Short-time implied volatility of additive normal tempered stable processes

Empirical studies have emphasized that the equity implied volatility is characterized by a negative skew inversely proportional to the square root of the time-to-maturity. We examine the short-time-to-maturity behavior of the implied volatility smile for pure jump exponential additive processes. An excellent calibration of the equity volatility surfaces has been achieved by a class of these additive processes with power-law scaling. The two power-law scaling parameters are $\beta$, related to the variance of jumps, and $\delta$, related to the smile asymmetry. It has been observed, in option market data, that $\beta=1$ and $\delta=-1/2$. In this paper, we prove that the implied volatility of these additive processes is consistent, in the short-time, with the equity market empirical characteristics if and only if $\beta=1$ and $\delta=-1/2$.


Introduction
Which characteristics of the implied volatility surface should be reproduced by an option pricing model?A stylized fact that characterizes the equity market is a downward slope in terms of strike, i.e. a negative skew, where the skew is the at-the-money (ATM) derivative of the implied volatility w.r.t. the moneyness. 2Specifically, the short-time 3 negative skew is proportionally inverse to the square root of the time-to-maturity.The first empirical study of the equity skew dates back to Carr and Wu (2003): they find that the S&P 500 short-time skew is, on average, asymptotic to −0.25/ √ t.Fouque et al. (2004) arrive at a similar conclusion considering only options with shorttime-to-maturity (i.e. up to three months).In this paper, we show that a pure jump additive process, which also calibrates accurately the whole equity volatility surface, reproduces the power scaling market skew.
A vast literature on short-time implied volatility and skew is available for jump-diffusion processes.Both the ATM (see, e.g.Alòs et al. 2007, Roper 2009, Muhle-Karbe and Nutz 2011, Andersen and Lipton 2013, Figueroa-López et al. 2016) and the OTM implied volatility (see, e.g.Tankov 2011, Figueroa-López and Forde 2012, Mijatović and Tankov 2016, Figueroa-López et al. 2018) are analyzed.For a jump-diffusion Lévy process, the ATM implied volatility is determined uniquely by the diffusion term; it goes to zero as the time-to-maturity goes to zero if there is no diffusion term, i.e. for a pure-jump process.For this reason, pure jumps Lévy processes are not suitable to reproduce the market short-time smile, because the short-time implied volatility is strictly positive in all financial markets.Muhle-Karbe and Nutz (2011) have shown that, for a relatively broad class of additive models, the ATM behavior at small-time is the same as the corresponding Levy.In this paper, we analyze the ATM implied volatility and skew for a class of pure jump additive processes that is consistent with the equity market smile, differently from the Lévy case: this is the main theoretical contribution of this study.
An additive process is a stochastic process with independent but non-stationary increments; a detailed description of the main features of additive processes is provided by Sato (1999).In this paper, we focus on a pure jump additive extension of the well-known Lévy normal tempered stable process (for a comprehensive description of this set of Lévy processes, see, e.g., Cont and Tankov 2003, Ch.4).Pure jump processes present a main advantage w.r.t.jump-diffusion models: they generally describe underlying dynamics more parsimoniously.In a jump-diffusion, both small jumps and the diffusion term describe little changes in the process (see, e.g.Asmussen and Rosiński 2001).Because both components of the jump-diffusion process are qualitatively similar, when calibrating the model to the plain vanilla option market, it is rather difficult to disentangle the two components and several sets of parameters achieve similar results.Recently, it has been introduced a class of pure jump additive processes, the power-law scaling additive normal tempered stable process (hereinafter ATS), where the two key time-dependent parameters -the variance of jumps per unit of time, k t , and the asymmetry parameter, η t -present a power scaling w.r.t. the time-to-maturity t.It has been shown the excellent calibrating performances of this class of processes (see, e.g.Azzone and Baviera 2021).On the one hand, this class of pure jump additive processes allows calibrating the S&P 500 and EURO STOXX 50 implied volatility surfaces with great accuracy, reproducing "exactly" the term structure of the equity market implied volatility surfaces.On the other hand, the observed reproduction of the skew term structure appears remarkable.Moreover, an interesting self-similar characteristic w.r.t. the time-to-maturity arises.Specifically, among all allowed power laws, the power scaling of k t , β, is close to one, while the power scaling of η t , δ, is statistically consistent with minus one half (see, e.g.Azzone and Baviera 2021).
