At-the-money short-time call-price asymptotics for new classes of exponential Lévy models

A t-the-money short-time call-price asymptotics for new classes of exp onen tial L ´ evy mo dels Allen Hoffmey er ∗ Christian Houdr ´ e †‡ Marc h 17, 2026 Abstract W e dev elop at-the-money call-price and implied volatilit y asymptotic expansions in time to maturity for a class of asset-price mo dels whose log returns follo w a L´ evy pro cess. Under mild assumptions placing the driving L´ evy process in the small-time domain of attraction of an α -stable la w with α ∈ (1 , 2), w e giv e first-order at-the-money call-price and implied v olatility asymptotics. A k ey observ ation is that both the stable domain of attraction and the finiteness of the centering constan t ¯ µ are preserved under the share measure transformation, so that all of the distributional input needed for the call-price expansion can b e read off from the regular v ariation of the L´ evy measure near the origin. When the L´ evy pro cess has no Bro wnian comp onen t, new rates of con vergence of the form t 1 /α ℓ ( t ) where ℓ is a slowly v arying function are obtained. W e provide an example of an exponential L´ evy mo del exhibiting this b eha vior, with ℓ not asymptotically constant, yielding a con vergence rate of ( t/ log(1 /t )) 1 /α . In the case of a L´ evy process with Brownian component, w e show that the jump con tribution is alwa ys low er order, so that the leading √ t behavior of the at-the-money call price is universal and driven entirely b y the Gaussian part of the c haracteristic triplet. AMS 2000 sub ject classifications : 60G51, 60F99, 91G20, 91G60. Keyw ords and phrases : Exp onen tial L ´ evy models; short-time asymptotics; A TM option pricing; implied v olatility; regular v ariation; stable domain of attraction 1 In tro duction The Blac k–Scholes paradigm is analytically elegant yet empirically inadequate: implied volatilit y is not flat across strikes or maturities and log returns are not Gaussian. In practice, smiles and skews p ersist even at short maturities, and tail risk is materially underpriced when returns are mo deled by a diffusion with exp onen tially decaying tails. Numerous extensions ha ve been prop osed. Lo cal-volatilit y mo dels [ 10 ] repro duce to da y’s smile b y design but are often unstable out of sample; sto chastic-v olatilit y mo dels (e.g. Heston/SABR) capture dynamics but typically miss the exact near-expiry smile without substan tial calibration effort. In this w ork w e fo cus on exp onen tial L ´ evy models, where the log-price is a L ´ evy pro cess and jumps pro duce the empirically observed smile and heavier tails while preserving analytic tractability in many cases (see the surv eys and references in [ 5 , 13 , 27 ]). A ma jor thread of research o ver the last tw o decades studies the smal l-time (near-expiry) b eha vior of option prices and implied volatilit y in exp onen tial L ´ evy mo dels. Early milestones include general at-the-money (A TM) price asymptotics and links to first absolute moments [ 25 ], small-time smile b eha vior in stable-like and temp ered settings [ 13 , 27 ], and mo del-specific expansions for CGMY and related classes [ 16 , 17 ]. Since then, the literature has clarified ho w jump activit y (e.g. the Blumenthal–Getoor index) and any Bro wnian component join tly determine A TM scalings and higher-order terms. Notably , [ 24 ] giv e a unifying view of short-maturity implied volatilit y in exp onen tial L´ evy mo dels with jumps, [ 16 ] deriv e high-order A TM price expansions for broad temp ered-stable classes, [ 15 ] obtain small-time expansions for distributions, densities, and option prices in sto chastic v olatility mo dels with L ´ evy jumps, and [ 18 ] extend close-to-the-money results in the presence of ∗ Pen um bra In vestmen t Group, New Y ork, NY 10020, USA ( allen.hoffmeyer@gmail.com ). † School of Mathematics, Georgia Institute of T echnology , Atlan ta, GA 30332, USA ( houdre@math.gatech.edu ). ‡ Research supp orted in part by grants #524678 and MP-TSM-00002660 from the Simons F oundation. 1 sto c hastic v olatility . Additional related dev elopmen ts include mo del-engineering and calibration persp ectiv es for exponential L´ evy surfaces [ 1 ], estimation adv ances for temp ered-stable mo dels of infinite v ariation [ 14 ], and short-time implied-v olatility limits for additiv e tempered-stable and allied pro cesses [ 3 ]. (F or comparison with the diffusion literature on near-expiry A TM behavior, see e.g. [ 11 ] and references therein.) Belo w, w e develop near-expiry asymptotic expansions for A TM call prices and the corresponding implied v olatility under exponential L´ evy dynamics, with an emphasis on stable domains of attraction and regular v ariation. Under mild structural assumptions placing the driving L ´ evy pro cess in the domain of attraction of a (possibly asymmetric) stable la w, we obtain first-order A TM expansions that unify and streamline existing results while cov ering case s beyond the standard temp ered-stable families. The form ulation highligh ts the role of regular v ariation of the L ´ evy measure near the origin. This article is organized as follows. Section 2 dev elops the domain-of-attraction and regular-v ariation framew ork needed for the sequel, and establishes that b oth the stable domain of attraction and the finite- ness of the cen tering constan t are preserv ed under the share measure transformation. Section 3 states and pro ves the main first-order A TM call-price and implied v olatility asymptotics, treating the pure-jump and Bro wnian-comp onen t cases separately . Section 4 presents an explicit exponential L ´ evy mo del whose first- order con v ergence rate inv olves a logarithmic correction to the classical scaling. Section 5 concludes and discusses directions for future work. Appendix A collects the regular-v ariation bac kground used throughout, and Appendix B contains the pro ofs. W e start b y briefly recalling the basic material on L´ evy processes and exponential-L ´ evy mo dels used in what follo ws; see [ 2 , 8 , 26 ] for comprehensiv e accoun ts. A sto c hastic process ( X t ) t ≥ 0 on (Ω , F , P ) with v alues in R is a L ´ evy pr o c ess if it has stationary and indep enden t incremen ts, c` adl` ag paths, and X 0 = 0 a.s. Ev ery L´ evy process is uniquely c haracterized b y its char acteristic triplet ( b, σ , ν ), where b ∈ R , σ ≥ 0, and ν is a p ositiv e Borel measure on ( R , B ( R )) without atom at the origin satisfying R R (1 ∧ x 2 ) ν ( dx ) < ∞ . Hence E [ e iuX t ] = e tψ ( u ) where the c haracteristic exponent is ψ ( u ) = iub − 1 2 σ 2 u 2 + Z R  e iux − 1 − iux 1 {| x |≤ 1 }  ν ( dx ) , u ∈ R . This is compactly written as X t ∼ L ( b, σ, ν ). Under mild conditions, X t can be written as X t = bt + σ W t + Z | x |≤ 1 x ˜ N ( t, dx ) + Z | x | > 1 x N ( t, dx ) , where W is a Bro wnian motion, N is a Poisson random measure with intensit y ν ( dx ) dt , and ˜ N is its comp en- sated v ersion. If P ∗ is an equiv alen t measure defined by d P ∗ d P    F t = exp  θ X t − tψ ( − iθ )  , then ( X t ) t ≥ 0 remains a L ´ evy pro cess under P ∗ , with transformed triplet ( b θ , σ, ν θ ) given b y the Esscher transform ν θ ( dx ) = e θx ν ( dx ) and b θ = b + σ 2 θ + R | x |≤ 1 x ( e θx − 1) ν ( dx ). Suc h c hanges of measure appear naturally in risk-neutral v aluation. In an exp onen tial L ´ evy mo del the asset price is S t = S 0 e X t , X t ∼ L ( b, σ, ν ) , so that E [ S t ] = S 0 e tψ ( − i ) , where ψ is the L´ evy–Khintc hine exp onen t asso ciated with the triplet ( b, σ, ν ) ev aluated at − i . Since ψ is a priori defined only for real arguments, ev aluating it at − i requires justification. F or the exponential moment E  e X t  to be finite it is necessary and sufficien t that Z {| y | > 1 } e y ν ( dy ) < ∞ , (1.1) whic h w e assume throughout. Under this condition, the L´ evy–Khin tchine integral con verges when u is replaced b y u + iz for any z ∈ [0 , 1], so ψ extends analytically to the strip { u ∈ C : − 1 ≤ ℑ ( u ) ≤ 0 } ; in particular 2 − i lies in this strip and ψ ( − i ) is well defined (see [ 26 ], Theorem 25.17). Under a risk-neutral measure the discoun ted asset e − rt S t m ust be a martingale, so E [ e X t ] = e rt and hence ψ ( − i ) = r . In this paper w e work in the zero-in terest (or forw ard) case r = 0, so that ψ ( − i ) = 0 and the drift parameter is fixed b y the martingale condition b = − σ 2 2 − Z ∞ −∞  e y − 1 − y 1 {| y |≤ 1 }  ν ( dy ) . (1.2) The abov e con ven tions (triplet ( b, σ , ν ), truncation x 1 | x |≤ 1 , and characteristic exp onen t ψ ) are main tained throughout the pap er. The asymptotics of at-the-money option prices and implied volatilit y are the main ob jects of study in this man uscript. F or this purpose, we discuss a few results that will b e necessary in deriving the first-order asymptotics. First, w e are interested in the short-time behavior of the at-the-money call price c ( t, 0) = E  e X t − 1  + , in terpreted under the risk-neutral measure. T o this end, we use a slightly more conv enient represen tation of the function c due to [ 7 ] (see also [ 13 ]). W e work with the share measure P ∗ obtained from the Essc her transform with parameter θ = 1 (so that d P ∗ /d P | F t = e X t when ψ ( − i ) = 0), whic h satisfies, for all Borel sets D ⊂ R , P ∗ ( D ) = E e X t 1 D . With this approach, [ 7 ] sho wed the follo wing. Theorem 1.1. Under P ∗ , let E b e a me an 1 exp onential r andom variable that is indep endent of ( X t ) t ≥ 0 . Then, 1 S 0 E ( S t − K ) + = P ∗  X t − E > log  K S 0  . (1.3) Corollary 1.2. L etting K = S 0 , the normalize d, at-the-money Eur op e an c al l option pric e has r epr esentation c ( t, 0) = 1 S 0 E ( S t − S 0 ) + = Z ∞ 0 e − x P ∗ ( X t ≥ x ) dx. (1.4) These last tw o results will b e used to derive the first-order asymptotic b eha vior of c ( t, 0) as t ↓ 0 for a wide class of L´ evy models. 2 Stable Domains of Attraction Stable domains of attraction play a central role in the short-maturit y b ehavior of at-the-money option prices. In muc h of the existing literature (see, e.g., [ 13 , 16 , 17 , 25 , 27 ]), the underlying L´ evy pro cess ( X t ) t ≥ 0 is assumed to satisfy a small-time stable limit of the form X t − A t B t ⇒ Z, t ↓ 0 , (2.1) where Z is an α -stable random v ariable with α ∈ (0 , 2], and ⇒ denotes conv ergence in distribution. In particular, in the pure-jump case with α ∈ (1 , 2) one t ypically has B t = t 1 /α up to a slowly v arying factor. Our goal in this section is to mak e precise the regular-v ariation assumptions under whic h ( 2.1 ) holds, and to describ e the associated scaling B t in sufficien t generalit y for the option-pricing applications that follow. 2.1 Preliminaries In this subsection w e collect the minimal regular v ariation notation that will be used throughout the rest of the paper. Our goal is to c haracterize when the small-time rescaling ( 2.4 ) of a L´ evy pro cess con verges to a 3 non-Gaussian stable la w and to iden tify the corresp onding scaling B t in terms of the b eha vior of the L´ evy measure near the origin. Regular v ariation pro vides the natural language for describing this lo cal behavior. A real-v alued function f is r e gularly varying of index ρ at ∞ if for ev ery λ ∈ (0 , ∞ ), lim x →∞ f ( λx ) f ( x ) = λ ρ . W e denote this as f ∈ RV ∞ ρ . Similarly , a real-v alued function ℓ is slow ly varying at ∞ if for every λ ∈ (0 , ∞ ), lim x →∞ ℓ ( λx ) ℓ ( x ) = 1 , and this is denoted by ℓ ∈ RV ∞ 0 . Finally , a function f is r e gularly varying at 0 (from the righ t) with index ρ if f  1 ·  ∈ RV ∞ − ρ . W e denote this by writing f ∈ RV 0 ρ . W e will sometimes drop the sup erscript and write, e.g., f ∈ RV ρ if it is clear from con text whether the function v aries regularly (or slo wly) at ∞ or 0. W e write f ( x ) ∼ g ( x ) as x → a to mean lim x → a f ( x ) /g ( x ) = 1. W e no w in tro duce some notation for X t ∼ L ( b, σ , ν ). F or x > 0, let γ ( x ) = γ + ( x ) + γ − ( x ) := ν ( { y > x } ) + ν ( { y < − x } ) , (2.2) and V ( x ) := Z | y |≤ x y 2 ν ( dy ) . (2.3) Here γ is the tw o-sided tail function of the L´ evy measure and V is the truncated second momen t of jumps of size at most x ; in particular, V enco des the con tribution of small jumps to the quadratic v ariation of X . Next, w e are mainly in terested in conditions under which X t − A t B t (2.4) con verges in distribution to an α -stable distribution, α ∈ (1 , 2), as t → 0 where A : [0 , ∞ ) → R and B : (0 , ∞ ) → (0 , ∞ ) are functions with lim t → 0 B t = 0. If the L ´ evy pro cess has finite second moment, then the Cen tral Limit Theorem (or the L´ evy-Khintc hine form ula) gives X t − t E X 1 √ t ⇒ N (0 , σ 2 ) , as t → 0. This is formalized as follo ws: Definition 2.1. A stochastic process ( X t ) t ≥ 0 on a probabilit y space (Ω , F , P ) is in the domain of attr action (DO A) of a stable random v ariable Z at a ∈ { 0 , ∞} if there exist functions A and B with A t ∈ R , B t > 0 for ev ery t ≥ 0 suc h that X t − A t B t ⇒ Z, (2.5) as t → a . Throughout, we are exclusiv ely concerned with the small-time case a = 0, so all domains of attraction will b e at 0. 2.2 Stable domains of attraction for L´ evy pro cesses W e now fo cus on domains of attraction in the L´ evy setting, following [ 22 ] and [ 19 ]. In particular, we recall the conditions from [ 22 ], which characterize when a L ´ evy pro cess is in the domain of attraction of a stable law at 0. Theorem 2.2. The fol lowing ar e e quivalent: 4 (i) ther e exist r e al-value d functions A and B with B t > 0 for t > 0 and lim t → 0 B t = 0 such that X t − A t B t ⇒ Z, as t → 0 , wher e Z is an a.s. finite, nonde gener ate r andom variable (in fact, Z is ne c essarily an α -stable r andom variable with α ∈ (0 , 2] ); (ii) either (a) the function γ ∈ RV 0 − α with α ∈ (0 , 2) and the limits p ± := lim x ↓ 0 γ ± ( x ) γ ( x ) , (2.6) exist or (b) V is slow ly varying at 0 . Note that in ( ii ) abov e, exactly one of ( a ) or ( b ) holds. The case where V is slowly v arying at 0 corresponds to the case where the cen tral limit theorem applies with B t = √ t and Z ∼ N  0 , σ 2  . Also, although the statemen t of the theorem does not require it, the function B can b e chosen monotone decreasing. Finally , the ab o v e theorem sho ws that the w eak conv ergence of expressions lik e ( 2.5 ) is necessarily tow ards an α -stable random v ariable, 0 < α ≤ 2, if ( X t ) t ≥ 0 is a L´ evy process. Additionally , the authors show ed that for x > 0, defining µ x := b − Z x ≤| y |≤ 1 y ν ( dy ) , (2.7) and U ( x ) := 2 Z x 0 y γ ( y ) dy , (2.8) the functions B t = inf  0 < x ≤ 1 : x − 2 U ( x ) ≤ 1 t  and A t = tµ B t , will do. In particular, A t = O ( t ) when sup 0 0 is a constan t. In what follo ws, w e require that the L ´ evy pro cess ( X t ) t ≥ 0 satisfies E  | X t | e X t  < ∞ for all t ≥ 0, equiv alently (see [ 26 ], Theorem 25.3) that Z | x | > 1 | x | e x ν ( dx ) < ∞ . (2.10) Additionally , we assume that ν has a densit y with resp ect to Lebesgue measure, i.e., that there exists a function ξ ≥ 0 suc h that for any Borel set D , ν ( D ) = Z D ξ ( x ) dx. Note that P ∗ , which is the natural measure under which at-the-money call prices take the form of exp ectations of X t -functionals, is well-defined b y ( 2.10 ) and Example 33.14 in [ 26 ]. It will be the k ey measure for our short-time option price asymptotics. W e assume throughout this section that σ = 0, so that X is a pure-jump L ´ evy process. Under P ∗ , the pro cess ( X t ) t ≥ 0 is again a L ´ evy pro cess with triplet ( b ∗ , 0 , ν ∗ ) where ν ∗ ( dx ) = e x ν ( dx ) = e x ξ ( x ) dx, (2.11) 5 and b ∗ = b + Z | x |≤ 1 ( ν ∗ − ν ) ( dx ) = b + Z | x |≤ 1 ( e x − 1) ξ ( x ) dx. (2.12) Throughout this section, w e define quan tities under both P and P ∗ , using the star notation to denote the corresp onding quantit y under P ∗ . F or instance, µ ε is defined in ( 2.7 ) and its share-measure counterpart is µ ∗ ε := b ∗ − Z ε ≤| x |≤ 1 x ν ∗ ( dx ) . The star-measure analogue of γ from ( 2.2 ) is γ ∗ ( x ) := Z | y | >x ν ∗ ( dy ) , for x > 0. W e no w define a few more functions and constan ts that we will need thereafter. F or x > 0, set ξ S ( x ) := ξ ( x ) + ξ ( − x ) , (2.13) while ξ ∗ S ( x ) := e x ξ ( x ) + e − x ξ ( − x ) . (2.14) W e are in terested in the quan tities ¯ µ = sup 0 <η < ∞ | µ η | and ¯ µ ∗ = sup 0 <η < ∞   µ ∗ η   , whic h w e will need to be finite. Note that sup 1 ≤ η < ∞ | µ η | = sup 1 ≤ η < ∞      b + Z 1 ≤| x |≤ η xν ( dx )      ≤ | b | + Z 1 ≤| x | < ∞ | x | ν ( dx ) < ∞ , and sup 1 ≤ η < ∞   µ ∗ η   = sup 1 ≤ η < ∞      b ∗ + Z 1 ≤| x |≤ η xν ∗ ( dx )      ≤ | b ∗ | + Z 1 ≤| x | < ∞ | x | ν ∗ ( dx ) < ∞ , since the L´ evy process ( X t ) t ≥ 0 has finite first momen t under b oth P and P ∗ . So, w e need only consider when the quan tities ¯ µ 0 = sup 0 <η ≤ 1 | µ η | and ¯ µ ∗ 0 = sup 0 <η ≤ 1   µ ∗ η   , are finite (e.g. when ν is symmetric, see [ 13 ]). 