Generalised arbitrage-free SVI volatility surfaces

In this article we propose a generalisation of the recent work of Gatheral and Jacquier on explicit arbitrage-free parameterisations of implied volatility surfaces. We also discuss extensively the notion of arbitrage freeness and Roger Lee's moment formula using the recent analysis by Roper. We further exhibit an arbitrage-free volatility surface different from Gatheral's SVI parameterisation.


Introduction
European option prices are usually quoted in terms of the corresponding implied volatility, and over the last decade a large number of papers (both from practitioners and academics) has focused on understanding its behaviour and characteristics. The most important directions have been towards (i) understanding the behaviour of the implied volatility in a given model (see [2], [3] [10], [15] for instance) and (ii) deciphering its behaviour in a model-independent way, as in [19], [20] or [22]. These results have provided us with a set of tools and methods to check whether a given parameterisation is free of arbitrage or not. In particular, given a set of observed data (say European calls and puts for different strikes and maturities), it is of fundamental importance to determine a methodology ensuring that both interpolation and extrapolation of this data are also arbitrage-free. Such approaches have been carried out for instance in [5], in [9] and in [23]. Several parameterisations of the implied volatility surface have now become popular, in particular [16], [18] and [11], albeit not ensuring absence of arbitrage.
In the recent paper [13], Gatheral and Jacquier proposed a new class of SVI implied volatility parameterisation, based on the previous works of Gatheral [11]. In particular they provide explicit sufficient and-in a certain sense-almost necessary conditions ensuring that such a surface is free of arbitrage.
We shall recall later the exact definition of arbitrage, and see that it can be decomposed into butterfly arbitrage and calendar spread arbitrage. This new class depends on the maturity and can hence be used to model the whole volatility surface, and not a single slice. It also depends on the at-the-money total implied variance θ t , and on a positive function ϕ such that the total variance w as a function of time-to-maturity t and log-(forward)-moneyness k is given by w(k, t) ≡ θ t SVI ρ (kϕ(θ t )), where SVI ρ is the classical (normalised) SVI parameterisation from [13], and ρ an asymmetry parameter (essentially playing the role of the correlation between spot and volatility in stochastic volatility models).
In this work, we generalise their framework to volatility surfaces parameterised as w(k, t) ≡ θ t Ψ(kϕ(θ t )) for some (general) functions ϕ, θ, Ψ. We obtain (Sections 3 and 4) necessary and sufficient conditions coupling the functions Ψ and ϕ that preclude arbitrage. This allows us to obtain (i) the exact set of admissible functions ϕ in the symmetric (ρ = 0) SVI case, and (ii) a constraint-free parameterisation of Gatheral-Jacquier functions satisfying the conditions of [13]. In passing (Section 4.4), we extend the class of possible functions by allowing for non-smooth implied volatility functions. Finally (Section 5), we exhibit examples of non-SVI arbitrage-free implied volatility surfaces.
We adopt here a general formulation for the implied volatility parameterisation. While this allows us to determine general classes of arbitrage-free volatility surfaces, it does however make some of the results fairly abstract and not readily tractable. We therefore provide some simpler (and thereby weaker) conditions ensuring no-arbitrage. Define M ∞ := lim u↑∞ uϕ(u) and assume that Ψ is asymptotically linear (Definition 3.4) and that θ ∞ = ∞. Corollary 4.1 and Proposition 5.1 each provide necessary conditions preventing butterfly arbitrage (Definition 2.3), namely , for some κ ≥ 0 and z large enough [Proposition 5.1].
Calendar spread arbitrage is easier to prevent and we refer the reader directly to Proposition 3.1 for necessary and sufficient conditions, that are easy to check in practice.