Consider an option price with strike K and time-to-maturity t.We define I t (x) the model implied volatility, where x := log K F 0 (t) is the moneyness and F 0 (t) is the underlying forward price with time-to-maturity t.In particular, we consider the moneyness degree y, s.t.x =: y √ t, introduced by Medvedev and Scaillet (2006).It has been observed that the moneyness degree y can be interpreted as the distance of the option moneyness from the forward price in terms of the Black Brownian motion standard deviation (see, e.g., Carr andWu 2003, Medvedev andScaillet 2006).The implied volatility w.r.t y is and its first order Taylor expansion w.r.t.y in y = 0 is We call ξt the skew term.We define σ0 and ξ0 as the the limits for t that goes to zero of σt and ξt .Their financial interpretation is straightforward: σ0 corresponds to the short-time ATM implied volatility, while ξ0 is related to the short-time skew, because it is possible to write the skew as In Figure 1, we present an example of the short-time implied volatility and the skew for the S&P 500 at a given date, the 22 nd of June 2020 (the business day after a quadruple witching Friday4 ).
On the left, we plot the one month (blue circles), two months (red squares), three months (orange stars), and four months (purple triangles) market implied volatility w.r.t. the moneyness degree y: we observe a positive and finite short-time σt .On the right, we plot the market skew w.r.t.
the time t: it appears to be well described by a fit O As already observed in some empirical studies (see, e.g.Carr andWu 2003, Fouque et al. 2004), equity market data are compatible with a positive and finite σ0 and a negative and finite ξ0 , that leads to a skew proportionally inverse to the square root of the time-to-maturity.We aim to present a pure-jump model with these features.
We study the behavior of σt and ξt for the ATS process, deriving, in (4, 5), an extension of the Hull and White (1987, p.4, Eq.( 7)) formula (see, e.g., Alòs et al. 2007, for another application of this formula to the short-time case).This formula leads to two results: on the one hand, we build some relevant bounds for σt ; on the other hand, we obtain an expression for ξt in (25) via the implicit function theorem (see, e.g.Loomis and Sternberg 1990, Th.11, p.164).
Three are the main contributions of this paper.First, we deduce for a family of pure-jump additive processes, the ATS, the behavior of the short-time ATM implied volatility σt and skew term ξt over the region of admissible parameters (see Theorem 2.3).Second, we prove that only the scaling parameters observed in market data (β = 1 and δ = −1/2) are compatible with a finite short-time implied volatility and a short-time skew proportionally inverse to the square root of the time-to-maturity.Third, we demonstrate it exists a pure-jump additive process (an exponential ATS) that presents the two features observed in market data: not only a finite and positive short-time implied volatility but also a power scaling skew.
The rest of the paper is organized as follows.Section 2 presents the ATS power scaling process and the extension of the Hull and White formula.Section 3 defines the implied volatility problem and analyzes the short-time ATM implied volatility σt .Section 4 computes the short-time limit of the skew term ξt .Section 5 presents the major result: the ATS process is consistent with the equity market if and only if β = 1 and δ = −1/2.Finally, Section 6 concludes.In the appendices, we report some technical lemmas: on basic properties in Appendix A and on short-time limits in Appendix B.

The ATS implied volatility
In this Section, we recall the characteristic function of the power-law scaling additive normal tempered stable process (ATS) and the notation employed in the paper.We also introduce a sequence of random variables (3) with the same distribution of the ATS for any fixed time t; we use these random variables to study the short-time implied volatility.We discuss the volatility smile at small-maturity produced by this forward model and determine the power laws of the ATS parameters that are consistent with the market data i.e. which choices of β and δ reproduce the market short-time features mentioned above.