2.3 Share measure inv ariance and the normalizing function First, w e inv estigate ho w the share measure transformation affects the regular v ariation property of the L ´ evy measure. In tuitively , this preserv ation of regular v ariation stems from the fact that the prop ert y dep ends only on the behavior of ν near the origin, and the transformed L´ evy measure ν ∗ ( dx ) = e x ν ( dx ) has the same lo cal b eha vior for x close to 0 because e x ≈ 1 there. Prop osition 2.3. If ν is r e gularly varying of index α at 0 , then ν ∗ is also r e gularly varying of index α at 0 . As a ma jor consequence of the pro of of Prop osition 2.3 , the constants p + and p − in ( B.5 ) and ( B.6 ) remain unc hanged under the measure transform. This fact, com bined with ( 2.9 ), gives the follo wing result. Prop osition 2.4. L et ( 2.5 ) hold with r esp e ct to P and thus with r esp e ct to P ∗ , wher e Z is an α -stable r andom variable with α ∈ (0 , 2) . Then Z has the same r epr esentation under b oth P and P ∗ . That is, the p ar ameters of the stable distribution Z ar e the same under b oth pr ob ability me asur es P and P ∗ . 6 In summary , the small-time stable domain of attraction of X is completely determined by the regular v ariation of the L´ evy measure near the origin, and this structure is pres erv ed by the share measure change. In particular, the same limiting α -stable random v ariable Z gov erns the fluctuations of X t under both P and P ∗ , so all of the distributional input needed for the at-the-money option price expansions can b e read off from the behavior of ν at zero. Next, we show that the finiteness of the constant ¯ µ is also a prop ert y that surviv es the share measure transformation. This quantit y will b e imp ortan t for Theorem 3.2 , whic h is one of the main new results of the pap er. Prop osition 2.5. ¯ µ < ∞ if and only if ¯ µ ∗ < ∞ , e quivalently ¯ µ 0 < ∞ if and only if ¯ µ ∗ 0 < ∞ . The previous prop ositions sho w that b oth the stable domain-of-attraction structure and the finiteness of ¯ µ are in v arian t under the share measure transformation. In particular, the same strictly α -stable random v ariable Z (and the same centering constant ¯ µ whenev er it is finite) go vern the small-time behavior of X t under b oth P and P ∗ . W e are now in a position to connect this structure to the behavior of the normalizing function B t and hence to the short-time asymptotics of the at-the-money call price. Previous results on A TM call prices (e.g., [ 13 ], [ 27 ], and [ 25 ]) considered only the case B t = κt 1 /α whenev er 1 < α < 2, where κ > 0 is a constant. As w e show in Section 3 , the framew ork developed here yields more general asymptotics for A TM call option prices. It is known (see [ 12 ], [ 19 ], and [ 23 ]) that the rate function B ∈ R V 0 1 /α whenev er the conv ergence is tow ards an α -stable random v ariable. Our goal for the remainder of this subsection is to further pin down the behavior of B when ( 2.5 ) is satisfied. Throughout, w e use the notation β t := 1 /B t for con v enience. Assume that ( 2.5 ) holds for the L´ evy process ( X t ) t ≥ 0 under the measure P (and hence also under P ∗ ). Th us, the L´ evy measures ν and ν ∗ are regularly v arying with index α > 0 at 0, and we further assume that α ∈ (1 , 2). Since γ ∗ is regularly v arying at 0 of order − α , it has represen tation γ ∗ ( x ) = x − α ℓ  1 x  , (2.15) for all x > 0, where ℓ is slowly v arying at ∞ (equiv alently , we can tak e γ ∗ ( x ) = x − α ˜ ℓ ( x ) where ˜ ℓ is slowly v arying at 0). W e deal with γ ∗ rather than γ as most of our calculations will be done with the quan tities under the measure P ∗ . The representation for γ ∗ is derived in the follo wing w ay . First, note that γ ∗ ( · ) ∈ RV 0 − α if and only if γ ∗  1 ·  ∈ RV ∞ α , and then the representation theorem for regularly v arying functions gives ( 2.15 ). Note that ℓ is asymptotically controlled near ∞ b y its slow v ariation, but its behavior near 0 is essen tially unconstrained apart from the monotonicity of γ ∗ and the integrabilit y condition R 1 − 1 x 2 ν ∗ ( dx ) < ∞ . W e can simplify ho w to lo ok at ( 2.5 ) b ecause we do not need the additive correction term. In [ 22 ], the authors sho w that A t can be tak en to b e O ( t ) (recall that α ∈ (1 , 2)). So, under P ∗ β t ( X t − A t ) ⇒ Z, as t → 0. Recall that β t ∈ RV 0 − 1 /α so that, again b y the representation theorem, β t = t − 1 /α ζ (1 /t ) , as t → 0, where ζ is slowly v arying at ∞ . Also, for some absolute constan t C > 0 we hav e | A t β t | ≤ C tβ t = C t 1 − 1 /α ζ (1 /t ) . The function s 1 /α − 1 ζ ( s ) is regularly v arying at ∞ with index 1 /α − 1 < 0. Standard results (e.g. Prop osition 1.3.6(v) in [ 4 ]) imply that s 1 /α − 1 ζ ( s ) → 0 as s → ∞ , whic h sho ws that A t β t → 0 as t → 0. So, we need only lo ok at the conv ergence β t X t ⇒ Z, (2.16) as t → 0 under P ∗ . The follo wing result makes precise the relationship betw een the normalizing function β t and the slowly v arying part of the share-measure tail γ ∗ . 7 Theorem 2.6. L et x 0 > 0 , and let x 2 ξ ∗ S ( x ) b e monotone (incr e asing or de cr e asing) for 0 < x < x 0 . Then lim t → 0 tβ α t ℓ ( β t ) = Λ α , wher e Λ α is a p ositive c onstant dep ending only on α . R emark 2.7 . The constan t Λ α encapsulates all the slowly v arying structure of ℓ ; once γ ∗ ( x ) = x − α ℓ (1 /x ) is fixed near 0, the normalization β t is determined up to this factor b y the relation t − 1 ∼ β α t ℓ ( β t ). This is the k ey link b et w een the regular v ariation of the L´ evy measure and the first-order at-the-money asymptotics obtained later. 3 First-Order A TM Asymptotics W e now state the main pricing results of the paper. W e main tain all of the assumptions and notation from Section 2 and add the follo wing: (A1) ν is regularly v arying of order − α , α ∈ (1 , 2), at 0. (A2) There exist C > 0 and x 1 > 0 (p ossibly depending on each other) such that R | y |≤ x y 2 e y ξ ( y ) dy ≤ C x 2 R | y | >x e y ξ ( y ) dy for all x ≥ x 1 . (A3) There exists x 0 > 0 suc h that x 2 ξ ∗ S ( x ) is monotone (either increasing or decreasing) for 0 < x < x 0 . W e first make some brief commen ts on the requiremen ts (A1) – (A2) . Assumption (A1) is the standard regular–v ariation condition placing X in the strictly α –stable domain of attraction and determining the small– time scaling B t . Assumption (A2) supplies the balance needed to apply the results of [ 21 ] under the share measure. Their result do es not require (A2) in general, but it is precisely the hypothesis ensuring that the truncated second moment of the share-measure densit y is negligible relative to its tail, whic h is the only part of their inequality used in the pro of. R emark 3.1 . The primary tec hnical input is Assumption (A3) , which imposes a lo cal regularity condition on the symmetrized share-measure densit y ξ ∗ S . W e assume that x 2 ξ ∗ S ( x ) is ev entually monotone as x ↓ 0, allo wing us to apply a monotone densit y theorem and deduce the k ey tw o-sided estimate ( B.16 ) linking ξ ∗ S with the tail γ ∗ S ( x ) := R ∞ x ξ ∗ S ( u ) du . Several alternativ e conditions yield the same conclusion. The classical