Notations: We consider here European option prices with maturity t ≥ 0 and strike K ≥ 0, written on an underlying stock S. Without loss of generality we shall always assume that S 0 = 1 and that interest rates are null, and hence the log (forward) moneyness reads k := log(K). We denote the Black-Scholes value for a European Call option C BS (K, w), for a strike K and total variance w, and more generally C(K, t) for (any) European Call prices with strike K and maturity t. For any k ∈ R, t ≥ 0, the corresponding implied volatility is denoted by σ(k, t) and the total variance w is defined by w(k, t) := σ(k, t) 2 t. With a slight abuse of language (commonly accepted in the finance jargon), we refer to the two-dimensional map (k, t) → w(k, t) as the (implied) volatility surface. Finally, for two functions g and h not null almost everywhere, we say that g(z) ∼ h(z) at z = 0 whenever lim z→0 g(z)/h(z) = 1.

Absence of arbitrage and volatility parameterisations
This preliminary section serves several purposes: we first recall the very definition of 'arbitrage freeness' and its characterisation in terms of implied volatility. We then state and prove a few results (which are also of independent interest) related to this notion of arbitrage. We finally quickly review the parameterisation proposed in [13] and introduce an extension, which is our new contribution.
2.1. Absence of arbitrage. As defined in [6], absence of static arbitrage corresponds to the existence of a non-negative martingale (on some probability space) such that European call options (written on this martingale) can be written as risk-neutral expectations of their final payoffs. This rigorous definition is however not easily tractable and often difficult to check. Roper [20] proved that, given a (twice differentiable in strike) volatility surface, absence of static arbitrage is satisfied as soon as the following three conditions holds: no calendar spread arbitrage, no butterfly arbitrage and the time-zero smile is null everywhere. We now define these terms precisely.
Definition 2.1. A volatility surface w is free of calendar spread arbitrage if ∂ t w(k, t) ≥ 0, for all k ∈ R and t > 0.
Absence of calendar spread arbitrage thus means that the total implied variance t → w(·, t) ≡ σ 2 (·, t)t is an increasing function. Define now the operator L acting on C 2,1 (R × (0, ∞) → R) functions by for all k ∈ R and t > 0. Even though the operator does not act on the second component of the function, we keep this notation for clarity. With the usual notations from the Black-Scholes formula: It is indeed well-known (under suitable differentiability assumptions) that the second derivative (with respect to the strike) of the call price function gives the density of the stock price (see [13,20] for details).
The following corollary [13,Lemma 2.2] provides an easy-to-check (at least in principle) condition ensuring no butterfly arbitrage: If σ loc represents the (Dupire) local volatility, the relationship σ 2 loc (k, t) = ∂ t w(k, t)/Lw(k, t), for all k ∈ R, t > 0 is now standard (see [9]). Therefore absence of static arbitrage implies that both the numerator and the denominator are non-negative quantities. The extra condition (from absence of buttefly arbitrage) is the 'Large-Moneyness Behaviour' (LMB) condition which is equivalent to call option prices tending to zero as the strike tends to (positive) infinity, as proved in [22,Theorem 5.3]. The following lemma however shows that other asymptotic behaviours of d + and d − hold in full generality. This was proved by Rogers and Tehranchi [22] in a general framework, and we include here a short self-contained proof.
Lemma 2.4. Let w be any positive real function. Then Proof. The arithmetic mean-geometric inequality  [20] show that the marginal law of the stock price at some fixed time t > 0 has no mass at zero if and only if lim K↓0 ∂ K C(K, t) = −1, which is a statement about a 'small-moneyness' behaviour. This can be fully recast in terms of implied volatility, and the above missing conditions then come naturally into play in the following proposition, the proof of which is postponed to Appendix A.1: (Symmetry under small-moneyness behaviour) Let a C 2 real function v satisfy (I) v(k) > 0 and Lv(k) ≥ 0 for all k ∈ R; Define the two functions p − and p + by k → p ± (k) := (2πv(k)) −1/2 exp − 1 2 d 2 ± (k, v(k)) Lv(k). Then (1) p + and p − define two densities of probability measures on R with respect to the Lebesgue measure, i.e. R p − (k)dk = R p + (k)dk = 1; (3) p − is the density of probability associated to Call option prices with implied volatility v, in the sense that p − (k) ≡ ∂ 2 KK C BS (K, v(log(K)))| K=e k , and k → p + (−k) is the density of probability associated to Call option prices with implied volatility k → w(k) := v(−k).