We define a sequence of positive random variables S t via its Laplace transform.
Notice that the characteristic function is the same as the power-law scaling additive normal tempered stable process (ATS) in Azzone and Baviera (2021, Eq.2); the notation has been slightly simplified, we report it at the end of the paper.We also define F 0 (t), the forward contract at time 0 with maturity t, and model the same forward contract at maturity as We report the known result on the existence of an additive process with characteristic function (1), cf.Azzone and Baviera (2021, Th.2.3).
Theorem 2.3.Power-law scaling ATS It exits an additive process with the same characteristic function of (1), where α ∈ [0, 1) and β, δ ∈ R with either where the second condition reduces to −β < δ ≤ 0 for α = 0 The region of admissible values for the scaling parameters β and δ is shown in Figure 2.
In particular, we mention that, ∀ α ∈ [0, 1), the scaling parameters observed in the market, {δ = −1/2, β = 1}, are always inside the ATS admissible region.In Figure 2, we plot the admissible region for the scaling parameters β and δ.In this paper, we prove that the ATS implied volatility at short-time is qualitatively different for different sets of scaling parameters.We separate the admissible region in five Cases: Case 1 (grey area): Case 2 (orange area): Case 3 (light green area): Case 4 (continuous dark green line): Case 5 (red dot): Notice that Case 3 includes all its boundaries, identified by the green circles, with the exception of the point {β = 1, δ = −1/2} (red); Case 1 includes just its upper boundary (but it does not include its lower boundary), identified by the grey squares.
The main objective of this paper is to prove that the 5 Cases correspond to different behaviors of the implied volatility in the short-time.A summary of the ATS short-time behavior, and in particular of the ATM value σ0 and of the skew term ξ0 , w.r.t. the different Cases is available in Table 1.
Figure 2: ATS admissible region for the scaling parameters.We separate the region in five Cases.
It is also useful to provide the same result dividing the region for the admissible values of Theorem 2.3, in terms of β and δ.A summary of the ATS short-time behavior, w.r.t. the scaling parameters β and δ in the additive process admissible region is reported in Table 2.
Table 2: Summary of ATS short-time implied volatility behavior, w.r.t. the scaling parameters β and δ in the additive process admissible region.The ATM implied volatility σ 0 is positive (and finite) when we report the value for ξ0 .
It can be proven that, for every time t, the random variable has the characteristic function in (1), where g is a standard normal random variable independent from S t .The proof is the same as in the Lévy case, but with time dependent parameters, and it is obtained by direct computation of E[e i u ft ], conditioning w.r.t.S t .The f t in (3) is then equivalent in law to the ATS process at maturity t; thus, we can use this expression of f t to compute European options.Consider a European call option discounted payoff B t F 0 (t) e ft − F 0 (t) e x + (and B t F 0 (t) e x − F 0 (t) e ft + the discounted payoff for the corresponding put) where t is option maturity, K option strike price, x := ln K F 0 (t) the asset moneyness and B t the deterministic discount factor between 0 and t.We can write the expected European call and put option price at time zero conditioning on S t where and N is the standard normal cumulative distribution.Equations ( 4) and ( 5) are crucial in the deduction of paper's key results: let us stop and comment.First, let us notice that we can consider option prices with F 0 (t) = 1 and B t = 1 without any loss of generality: we are interested in the implied volatility and these two quantities cancel out from both sides of the implied volatility equation.Second, let us emphasize that the quantity inside the expected values are, in both equations ( 4) and ( 5), positive.Finally, it can be useful to mention that a similar result has been obtained by Hull and White (1987) for options on an asset with stochastic volatility.We can re-write equations ( 4) and ( 5) w.r.t to the moneyness degree y (cf.Introduction) and we can define c t (S t , y) and p t (S t , y) such that Black (1976) option prices w.r.t.y are where I t (y) is the implied volatility w.r.t. the moneyness degree.
The implied volatility equation for the call options is and the one for the put option is In the following lemma, we prove that σt √ t goes to zero at short-time following an approach similar to Alòs et al. (2007, Lemma 6.1, p.580), who considered a generalization of the Bates model.