monotone densit y theorem ([ 4 ], Theorem 1.7.2) shows that if γ ∗ S ∈ R V − α and ξ ∗ S is ultimately monotone, then ξ ∗ S ∈ R V − α − 1 (pro vided α > 0). More generally , the O–version of the monotone densit y theorem ([ 4 ], Prop. 2.10.3) replaces monotonicity with bounded increase/decrease or finite Matuszewsk a index assumptions and still yields ξ ∗ S ( x ) ≍ γ ∗ S ( x ) x , x ↓ 0 , whic h, when γ ∗ S ∈ R V − α , implies precisely the estimate ( B.16 ) (where for functions f and g , f ≍ g if and only if f = O ( g ) and g = O ( f )). One ma y also inv oke the smooth-v ariation construction ([ 4 ], Section 1.8), whic h sho ws that an y regularly v arying tail is asymptotically equiv alent to a smoothly v arying representativ e whose deriv ative is regularly v arying of index − α − 1. Th us (A3) is simply a conv enien t sufficien t condition: the pro of requires only an estimate of the form ( B.16 ) for ξ ∗ S near the origin, and any of the monotone, O–regular v ariation, or smooth-v ariation assumptions would suffice. Theorem 3.2. Along with the c onditions (A1) – (A3) , assume that ¯ µ < ∞ , (3.1) and let Z b e the α -stable r andom variable fr om ( 2.16 ) . Then an A TM Eur op e an c al l option has asymptotic exp ansion E ( S t − S 0 ) + = ( S 0 E ∗ Z + ) B t + o ( B t ) , (3.2) as t → 0 . 8 Corollary 3.3. Under the assumptions of The or em 3.2 , ˆ σ , the implie d volatility of an A TM c al l option, is such that ˆ σ ( t ) = √ 2 π B t √ t E ∗ Z + + o  B t √ t  , (3.3) as t → 0 . W e no w consider exp onen tial L ´ evy mo dels whose driving pro cess has a nonzero Gaussian com ponent. Let X b e a L´ evy process with triplet ( b, σ, ν ) under the risk–neutral measure, so that ( S t ) t ≥ 0 =  S 0 e X t  t ≥ 0 is a martingale. When σ > 0 the path decomp osition X t L = σ W t + L t , holds for ev ery fixed t ≥ 0, where W is a standard Brownian motion and L is a pure–jump L´ evy pro cess with triplet ( b, 0 , ν ). The equalit y in law does not imp ose an y indep endence assumption b et w een W and L and is used only to separate the diffusiv e and jump con tributions at small times. The martingale condition ψ ( − i ) = 0 fixes the drift b so that E [ e X t ] = 1. Theorem 3.4. L et ( X t ) t ≥ 0 b e a L´ evy pr o c ess with triplet ( b, σ, ν ) under the risk–neutr al me asur e, wher e σ > 0 . Then for every t ≥ 0 the pr o c ess admits the distributional de c omp osition X t L = bt + σ W t + L t , wher e W is a standar d Br ownian motion, L is a pur e–jump L´ evy pr o c ess with triplet (0 , 0 , ν ) , and b is chosen so that E [ e X t ] = 1 . Supp ose that L satisfies the hyp otheses of The or em 3.2 . Then the at-the-money c al l pric e is such that E ( S t − S 0 ) + = S 0 σ √ t E ∗ ( W 1 ) + + o ( √ t ) , t ↓ 0 . Theorems 3.4 and 3.5 sho w that, under the assumptions of Theorem 3.2 , the jump comp onen t is alw ays o ( √ t ) regardless of its structure, so the leading √ t behavior of the A TM call price is universal and driv en en tirely by the Gaussian part of the triplet. Theorem 3.5 b elo w gives a self-con tained form ulation of this principle. Theorem 3.5. L et L = ( L t ) t ≥ 0 b e a L´ evy pr o c ess with triplet ( b, 0 , ν ) such that E  | L 1 | e L 1  < ∞ and let S t = S 0 e L t . L et ther e exist B t > 0 with B t → 0 as t → 0 , α ∈ (1 , 2) , and a pr ob ability me asur e P ∗ such that 1 B t E ( S t − S 0 ) + → E ∗ Z + , and P ∗ ( L t ≥ B t u ) → P ∗ ( Z ≥ u ) , for every u ≥ 0 , wher e Z is an α -stable r andom variable under P ∗ . F or t ≥ 0 , let ( X t ) t ≥ 0 b e given by X t = σ W t + L t wher e W = ( W t ) t ≥ 0 is a Br ownian motion indep endent of L under P ∗ . If B t √ t → 0 , as t → 0 , and if ( S t ) t ≥ 0 =  S 0 e X t  t ≥ 0 is a martingale with r esp e ct to its own filtr ation, then c ( t, 0) = σ E ∗ ( W 1 ) + √ t + o  √ t  , (3.4) as t → 0 . Finally , w e obtain the implied volatilit y asymptotics in a similar w ay to the case without Bro wnian com- p onen t. Corollary 3.6. Under the hyp otheses of The or em 3.4 , ˆ σ , the implie d volatility, is such that ˆ σ ( t ) = σ √ 2 π E ∗ ( W 1 ) + + o (1) , as t → 0 . 9 4 Example: Beyond P o w er-La w Con v ergence Rates The results of Section 3 sho w that new first-order dynamics are p ossible; to illustrate this, we no w present a simple example that satisfies the assumptions of Theorem 3.2 and for whic h w e can compute the rate of con vergence for the call option price and implied volatilit y explicitly . Consider the L´ evy measure on R ν ( dx ) = ( 0 , x < − 1 or x = 0 | x | − α − 1 e − x | ln | x || dx, x ≥ − 1 and x  = 0 , (4.1) with α ∈ (1 , 2), and let ( X t ) t ≥ 0 b e the L ´ evy process with triplet ( b, 0 , ν ), with b c hosen to satisfy the martingale condition ( 1.2 ). Then, γ ∗ ( x ) =      2 α x − α  ln  1 x  − 1 α  + 3 α 2 , 0 < x < 1 1 α 2 , x = 1 1 α x − α  ln x + 1 α  , x > 1 , (4.2) and so γ ∗ is regularly v arying of order − α at both 0 and ∞ . F rom the form of γ ∗ and the fact that γ ∗ is regularly v arying at both 0 and ∞ , w e immediately v erify conditions (A1) and (A2) . The requirement ( 3.1 ) holds simply because the L ´ evy measure is symmetric about 0 for | x | < 1. Finally , the expression in (A3) b ecomes, for 0 < x < 1, 2 x 1 − α ln  1 x  , whic h is easily seen to b e monotone. It is worth noting that verifying (A3) is not strictly necessary here, since its role in Theorem 3.2 is to establish the form of the L ´ evy densit y near the origin, which we hav e directly from ( 4.1 ). All the hypotheses of Theorem 3.2 are now satisfied, and w e pro ceed to determine the b ehavior of β t as t → 0. Again, B t = 1 /β t can b e expressed as t 1 /α ˜ ℓ (1 /t ) for t > 0, where ˜ ℓ is slo wly v arying at ∞ . F urther, there exists Λ > 0 suc h that tβ α t ℓ ( β t ) → Λ , as t → 0, where ℓ is the slowly v arying part of γ ∗ near 0. The functions ℓ and β are defined up to asymptotic equiv alence, so we use asymptotic versions that are simpler to manipulate. F rom ( 4.2 ), the slowly v arying factor satisfies ℓ ( y ) ∼ 2 α ln y as y → ∞ , so Theorem 2.6 giv es tβ α t ln β t → Λ , for some Λ > 0 as t → 0, whic h is equiv alent to tβ α t ln β α t → α Λ . (4.3) W riting f ( x ) = x log x , w e rewrite ( 4.3 ) as tf ( β α t ) → α Λ for t → 0. F urthermore, w e ha ve that β α t ∼ f − 1 ( α Λ /t ), as t → 0 (the function f has an inv erse for x large enough and β α t gro ws large as t → 0). Inv erting f requires the Lam b ert W function: if f ( x ) = x log x , then f − 1 ( y ) = y /W ( y ) where W denotes the principal branc h of the Lambert function (we use upper-case W here only; it will not b e confused with Bro wnian motion in this section since the example has no Gaussian comp onen t). Since W ( y ) ∼ log y as y → ∞ (see e.g. [ 9 ]), f − 1 ( y ) ∼ y log y , y → ∞ . Applying this to β α t ∼ f − 1 ( α Λ /t ) giv es β α t ∼ α Λ /t log( α Λ /t ) , 10 and therefore B t = 1 β t = α Λ t log  α Λ t  ! 