The strict positivity of the function v in Assumption (I) ensures that the support of the underlying distribution is the whole real line. One could bypass this assumption by considering finite support as in [22]. In the latter-slightly more general-case, the statements and proofs would be very analogous but much more notationally inconvenient. Symmetry properties of the implied volatility have been investigated in the literature, and we refer the interested reader to [4,14,21] This proposition has been intentionally stated in a maturity-free way: it is indeed a purely 'marginal' or cross-sectional statement, which does not depend on time. A natural question arises then: can such a function v, satisfying the assumptions of Proposition 2.5, represent the total implied variance smile at time 1 associated to some martingale (issued from 1 at time 0)? The answer is indeed positive and this can be proved as follows.
Consider the natural filtration B of a standard (one-dimensional) Brownian motion, (B t ) t≥0 . Let P be the cumulated distribution function associated to p − characterised in Proposition 2.5, and let N be the Gaussian cumulative distribution function. Then the random variable X := P −1 (N (B 1 )) has law P , and E(X) = 1. Set now M s := E(X|B s ), then M is a martingale issued from 1. Note that M is even a Brownian martingale and therefore a continuous martingale. The associated Call option prices E[(M s − K) + ] uniquely determine a total implied variance surface (t, k) → w(k, t) such that v = w(1, ·).

Volatility parameterisations.
In [11], Gatheral proposed a parameterisation for the implied volatility, the now famous SVI ('Stochastic Volatility Inspired). However, finding necessary and sufficient conditions preventing static arbitrage have been inconclusive so far. Recently, Gatheral and Jacquier [13] extended this approach and introduced the following parameterisation for the total implied variance w: with θ t > 0 for t > 0 and ϕ is a smooth function from (0, ∞) to R + and ρ ∈ (−1, 1). The main result in their paper (Corollary 5.1) is the following theorem, which provides sufficient conditions for the implied volatility surface w to be free of static arbitrage: Theorem 2.6. The surface (2.3) is 'free of static arbitrage' if the following conditions are satisfied: (1) ∂ t θ t ≥ 0 for all t > 0; (2) ϕ(θ) + θϕ ′ (θ) ≥ 0 for all θ > 0; (3) ϕ ′ (θ) < 0 for all θ > 0; (4) θϕ(θ)(1 + |ρ|) < 4 for all θ > 0; A few remarks are in order here: (1) the conditions in Theorem 2.6 are sufficient, but may be far from necessary; (2) the full characterisation of the functions ϕ guaranteeing absence of (static or not) arbitrage in the symmetric SVI case ρ = 0 is left open; (3) it would be useful to 'parameterise' the set of functions ϕ satisfying the conditions of Theorem 2.6.
This could lead to easy-to-implement calibration algorithms among the whole admissible class, without being tied to a particular family as in [13].
In this paper, we try to settle all these points, and state our results in a more general framework, not tied to the specific shape of the SVI model, i.e. we consider implied volatility surfaces of the form with the following assumptions: Assumption 2.7.
The time-dependent function θ models the at-the-money total variance; the assumption on its behaviour at the origin is thus natural. A constant function Ψ corresponds to deterministic time-dependent volatility, a trivial case we rule out here. Likewise, were θ assumed to be constant, it would be null everywhere, which we shall also not consider. Assumption (iv) ensures that at maturity, European Call option prices are equal to their payoffs. We can recast it in terms of assumptions on ϕ and Ψ, for example: Assumption (iv)(bis): ϕ(θ) converges to a non-negative constant as θ ↓ 0.