Lemma 2.4.For the ATS, at short-time, Proof.For an ATM put (i.e. when y = 0), the left-hand side of equation ( 8) is equal to In the region of admissible scaling parameters, f t goes to zero in distribution because its characteristic function in (1) goes to one.Hence, by the dominated convergence theorem, E 1 − e ft + goes to zero at short time.For y = 0, the right-hand side of equation ( 8) becomes that goes to zero if and only if σt √ t goes to zero.
Thanks to this lemma, ATM and at short-time, we can rewrite the right hand side of ( 7) and ( 8) as where the asymptotic expansion holds because N ′ (0) = 1.If the model price (left hand side in ( 7) and ( 8)) goes to zero faster than √ t , then σ0 = 0 (Case 1).
2. If the model price goes to zero slower than √ t, then σ0 = ∞ (Case 2).
The idea of the proofs is the following.In Case 1 we bound from above the model price and we prove that it is o √ t .In Case 2 we bound from below the model price and we demonstrate that it goes to zero slower than √ t.Finally in the remaining Cases we build upper and lower bounds on the model price and prove that both bounds are O √ t .Furthermore, the proofs are divided in some sub-cases that correspond to particular ranges of the parameters β and δ : we indicate with bold characters the range at the beginning of each sub-case.
For Case 1: In the equality we have just added and subtracted the quantity . The inequality holds because, by definition of standard normal cumulative distribution function, and because we bound from above the product e ϕtt−tσ with the (positive) maximums of both factors.We bound the expected value of c t (S t , 0) as The first equality holds because Proof.
We divide the proof in two sub-cases.
Consider the left hand side of equation ( 7).We compute the derivative of c t (z, y) w.r.t.z in y = 0.
At short-time, for a given z ∈ 0, ϕt σ2 ηt , because the first term goes to zero as tη t , while the second and the third terms go to zero as N ′ ( √ t η t ) (i.e. as a negative exponential).Thus, for sufficiently small t, c t (z, 0) is decreasing w.r.t.z in (0, ϕt σ2 ηt ).We emphasize that the right extreme of the interval is increasing to one for sufficiently small t, see Lemma A.6 points 2 and 3. Fix τ > 0 and S * ∈ (0, ϕτ σ2 ητ ); for any t < τ The first inequality holds because c t (z, 0) is positive for any z ≥ 0 and because we bound from below the expected value with its minimum in the interval (0, S * ) multiplied by the probability of the interval, P (S t ≤ S * ).The second inequality is due to the fact that e x ≥ x + 1.Finally, the last inequality holds because, by definition of the standard normal cumulative distribution function, Recall that N l S * t + σ √ S * t 2 = 1 + o(1); notice that P (S t ≤ S * ) is constant for β = 1 and goes to one, by Lemma A.4 point 1, for β < 1.This proves the last equality.
Notice that t η t goes to zero slower than √ t (δ < −0.5), then the ATM call price goes to zero slower than √ t.
It exits q such that (β − 1)/2 < q < −δ − 1/2.We bound the ATM put price (5) from below for a sufficiently small t E[p t (S t , 0)] The first inequality holds because p t (S t , 0) is non negative and because we have added and sub- . The second because the difference between the standard normal cumulative distribution functions is non negative, analogously to (13).The third because, for S t ∈ [1, ∞), 1 − e ϕtt−tσ 2 ηtSt is positive and non decreasing in S t ; moreover, for a sufficiently small t, N −l The quantity M t is defined in (14).At short-time M t = 1/6 + o(1) because i) by Lemma B.3, P(S t ≥ 1 + t q ) goes to 1/2 as t goes to zero, and ii) by Lemma A.6 point 1, 1 − e ϕtt−tσ 2 ηt(1+t q ) = t 1+q σ2 η t + tσ 4 η 2 t k t /2 (1 + o(1)) .
Notice that t 1+q η t goes to zero slower than √ t, then ATM put price goes to zero slower than √ t. Case , the lower bounds on the ATM option prices in ( 7) and ( 8 Proof.