1 /α . Ignoring the constants, the first-order rate of con v ergence is therefore  t log (1 /t )  1 α . Note that we could get even more in teresting b eha vior by in tro ducing further slowly v arying function b eha vior near the origin (e.g., b ounded oscillatory terms lik e cos(log(1 /x ))). This example illustrates the basic mechanism for generating new conv ergence rates: once γ ∗ ( x ) is sp ecified near the origin with a slo wly v arying mo difier, the normalization B t and hence the first-order call price asymp- totics are determined b y solving the relation t − 1 ∼ B − α t ℓ (1 /B t ), whic h in turn can inv olv e logarithmic, log–log, oscillatory , or mixed slow v ariation. By choosing different slowly v arying factors ℓ (such as additional iterated logarithms or b ounded oscillatory terms), one can construct L ´ evy models whose at-the-money conv ergence rates differ from the classical t 1 /α b eha vior in a con trolled w ay . The de Bruijn conjugate metho dology then pro vides the precise form of B t in eac h case. 5 Conclusion and F uture W ork In this article w e developed a regular–v ariation framework for the short-maturity b eha vior of at-the-money call options in exp onen tial L´ evy mo dels and used it to obtain new orders of conv ergence. Section 2 established that the stable domain-of-attraction structure and the finiteness of the cen tering constan t ¯ µ are preserv ed under the share measure, and Section 3 used these facts to sho w that, under mild assumptions placing the driving L´ evy pro cess in the domain of attraction of an α -stable law with α ∈ (1 , 2), the at-the-money call price admits the expansion E ( S t − S 0 ) + =  S 0 E ∗ Z +  B t + o ( B t ) , t ↓ 0 , where B t is the normalizing function from the stable limit and Z is the limiting stable random v ariable. All of the distributional input needed for this expansion can b e read off from the regular v ariation of the L´ evy measure near zero. Theorem 2.6 links the small-time scale B t to the de Bruijn conjugate of the slowly v arying factor in the share-measure tail, and Theorem 3.2 then translates this in to a general A TM asym ptotic v alid for a broad class of pure-jump exp onen tial L´ evy mo dels. When a nonzero Bro wnian comp onen t is present, Theorems 3.4 and 3.5 show that the jump contribution is alw ays low er order, so that the leading √ t b eha vior of the A TM call price is universal and driv en en tirely b y the Gaussian part of the triplet. Within this framework we constructed an explicit example for whic h the first-order scale B t deviates from the classical t 1 /α rate b y a logarithmic correction. By sp ecifying the share-measure tail γ ∗ ( x ) = x − α ℓ (1 /x ) with a slowly v arying factor ℓ in volving log(1 /x ), w e obtained a normalization of the form B t ∼  α Λ t log( α Λ /t )  1 /α , and hence an A TM con v ergence rate driven by t 1 /α up to a precise logarithmic p enalt y . This example illustrates ho w stable domains of attraction combined with regular v ariation and de Bruijn conjugates can b e used to design mo dels with con trolled, nonstandard short-maturit y b eha vior while still retaining analytic tractabilit y . F uture W ork Sev eral extensions of the presen t w ork appear natural. • Higher-or der and off-A TM exp ansions. The domain-of-attraction framework used here is well suited to studying higher-order corrections and small-log-moneyness regimes beyond pure at-the-money options. Extending the analysis to join t expansions in time and moneyness—for example, in the regime where log-moneyness scales with B t —w ould connect the present results to the full short-maturity smile. 11 • R elaxing structur al assumptions. Assumptions such as the monotonicit y of x 2 ξ ∗ S ( x ) and the specific in te- grabilit y condition (A2) are technically con venien t but p oten tially stronger than necessary . It w ould be of in terest to replace these b y w eaker local regularity or oscillation conditions while still retaining con trol of B t and the A TM asymptotics, thereby enlarging the admissible class of L´ evy measures. A Regular V ariation W e giv e a brief ov erview of regular v ariation and direct the reader to [ 4 ] for a more comprehensiv e treatment. First, w e in troduce the notion of a slowly v arying function. Definition A.1. A nonnegative, measurable function ℓ defined on some neighborho od [ M , ∞ ) of infinit y ( M > 0) is slow ly varying at ∞ if for every λ > 0, w e hav e lim x →∞ ℓ ( λx ) ℓ ( x ) = 1 . (A.1) The next theorem allo ws us to ha ve a con venien t functional form for any slowly v arying function. Theorem A.2. (R epr esentation The or em) A function ℓ is slow ly varying if and only if it c an b e written in the form ℓ ( x ) = h ( x ) exp  Z x a ε ( u ) du u  , for x ≥ a wher e a > 0 , h is me asur able with lim x →∞ h ( x ) = h ∈ (0 , ∞ ) , and ε is such that lim x →∞ ε ( x ) = 0 . Prop osition A.3. L et ℓ, ℓ 1 , and ℓ 2 b e slow ly varying functions. Then: (i) The function ℓ α is slow ly varying for every α ∈ R . (ii) The functions ℓ 1 ℓ 2 , ℓ 1 + ℓ 2 , and (if ℓ 2 ( x ) → ∞ as x → ∞ ) ℓ 1 ◦ ℓ 2 ar e al l slow ly varying. (iii) F or any α > 0 , x α ℓ ( x ) → ∞ and x − α ℓ ( x ) → 0 , (A.2) as x → ∞ . W e are no w in a position to in troduce the concept of a regularly v arying function. Definition A.4. Let f b e a positive, measurable function. W e say that f is r e gularly varying at ∞ if an y one of the follo wing conditions holds. (i) The limit lim x →∞ f ( λx ) f ( x ) = g ( λ ) ∈ (0 , ∞ ) , (A.3) exists for every λ ∈ (0 , ∞ ). (ii) The limit ( A.3 ) exists for every λ ∈ S where S is either a p ositiv e-measure subset of (0 , ∞ ) or a dense subset of (0 , ∞ ). (iii) The limit ( A.3 ) exists and equals g ( λ ) = λ ρ for some ρ ∈ R . (iv) The function f has representation f ( x ) = x ρ ℓ ( x ) , (A.4) where ρ ∈ R and ℓ is slo wly v arying at ∞ . If an y of the ab o ve hold, we write f ∈ R V ∞ ρ where ρ is the real found in ( A.4 ). W e use the notation ℓ ∈ R V ∞ 0 if the function ℓ is slo wly v arying at ∞ . 12 The regular v ariation prop erty of a function defines how f behav es near ∞ . Since only the asymptotic b eha vior as x → ∞ (or x → 0) matters, the v alues of f on any fixed compact in terv al ma y b e c hosen arbitrarily , and we assume f is locally b ounded on [0 , ∞ ). F or our purposes, w e will also be concerned with functions that are regularly v arying at the origin. Definition A.5. A function f is r e gularly varying at 0 (from the right) with index ρ if f  1 ·  ∈ RV ∞ − ρ . W e denote this by writing f ∈ R V 0 ρ . Com bining ( A.4 ) with the representation theorem for slowly v arying functions abov e giv es the standard represen tation for a regularly v arying function. That is, f is regularly v arying at ∞ if and only if it has represen tation f ( x ) = x ρ h ( x ) exp  Z x a ε ( u ) du u  , (A.5) for x ≥ a where ρ ∈ R , h ≥ 0 is a measurable function with a p ositive, finite limit at ∞ , and ε is a measurable function suc h that lim x →∞ ε ( x ) = 0. The follo wing corollary is clear. Corollary A.6. L et f b e r e gularly varying with index ρ ∈ R at ∞ . Then, lim x →∞ f ( x ) = ( ∞ , if ρ > 0 0 , if ρ < 0 . There are imp ortan t results concerning regularly v arying functions that w e will need to exploit. These results can b e found in [ 4 ], and w e list them here without pro of. Theorem A.7. (Potter’s The or em) (i) If ℓ is slow ly varying, then for any given c onstants A > 1 and δ > 0 ther e exists x 0 = x 0 ( A, δ ) such that ℓ ( y ) ℓ ( x ) ≤ A max   y x  δ ,  y x  − δ  , for al l x, y ≥ x 0 . (ii) If f ∈ RV ∞ ρ , then for any given A > 1 and δ > 0 ther e exists x 0 = x 0 ( A, δ ) such that f ( y ) f ( x ) ≤ A max   y x  ρ + δ ,  y x  ρ − δ  , for al l x, y ≥ x 0 . Theorem A.8. (Kar amata’s The or em) (i) (Dir e ct Half ) If ℓ is slow ly varying, ℓ is lo c al ly b ounde d on [ x 0 , ∞ ) for some x 0 > 0 , and α > − 1 , then lim x →∞ R x x 0 u α ℓ ( u ) du x α +1 ℓ ( x ) = 1 ( α + 1) . (ii) (Converse Half ) L et f b e p ositive and lo c al ly inte gr able on [ x 0 , ∞ ) . • If for some ζ > − ( ρ + 1) , lim x →∞ x ζ +1 f ( x ) R x x 0 u ζ f ( u ) du = ζ + ρ + 1 , then f varies r e gularly with index ρ . • If for some ζ < − ( ρ + 1) we have lim x →∞ x ζ +1 f ( x ) R ∞ x u ζ f ( u ) du = − ( ζ + ρ + 1) , then f varies r e gularly with index ρ . 