not enough for the upper bound price of the vanilla variance swap to be finite. We show that given an extra piece of information, namely the price of an additional asset, a finite bound can be explicitly determined. 1 Introduction Questions concerning when traded options are priced correctly have been asked many times in the field of financial mathematics; Black & Scholes [2], Merton [12], Hobson [1], Davis & Hobson [6], Cox and Ob lój [5] to name a few. We extend this line of study to include options on realised variance. We build on the work of Davis & Hobson [6], where absence of arbitrage conditions for a strip of co-maturing European calls are found, by adding a weighted variance swap to the set of traded derivatives. Weighted realised variance is measured in two ways, by discrete monitoring and continuous monitoring. In both cases a weight function w : [, ) [, ) is specified. hen one obloj@maths.ox.ac.uk; web: Research partially supported by the Oxford-Man Institute of Quantitative Finance and by a Marie Curie Intra- European Fellowship at Imperial College London within the 6 th European Community Framework Programme. vimal.raval@imperial.ac.uk; web: www2.imperial.ac.uk/ vr42/ Research supported by the Engineering & Physical Sciences Research Council. 1

2 specification of discretely monitored variance is 1 N w ( Sti 1 F ti 1 ) ( ) ln 2 Sti 1, F ti 1 where = t < t 1 <... < t N = are sampling dates, S t denotes the price of the underlying asset and F t is the t-forward price of the asset. In a model where the asset price is a continuous semimartingale, continuously monitored weighted variance is defined as 1 w(s t /F t )d ln S t, where ln S is the quadratic variation of the semimartingale (ln S t ) t [, ]. A -maturing w-weighted variance swap, is simply a forward on the corresponding realised variance with settlement at. he most liquid among these swaps is the plain/vanilla variance swap, which corresponds to the weight function w 1. he holder of the swap gains a variance exposure without at the same time incurring exposure to other risk factors. When w(x) = 1 (a,b) (x) or w(x) = x the variance accrued is contingent on the asset price level. Forwards with these weights are called corridor variance swaps and gamma swaps respectively. he corridor swap is cheaper than the realised variance swap and under the gamma swap variance accumulates proportionally to the asset price level. hese contracts belong to the larger class of volatility derivatives, a general discussion of which can be found in Gatheral [9, Chapter XI]. In this paper we determine necessary and sufficient conditions for the price of a continuously monitored weighted variance swap to be consistent with absence of arbitrage when finitely many European calls and puts are also traded on the underlying with the same maturity as the swap. Our main assumptions are threefold. Continuous price trajectories of the underlying asset and continuous monitoring of variance, are two. he third is the specification of continuously monitored variance in that the weight function is evaluated at the ratio of the asset and forward price, S t /F t. Whilst this is in line with the recent paper by Carr & Lee [11], it is not necessarily the industry standard. Further, in practice, traded swaps are written on discretely monitored variance. Recent work concerning the convergence rate of the price of a discretely monitored variance swap (w 1) to that of the continuously monitored, under the Heston model with and without jumps, can be found in Broadie & Jain [3]. In this paper however, we do not address the discrete case. Within the framework of continuous prices of the reference asset, our results are model independent. In assuming only finitely many European options are traded in the market our treatment of this problem differs to the existing literature in the field. It is well known, see for instance Derman et al [8], that if options are traded over all strikes in [, ) then the realised variance swap (case w 1) has a uniquely specified price consistent with no arbitrage. his argument hinges upon a simple application of Ito s formula to express realised variance, ln S, as the terminal value of a portfolio that dynamically trades the underlying and holds a European log-contract, the role of which as a volatility hedging instrument is discussed in Neuberger [13], paying ln S at, i.e. ln S = 2 ln(s /S ) + 2 Via local time arguments, weighted realised variance admits a similar decomposition into terminal gains from dynamic trading and the payoff from a European claim with convex payoff. Carr 2 ds t S t.

3 & Lee use this in [11], and for completeness we state the result in Section 2.1. Consequently, this forms the building blocks of our analysis as the problem of determining arbitrage bounds for the price of a weighted variance swap transforms into doing so for convex payoffs. he remainder of this paper is organised as follows: in Section 2 we detail the standing assumptions required throughout and recall essential results on continuously monitored weighted realised variance. Section 3 contains our contributions, namely arbitrage bounds for differentiable convex payoffs, Propositions 3.1 and 3.2, and subsequently, for the corresponding variance swap in heorem 3.4, which is our main result. We illustrate our findings in Section 4 with the vanilla variance swap, corridor variance swap and gamma swap as examples. In section 5 we find that prices of European call/puts are not enough for the upper bound price of the variance swap to be finite. We show that given an extra piece of information, namely the price of an additional asset, a finite bound can be explicitly determined. Finally, in section 6 we finish with some concluding remarks. 2 Problem Formulation We consider a market over the time horizon [, ], with >, trading a financial asset S. For t [, ], let S t denote the asset price at time t, so S is the spot price. We also assume that a certain number of derivative prices are quoted in the market at time. We make the following standing assumptions on the market, the underlying asset S and the options written on S: Assumption A1. he asset S at any time t [, ], and the quoted options at time only can be traded long or short in arbitrary amounts with no transaction costs. he interest rate for borrowing and lending is the same. here is no interest rate volatility so that discount factors D t for $1 paid at time t are known for all t [, ]. he asset S pays dividends at a deterministic yield; let Γ(t) denote the number of shares that would be owned by time t if dividend income is fully re-invested in shares 1. As functions of time, D t and Γ(t) are assumed to be of bounded variation 2. Suppose a finite number, n N, of European put options on S are traded at time, maturing at time for strikes < K 1 <... < K n <. Let p i denote the price of the put struck at K i. Further, a w-weighted variance swap maturing at is also traded on the asset S, with swap rate k w. P(1 t ) = D t, P(Γ(t)S t 1 t ) = S, P(K i S ) + = p i, i = 1,..., n, P(V w k w ) =, and P acts linearly on the combinations of the above, (1) 1 i.e. Γ(t) is an increasing function of t with Γ() = 1. If the dividend yield is constant, then Γ(t) = exp(qt). 2 his implies the forward price of S is a bounded variation function of time. 3