We split the proof in three sub-cases.For each sub-case we build an upper and a lower bound, on the model price, and we demonstrate that both bounds are O √ t and then, that σ0 is finite.
Let us split the expected value of the ATS call in two parts We prove that both parts are bounded from above by quantities O √ t .The first expected value is s.t.
where the inequality holds true because of ( 11) and ] goes to one as t goes to zero.Let us study the term A 2 (t).
, where P St is the law of S t .The first inequality is true because the quantity inside the expected value is positive on 0, ϕt σ2 ηt and negative elsewhere.The equality is obtained by adding and subtrancting the same expected value for a Gaussian random variable.We prove the second inequality in two steps, showing that both ( 16) and ( 17) are bounded by O t δ+(β+1)/2 .First, we consider ( 16) where A t ≡ w ∈ R : − t kt < w < (ϕ t /(σ 2 η t ) − 1) t kt .Equality ( 18) is due to a change of the integration variable w := (z − 1)/ k t /t, equality (19) to the fact that, by Lemma A.6, e ϕtt−tσ 2 ηt < 1. Equality (20) The first inequality is due to integration by part and to the triangular inequality.The second inequality is a consequence of Jensen inequality and of Lemma B.3.

Lower bound.
As discussed in the proof of Proposition 3.2, for a sufficiently small t, c t (S t , 0) is decreasing for The first inequality is because c t (S t , 0) is non negative and the second is because we bound the expected value from below with the minimum of c t (S t , 0) multiplied by the probability of the interval 0, ϕt σ2 ηt .The equality holds because, by Lemma B.3, Upper bound.
The upper bound on the ATS call price is the same to the one of the previous sub-case −β/2 ≤ δ < −1/2, β > 1.
Lower bound.
We bound the put price from below.It exist H > 1 such that for a sufficiently small t The first inequality holds because p t (S t , 0) is non negative.The second because e ϕtt−tσ 2 ηtSt < 1 in [1, H].The third inequality is due to the fact that we bound from above the difference with the standard normal law evaluated in the maximum between the two (positive) arguments multiplied by the difference of the two arguments.Notice that σ√ S t t is a positive quantity almost surely.The last inequality holds because, by Lemma B.4, it exists H > 1 s.t. the quantity inside the expected value is increasing in [1, H] for a sufficiently small t.The equality is because, by Lemma B.3, P(S t ∈ [1, H]) goes to 1/2 as t goes to zero and We can bound c t (S t , 0) from above as in (10).We bound the ATS option price as The last inequality holds because, by Jensen inequality with concave function Lower bound.
To bound c t (z, 0) from below we have to study its derivative in (12).Notice that, at short-time, 1), due to Lemma A.6 point 1, and to the fact that δ > −1/2.Moreover, again due to Lemma A.6 point 1, e ϕtt−tσ 2 ηtz = 1 + O (tη t ).Then, we have i) The negative first term at short-time is o ii) The second term at short-time is o iii) The positive third term at short-time is Summarizing, the leading term in (12), at short-time, is the third one, which is positive.Hence, for a fixed z > 0 and for sufficiently small t, c t (z, 0) is increasing; thus, we can bound the expected value from below The first inequality holds because c t (S t , 0) is non negative.The second because, for a sufficiently small t, c t (S t , 0) is increasing.The third is true because, for sufficiently small t, e ϕtt−tσ 2 ηt/2 > 1, by Lemma A.6 point 3.The forth is due to the fact that the difference of the standard normal cumulative distribution functions can be bounded from below by the (positive) maximum of the two arguments multiplied by the (positive) difference of the two arguments.The equality is due to the fact that P S t ∈ 1 2 , 3 2 is constant if β = 1 and goes to 1 at short-time if β > 1 because, by Lemma A.4 point 2, S t goes to one in distribution at short-time.
Summing up, in all sub-cases the upper bound and the lower bounds of the ATS option prices in ( 7) and ( 8) are O( √ t).Moreover, from ( 9) we have that the Black price is O σt √ t .Thus, σ0 is finite Proposition 3.4.