Indeed, (iv)(bis), together with (iii), clearly implies (iv). We shall present another alternative below with the help of the 'asymptotic linear' property of Ψ (Definition 3.4 and Assumption 3.5). Assumption (iii) may look strong from a purely theoretical point of view, but is always satisfied in practice. In Section 4.4 though, we partially relax it (Assumption 4.8) to allow for possible kinks. The main goal here is to provide sufficient conditions on the triplet (θ, ϕ, Ψ) that will guarantee absence of static arbitrage. Note that the SVI parameterisation (2.3) corresponds to the case Ψ(z) ≡ 1 2 (1 + ρz + z 2 + 2ρz + 1). In the sequel, we shall refer to this case as the SVI case. The next sections provide necessary and sufficient conditions on θ, ϕ and Ψ in order to prevent static arbitrage.

Elimination of calendar spread arbitrage
We first concentrate here on determining (necessary and sufficient) conditions on the triplet (θ, ϕ, Ψ) in order to eliminate calendar spread arbitrage.
3.1. The first coupling condition. The quantity ∂ t w(k, t) in Definition 2.1 is nothing else than the numerator of the local volatility expressed in terms of the implied volatility, i.e. Dupire's formula (see [12]).
Define now the functions F : R → R and f : R → R by They will play a major role in our analysis, and Assumption 2.7 (iii) implies that F (z) ∼ Ψ ′ (0)z/Ψ(0) at the origin and F (0) = 0. Note that Ψ and ϕ can be recovered through the identities for some arbitrary constant r > 0. The following proposition gives new conditions for absence of calendar spread arbitrage. (i) θ is non-decreasing; Proof. By Definition 2.1, the surface defined by (2.4) is free of calendar spread arbitrage if and only if ≥ 0 for all z ∈ R and t > 0, where the functions F and f are defined in (3.1). For k = 0 we get θ ′ t ≥ 0 for all t > 0. Otherwise (ii) is necessary and sufficient for the surface to be free of calendar spread arbitrage.
Remark 3.2. We do not assume here that θ ∞ is infinite. In most popular stochastic volatility models with or without jumps, θ ∞ is infinite. Rogers and Tehranchi [22] showed that for a non-negative martingale (S t ) t≥0 the equality θ ∞ = ∞ is equivalent to the almost sure equality lim t↑∞ S t = 0 (where the limit exists by the martingale convergence theorem). However, it may occur that θ ∞ < ∞. As a corollary of coupling properties of stochastic volatility models, Hobson [17] provides instances where such a phenomenon appears, for example the SABR [16] model with β = 1.
Motivated by the celebrated moment formula in [19] (see also Theorem B.1), which forces the function Ψ to be at most linear at (plus/minus) infinity, let us propose the following definition: Definition 3.4. We say that the function Ψ is asymptotically linear if and only if Ψ ′ has a finite non-zero limit α ± when z tends to ±∞.
We now obtain a necessary condition on the behaviour of the function ϕ in (2.4).
Since ϕ is a strictly positive function by Assumption 2.7 (ii), the proposition follows from the definition of the function f in (3.1).
Note that if lim z→±∞ Ψ ′ (z) = 0 then the limit of the function F at (plus or minus) infinity does not necessarily exist. Whenever it does, since z → Ψ(z)/z is decreasing as z → ±∞, the limit can take any value in (−∞, 1).
with |ρ| < 1, so that Ψ is asymptotically linear with α + = ρ + 1 and α − = ρ − 1. Therefore Proposition 3.6 applies, and a necessary condition is that u → uϕ(u) is not decreasing. In [13,Theorem 4.1], this condition, together with ϕ being non-increasing, are shown to be sufficient to avoid calendar spread arbitrage. In the case of the symmetric SVI model, the following corollary relates our conditions to those in [13].
Corollary 3.7. In the symmetric SVI case, the necessary condition of Proposition 3.6 is also sufficient.
Proof. In the symmetric case ρ = 0, we can compute explicitly It is then clear that the even function F is strictly increasing on (0, ∞) and strictly decreasing on

Elimination of butterfly arbitrage
We now consider butterfly arbitrage which, probably not surprisingly, is more subtle to handle. We first start with a general result (Section 4.1), which is unfortunately not that tractable in practice.