4 where Vt w = t w(s u/f u )d ln S u denotes the w-weighted realised variance 3 at time. Let X denote the set of traded options on S in which the options are identified by the payoffs, i.e. X = {(K i S ) + ; i = 1... n, V w k w } he purpose of this paper is to understand when P, which is our market input, on X is free of arbitrage in the sense that a model can be built in which pricing, i.e. the discounted risk neutral expectation, extends P. o formalise this question we need to define what is a portfolio and when it involves an arbitrage, and what a model is. We start with a crucial assumption to derive hedging strategies for weighted realised variance. Assumption A2. he stock price S t, on [, ], is a positive and continuous function of t. Note that since S we can assume that a put with strike K = is traded at price p = P( S ) + =. Denoting C([, ], [, )) the space of positive-valued continuous functions on [, ], one can regard elements f( ) of this set with f() = S as price trajectories of the asset S. Portfolios holding the underlying asset, the options and bonds are now defined. Since the notion of a model has not yet been introduced, in which a portfolio has sufficient structure to make sense of dynamic strategies, a portfolio is, for now, a model-free entity. Definition 2.1. A portfolio is a triple (π, φ, ψ), where: π = (π 1,..., π n+1 ) R n+1, in which π i denotes the number of units of the put option with strike K i that is held at time, for i = 1,..., n and π n+1 denotes the number of units of the weighted variance swap held at time zero. his forms the static component of the portfolio. For t [, ], φ t R denotes the number of units of the asset S held at time t and is determined by market observations up to time t. Finally, ψ t R denotes the number of units of the risk-free asset held at time t for t [, ] and is dependent on market observations up time t. he payoff of a portfolio (π, φ, ψ) at time is denoted by X (π,φ,ψ). A portfolio (π, φ, ψ) is termed simple if the position in the underlying asset, φ, is modified only at a finite number of times in [, ]. hese could be deterministic times or stopping times 4. A simple portfolio (π, φ, ψ) is called self-financing if no further funds are required after the initial investment to follow the portfolio s strategy. Remark 2.2. Observe that at time, the payoff from a portfolio (π, φ, ψ) is given by X (π,φ,ψ) = π i (K i S ) + + π n+1 (V w k w ) + φ S + ψ D 1. 3 Formally speaking a model is required to define this version of realised variance, we do this in due course. 4 Note that, as in Cox and Ob lój [5], we do not need to have a probability space to talk about stopping times. Indeed, stopping times here are simply mappings from C([, ], [, )) to [, ] that are appropriately measurable w.r.t. the filtration generated by the canonical process or equivalently by the one-dimensional projections. 4

5 he price of entering the portfolio at time is specified from the market data as: X (π,φ,ψ) = π i p i + φ S + ψ. If the portfolio is self-financing, then PX (π,φ,ψ) = X (π,φ,ψ). Definition 2.3. A model-independent arbitrage 5 is a simple self-financing portfolio (π, φ, ψ) with PX (π,φ,ψ) < and X (π,φ,ψ). Let us examine some simple necessary conditions for absence of model-independent arbitrage (MIA). Under the frictionless markets assumption and linearity of P, taking all traded European options to be puts only is justified by the conversion of calls to puts via the put-call parity. It follows that no MIA implies P(S K i ) + = S Γ( ) 1 D K i + p i, where recall Γ(t) is defined in A1. Further, standard arguments show that a traded forward on S maturing at t and entered at u [, t] has a unique price given by S u F t /F u, where F t = S Γ(t) 1 Dt 1, otherwise there is a model-independent arbitrage. Hence, in all generality, we have ( ) F τ P S τ S ρ =, (2) F ρ where ρ < τ are two stopping times. When forwards on S are correctly priced, another example of a model-independent arbitrage is the butterfly spread which is realised when the linear interpolation of the given put option prices fails to be convex. Definition 2.4. A model, M, for the asset price is a filtered probability space (Ω, F, (F t ) t [, ], Q) with a positive and continuous (F t, Q)-semimartingale (S t ) t [, ] with S almost surely equal to the spot price. he filtration satisfies the usual hypotheses and F is trivial. A model is called a (P, X)-market model if (S t /F t ) is an (F t, Q)-martingale and PX = D E Q [X] for all market quoted options X X. When a model is fixed, E = E Q = E M denotes the expectation under Q. We will denote the set of models, as defined above, by M. Note that the condition that S t /F t is a martingale is equivalent to saying that forwards are priced correctly, i.e. that = P[ ] = E Q [ ] for all forward transactions in (2), see Revuz & Yor [15, Chapter 2, Proposition 3.5]. We would like to have statements of the form: the market does not admit an arbitrage if and only if the market prices satisfy certain relations, if and only if a market model exists. Given that a model is required to define realised variance as above, the type of arbitrage the statements concern will not be model-independent. However, as shown by Davis and Hobson [6] or Cox and Ob lój [5], the notion of model-independent arbitrage may not be sufficient to determine such relations even in the market with no variance option and weaker notions of arbitrage are necessary. o define this notion we first need to establish admissibility criteria of a portfolio. 5 also called model-free arbitrage 5

6 Definition 2.5. Given a model M M, a portfolio (π, φ, ψ) is M-admissible if: are predictable processes satisfying φ, ψ : [, ] Ω R φ 2 t d S t /F t t < a.s, φ t S t Γ 1 t dγ t < a.s, and there exists 6 Y L 1 (M) and constant c R such that ψ t d(dt 1 ) < a.s. t (φ u c)γ 1 u d(s u /F u ) Y for all t a.s. Moreover, (π, φ, ψ) is an M-admissible self-financing strategy if almost surely for all t [, ] φ t S t + ψ t D 1 t φ S ψ = or equivalently : D t (φ t S t + ψ t D 1 t t φ u ds u + t ) = φ S + ψ + S t φ u S u Γ 1 u dγ u + t Γ 1 u φ u d(s u /F u ). ψ u d(d 1 u ), Remark 2.6. he role of the constant c appearing in the admissibility conditions is to ensure that the strategy φ t = cγ t, or such a component of a strategy, in which c units of the stock is held with dividends re-invested in to the stock is admissible. his becomes relevant immediately in Lemma We now have equipped ourselves with enough tools to define our analogue of weak-arbitrage. Definition 2.7. We say that the market prices (P, X) admit a weak-arbitrage (WA) if in any model M M, there exists an M-admissible self-financing strategy (π, φ, ψ) satisfying: X (π,φ,ψ) almost surely, Q[X (π,φ,ψ) > ] > and PX (π,φ,ψ). We say that the WA is simple if the strategy can always be chosen simple. he notion of WA was defined in Davis & Hobson [6]. More precisely, the authors there looked only at simple WA and defined it to be mutually exclusive with model-independent arbitrage. Here, following Cox and Ob lój [5], we find it more convenient to define WA as a weaker notion of arbitrage, i.e. MIA is in particular a WA. As we will see WA is the appropriate concept for our setup. If there were infinitely many European options traded then weak free lunch with vanishing risk would likely prove to be useful (see Cox and Ob lój [5]). For completeness we state Lemma 2.8. Suppose P is finite on X. If there exists a (P, X) market model then (P, X) do not admit a weak arbitrage. In particular the prices do not admit a model-independent arbitrage. Proof. It suffices to observe from Fatou s lemma that in the market model, using (3), for any admissible self-financing strategy (π, φ, ψ) the discounted process (D t φ t S t + ψ t ) t [, ] is a supermartingale. Hence a strategy with non-negative payoff has a non-negative price, because it holds that E M [D X (π,φ,ψ) ] X (π,φ,ψ) = PX (π,φ,ψ), where the last equality follows because the portfolio is self-financing. hus in this model there is no weak-arbitrage portfolio. 6 L 1 (M) denotes the equivalence class of integrable random variables on the probability space of the model M. (3) (4) 6