For Cases 4 and 5: We can bound c t (S t , 0) from above as in ( 10).
We bound the ATS option price as The equality holds because, by Jensen inequality with concave function Lower bound.
We bound c t (S t , 0) from below as: The first inequality is because c t (S t , 0) is non negative, because e x ≥ x + 1, and because the normal cumulative distribution function evaluated in a positive quantity is above 1/2.The second holds because the difference between the two normal cumulative function is non negative.
The last equality is due to the fact that can be bounded from below with a positive constant for sufficiently small t.This fact can be deduced for β ≤ 1.We prove it separately for the two cases β < 1 and β = 1.For β < 1, let us observe that, at short-time, For β = 1, we remind that the law of S t does not depend from t and we observe that the limit of (24) for t that goes to zero is positive lim where the last inequality is due to the fact that S t has unitary mean and finite variance k.
Summarizing, as in Proposition 3.3 the upper and lower bounds of the ATM prices in ( 7) are O √ t .From ( 9), we have that the Black price is O σt √ t .Thus, σ0 is finite In the propositions above, we have proven that σ0 is finite only in Cases 3, 4 and 5.Only for these Cases we study the short-time skew in the next Section.

Short-time skew
In this Section, we focus on the skew term ξt for the ATS when σ0 is finite.We obtain an expression of ξt in Lemma 4.1 and study its short-time limit.
In the introduction, we have mentioned that the implied volatility skew observed in the equity market is negative and it goes to zero as one over the square root of t.This behavior is equivalent to a negative and finite ξ0 .In this Section, we prove that ξ0 is zero in Case 3 (Proposition 4.2) and is negative and finite in Cases 4 and 5 (Proposition 4.3).Moreover, Case 5 identifies the unique parameters' set where ξ0 can be a generic value that it is possible to calibrate from market data.
Lemma 4.1.The skew term ξt is Proof.Applying the implicit function theorem to the implied volatility equation for the call option (7) we obtain the derivative of the implied volatility w.r. .
We prove the thesis by computing the three partial derivatives separately.
Notice that it is possible to exchange the expected value w.r.t.S t and the derivative w.r.t.y using the Leibniz rule because the law of S t does not depend from y.By substituting y = 0 and reminding that I t (0) = σt , we get (25) Notice that, because of lemma 2.4, the denominator of ξt in (25), , goes to 1 √ 2π at short-time.To study the short-time behavior of ξt it is sufficient to consider only the numerator of equation ( 25) Proposition 4.2.
We divide the proof in two sub-cases.
We compute the limit thanks to the dominated convergence theorem because the law of S t does not depend on t and l z t = o(1) in this sub-case.
The equality holds because where P St is the distribution of S t .We study the quantities in ( 27) and ( 28) separately.First, we consider ( 27) The first equality is obtained via a change of the integration variable (w := √ t(z − 1)/ √ k t ).The second equality is due to the asymptotic of ϕ t t in Lemma A.6 point 1.The third equality holds because of the dominated convergence theorem.The last is trivial because Second, we consider (28) The first equality is due to integration by part.The second to the fact that i) P(S t < 0) = 0, ii) N − t kt go to zero as t goes to zero, and iii) where the inequality is due to Lemma B.3 and the first equality is due the fact that

This proves (26).
It is now possible to compute the short-time limit of the skew term For Case 4: β < 1 and δ = −1/2, the skew term is For Case 5: β = 1 and δ = −1/2 the skew term is where r(S t ) := Proof.
We prove separately the two Cases.
Thanks to Lemma B.2, the limit of the numerator of ξt in (25) can be computed simply, We compute the limit in t = 0 of the numerator of ξt in (25) We obtain the equality thanks to the dominated convergence theorem, because the law of S t is constant in time.We recall that erf (z) = 2N(z/ √ 2) − 1, substituting in (25), we obtain (29) Equation ( 29) is one of the major results of the paper.Let us stop and comment.First, let us notice that ξ0 in (29), is a generic function of the couple of positive parameters σ η and k; in particular the erf function is odd in its argument and r : R + → R. Moreover ξ0 depends on the parameter α ∈ [0, 1) that selects the truncated additive process of interest.Second, i.e. the minimum value for the skew term is − π/2, its value in Case 4.