When the function Ψ is asymptotically linear, however, more elegant formulations are available, and we is a necessary condition for absence of butterfly arbitrage. In particular M ∞ ≤ 2/ sup{|α + |, |α − |}.
4.1. The second coupling condition. We consider here the positivity condition Lw(k, t) ≥ 0 from Definition 2.3, and reformulate the butterfly arbitrage condition in our setting. We first start with a general formulation, and then consider the asymptotically linear case (for the function Ψ), which turns out to be more tractable. For u > 0 define the set

4.2.
The asymptotically linear case. We now consider the case where Ψ is asymptotically linear (Definition 3.4). Define the sets as well as the constant , otherwise; (ii) for any u ∈ (0, θ ∞ ),
Little work on this proposition yields the following sufficient condition preventing butterfly arbitrage, which is easier to check in practice.
Corollary 4.4. Assume that Ψ is asymptotically linear and that there is no calendar spread arbitrage.
Assume further that for any u ∈ (0, θ ∞ ), the inequality in Proposition 4.3 (ii) is strict. Then the corresponding implied volatility surface is free of static arbitrage.
Proof. In our setting (with Ψ being asymptotically linear) lim k↑∞ (w(k, t)/k) = θ t ϕ(θ t )α + , so that all we need to prove is that θ t ϕ(θ t ) < 2/α + , since lim k↑∞ w(k, t)/k < 2 clearly implies the LMB condition. For any z ∈ Z + (defined in (4.3)), note that Applying this relation to a sequence in Z + diverging to infinity leads to (θ t ϕ(θ t )) 2 < 4/α 2 + and the proposition follows.  Of course we only define these functions on their effective domains, the forms of which we omit for clarity.
The following proposition makes the conditions of Proposition 4.3 explicit in the uncorrelated SVI case. Proof. Define y z := √ 1 + z 2 ; then Since Ψ(z) > 0 for all z ∈ R, the first equation implies that For any fixed u, the function appearing on the right-hand side of Proposition 4.3 (ii) simplifies to A(y z , u) given in (4.3). In particular A(2, u) = 48 and lim y↑∞ A(y, u) = 16. For any u ≥ 0, we have where B u (y) := 1 − u 4 y 2 − 4y − 2. When u ≥ 4, B u is concave on (2, ∞) with B u (2) = −(6 + u) < 0, and hence the map y → A(y, u) is decreasing on (2, ∞) and its infimum is equal to lim y↑∞ A(y, u) = 16.
For u ∈ [0, 4), the strict convexity of B u and the inequality B u (2) = −(6 + u) < 0 implies that the equation B u (y) = 0 has a unique solution in (2, ∞), which in fact is equal to Y (u) given in (4.6). Then the map y → A(y, u) is decreasing on (2, Y (u)) and increasing on (Y (u), ∞). Its infimum is attained at Y (u) and is equal to A * (u) defined in (4.7).
Remark 4.6. In [13], the authors prove that the two conditions (altogether) uϕ(u) < 4 and uϕ(u) 2 < 4 (for all u ≥ 0) are sufficient to prevent butterfly arbitrage in the uncorrelated (ρ = 0) case. These two conditions can be combined to obtain (uϕ(u)) 2 < 16 min(1, ϕ(u) −2 ). A tedious yet straightforward computation shows that A * is increasing on [0, 4) and maps this interval to [0, 16). Notwithstanding the fact that our condition is necessary and sufficient, it is then clear that (i) for u ≥ 4, it is also weaker than the one in [13] whenever ϕ(u) < 1; (ii) for u < 4 (which accounts for most practically relevant cases) it is weaker whenever 16/ϕ(u) < A * (u). C(e k , t) = C BS (e k , w(k, t)), for all k ∈ R, t ≥ 0.