7 Here onwards we re-scale the units for the stock and option prices via S t = S t F t, r i = p i D F, k i = K i F, and let 7 ˉn := inf{i : r i = k i 1} n. (5) so in particular S = 1. When the weighted variance swap is not traded, the question of whether an arbitrage exists in the market was answered in [6]. Recall that X is the set of traded options on S and let X E denote the set of traded vanilla European options and cash, i.e. X E = {(K i S ) + : i =,..., n}, { } X = X E w(s t /F t )d ln S t k w. (6) heorem 2.9 (Davis and Hobson [6]). Under A1 and A2 the following statements are equivalent: 1. he market prices (P, X E ) do not admit a weak-arbitrage he forwards are correctly priced, i.e. (2) holds. he option prices satisfy: r =, r i (k i 1) + i, and the piecewise linear interpolation over [, kˉn ] of the points (k, r ),..., (kˉn, rˉn ) is increasing, convex and with slope strictly bounded by here exists a (P, X E ) market model. Proof. his follows from heorem 3.1 in [6] and Lemma 2.8. Henceforth, we define the set of market models for European puts by M E := {M M : M is a (P, X E ) market model}. (7) Remark 2.1. Note that for M M E, S Kˉn Q-a.s. if rˉn = kˉn 1, where ˉn is given in (5). Secondly, a market model has to match only the first ˉn n put prices to match all n put prices. In the remainder of the paper we study swap rates k w in (1) under which there exists a (P, X) market model. We compute the bounds LB := [ ] inf E M w( S t )d ln S t M M E and UB := [ ] sup E M w( S t )d ln S t, (8) M M E and show that there exists a (P, X) market model if k w (LB, UB) and there is a weak arbitrage if k w / [LB, UB]. When k w {LB, UB} then either there is a weak arbitrage or there exists a market model and we provide criteria to decide which is the case. 2.1 Weighted realised variance We specify now how the path-dependence in the payoff from a variance swap can be hedged by dynamic trading in the underlying asset, the residual randomness being hedged by holding a particular European contingent claim with convex payoff. We start by imposing some minimal assumptions on the weighting function. 7 with the convention inf =. 8 Here any portfolio is required to have π n+1 = since the weighted variance swap is not traded. 7

8 Assumption A3. he weighting function w : (, ) [, ) is a measurable map satisfying w(a) a 2 da < for all compact A (, ). A he following lemma is a straightforward application of the Meyer-Itô formula, see e.g. Rogers & Williams [16, Section IV.45, heorem 45.1]. We remind the reader here that F t is of bounded variation. Lemma 2.11 (Hedging Weighted Variance). For w satisfying Assumption A3 and λ w a convex function defined by λ w(a) = w(a) we have, in any model M M and stopping time τ, a 2 τ w( S u )d ln S u = 2λ w ( S t ) 2λ w (1) 2 τ λ w( S u )d S u a.s. (9) he right hand side of (9) is zero for affine functions and consequently, given a weight w, any normalisation can be chosen. For three common specifications of the weight w, all of which satisfy Assumption A3, we specify functions λ w explicitly as: 1. Realised variance swap - w 1: λ w (x) = ln(x). 2. Corridor variance swap - w(x) = 1 (,a) (x) or w(x) = 1 (a, ) (x), where < a < : ( ( x ) λ w (x) = ln + x ) a a 1 w(x). 3. Gamma swap - w(x) = x: λ w (x) = x ln(x) x. Lemma 2.11 shows that half the realised variance, with weight w, at time is replicated by the following strategy: holding a European claim paying λ w ( S ), dynamic trading in the underlying with the strategy, which using (4), has Γ t λ w( S t ) units of the stock at time t, and λ w (1) cash at time. For no arbitrage then, the price k w of a weighted variance swap is the forward price of the above portfolio. he task of determining no-arbitrage bounds for the price of the w-variance swap is basically reduced to doing so for the price of a European option with payoff λ w. his is made precise in Lemma 2.13, but first we show that the dynamic strategy in the underlying asset, (Γ t λ w( S t )) t [, ], is admissible in any market model where the weighted variance swap has finite price. Lemma Let M M. For w satisfying A3, if [ ] E w( S t )d ln S t <, (1) then the strategy (Γ t λ w( S t )) t [, ] is admissible in this model. 8

9 Proof. he main point to prove is that there exists an integrable random variable Y and constant c R such that almost surely for each t [, ] it holds t (λ w( S u ) c)d S u Y. his however falls out immediately on noting that since t w( S u )d ln S u increases in t, the relation (9) and convexity of λ w imply for any t t t λ w( S u )d S u λ w ( S t ) λ w (1) λ w(1)( S t 1) (λ w( S u ) λ w(1))d S u w( S u )d ln S u. w( S u )d ln S u w( S u )d ln S u hus the constant is c = λ w(1) and the random variable bounding the strategy from below is the realised variance at, which is integrable by assumption. Lemma 2.13 ( Price Relation). Let ( S t ) be a continuous martingale on a filtered probability space (Ω, F, (F t ), P) satisfying the usual assumptions, and let σ be a bounded stopping time. For w satisfying A3 it holds [ σ ] E w( S t )d ln S t = 2E[λ w ( S σ )] 2λ w (1) (11) and in particular [ σ ] E w( S t )d ln S t < E[ λ w ( S σ ) ] <. Proof. Let τ n be a localising sequence for t λ w( S u )d S u and such that S u n, u τ n. Stopping (9) at σ τ n and taking expectations we see that E σ τn w( S u )d ln S u = 2Eλ w ( S σ τn ) 2λ w (1). (12) As λ w is convex, its negative part λ w(s) is bounded above by an affine function and hence Eλ w( S t ) < for all t σ τ n. he positive part λ + w(s) is also a convex function and Jensen s inequality together with Optional Sampling and Fatou s lemma yield lim sup Eλ + w( S σ τn ) Eλ + w( S σ ) lim inf Eλ + n n w( S σ τn ). In consequence Eλ + w( S σ ) = lim Eλ + w( S σ τn ) and the lemma follows by taking limits as n in (12). Remark Using Revuz & Yor [15, Chapter 2, Proposition 3.5], a corollary to Lemma 2.13 is that when S ( t t is as in Lemma 2.13, the local-martingale λ w( S u )d S u is a (F t, Q)- )t [, ] martingale, since upon re-arranging (9) one finds for any stopping time τ, [ τ ] E λ w( S u )d S u =. his however does not play a role in our developments. 9