13 Theorem A.9. (de Bruijn Conjugate) If f ∈ R V ∞ α with α > 0 , then ther e exists g ∈ R V ∞ 1 /α such that lim x →∞ f ( g ( x )) x = lim x →∞ g ( f ( x )) x = 1 , i.e. f and g ar e asymptotic al ly invertible. A natural question is to kno w whether or not the deriv ativ e of a regularly v arying function is itself regularly v arying. The following result sho ws that monotonicity of the deriv ativ e in a neighborho od of ∞ suffices. Theorem A.10. (Monotone Density The or em) L et H ( x ) = R x 0 h ( u ) du wher e h : [0 , ∞ ) → R is me asur able. If H ( x ) ∼ cx ρ ℓ ( x ) as x → ∞ wher e c, ρ ∈ R and ℓ ∈ RV ∞ 0 and if ther e exists x 0 > 0 such that h ( x ) is monotone on ( x 0 , ∞ ) , then lim x →∞ h ( x ) cρx ρ − 1 ℓ ( x ) = 1 . It is important to note that Theorem A.10 do es not imply that H or h is regularly v arying, as the quan tities c and cρ are p oten tially negative; how ever, if c > 0, then H ∈ R V ∞ ρ , but w e still do not necessarily ha ve h ∈ R V ∞ ρ − 1 . Additionally , though we use Theorem A.10 , there are wa ys to w eaken the h yp otheses of our main result to not require even tual monotonicity . F or this we refer the reader to the O-version of Theorem A.10 (whic h requires bounded increase and b ounded decrease assumptions) in [ 4 ] Chapter 2.10 (e.g., Prop 2.10.3). Most of the corresp onding definitions and theorems extend straigh tforwardly to regular v ariation at 0 b y considering f (1 /x ) as x → ∞ , and we omit these routine adaptations. B Pro ofs Pr o of of Pr op osition 2.3 . Let ν be regularly v arying of index α at 0, i.e., lim r → 0 ν ( | x | > r t ) ν ( | x | > r ) = t − α , for all t > 0, or equiv alently lim r → 0 R ∞ rt ξ S ( x ) dx R ∞ r ξ S ( x ) dx = t − α . First, w e sho w that for an y fixed δ > 0 and t > 0 w e also ha v e lim r → 0 R δ rt ξ S ( x ) dx R δ r ξ S ( x ) dx = t − α . Recall that γ ( x ) = R ∞ x ξ S ( z ) dz → ∞ as x → 0 b y the representation theorem for regularly v arying functions (see [ 4 ]). So for fixed δ > 0, lim r → 0 R δ rt ξ S ( x ) dx R δ r ξ S ( x ) dx = lim r → 0 R ∞ rt ξ S ( x ) dx − R ∞ δ ξ S ( x ) dx R ∞ r ξ S ( x ) dx − R ∞ δ ξ S ( x ) dx = lim r → 0 R ∞ rt ξ S ( x ) dx  1 − R ∞ δ ξ S ( x ) dx R ∞ rt ξ S ( x ) dx  R ∞ r ξ S ( x ) dx  1 − R ∞ δ ξ S ( x ) dx R ∞ r ξ S ( x ) dx  = t − α . (B.1) Con tinuing, we let 0 < ε < 1 and choose δ > 0 suc h that 1 − ε ≤ e x ≤ 1 + ε, for all x ∈ ( − δ, δ ). Now, recalling ( 2.14 ), w e ha ve ν ∗ ( | x | > r t ) ν ∗ ( | x | > r ) = R | x | >rt e x ξ ( x ) dx R | x | >r e x ξ ( x ) dx = R ∞ rt ( e x ξ ( x ) + e − x ξ ( − x )) dx R ∞ r ( e x ξ ( x ) + e − x ξ ( − x )) dx = R δ rt ξ ∗ S ( x ) dx + C ∗ δ R δ r ξ ∗ S ( x ) dx + C ∗ δ , (B.2) 14 where C ∗ δ = R ∞ δ ξ ∗ S ( x ) dx < ∞ . F or 0 < x < δ , define γ ∗ δ ( x ) = R δ x ξ ∗ S ( y ) dy and γ δ ( x ) = R δ x ξ S ( y ) dy as the truncated tail functions. Note that γ δ ( x ) → ∞ as x → 0 b y the representation theorem for regularly v arying functions. W e estimate ( B.2 ) from ab o ve as R δ rt ξ ∗ S ( x ) dx + C ∗ δ R δ r ξ ∗ S ( x ) dx + C ∗ δ ≤ R δ rt e x ξ S ( x ) dx + C ∗ δ R δ r e − x ξ S ( x ) dx + C ∗ δ ≤ (1 + ε ) R δ rt ξ S ( x ) dx + C ∗ δ (1 − ε ) R δ r ξ S ( x ) dx + C ∗ δ = γ δ ( r t ) γ δ ( r )   1 + ε + C ∗ δ γ δ ( rt ) 1 − ε + C ∗ δ γ δ ( r )   , (B.3) and from ( B.3 ) obtain lim sup r → 0 ν ∗ ( | x | > r t ) ν ∗ ( | x | > r ) ≤ t − α 1 + ε 1 − ε . W e estimate ( B.2 ) from below R δ rt ξ ∗ S ( x ) dx + C ∗ δ R δ r ξ ∗ S ( x ) dx + C ∗ δ ≥ R δ rt e − x ξ S ( x ) dx + C ∗ δ R δ r e x ξ S ( x ) dx + C ∗ δ ≥ γ δ ( r t ) γ δ ( r )   1 − ε + C ∗ δ γ δ ( rt ) 1 + ε + C ∗ δ γ δ ( r )   , (B.4) and from ( B.4 ) obtain lim inf r → 0 ν ∗ ( | x | > r t ) ν ∗ ( | x | > r ) ≥ t − α 1 − ε 1 + ε . Th us, t − α 1 − ε 1 + ε ≤ lim inf r → 0 ν ∗ ( | x | > r t ) ν ∗ ( | x | > r ) ≤ lim sup r → 0 ν ∗ ( | x | > r t ) ν ∗ ( | x | > r ) ≤ t − α 1 + ε 1 − ε , and letting ε → 0 giv es the result. Next, w e need to sho w the existence of the limits lim x → 0 γ ∗ ± ( x ) γ ∗ ( x ) , giv en the existence of the limits lim x → 0 γ ± ( x ) γ ( x ) . Assume that lim x → 0 γ + ( x ) /γ ( x ) = p + and lim x → 0 γ − ( x ) /γ ( x ) = p − . In fact, we show that the limits abov e are equal, i.e. lim x → 0 γ ∗ + ( x ) γ ∗ ( x ) = lim x → 0 γ + ( x ) γ ( x ) = p + , (B.5) and lim x → 0 γ ∗ − ( x ) γ ∗ ( x ) = lim x → 0 γ − ( x ) γ ( x ) = p − . (B.6) By an argumen t similar to the one dev elop ed in the b eginning of this pro of, the limits can also b e computed via lim x → 0 ν ( x < y < δ ) ν ( x < | y | < δ ) = p + , and lim x → 0 ν ( − δ < y < − x ) ν ( x < | y | < δ ) = p − , 15 where δ > 0. W e no w sho w that the same limits hold for γ ∗ . First, let 0 < p + < 1 (note that p + + p − = 1). In this case, 0 < p − < 1, and so lim x → 0 γ ∗ + ( x ) = lim x → 0 γ ∗ − ( x ) = ∞ , since lim x → 0 γ ∗ ( x ) = ∞ . Again, let ε > 0 and c ho ose δ > 0 such that 1 − ε ≤ e x ≤ 1 + ε for all x ∈ ( − δ, δ ). Con tinuing ν ∗ ( y > x ) ν ∗ ( | y | > x ) = R ∞ x ξ ∗ ( y ) dy R ∞ x ξ ∗ S ( y ) dy = R δ x e y ξ ( y ) dy + R ∞ δ e y ξ ( y ) dy R δ x ξ ∗ S ( y ) dy + R ∞ δ ξ ∗ S ( y ) dy = R δ x e y ξ ( y ) dy + D ∗ δ R δ x ξ ∗ S ( y ) dy + C ∗ δ , (B.7) where D ∗ δ = R ∞ δ e y ξ ( y ) dy and, again, C ∗ δ = R ∞ δ ξ ∗ S ( y ) dy are constants dep ending only on δ . Estimating ( B.7 ) ab o v e b y R δ x e y ξ ( y ) dy + D ∗ δ R δ x ξ ∗ S ( y ) dy + C ∗ δ ≤ R δ x e y ξ ( y ) dy + D ∗ δ R δ x e − y ξ S ( y ) dy + C ∗ δ ≤ (1 + ε ) R δ x ξ ( y ) dy + D ∗ δ (1 − ε ) R δ x ξ S ( y ) dy + C ∗ δ = R δ x ξ ( y ) dy R δ x ξ S ( y ) dy   1 + ε + D ∗ δ R δ x ξ ( y ) dy 1 − ε + C ∗ δ R δ x ξ S ( y ) dy   , (B.8) whic h, taking the limsup, giv es lim sup x → 0 ν ∗ ( y > x ) ν ∗ ( | y | > x ) ≤ p +  1 + ε 1 − ε  , (since lim x → 0 R δ x ξ ( y ) dy = lim x → 0 R δ x ξ S ( y ) dy = ∞ also for ev ery δ > x ). W e estimate ( B.7 ) from below as R δ x e y ξ ( y ) dy + D ∗ δ R δ x ξ ∗ S ( y ) dy + C ∗ δ ≥ R δ x e y ξ ( y ) dy + D ∗ δ R δ x e y ξ S ( y ) dy + C ∗ δ ≥ R δ x ξ ( y ) dy + D ∗ δ (1 + ε ) R δ x ξ S ( y ) dy + C ∗ δ = R δ x ξ ( y ) dy R δ x ξ S ( y ) dy   1 + D ∗ δ R δ x ξ ( y ) dy 1 + ε + C ∗ δ R δ x ξ S ( y ) dy   , (B.9) and taking the liminf gives lim inf x → 0 ν ∗ ( y > x ) ν ∗ ( | y | > x ) ≥ p +  1 1 + ε  . Com bining these estimates, we obtain p +  1 1 + ε  ≤ lim inf x → 0 ν ∗ ( y > x ) ν ∗ ( | y | > x ) ≤ lim sup x → 0 ν ∗ ( y > x ) ν ∗ ( | y | > x ) ≤ p +  1 + ε 1 − ε  , and letting ε → 0 giv es the first limit. An iden tical argumen t shows that lim x → 0 ν ∗ ( y < − x ) ν ∗ ( | y | > x ) = p − . W e no w deal with the remaining cases, i.e. p + = 0 and p + = 1, and assume without loss of generalit y that p + = 0. This implies that lim x → 0 γ + ( x ) γ ( x ) = 0 , and lim x → 0 γ − ( x ) γ ( x ) = 1 , 16 whic h in turn implies that lim x → 0 γ − ( x ) = ∞ . There are tw o distinct p ossibilities for γ + , either lim x → 0 γ + ( x ) < ∞ , (B.10) or lim x → 0 γ + ( x ) = ∞ . (B.11) If ( B.11 ) holds true, then b oth tails ha v e infinite mass and a pro of similar to the one for 0 < p + < 1 gives the result. In the case ( B.10 ), lim x → 0 γ ∗ + ( x ) γ ∗ ( x ) = 0 , since γ ∗ ( x ) → ∞ as x → 0. Indeed, w e estimate γ ∗ ( x ) = Z ∞ x ξ ∗ S ( y ) dy = Z δ x ξ ∗ S ( y ) dy + C ∗ δ ≥ Z δ x e − y ξ S ( y ) dy + C ∗ δ ≥ (1 − ε ) Z δ x ξ S ( y ) dy + C ∗ δ → ∞ , as x → 0. Thus, lim x → 0 γ ∗ − ( x ) γ ∗ ( x ) = lim x → 0 γ ∗ ( x ) − γ ∗ + ( x ) γ ∗ ( x ) = 1 . Pr o of of Pr op osition 2.5 . First, we assume ¯ µ 0 < ∞ . Observe that sup 0 <η ≤ 1 | µ η | < ∞ implies that sup µ η < ∞ and inf µ η > −∞ . Th us, −∞ < inf 0 <η ≤ 1 b − Z η < | y |≤ 1 y ν ( dy ) ! = b + inf 0 <η ≤ 1 − Z η < | y |≤ 1 y ν ( dy ) ! = b − sup 0 <η ≤ 1 Z η < | y |≤ 1 y ν ( dy ) , whic h implies sup 0 <η ≤ 1 R η < | y |≤ 1 y ν ( dy ) < ∞ . A similar argument implies that inf 0 <η ≤ 1 Z η < | y |≤ 1 y ν ( dy ) > −∞ , and so we know sup 0 <η ≤ 1      Z η < | y |≤ 1 y ν ( dy )      < ∞ . F or fixed 0 < η ≤ 1,   µ ∗ η   ≤ | b | +      Z η < | y |≤ 1 y ν ∗ ( dy )      = | b | +      Z η < | y |≤ 1 y e y ν ( dy )      ≤ | b | + e      Z η < | y |≤ 1 y ν ( dy )      ≤ | b | + e sup 0 <η ≤ 1      Z η < | y |≤ 1 y ν ( dy )      < ∞ . T aking the suprem um giv es the implication. The con v erse can b e pro ven by noting that      Z η < | y |≤ 1 y e y ν ( dy )      ≥ 1 e      Z η < | y |≤ 1 y ν ( dy )      , and taking the suprem um since the left-hand side is bounded when η → 0 b y our assumption. 