More precisely, Roper uses the regularity assumption in k of w in order to define pointwise the second derivative of C with respect to the strike. He then proves that the latter is positive, henceforth obtaining the convexity of the price with respect to the strike (Assumption A.1 of Theorem 2.1). It turns out that the same result can be obtained without this regularity assumption: assume that for any t, the function k → w(k, t) is continuous and almost everywhere differentiable. Since ∂ k w is defined almost everywhere, and since the term in ∂ kk w is linear, Lw can be defined as a distribution and we replace the assumption Lw(k, t) ≥ 0 everywhere by Lw(k, t) ≥ 0 in the distribution sense. In order to do so on R \ {0}-saving us from dealing with the boundary behaviour at the origin-we can assume additionally the SMB Condition (Proposition 2.5 (II)) and work on (0, ∞). To conclude, we need to prove that the Call options C defined in (4.8) are convex function in K = e k . Since a continuous function is convex if and only if its second derivative as a distribution is a a positive distribution, the only remaining point to check is that the second derivative as a distribution of C is Lw. But this the same computation already carried out in [20], and the result hence follows. Let us finally note that our assumptions on w are indeed minimal: conversely, if we start from option prices convex in K, their first derivative are defined almost everywhere, and so are those of w (in K or k) since the Black-Scholes mapping in total variance is smooth.
Proof. As in the proof of 4.2, in the distribution sense, the equality holds pointwise outside A, and the first part of the proposition follows. The remaining part of the distribution Lw(k, t) is the sum of the disjoint Dirac masses (θ t ϕ(θ t )) 2 α i δ i . By localisation it is clear , otherwise; (ii) for any u ∈ (0, θ ∞ ),

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, and the jumps of Ψ ′ are non-negative.

The quest for a non-SVI Ψ function
In order to find examples of pairs (ϕ, Ψ), with Ψ different from the SVI parameterisation (2.3), observe first that the second coupling condition (Proposition 4.2) is more geared towards finding out ϕ given Ψ than the other way round. We first start with a partial result (proved in Appendix A.3) in the other direction, assuming that Ψ is asymptotically linear.
Using this proposition, we now move on to specific examples of non-SVI families. A few remarks are in order: • the function ϕ is continuous on R + ; • θ ∞ = ∞; • the map u → uϕ(u) is increasing and its limit is M ∞ = 1; • the function Ψ-directly inspired from the computations in Proposition 5.1-is symmetric and continuous on R. It is also C ∞ on R \ {0}, and asymptotically linear. Its derivative is therefore C 1 piecewise, moreover it has a positive jump at the origin, so we are in the framework of Propositions 4.9 and 4.10.
With these functions, the total implied variance (2.4) reads for all k ∈ R, and the following proposition (proved in Appendix A.5) is the main result here: Proposition 5.2. The surface w is free of static arbitrage.
where ν is a real number in (1, ∞) and α ∈ (0, α) with α ≈ 1.33. Note that when ν = 2, modulo a constant, the function Ψ 2 corresponds to SVI. We could in principle let α depend on ν. The reason for the construction above is that we want to show that the corresponding implied volatility surface is free of static arbitrage for all ν > 1. The same remarks as in the example in Section 5.1 hold: ϕ is continuous Consider first p − . It is readily seen that the function D(k) ≡ ∂ K C BS (K, v(log(K)))| K=e k is a primitive of p − . We now proceed to prove that p − is indeed a density. Let N denote the cumulative distribution function of the standard Gaussian distribution. An explicit computation yields (the reverse one can be found in [13,Lemma 2.2]) where d ± and their derivatives are evaluated at (log(K), v log(K)), and where we have used the equality d − (log(K), v(log(K))) = d + (log(K), v(log(K)))− v(log(K)). Evaluating the right-hand side at K = e k , and using the fact that −k − 1 2 d 2 Therefore if where we have used the SMB Condition in Assumption (II). Let us first deal with the case when k tends to (positive) infinity. From Lemma 2.4, exp − 1 2 d − (k, v(k)) 2 tends to zero. The key point is that D is the primitive of a non-negative function, therefore it is non-decreasing and has a (generalised) limit L ∈ (−∞, ∞] as k ↑ ∞. Since N (d − (k, v(k))) converges to zero by Lemma 2.4 (ii), we deduce Proof of Theorem 5.3], the inequality v ′ (k) < 2v(k)/k holds for any k > 0 so that L is necessarily non-positive. Assume that L is negative; √ v is eventually decreasing. Since it is bounded from below by 0, there exists a sequence (k n ) n≥0 going to infinity such that ∂ k v(k n ) converges to zero by the mean value theorem, and hence L = 0.