10 3 General Results Hedging of w-weighted realised variance is achieved by dynamically trading the underlying asset and holding a claim paying λ w ( S ) at time. he key observation being that the claim depends only on S, the asset price at time, which is when the traded European options also expire. Lemma 2.13 yields the immediate relations [ ] inf E M w( S t )d ln S t M M E [ ] sup E M w( S t )d ln S t M M E [ = 2 inf E M λ w ( S ] ) 2λ w (1) M M E and = 2 sup M M E E M [ λ w ( S ] ) 2λ w (1). Consequently, the problem of determining no-arbitrage bounds for the price of the weighted variance swap reduces to proving the bounds for the corresponding European claim. 3.1 Arbitrage bounds for convex payoffs In this section we work under assumption A1 and in addition suppose a European claim is written on S paying λ( S ) at maturity, where the payoff function λ : (, ) R is convex. he forward price of this claim, under any market model, is determined by the law of S. Recalling ( S t ) is a martingale with S = 1 and that the normalised put price for moneyness-strike k i is r i, the set of feasible laws of S is given by { } M E = μ : μ([, )) = 1, xμ(dx) = 1, (k i x) + μ(dx) = r i for i = 1,..., n { = L M ( S ) : M M E} (13), where L M ( S ) is the law of S under a model M. he equivalence of definitions follows by arguments based on Skorokhod embeddings (cf. Ob lój [14]), as in Davis and Hobson [6] or Cox, Hobson and Ob lój [4]. More precisely, given a marginal μ in the set in the first line of (13) we can find a stopping time τ, for a Brownian motion (B t ) defined on some probability space, B = S, such that B τ μ and B t τ is an uniformly integrable martingale. hen S t := B t t τ defines a (P, X E ) market model in which S μ. Hinging on the convexity of the payoff λ, the upper bound price for the option is found in Proposition 3.2, but first we prove below in Proposition 3.1 that the lower bound price is determined by the solution of a dynamic programming recursion. Proposition 3.1 (Lower bound). Under A1 and A2, assume that prices of European options (P, X E ) do not admit weak arbitrage. hen for a differentiable convex function λ : (, ) R, > λ(x)μ(dx) = λ(x)μ (dx) λ(1), inf μ M E in which recall M E, defined in (13), denotes the set of calibrated marginals for S and μ is an atomic probability measure with at most n + 1 atoms described in the proof. 1

11 Moreover, there exists a static portfolio 9 (π, φ, ψ ) in the n European options, the underlying asset S and cash respectively, with π n+1 =, such that X (π,φ,ψ ) = π i (K i S ) + + φ S + ψ λ(s /F ) and the price of which satisfies ( P X (π,φ,ψ ) ) = π i p i + D φ F + D ψ = D λ(x)μ (dx). (14) In particular, if Pλ(S /F ) < D λ(x)μ (dx) then market prices (P, X E {λ(s /F )}) admit a model independent arbitrage. Proof. Note firstly that the infimum is finite because the set M E contains marginals with compact support on which λ is bounded. Further, a convex function dominates any tangential line. Choosing a tangent to λ(x) at x = 1, the mean of any calibrated marginal μ M E, it follows that for a μ with λ(x) μ(dx) <, λ(x)μ(dx) λ(1) and so inf M M E λ(x)μ(dx) λ(1). We now prove that the minimising probability measure μ is obtained by a dynamic programming algorithm. Formally, the optimisation is to determine μ([, )) = 1 (Probability measure) inf λ(x)μ(dx), subject to xμ(dx) = 1 (Forward price) (15) μ (ki x) + μ(dx) = r i, for i = 1,..., ˉn. (Put prices) Recall ˉn = inf{i : r i = k i 1} n. If inf{i : r i = k i 1} <, then in any marginal μ M E it holds μ([, kˉn ]) = 1, which is why only the first ˉn puts appear in the constraints above. Now, we partition [, ) into intervals I 1,..., Iˉn+1 defined by I i = [k i 1, k i ) for i = 1,... ˉn and Iˉn+1 = [kˉn, ). Let μ M E be such that λ(x) μ(dx) < and define I μ = {i ˉn + 1 μ(i i ) > }. (16) From conditional Jensen s inequality observe that λ(x)μ(dx) λ(x)μ (dx), (17) R + R + where measure μ is defined by μ = i I μ μ(i i )δ xi, 9 Note that a static portfolio is universally well-defined since it corresponds to a buy-and-hold strategy. 11

12 in which δ x denotes the Dirac measure at x, and for an index i I, the point x i = hen by direct computation, μ is a probability measure that satisfies xμ (dx) = 1 and (k i x) + μ (dx) = r i R + R + I i xdμ(x) μ(i i ). for i =,..., ˉn and so is of the form (15). his implies that in (15), it suffices to search over atomic marginals with at most one atom per interval I i. hese measures are specified by the cumulative mass up to each moneyness-strike k i. Indeed, let us denote a vector of such masses by ζ = (ζ 1,..., ζˉn ). For the resulting measure to be an element of the set of feasible marginals, M E, consistency of the put prices r 1,..., r n with absence of arbitrage - heorem dictates that ζ i A i := [ ri r i 1, r ] i+1 r i k i k i 1 k i+1 k i for i < ˉn and ζˉn Aˉn := [ ] rˉn rˉn 1, 1. (18) kˉn kˉn 1 he hypotheses that each interval I i has at most one point of support gives rise to a system of ˉn + 1 linear equations solved by the atoms, denoted χ i I i, to match the given put prices and forward price, i.e. letting ζ =, j (ζ i ζ i 1 )(k j χ i ) = r j, j = 1,..., ˉn ˉn+1 (ζ i ζ i 1 )χ i = 1. Note that the first ˉn atoms are determined by the European option prices, and the forward (or martingale) condition only plays a role in the last atom. In particular, if inf{i : r i = k i 1} < so that ζˉn = 1, the martingale condition is automatically satisfied and so there is no (ˉn + 1) th atom. Solving, one finds when ζ i 1 < ζ i χ i = χ i (ζ i 1, ζ i ) = k i + ζ i 1(k i k i 1 ) (r i r i 1 ) ζ i ζ i 1 χˉn+1 = χˉn+1 (ζˉn ) = kˉn rˉn kˉn 1 ζˉn. he measure corresponding to policy ζ is thus ˉ for i = 1,..., ˉn (ζ i ζ i 1 )δ χi (ζ i 1,ζ i ) + (1 ζ n )δ χn+1 (ζ n). (19) he minimisation problem (15) can therefore be written as inf... inf ζ 1 A 1 ζˉn Aˉn { ˉ For notational convenience in what follows, define G i (ζ i 1, ζ i, ζ i+1,..., ζˉn ) = (ζ i ζ i 1 )λ(χ i (ζ i 1, ζ i )) + (1 ζˉn )λ(χˉn+1 (ζˉn )) g i (ζ, ζ ) = (ζ ζ)λ(χ i (ζ, ζ )) for i ˉn }. (2) gˉn+1 (ζ) = (1 ζ)λ(χˉn+1 (ζ)) (21) ˉ j=i g j (ζ j 1, ζ j ) + gˉn+1 (ζˉn ) for i ˉn. 12