To show the upper bound, we can rewrite where the second equality is due to the change of variable w = 1/z, and second to r(1/w) = −r(w) and to the fact that erf (z) is odd.We also observe that erf (σ η r(z)) > 0 in (0, 1).
For the two cases where the distribution of S t is known analytically α = 0 (VG) and α = 1/2 (NIG), we can prove that the skew term ξ0 in ( 29) is negative for non zero σ η and k (for the expression of the Gamma and Inverse Gaussian laws see, e.g., Cont and Tankov 2003, Ch.4, p.128).In both cases we can prove that P St (z) − > 0 in (0, 1); recall that P St (z) does not depend from time because β = 1.In the α = 0 case, S t has the law of a Gamma random variable where the inequality is true in (0, 1) because 1 − e −1/ k (1/z−z) z 2/ k > 0 or equivalently 1/z −z + 2 ln z > 0. The last inequality is trivial ∀z ∈ (0, 1), because it is equal to zero for z = 1 and its derivative is negative.
In all other cases, we compute numerically the skew term ξ0 for different admissible values of k, σ η ∈ R + and α ∈ [0, 1), by means of inversion of the characteristic function of S t , showing that it is either negative or equal to zero.In Figure 3, we plot the numerical estimation of the skew term for σ η and k below 3 (an interval in line with the situation generally observed in market data) and for a grid of four values of α (α = 0, 1/4, 1/2, 3/4); in all cases the skew term ξ0 looks rather similar: equal to zero on the boundaries ( k = 0 and σ η = 0), a negative quantity in all other cases and a decreasing function w.r.t.both k and σ η.In Figure 4, we plot also the skew term for the same four values of α, varying k with σ η = 1 (on the left) and varying σ η for k = 1 (on the right): all plots look rather similar with a decreasing ξ0 .Finally, let us emphasize that the limits of ξ0 are zero for σ η and k that go to zero.
On the one hand, recall that the law of S t , P St , does not depend of σ η.By the dominated convergence theorem with bound P St , we have that lim σ η→0 E [erf (σ η r(S t ))] = 0 .
On the other hand, by Kijima (1997, Th.B.9, p.308), we have that S t converges in distribution to 1 as k goes to zero because We are computing the expected value of a bounded function of S t that does not depend of k.Thus, by definition of convergence in distribution, lim k→0 E erf σ η√ 2 1/ S t − S t = 0 .

Main Result
In the following theorem, we present the main results of this paper.We prove that if and only if β = 1 and δ = − 1 2 the ATS has a positive and constant short-time implied volatility σ0 and a negative and constant short-time skew ξ0 .We point out that a finite skew w.r.t.y correspond to a skew that goes as 1 √ t at short-time w.r.t. the moneyness x.The proof is based on the propositions of Sections 3 and 4.
Theorem 5.1.The ATS short-time implied volatility behaves as described in Table 1.

Conclusions
In this paper, we have analyzed the short-time behavior of the implied volatility of a class of pure jumps additive processes, the ATS family.An excellent calibration of the equity implied volatility surface has been achieved by the ATS, a class of power-law scaling additive processes (see, e.g.Azzone and Baviera 2021).This class of processes builds upon the power-law scaling parameters β, related to the variance of jumps, and δ related to the smile asymmetry.