Let us now consider the case where k tends to negative infinity. Using similar arguments, the quantity Then v is decreasing for k small enough. Since v is positive, this implies that v(k) > ε for some ε > 0 and k small enough. In [22,Theorem 5.1], then for k small enough, the inequalities −4 < v ′ (k) ≤ 0 hold, and the term outside the exponential in (A.1) is bounded.
Since the exponential converges to zero by Lemma 2.4 (ii), we obtain M = 0. If M > 0, then v is increasing for k small enough. We conclude as above by the mean value theorem since √ v is increasing and bounded from below. Therefore M = 0 and the limit (A.1) holds.
So far we have proved that p − is the density of probability associated to Call option prices with implied and it follows by inspection that Lw(k) = Lv(−k) ≥ 0. Consider the function p − associated to w, i.e.
Now d − (k, w(k)) = −d + (−k, v(−k)), so that p − (k) = p + (−k). In order for p − to be a genuine density, we need to check conditions symmetric to those ensuring that p − is a density. The condition symmetric to the SMB assumption (II) is precisely Condition (i) in Lemma 2.4, and the condition symmetric to the Lemma 2.4 (ii) is precisely the LMB assumption (III). Therefore k → p + (−k) is also a density, associated to Call option prices with implied volatility w. Finally the identity p + (k) = e k p − (k) follows immediately from the equality −k − 1 2 d 2 A.2. Proof of Proposition 4.3. Assume that Ψ is asymptotically linear and that there is no calendar spread arbitrage. The proof relies on the decomposition of the real line into the following disjoint unions, for any u > 0: . As in the proof of Proposition 4.2, butterfly arbitrage is clearly precluded on R \ Z + (u), so that we are left with two cases to consider: Z + and Z − ∩ Z + (u).
Consider first the set Z − ∩Z + (u), which corresponds to case (i) in the proposition. The map u → uϕ(u) is non-decreasing on (0, ∞) by Proposition 3.6. On is clearly also non-decreasing on (0, ∞). Therefore Z − (Z + (u)) u>0 is a non-decreasing family of sets and therefore, in view of (4.2), absence of butterfly arbitrage (Lw ≥ 0) on this set is equivalent to

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, for all u ∈ (0, θ ∞ ), when θ ∞ < ∞, which in turn is equivalent to When θ ∞ = ∞, analogous arguments show that absence of butterfly arbitrage is equivalent to follows immediately from Proposition 4.2 and the computation above. Let us now prove that Z + is not empty. If it is, the asymptotic linearity of Ψ allows us to choose a > 0 such that Ψ ′ (z) > 0 for all z > a. Therefore 1/Ψ(z) ≤ 2Ψ ′′ (z)/Ψ ′ (z) 2 for all z > a, which in turn yields The integral diverges to infinity as z ↑ ∞ since Ψ(z) ∼ α + z whereas the right-hand side is bounded by Definition 3.4. The same argument shows that Z + is not bounded from above.
From the proof of Proposition 4.3, we know that when θ ∞ = ∞, the first condition simplifies to .
Appendix B. Lee's moment formula in the asymptotically linear case In Section 2 we stressed that, following Roper or the variant in Proposition 2.5, the positivity of the operator L in (2.1) guarantees the existence of a martingale explaining market prices. As a consequence, the celebrated moment formula [19] holds: Theorem B.1 (Roger Lee's moment formula [19]). Let S t represent the stock price at time t, assumed to be a non-negative random variable with positive and finite expectation. Letp := sup{p ≥ 0 : ES 1+p t < ∞} and β := lim sup k↑∞ w(k,t) x . Then β ∈ [0, 2] andp = 1 2 β 4 − 1 + 1 β .