13 hen (2) is inf... inf {G 1 (, ζ 1,..., ζ n )} ζ 1 A 1 ζ n A n { } = inf g 1 (, ζ 1 ) + inf... inf {G 2 (ζ 1, ζ 2,..., ζ n )}, (22) ζ 1 A 1 ζ 2 A 2 ζ n A n which shows that the Bellman equation is satisfied and the infimum is obtained at the ˉn th step of a dynamic program. Since λ is convex, the g-functions continuous and intervals A i compact, the infimum is attained 1 by a policy ˉζ = (ˉζ, ˉζ 1,..., ˉζˉn ) {} A 1... Aˉn. he minimising probability measure is therefore given by μ = ˉζ 1 δ χ1 (,ˉζ 1 ) + ˉ i=2 (ˉζ i ˉζ i 1 )δ χi (ˉζ i 1,ˉζ i ) + (1 ˉζˉn )δ χˉn+1 (ˉζˉn). It now remains to prove there exists a static sub-hedging portfolio (μ, φ, ψ ) in the European options, the underlying and cash with forward cost equal to λ(x)μ (dx). Of significance are the atoms of μ. By construction, the set of indices corresponding to these are given by I μ = {i {1,..., ˉn + 1} μ (I i ) > }, where recall I i = [k i 1, k i ) in which we define k n+1 = +. For notational convenience, the atoms of μ are abbreviated as follows ˉχ i := χ i (ˉζ i 1, ˉζ i ) for i ˉn and ˉχˉn+1 := χˉn+1 (ˉζˉn ). We construct a vector (π, φ, ψ) R n+2 such that for any x >, { Π(x) := π i (k i x) + = λ(x) if x {ˉχ i : i I + φx + ψ μ } < λ(x) otherwise, (23) where φ = if ˉζˉn = 1, and if ˉn < n then πˉn+1 =... = π n =. he portfolio (μ, φ, ψ ) is then obtained by μ i = π i/f for i = 1,..., n, φ = φ/f and ψ = ψ. Indeed, if (23) holds, then λ(s /F ) dominates the portfolio payoff Π(S /F ) (since k i = K i /F ), and in particular, μ ({x : λ(x) = Π(x)}) = 1, which implies the forward setup cost satisfies μ i r if + φ F + ψ = π i r i + φ + ψ = Π(x)μ (dx) = λ(x)μ (dx). he requirement is thus to determine the vector (π, φ, ψ) so that the corresponding function Π satisfies (23) and is continuous. o this end, recall firstly that ˉζ minimises G 1 (ζ 1,..., ζˉn ) over A 1... Aˉn. Using the same recipe to prove (17), one can show that G 1 is convex on the interior of this domain. ( Now, if for some i, ˉζ i 1 < ˉζ i < ˉζ i+1, then ˉζ i is contained in the interior of A i, i.e. ˉζi ri r i 1 k i k i 1, r i+1 r i k i+1 k i ), otherwise both inequalities cannot hold. Convexity of G 1 then implies ζ i G 1 (ˉζ) =. 1 his is justified by repeated application of Proposition 7.32 of Bertsekas & Shreve [1]. 13

14 Computing the partial derivative and evaluating, one obtains φˉχi (k i ) = φˉχi+1 (k i ), where φ y (x) = λ(y) + λ (y)(x y) is the tangent to λ at y evaluated at x. hat is, supporting hyperplanes to λ at atoms in consecutive intervals, (k i 1, k i ) and (k i, k i+1 ), meet at the separating strike k i. One can therefore hold a combination of puts, the underlying and cash so that the normalised payoff, Π(x), is given by: if i I μ, then Π(x) = φˉχi (x) for x [k i 1, k i ] and linearly interpolated over remaining intervals. hen, given the above, Π(x) is continuous and satisfies (23). We now turn attention to the least upper bound for the price of a contingent claim paying λ(s /F ) at time. Proposition 3.2 (Upper Bound). Under A1 and A2 assume the European prices (P, X E ) do not admit a weak arbitrage. For a convex function λ : (, ) R, if: λ(x) = O(1) as x, and λ(x) = O(x) as x, (24) then the static portfolio (π, φ, ψ ) in the n European options, the underlying asset S and cash respectively, with πn+1 =, defined by φ F = λ(y) λ(kˉn ) lim y y kˉn ψ = λ(kˉn ) φ F kˉn π ˉnF = φ F λ(kˉn) λ(kˉn 1 ) kˉn kˉn 1 if inf{i : r i = k i 1} = or otherwise π i F = λ(k i+1) λ(k i ) k i+1 k i λ(k i) λ(k i 1 ) k i k i 1 for i = 1,..., ˉn 1. where if ˉn < n, then π ˉn+1 =... = π n =, has payoff πi (K i S ) + + φ S + ψ λ(s /F ) and the price which satisfies ( P X (π,φ,ψ ) ) = πi p i + D φ F + D ψ = D sup μ M E λ(x)μ(dx). (25) In particular, if Pλ(S /F ) > PX (π,φ,ψ ) model independent arbitrage. If (24) does not hold, then sup μ M E then market prices (P, X E {λ(s /F )}) admit a λ(x)μ(dx) = +. (26) Proof. Assume (24) holds so that λ(x) is bounded near and is at most linear near. herefore, the limit λ(y) λ(kˉn ) lim y y kˉn 14