Several are the results achieved in this study.First, we have deduced for this family of pure-jump additive processes the behavior of the short-time ATM implied volatility σt and the skew term ξt over the region of admissible parameters (cf.Theorem 2.3).We get this result constructing some relevant bounds for σt and obtaining the expression of ξt , cf. equation ( 25), via the implicit function theorem.Second, we have proven that only the scaling parameters observed in empirical analysis (β = 1 and δ = −1/2) are compatible with the implied volatility observed in the equity market (cf.Theorem 5.1 ).Hence, we have demonstrated that it exists a pure-jump additive process (an exponential ATS) that, differently from the Lévy case, presents the two features observed in market data: not only a finite and positive short-time implied volatility but also a short-time skew proportionally inverse to the square root of the time-to-maturity.3.If β = 1 the distribution of S t does not depend from t.
We compute the limit of S t Laplace transform for β < 1.By using the fact that k t /t goes to infinity as t goes to zero we obtain Thus, S t converges in distribution to the constant zero.This proves point 1.
We compute the limit of S t Laplace transform for β > 1.By using the fact that k t /t goes to zero as t goes to zero we obtain Thus, S t converges in distribution to the constant one.This proves point 2.
Point 3 follows from the fact that, if β = 1, L t (u; k t , α) is constant in t Lemma A.5.
where D is a positive constant.
Proof.Recall that S t is a positive r.v. and E[S t ] = 1.Then, its moment of order 1/2 is finite.By Lemma A.1 This proves point 1.
We prove that ϕ t /(σ 2 η t ) ≤ 1.We substitute the definition of ϕ t in (2), for α > 0, in (36) and we get We define z := The first expected value goes to zero because g t (z) converges uniformly to g(z) on [0, K], as proven above via Ascoli-Arzelá theorem.It exists K s.t. it is possible to bound the second with ǫ because h(z) goes to zero as z goes to infinity.Moreover, g(z) is bounded because it is the limit of a uniformly bounded sequence and lim

Figure 1 :
Figure1: Example of the S&P 500 short-time implied volatility and skew on the 22 nd of June 2020.On the left, we plot the one month (blue circles), two months (red squares), three months (orange stars), and four months (purple triangles) market implied volatility w.r.t. the moneyness degree y.We observe a positive short-time σt .On the right, we plot the market skew w.r.t. the time t and the fitted ≈ (−) 1 t .

t→0E
[|g(X t ) − g(X)|] = 0 .by definition of convergence in distribution, because g(z) is bounded.We have that 0 ≤ lim t→0 E[|g t (X t ) − g(X)|] ≤ lim t→0 {E[|g t (X t ) − g(X t )|] + E [|g(X t ) − g(X)|]} = 0 , /arxiv.org/ps/2108.02447v1 The random variable S t appears in the definition of the random variable f t , that is used to model a forward contract of the underlying of interest.Definition 2.1.Definition of {S t } t≥0 Let {S t } t≥0 be a sequence of positive random variables with a Laplace transform s.t.
We observe that h(0; s, t) = 0 and the first order derivative is non negative ∀u > 0 because k t /t is non decreasing in t, if β > 1, and is constant in t, if β = 1.Thus, h(u; s, t) ≥ 0, ∀u ≥ 0, and L t (u; k t , α) is non decreasing w.r.t.t.This proves point 2 Lemma A.4.1.If β < 1 S t goes to zero in distribution as t goes to zero.2.If β > 1 S t goes to one in distribution as t goes to zero.
which is a well known inequality.This proves point 2.Point 3 is straightforward, given point 1, because, if δ > − min(1, β), η t k t goes to zero as t goes to zero B Short-time limits Lemma B.1.Consider a family of positive random variables X t s.t.lim t→0 X t = X in distribution and a sequence of functions g t (z) ≥ 0 and uniformly bounded s.t.lim t→0 g t (z) = g(z).If ∃ τ > 0 s.t. for t ∈ (0, τ ) i) g t (z) is Lipschitz continuous with bounded Lipschitz constant, It is possible to apply the Ascoli-Arzelá theorem (see, e.g.,,Rudin 1976, Th.7.25, p.158)on every compact set [0, K], K > 0, because a sequence of Lipschitz continuous functions with bounded Lipschitz constant is equicontinous on any compact set.Thus, a sub-sequence of g t (z) converges uniformly to g(z) in any [0, K].For every ǫ > 0, ∃ K s.t.