15 exists. he payoff from the portfolio (on the normalised scale) is simply the linear interpolation of points (k i, λ(k i )) for i =,..., ˉn on [, kˉn ] and then linear with slope γ on [kˉn, ). Consequently, for any measure μ M E, the convexity of λ implies this payoff dominates λ(s /F ) μ-a.e. o proceed, define β =, β i = r i r i 1 k i k i 1 for i = 1,..., ˉn and βˉn+1 = 1. Letting π i = π i F for i = 1,..., n, φ = φ F and ψ = ψ and substituting from definition of π in the heorem, observe the forward cost of the portfolio can be written as πi F r i + φ F + ψ = = π i r i + φ + ψ ˉ i= (β i+1 β i )λ(k i ) + (1 + rˉn kˉn )φ. (27) Since any model M M E does not admit model independent arbitrage, we see that (27) is an upper bound for λ(x)μ(dx) for μ M E - we now show that in fact the bound is the least upper bound. Indeed, if rˉn = kˉn 1, then as defined in the Proposition, φ = φ = and moreover μ defined by μ ({k i }) = β i+1 β i for i =,... ˉn satisfies μ M E. Substituting φ = in to (27) one sees that πi F r i + φ F + ψ = λ(x)μ (dx) and so the upper-bound identified is attained by market models with marginal μ for S. On the other hand if rˉn > kˉn 1, or equivalently r n > k n 1, we prove the upper bound is a limit point of the set { λ(x)μ(dx) μ M E }. For y > kˉn define measure μ y by μ y ({, k 1, k 2,..., kˉn, y) = 1. μ y ({k i }) = β i+1 β i for i =,... ˉn 1. μ y ({y}) = 1+rˉn kˉn y kˉn. hen by direct computation one can check μ y M E λ(x)μ y (dx) = ˉ i= (β i+1 β i )λ(k i ) + (1 + rˉn kˉn ) for y sufficiently large. Moreover, λ(y) λ(kˉn) y kˉn, which obviously converges to the forward cost of the portfolio as y. If (24) does not hold then a similar construction yields a sequence of measures μ n M E that for each n, λ(x) μ n (dx) < however λ(x)μ n (dx) as n. such Propositions 3.1 and 3.2 specified bounds for the price of European option with payoff λ but did not make precise what types of arbitrage are possible, if any, at the bounds. his is now complemented. Observe that Lemma 2.8 implies that if there exists a (P, X E {λ(s /F )}) market model then (P, X E {λ(s /F )}) do not admit a weak arbitrage. 15

16 heorem 3.3. Let (P, X E ) be the market input (1), (6). Under A1 and A2 assume the European prices (P, X E ) do not admit a weak arbitrage. For a differentiable convex function λ : (, ) R let LB λ := D inf μ M E λ(x)μ(dx) and UB λ := D sup μ M E λ(x)μ(dx) (28) and recall that these are given in (14), Proposition 3.1 and in (25)-(26), Proposition 3.2 respectively. (i) If Pλ(S /F ) (LB λ, UB λ ) then there exists a (P, X E {λ(s /F )}) market model. (ii) If Pλ(S /F ) = LB λ then there exists a (P, X E {λ(s /F )}) market model if and only if r n = k n 1 or if r n > k n 1 but μ ([, k n ]) < 1, where μ is the minimising probability measure from Proposition 3.1 such that LB λ = D λ(x)μ (dx). Otherwise (P, X E {λ(s /F )}) admit a weak arbitrage. (iii) If Pλ(S /F ) = UB λ < then there exists a (P, X E {λ(s /F )}) market model if and only if r n = k n 1. Otherwise (P, X E {λ(s /F )}) admit a weak arbitrage. (iv) If Pλ(S /F ) / [LB λ, UB λ ] then (P, X E {λ(s /F )}) admit a model-independent arbitrage. Proof. (i) By definition of LB λ and UB λ there exist sequences (μ n) n N, (μ n) n N M E that λ(x)μ n(dx) LB λ and λ(x)μ n(dx) UB λ. such For some n it therefore holds that λ(x)μ n(dx) < Pλ(S /F ) < λ(x)μ n(dx). Noting that M E is a convex set and recalling (13), a (P, X E {λ(s /F )}) market model exists. (ii) Recall from Proposition 3.1 the measure μ is such that LB λ = D λ(x)μ (dx). Moreover, r n = k n 1 or r n > k n 1 and μ ([, k n ]) < 1 are equivalent to μ M E, which is equivalent to existence of a (P, X E {λ(s /F )}) market model by (13). When the given conditions do not hold, i.e. μ ([, k n ]) = 1 and r n > k n 1, we now prove there is weak arbitrage. We do this by showing there exists a weak arbitrage portfolio in a model with marginal law μ for S when μ([, k n ]) < 1 or μ([, k n ]) = 1, which thereby shows the market admits weak arbitrage. If μ([, k n ]) < 1 then being long the convex claim paying λ(s /F ) and short the portfolio (π, φ, ψ ), defined in the proof of Proposition 3.1, satisfies the criteria for a weak-arbitrage portfolio amongst these models. On the other hand if μ([, k n ]) = 1, then in the model it almost surely holds = (S K n ) + = K n S (K n S ) +. However, r n > k n 1 is equivalent to P n > D K n S Γ 1 and so the price of the portfolio satisfies P( (S K n ) + ) = P() = D K n S Γ 1 P n <. 16

17 hus the portfolio with payoff K n S (K n S ) + + c, where c (, P n D K n + S Γ 1 ), forms a weak arbitrage portfolio in this class of models. (iii) he first part of this statement is proven in Proposition 3.2. he weak arbitrage when UB λ < and r n = k n 1 is given by holding the portfolio (π, φ, ψ ) defined in Proposition 3.2 and being short the convex payoff λ(s /F ) for models in which S > k n is not a null set. For models in which S k n a.s. the weak arbitrage portfolio is the same as that constructed in this case in part (ii). (iv) See Propositions 3.1 and Arbitrage bounds for Weighted Variance Swaps We now return to the original problem: determining if the market trading the underlying asset, the put options and the variance swap admits arbitrage. he w-weighted variance swap maturing at has payoff w( S t )d ln(s) t k w, where k w is fixed so that (1) holds, i.e. there is no time zero cost of entering the contract. he answer is immediate given the representation (9) and heorem 3.3. We assume European put prices (P, X E ) admit no weak arbitrage, which by heorem 2.9, is equivalent to existence of a market model. heorem 3.4. Let (P, X) be the market input (1), (6). Under Assumptions A1 A3, suppose (P, X E ) do not admit a weak arbitrage. Let λ w denote the convex function given by Lemma hen the following are equivalent: 1. here exists a (P, X) market model. 2. he market prices (P, X) do not admit a weak-arbitrage. 3. he market prices (P, X E {λ w (S /F )}), with Pλ w (S /F ) := D ( λw (1) + k w 2 ), do not admit a weak-arbitrage. In particular if k w ( 2D 1 LB λ w 2λ w (1), 2D 1 UB λ w 2λ w (1) ) then there exists a (P, X) market model and if k w / [ 2D 1 LB λ w 2λ w (1)2D 1 UB λ w 2λ w (1) ] then market prices admit weak arbitrage, where LB λw, UB λw in (28) are given in (14) and in (25)-(26). Proof. o see that 2 implies 3, note that if 2 holds and 3 does not then a contradiction is obtained. his is because if 3 does not hold then in any model M M there exists a M-admissible selffinancing strategy, holding the convex payoff, that [ is a weak-arbitrage portfolio. In particular this must be true for any model M in which E M w( S ] t )d ln S t <. From the representation (9) and Lemma 2.12, however, it holds that the option with convex payoff λ w ( S ) is replicated by a M-admissible self-financing strategy involving the weighted variance swap. hus there exists a weak-arbitrage portfolio involving the weighted variance swap. his contradicts the assumption of statement 2. Statement 3 implies that there exists a (P, X E {λ w (S [/F )}) market model M, with Pλ w (S /F ) := λ w (1) + kw 2. From Lemma 2.13 we have E M w( S )d ln S ] t = 17

18 2E M [λ w ( S )] 2λ w (1) = k w and hence M is also a (P, X) market model, which gives 1. We have 1 = 2 by Lemma Example o illustrate an application of our results, consider a fictional market with an index S that trades 3 European put options maturing in 1 year. he data are S = 1, F = 15, D = exp(.3), K i = 5, 1 and 15, p 1 = 1.127, p 2 = 18.6 and p 3 = Using heorem 3.4, the range of (weak) arbitrage-free prices for a vanilla variance swap, corridor variance swap and gamma swap have been determined and summarised in able 1. Of course, from the hedging relation (9) and price relation, Lemma 2.13, in finding the bounds for a w-weighted variance swap one considers an option with payoff λ w (S /F ). Recall the portfolios identified in Propositions 3.1 and 3.2 respectively sub and super-replicate λ w (S /F ) whilst admitting the lower and upper bound price. he payoffs for each weight from the corresponding option and hedging portfolios are illustrated in Figures 1, 2 and 3. Figure 1 illustrates the log contract payoff ln(s /F ) (blue line) and the consequent subhedging portfolio (black line). he portfolio is given by π1 =.176, π 2 =.472, π 3 =.259, φ =.536 and ψ = Figure 2 illustrates the payoff [ ln( S 75 ) + S 75 1]1 [ ) 75 (S, /F ) (blue line), the consequent sub-hedging portfolio (black line) and superhedging portfolio (red line). he portfolios are given by π 1 =, π 2 =.66, π 3 =.66, F φ = and ψ =.3293 for the sub-hedge, and for the super-hedge: π1 =.91, π 2 =.431, π3 =.811, φ =.1333 and ψ = Figure 3 illustrates the payoff S F ln (blue line) and the consequent sub-hedging portfolio (black line). ( ) S F he portfolio is given by π 1 =.772, π 2 =.571, π 3 =.225, φ = and ψ = Using these, and the fact that p 3 = > = D (K 3 F ), one can deduce why the intervals stated in able 1 are as such. Indeed, the analysis of section 5 will show that the interval for the vanilla variance swap is half closed and unbounded. For the corridor variance swap, the sub-hedging strategy does not hold the underlying index and so indicating the lower bound is not attained. Note that for the choice of corridor in the weight of the corridor variance swap, the swap price has a finite upper bound. In particular the upper bound is not attained due to the inequality p 3 > D (K 3 F ). he Gamma swap has no finite upper bound and the lower bound is not an admissible price, which again is indicated by the corresponding sub-hedge for the convex payoff not holding the index. S F able 1: Arbitrage bounds for the price, k w, of a w-weighted variance swap maturing in 1-year when the market trades 3 European put options on an index with the market data as specified in the example. Variance Swap w(x) λ w (x) Arbitrage bounds for k w ype ($) Vanilla Swap 1 ln(x) [.224, ) Corridor Variance Swap 1 [ 75 (x), F [ ln( xf 75 ) + F x 75 1]w(x) (.38,.34) Gamma Swap x x ln(x) x (.125, ) 18

19 Figure 1: Illustration of the log contract payoff ln(s /F ) (blue line) and the consequent sub-hedging portfolio (black line) S 5 he Variance Swap (w 1) Here we demonstrate that for the vanilla variance swap, given by weight w 1 and corresponding payoff ln S k vs, the interval of arbitrage-free swap rates found by heorem 3.4 is never bounded above and half closed, irrespective of the solution to the dynamic program μ. Moreover in section 5.1, we demonstrate that the upper bound can be made finite if an additional option is traded in the market. he model independent relation (9) reads ln S = d S t S t ln S. By applying heorem 3.4 together with heorem 3.3, the claim to prove is that the interval of weak arbitrage-free variance swap prices is [ ) 2 π i F r i + φ F + ψ,. (29) Since ln(x) is not bounded near x = there is no upper bound. On the other hand the lower bound is an admissible price. When r n = k n 1 this is always the case and otherwise, by (ii) in 19

20 Figure 2: Illustration of the payoff [ ln( S 75 )+ S 75 1]1 [ ) 75 (S, /F ) (blue line), the consequent F sub-hedging portfolio (black line) and super-hedging portfolio (red line) S heorem 3.3, we have to show that μ ([, k n ]) < 1. Recall that, with ˉn = n, μ = ˉζ 1 δ χ1 (,ˉζ 1 ) + (ˉζ i ˉζ i 1 )δ χi (ˉζ i 1,ˉζ i ) + (1 ˉζ n )δ χn+1 (ˉζ n) i=2 where the atoms are given by (19) and ˉζ = (ˉζ 1,..., n) attains the minimum of, with ζ =, in which G(ζ 1,..., ζ n ) := ζ i (ζ i ζ i 1 ) ln(χ i (ζ i 1, ζ i )) (1 ζ n ) ln(χ n+1 (ζ n )), [ ri r i 1, r ] i+1 r i k i k i 1 k i+1 k i for i < n and ζ n [ ] rn r n 1, 1. k n k n 1 Recall from the proof of Proposition 3.1 that G is convex on the interior of its domain. Since μ ([, k n ]) = ˉζ n, we now prove that for ˉζ to be the minimiser, ˉζ n < 1 is optimal. Indeed, there are two possibilities: 1. ˉζn 1 = rn r n 1 k n k n 1 ; in which case ˉζ n = ˉζ [ ] n 1 < 1, because for all ζ n rn r n 1 k n k n 1, 1, convexity of ln(x) implies 2. Alternatively, ˉζ n 1 < rn r n 1 k n k n 1 ; but then ζ n G(ˉζ 1,..., ˉζ n 1, ζ n ). ζ n G(ˉζ 1,..., ˉζ n 1, ζ n ) +. 2

21 Figure 3: Illustration of the payoff S F ln portfolio (black line). ( ) S F S F (blue line) and the consequent sub-hedging S as ζ n 1 and so for the minimiser it must be that ˉζ n < 1. he same method of proof shows μ ([, k n ]) < 1 for any convex function λ that is decreasing with lim x λ(x) =. 5.1 Left-Wing Information Variance swap prices are not bounded above because, recalling the criteria of (24), the convex function ln x is not bounded near x =, however the right-tail growth is sufficiently slow. One interpretation is simply that the information about the risk-neutral distribution of S given by put prices alone, the payoff from which are bounded if S =, is not enough to restrict the mass on its left wing in all arbitrage-free models. rading an additional contingent claim should therefore make the upper bound, for the variance swap, finite. o demonstrate this we provide a class of claims, the price of which then implies an explicitly solvable least upper bound for the variance swap price. In what follows, I will always denote an interval in R. Definition 5.1. Let f, g C 2 (I, R). We call f auxiliary to g, if f (x) g (x) for all x I. his relationship between functions will form the basis of this section; the next result motivates why. 21

22 Lemma 5.2. Let f, g C 2 (I, R) be two convex functions such that f is auxiliary to g. One can then construct functions F, G C 2 (I, [, )) such that for x I: F (x) G(x), F (x) = f (x) and G (x) = g (x). Consequently F is auxiliary to G and both functions are convex. Proof. Let a I and simply define F (x) = f(x) f(a) f (a)(x a) and G(x) = g(x) g(a) g (a)(x a) for x I. his choice of F and G suffices by convexity of f and g, and because f is auxiliary to g. Here onwards, suppose that in addition to the n European put options written on S, an auxiliary f-call struck at k 1 is traded in the market today with European payoff at time H f k 1 := (f( S ) f(k 1 )) +, X f E := X E {H f k 1 }, X f := X {H f k 1 }, (3) where f C 2 ((, k 1 ], R) is monotone, auxiliary to ln(x), and hence convex. Let denote the un-discounted traded price of the claim and h f k 1 := D 1 PHf k 1 (31) M E h = {M ME : h f k 1 = E M (f( S ) f(k 1 )) + }. (32) the set of calibrated models. he role of the auxiliary call will eventually be to super-hedge the log-contract component of the variance swap payoff. Before detailing how, we ensure the prices of the European puts and the additional call are consistent with absence of arbitrage. Proposition 5.3. Under A1 and A2, assume that (P, X E ) do not admit a weak arbitrage. hen (P, X f E ) do not admit a weak arbitrage if and only if ( r2 k 1 r 1 k 2 h f k 1 r [ 2 r 1 f k 2 k 1 r 2 r 1 ) ] f(k 1 ), (33) and then there exists a (P, X f E ) market model, i.e. ME h. If (33) doesn t hold, then (P, Xf E ) admits MI arbitrage. Remark 5.4. Note that x = r 2k 1 r 1 k 2 r 2 r 1 (k 1, r 1 ) and (k 2, r 2 ) occurs. is the point where the zero of the linear interpolation of Proof. o prove the only if and arbitrage statements, we form a portfolio holding πi units of the puts struck at K i, for i = 1, 2 that sub-hedges the auxiliary call at maturity but has forward price B := r [ ( ) ] 2 r 1 r2 k 1 r 1 k 2 f f(k 1 ). k 2 k 1 r 2 r 1 Let Π(x) denote the payoff from the portfolio in normalised units, i.e. Π( S ) = π 1 (k 1 S ) + + π 2 (k 2 S ) +, where π i = F π i for i = 1, 2. Choose π 1 and π 2 so that 22

23 over [, k 1 ], the payoff Π(x) is the supporting hyperplane to f(x) f(k 1 ) at x = r 2k 1 r 1 k 2 r 2 r 1, over [k 2, ), Π(x) = and Π(x) is continuous. hen convexity of f implies (f(x) f(k 1 )) + Π(x) for all x > and so the portfolios cost is a lower bound for the price of the auxiliary call. Consider a feasible marginal law μ for S such that ({ }) r2 k 1 r 1 k 2 μ = r 2 r 1, r 2 r 1 k 2 k 1 then μ([k 1, k 2 ]) =, existence of which is guaranteed when the European option prices are consistent with absence of arbitrage, i.e. statement 2 of heorem 2.9 holds. It follows that the forward setup cost of the portfolio satisfies π 1 r 1 + π 2 r 2 = Π(x)μ(dx) = (f(x) f(k 1 )) + μ(dx) = B, which proves B is a lower bound, and (π1, π 2 ) is the portfolio for the model independent arbitrage if the bound is violated. o prove the if statement, assume that (33) holds. We construct a marginal distribution for S, which concentrates mass in the interval [, k 1 ] on one point, that is consistent with the given price data. Indeed, since f is monotone decreasing the payoff from the call can be re-written as (f( S ) f(k 1 ))1 S k 1, it suffices to restrict attention to pricing the first put and the auxiliary call. Observe now that from the construction in the proof of Lemma 5.2, it follows that f(x) f(k 1 ) f (k 1 )(x k 1 ) ln(x) + ln(k 1 ) + 1 k 1 (x k 1 ) and so f(x) as x. Moreover, convexity of f implies the function m [ f ( k 1 r 1 m ) f(k1 ) ] is monotone decreasing as m increases. herefore any price h > B can be realised by a market model in which the law of S over [, k 1 ] has the form m(h)δ k1 r 1, m(h) ( ] where m(h) r1 k 1, r 2 r 1 k 2 k 1 uniquely solves [ ( h = m(h) f k 1 r ) ] 1 f(k 1 ). (34) m(h) We recall that market input is now taken to be P in (1) and (31) defined on X f in (6) and (3), where f C 2 ((, k 1 ], R) is auxiliary to ln x on (, k 1 ]. Further, we re-define the static component π of a portfolio (π, φ, ψ) to be a vector π R n+2 where the extra component, π n+2, denotes the number of units of the auxiliary call option bought at time. heorem 5.5 (Variance Swap Bounds with Left-Wing Information). Let (P, X f ) be the market input. Under Assumptions A1 A3, suppose (P, X f E ) do not admit a weak arbitrage. hen the following are equivalent: 23