17 Pr o of of The or em 2.6 . W e kno w β t X t ⇒ Z, (B.12) as t → 0, under P ∗ where Z is an α -stable random v ariable. Recall that [ 19 ] and [ 4 ] b oth giv e the representation φ Z ( u ) = exp  − c α | u | α (1 − i ( p + − p − ) sgn ( u ) tan ( π α/ 2))  , (B.13) where p ± are defined in ( 2.9 ). W e need further information concerning the slowly v arying part of γ ∗ . So, w e examine the characteristic functions of b oth β t X t and Z , which we know must b e equal when t → 0 by ( 2.16 ). First, w e will need to determine the b eha vior of ξ ∗ S ( x ). T o this end, w e will use the Monotone Densit y Theorem (Theorem A.10 ). W e kno w γ ∗ ( x ) = R ∞ x ξ ∗ S ( y ) dy = x − α ℓ (1 /x ), for x > 0. Hence, x − α ℓ ( x ) = γ ∗ (1 /x ) = Z ∞ 1 /x ξ ∗ S ( y ) dy = − Z 0 x ξ ∗ S (1 /u ) du u 2 = Z x 0 ξ ∗ S (1 /u ) du u 2 = Z x 0 s ( u ) du, (B.14) where s ( u ) := ξ ∗ S (1 /u ) u − 2 . Note that x 2 ξ ∗ S ( x ) = s (1 /x ) and x 2 ξ ∗ S is monotone for x close enough to 0 (i.e. s ( u ) is monotone for u large enough). Now, we use the Monotone Densit y Theorem to get that s ( x ) ∼ αx α − 1 ℓ ( x ), as x → ∞ . That is, for y close to 0, we hav e s (1 /y ) = y 2 ξ ∗ S ( y ) ∼ αy − α +1 ℓ (1 /y ) whic h implies ξ ∗ S ( y ) ∼ αy − α − 1 ℓ (1 /y ) for y positive and near 0. Now, the exponent of the c haracteristic function of β t X t is giv en b y log  E ∗ e iuβ t X t  = t Z ∞ −∞ (exp ( iuβ t y ) − 1 − iuβ t y ) ξ ∗ ( y ) dy . (B.15) Fix an y 0 < ε < 1 and let w 0 ( ε ) > 0 be suc h that (1 − ε ) αx − α − 1 ℓ (1 /x ) ≤ ξ ∗ S ( x ) ≤ (1 + ε ) αx − α − 1 ℓ (1 /x ) , (B.16) for all 0 < x < w 0 . The real part of ( B.15 ) con verges to − c α | u | α where c α > 0 as t → 0 (again see [ 19 ], [ 4 ], and [ 22 ]). First, w e need to rewrite ( B.15 ) in a nicer form. Let g ( u, y ) = exp ( iuβ t y ) − 1 − iuβ t y and rewrite log  E ∗ e iuβ t X t  = t Z ∞ −∞ g ( u, y ) ξ ∗ ( y ) dy = t Z ∞ 0  g ( u, y ) ξ ∗ ( y ) + g ( u, y ) ξ ∗ ( − y )  dy . (B.17) Note that the real part of g is ℜ ( g ( u, y )) = cos ( uβ t y ) − 1, and the real part of ( B.17 ) is ℜ  log  E ∗ e iuβ t X t  = t Z ∞ 0 ℜ g ( u, y ) ( ξ ∗ ( y ) + ξ ∗ ( − y )) dy = t Z ∞ 0 (cos ( uβ t y ) − 1) ξ ∗ S ( y ) dy = t | u | β t Z ∞ 0 (cos (sgn ( u ) w ) − 1) ξ ∗ S  w | u | β t  dw = t | u | β t Z ∞ 0 (cos ( w ) − 1) ξ ∗ S  w | u | β t  dw . (B.18) Con tinuing, we break ( B.18 ) in to tw o parts by writing for L > 0 ℜ  log  E ∗ e iuβ t X t  = t | u | β t Z ∞ 0 (cos ( w ) − 1) ξ ∗ S  w | u | β t  dw = t | u | β t Z ∞ 0 (cos ( w ) − 1) ξ ∗ S  w | u | β t  1 { w | u | β t ≤ L } dw (B.19) + t | u | β t Z ∞ 0 (cos ( w ) − 1) ξ ∗ S  w | u | β t  1 { w | u | β t >L } dw . (B.20) 18 It is easy to see that ( B.20 ) go es to 0, as t → 0, since t | u | β t     Z ∞ 0 (cos ( w ) − 1) ξ ∗ S  w | u | β t  1 { w | u | β t >L } dw     = t     Z ∞ 0 (cos ( uβ t z ) − 1) ξ ∗ S ( z ) 1 { z ≥ L } dz     ≤ 2 t Z ∞ L ξ ∗ S ( z ) dz . No w, w e estimate ( B.19 ) to get the desired result. First, we sho w a preliminary result of use later. Namely , w e sho w that there exists M > 0 with 1 / M ≤ w 0 suc h that Z ∞ 0 (cos ( w ) − 1) w 1+ α ℓ  | u | β t w  ℓ ( β t ) 1 { w | u | β t ≤ 1 M } dw → Z ∞ 0 (cos ( w ) − 1) w 1+ α dw < ∞ , (B.21) as t → 0. Recall that ℓ slo wly v arying implies that ℓ ( λx ) /ℓ ( x ) → 1, for an y λ > 0, as x → ∞ . Letting x = β t and λ = | u | /w implies that ℓ  | u | β t w  ℓ ( β t ) → 1 , as t → 0. W e recall the Potter bounds from Theorem A.7 for ℓ , that is for any A > 1 and δ > 0 there exists M > 0 such that for x, y ≥ M , ℓ ( y ) ℓ ( x ) ≤ A   y x  δ ∨  y x  − δ  . Cho ose A = 2 and δ > 0 such that α + 1 ± δ ∈ (2 , 3) and let M 0 > 0 b e the M in the ab ov e statement for the giv en A and δ . Note if w ≤ | u | β t w 0 and 1 ≥ w 0 M 0 then (cos w − 1) w − α − 1 ℓ  | u | β t w  ℓ ( β t ) 1 { w | u | β t ≤ w 0 } ≤ (cos w − 1) w − α − 1 A max (  | u | w  δ ,  | u | w  − δ ) . ≤ A (cos w − 1)  | u | δ w − α − 1 − δ + | u | − δ w − α − 1+ δ  . (B.22) In ( B.22 ) the first term is integrable on [0 , ∞ ) since α + 1 + δ < 3, and so is the second term since α + 1 − δ > 2. Applying Leb esgue’s Dominated Con v ergence Theorem gives ( B.21 ). If w 0 M 0 > 1, then w e apply similar argumen ts on the set {| u | β t ≥ w M 0 } . In either case, there exists M > 0 suc h that 1 / M ≤ w 0 and ( B.21 ) holds. Equation ( B.20 ) conv erging to 0 as t → 0 implies (as stated b efore) that ( B.19 ) con verges to − c | u | α as t → 0. If we let L = min  1 M 0 , w 0  , then w e are no w in a p osition to analyze the conditions under whic h ( B.19 ) con v erges to − c α | u | α . Using ( B.16 ) w e obtain t | u | β t Z ∞ 0 (cos w − 1) ξ ∗ S  w | u | β t  1 { w | u | β t ≤ L } dw ≤ t (1 + ε ) | u | β t Z ∞ 0 (cos w − 1)  w | u | β t  − α − 1 ℓ  | u | β t w  1 { w | u | β t ≤ L } dw = (1 + ε ) | u | α tβ α t Z ∞ 0 (cos w − 1) w α +1 ℓ  | u | β t w  1 { w | u | β t ≤ L } dw = (1 + ε ) | u | α tβ α t ℓ ( β t ) Z ∞ 0 (cos w − 1) w α +1 ℓ  | u | β t w  ℓ ( β t ) 1 { w | u | β t ≤ L } dw . (B.23) Similarly , w e obtain the low er bound t | u | β t Z ∞ 0 (cos w − 1) ξ ∗ S  w | u | β t  1 { w | u | β t ≤ L } dw ≥ (1 − ε ) | u | α tβ α t ℓ ( β t ) Z ∞ 0 (cos w − 1) w α +1 ℓ  | u | β t w  ℓ ( β t ) 1 { w | u | β t ≤ L } dw . (B.24) 19 Dividing each side of both ( B.23 ) and ( B.24 ) by tβ α t ℓ ( β t ), letting − ς = R ∞ 0 (cos ( w ) − 1) w − α − 1 dw , and letting t → 0 giv es − (1 + ε ) ς | u | α ≤ − c α | u | α lim t → 0 tβ α t ℓ ( β t ) ≤ − (1 − ε ) ς | u | α . Letting ε → 0 implies that lim t → 0 tβ α t ℓ ( β t ) = Λ ∈ (0 , ∞ ) , (B.25) where Λ = c α /ς (note that ς > 0, for 1 < α < 2). In fact, e.g. see [ 26 ], ς = π 2Γ (1 + α ) sin  π α 2  , where Γ here is Euler’s Gamma function. Lemma B.1. L et the shar e–me asur e L ´ evy me asur e ν ∗ satisfy Z | x | > 1 | x | ν ∗ ( dx ) = Z | x | > 1 | x | e x ν ( dx ) < ∞ . Then, for every R 0 > 0 , Z | y | >R 0 γ ∗ ( y ) dy < ∞ , wher e γ ∗ ( y ) := ν ∗  {| x | > y }  . Pr o of. By symmetry of the absolute v alue, it suffices to show 2 R ∞ R 0 γ ∗ ( y ) dy < ∞ . T onelli’s theorem giv es, for an y R 0 > 0, Z ∞ R 0 γ ∗ ( y ) dy = Z ∞ R 0 Z {| z | >y } ν ∗ ( dz ) dy = Z {| z | >R 0 } ( | z | − R 0 ) ν ∗ ( dz ) . Since ν ∗ ( dz ) = e z ν ( dz ), Z {| z | >R 0 } ( | z | − R 0 ) ν ∗ ( dz ) ≤ Z {| z | >R 0 } | z | e z ν ( dz ) = I 1 + I 2 , with I 1 := Z {| z | > 1 } | z | e z ν ( dz ) , I 2 := Z R 0 < | z |≤ 1 | z | e z ν ( dz ) . By ( 2.10 ), I 1 < ∞ . F or I 2 , note that I 2 ≤ e Z R 0 < | z |≤ 1 ν ( dz ) < ∞ . Therefore R ∞ R 0 γ ∗ ( y ) dy < ∞ , and the claim follo ws. One imp ortan t quantit y in the estimation of call option prices in exp onential L ´ evy mo dels is the expression P ( X t ≥ y ) where y ≥ 0 and ( X t ) t ≥ 0 is a L´ evy pro cess on R with triplet ( b, 0 , ν ). F or estimating these tail quan tities, w e need concentration inequalities similar to those found in [ 6 ] and [ 20 ]. The next lemma provides the estimation w e need. F or no w, we will allo w γ to b e an y nonnegativ e function, but our future application of this lemma will use the definition of γ in ( 2.2 ); ho wev er, for the pro of of the main theorem we do require the function V defined in ( 2.3 ) and µ y defined in ( 2.7 ). Lemma B.2. L et γ : R + → R + b e such that for al l R > 0 , (i) R | x | >R ν ( dx ) ≤ γ ( R ) , (ii) and ther e exists C > 0 indep endent of R such that V ( R ) ≤ C R 2 γ ( R ) . 20 Then, for every y > 0 and for every 0 < t < y / 4  µ y / 4  + (with y / 0 = ∞ ), P ( X t ≥ y ) ≤  1 + C e 2  tγ  y 4  . Pr o of. In the traditional manner (e.g. see [ 6 ] or [ 21 ]), we break X = ( X t ) t ≥ 0 up into tw o parts, X ε = ( X ε t ) t ≥ 0 whic h consists of all jumps smaller than ε and e X ε =  e X ε t  t ≥ 0 consisting of all jumps larger than ε . F or eac h t > 0, w e can represent X t as X t = bt + Z t 0 Z | x |≤ 1 x ( µ − ¯ µ ) ( dx, ds ) + Z t 0 Z | x |≥ 1 xµ ( dx, ds ) (B.26) where µ is a Poisson random measure on R \ { 0 } with mean measure ¯ µ ( dx, dt ) = ν ( dx ) dt . Let f ε ( x ) = 1 [ − ε,ε ] and ¯ f ε = 1 − f ε . W e can define the pro cesses for each t > 0 b y e X ε t = Z t 0 Z R x ¯ f ε ( x ) µ ( dx, ds ) and X ε t = X t − e X ε t . (B.27) The process e X ε is a comp ound Poisson pro cess with intensit y λ ε = R ¯ f ε ( x ) ν ( dx ) and jump distribution ¯ f ε ( x ) ν ( dx ) λ ε , and X ε is a L´ evy process with characteristic triplet ( b ε , 0 , f ε ν ) where b ε = b − Z | x |≤ 1 x ¯ f ε ( x ) ν ( dx ) . W e will need the fact that E X ε t = tµ ε where µ ε is defined by ( 2.7 ). F or a fixed y > 0, w e ha v e P ( X t ≥ y ) ≤ P ( X ε t ≥ y / 2) + P  e X ε t ≥ y / 2  ≤ P ( X ε t − E X ε t ≥ y / 2 − E X ε t ) + P  e X ε t  = 0  ≤ P ( X ε t − E X ε t ≥ y / 2 − E X ε t ) + tγ ( ε ) = P ( X ε t − E X ε t ≥ y / 2 − tµ ε ) + tγ ( ε ) . (B.28) Using a general concentration inequality (e.g. Corollary 1 in [ 20 ]) and writing V ε := V ( ε ), w e obtain for z > 0 P ( X ε t − E X ε t ≥ z ) ≤ exp  z ε −  z ε + tV ε ε 2  log  1 + εz tV ε  ≤ exp  z ε − z ε log  1 + εz C tε 2 γ ( ε )  = exp  z ε − z ε log  1 + z C tεγ ( ε )  = exp  z ε   1 + z C tεγ ( ε )  z /ε . (B.29) W e now need to choose ε in suc h a w ay that b oth terms in ( B.28 ) are of the same order. W e choose ε = y / 4 and w e consider t wo cases. First, consider the case where µ y / 4 ≥ 0 and further assume that 0 < t < y / 4 µ y / 4 (equiv alently y / 2 − tµ y / 4 > y / 4). Then, P  X y / 4 t − E X y / 4 t ≥ y / 2 − tµ y / 4  ≤ P  X y / 4 t − E X y / 4 t ≥ y / 4  = exp  y / 4 y / 4   1 + y / 4 C t ( y / 4) γ ( y/ 4)  y/ 4 y/ 4 = e  1 + 1 C tγ ( y / 4)  ≤ C e 2 tγ  y 4  . (B.30) 21 Next, consider the case where µ y / 4 < 0. Then, for all t > 0, P  X y / 4 t − E X y / 4 t ≥ y / 2 − tµ y / 4  ≤ P  X y / 4 t − E X y / 4 t ≥ y / 2  ≤ exp  y / 2 y / 4   1 + y / 2 C t ( y / 4) γ ( y/ 4)  y/ 2 y/ 4 = e 2  1 + 2 C tγ ( y / 4)  2 ≤ e 2  1 + 2 C tγ ( y / 4)  ≤ C e 2 tγ  y 4  . (B.31) Notice that the terms in ( B.30 ) and ( B.31 ) are the same. Com bining this result with ( B.28 ) gives P ( X t ≥ y ) ≤ C e 2 tγ  y 4  + tγ  y 4  = (1 + C e 2 ) tγ  y 4  , for all y > 0 and 0 < t < y / 4( µ y / 4 ) + . Prop osition B.3. L et ( X t ) t ≥ 0 b e a L´ evy pr o c ess in the domain of attr action of an α -stable r andom variable with α ∈ (1 , 2) . Then for any 0 < x < y < ∞ , sup x ≤ R ≤ y V ( R ) R 2 γ ( R ) < ∞ . Pr o of. By Theorem 2.2 , γ ( R ) = R − α ψ ( R ) where ψ is slo wly v arying and α ∈ (1 , 2). In tegration by parts gives 0 < V ( z ) = − z 2 γ ( z ) + 2 Z z 0 ξ γ ( ξ ) dξ = − z 2 γ ( z ) + 2 Z z 0 ξ 1 − α ψ ( ξ ) dξ (B.32) whic h is w ell-defined since 1 − α ∈ ( − 1 , 0). So, V ( z ) z 2 γ ( z ) = − z 2 γ ( z ) + 2 R z 0 ξ 1 − α ψ ( ξ ) dξ z 2 γ ( z ) = − 1 + 2 R z 0 ξ 1 − α ψ ( ξ ) dξ z 1 − α ψ ( z ) . The numerator is contin uous and the denominator is piecewise contin uous and bounded a wa y from 0 in any compact in terv al of R + not including 0 (the function γ is nonincreasing on (0 , ∞ ) so it can only ha ve jump discon tinuities). Th us, the supremum is b ounded and the result follo ws. Prop osition B.4. L et ( X t ) t ≥ 0 b e a L ´ evy pr o c ess in the domain of attr action of an α -stable r andom variable, 0 < α < 2 . F urther, let ther e exist R 0 > 0 and C > 0 p ossibly dep ending on R 0 such that for al l R > R 0 , V ( R ) ≤ C R 2 γ ( R ) wher e V and γ ar e define d in ( 2.3 ) and ( 2.2 ) , r esp e ctively. Then, for every y > 0 and for every 0 < t < y / 4  µ y / 4  + (with y / 0 = ∞ ), P ( X t ≥ y ) ≤  1 + C e 2  tγ  y 4  . Pr o of. By assumption, X is in the domain of attraction of an α -stable random v ariable. When γ is defined as in ( 2.2 ), condition (i) of Lemma B.2 is then satisfied trivially . Moreo ver, it is sho wn in [ 4 ] (Chapter 8.1), [ 12 ] (VII.9), and [ 22 ] that for pro cesses in the domain of attraction of an α -stable la w, condition (ii) of Lemma B.2 holds true automatically for small R > 0: there exist R 1 > 0 and C 1 > 0 suc h that V ( R ) ≤ C 1 R 2 γ ( R ) , 0 < R ≤ R 1 . On the other hand, Prop osition B.3 ab o ve shows that for any 0 < x < y < ∞ , sup x ≤ R ≤ y V ( R ) R 2 γ ( R ) < ∞ . 22 In particular, choosing x := R 1 and y := R 0 w e obtain a constant C 2 > 0 suc h that V ( R ) ≤ C 2 R 2 γ ( R ) , R 1 ≤ R ≤ R 0 . By h yp othesis of the present prop osition, there exists C > 0 such that V ( R ) ≤ C R 2 γ ( R ) , R > R 0 . Com bining these three b ounds and letting e C := max { C 1 , C 2 , C } , w e get a uniform estimate V ( R ) ≤ e C R 2 γ ( R ) , for all R > 0 . Th us b oth conditions (i) and (ii) of Lemma B.2 hold with the same constan t e C . Applying that lemma with this constan t e C yields, for every y > 0 and ev ery 0 < t < y / 4 ( µ y / 4 ) + , P ( X t ≥ y ) ≤  1 + e C e 2  t γ  y 4  . Pr o of of The or em 3.2 . Recalling ( 1.4 ) and β t = 1 /B t and using M 0 from the Potter bound argument in Theorem 2.6 , we obtain c ( t, 0) B t = 1 B t Z ∞ 0 e − x P ∗ ( X t ≥ x ) dx = Z ∞ 0 e − B t u P ∗ ( X t ≥ B t u ) du = Z ∞ 0 e − B t u P ∗ ( X t ≥ B t u )  1 { 1 M 0 ≥ B t u 4 ≥ t ¯ µ ∗ } + 1 { B t u 4 > 1 M 0 } + 1 { B t u 4 1 and ℓ is slo wly v arying. Therefore w e only need to deal with the integral of A 1 ( t ) and A 2 ( t ). Using { B t u > 4 t ¯ µ ∗ } ⊆ { B t u > 4 tµ B t u/ 4 } and the estimate from Prop osition B.4 , for some constant C > 0 and for any u ∈ I ⊂ { B t u > 4 t ¯ µ ∗ } (B.36) with I measurable and t > 0 fixed, P ∗ ( X t ≥ B t u ) ≤   1 + C e 2  tγ ∗  B t u 4  ∨ 1  =   1 + C e 2  tγ ∗  u 4 β t  ∨ 1  ≤ " (1 + C e 2 ) t  u 4 β t  − α ℓ  4 β t u  ! ∨ 1 # =  κtβ α t u − α ℓ  4 β t u  ∨ 1  , (B.37) 23 where κ > 0 is a collection of all the constan ts. In what follows, we use κ to represent a p ositiv e constant whose v alue migh t c hange from line to line. Recall from our argument in Theorem 2.6 that α ± δ ∈ (1 , 2). W e also choose t 0 > 0 suc h that for all 0 < t < t 0 , Λ 2 ℓ ( β t ) < tβ α t < 3Λ 2 ℓ ( β t ) . Con tinuing ( B.37 ) for 0 < t < t 0 and u ∈ I , P ∗ ( X t ≥ B t u ) ≤ κu − α ℓ  4 β t u  ℓ ( β t ) ∨ 1 . First, w e sho w that the integral of A 2 ( t ) goes to 0 as t → 0. Indeed, choosing I = { 4 β t < M 0 u } in ( B.36 ) and c hanging v ariables giv es Z ∞ 0 A 2 ( t ) du ≤ Z ∞ 0   κu − α ℓ  4 β t u  ℓ ( β t ) ∨ 1   1 { 4 β t u 0, A 1 ( t, u ) ≤   κu − α ℓ  4 β t u  ℓ ( β t ) ∨ 1   1 { M 0 ≤ 4 β t u < 1 t ¯ µ ∗ } ≤ max  κ 4 δ u − α − δ ∨ 1 , κ 4 − δ u − α + δ ∨ 1  ∈ L 1 ([0 , ∞ )) , (B.39) so that we are able to apply Lebesgue’s Dominated Conv ergence theorem to Z ∞ 0 A 1 ( t, u ) du = Z ∞ 0 e − B t u P ∗ ( X t ≥ B t u ) 1 { 1 M 0 ≥ B t u 4 >t ¯ µ ∗ } du. Com bining this fact with ( B.38 ) and ( B.39 ) gives lim t → 0 c ( t, 0) B t = lim t → 0 Z ∞ 0 A 1 ( t, u ) du = Z ∞ 0 P ∗ ( Z ≥ u ) du = E ∗ Z + , whic h can be rewritten as lim t → 0 E ( S t − S 0 ) + = S 0 B t E ∗ Z + + o ( B t ) , pro ving the theorem. 24 Pr o of of Cor ol lary 3.3 . W e pro ceed as in the pro of of Prop osition 3.7 in [ 17 ]. W e know that the Black-Sc holes call price asymptotics are S 0 c B S , where c B S ( t, σ ) = σ √ 2 π √ t + o  √ t  , (B.40) as t → 0, since c B S = N  σ √ t  where N ( θ ) := Z θ 0 Φ ′  u 2  du = 1 √ 2 π Z θ 0 exp  − u 2 8  du, where Φ is the standard normal cum ulativ e distribution function, and where N has asymptotic b ehavior N ( θ ) = 1 √ 2 π θ + o ( θ ) , as θ → 0. W e need an expression similar to ( B.40 ) where the constan t σ is replaced b y the implied v olatilit y function ˆ σ ( t ). Now, ˆ σ ( t ) → 0 as t → 0, so a substitution in c B S giv es c B S ( t, ˆ σ ( t )) = ˆ σ ( t ) √ 2 π √ t + o  ˆ σ ( t ) √ t  = ˆ σ ( t ) √ 2 π √ t + o  √ t  , (B.41) as t → 0, since ˆ σ ( t ) = o (1) as t → 0. Equating ( B.41 ) with ( 3.2 ), leads to ˆ σ ( t ) √ 2 π √ t ∼ B t E ∗ Z + , as t → 0, i.e. ˆ σ ( t ) ∼ √ 2 π B t √ t E ∗ Z + , as t → 0, giving the result. Lemma B.5. L et ( S, Σ , µ ) b e a me asur e sp ac e. L et f , g : S × [0 , ∞ ) → [0 , ∞ ) and h : S → [0 , ∞ ) b e me asur able and such that f ( · , t ) , g ( · , t ) , h ∈ L 1 ( S ) for almost every t ≥ 0 . Also, supp ose (C1) f ( s, t ) → ¯ f ( s ) ∈ L 1 ( S ) as t → 0 , (C2) f ( s, t ) ≤ h ( s ) + g ( s, t ) for µ -a.e. s ∈ S and almost every t ≥ 0 , (C3) g ( s, t ) → 0 as t → 0 µ -a.e s ∈ S , (C4) R S g ( s, t ) µ ( ds ) → 0 as t → 0 . Then lim t → 0 Z S f ( s, t ) µ ( ds ) = Z S ¯ f ( s ) µ ( ds ) . Pr o of. First, w e choose a sequence ( t n ) n ≥ 1 suc h that t n → 0, as n → ∞ , and lim n →∞ Z S f ( s, t n ) µ ( ds ) = lim inf t → 0 Z S f ( s, t ) µ ( ds ) . (B.42) W e apply F atou’s lemma to the nonnegative functions f ( · , t n ), since f ≥ 0, to obtain Z S ¯ f ( s ) µ ( ds ) = Z S lim inf t → 0 f ( s, t ) µ ( ds ) ≤ Z S lim inf n →∞ f ( s, t n ) µ ( ds ) ≤ lim inf n →∞ Z S f ( s, t n ) µ ( ds ) = lim inf t → 0 Z S f ( s, t ) µ ( ds ) . (B.43) 25 Next, w e c hoose another sequence  t ′ n  n ≥ 1 suc h that t ′ n → 0, as n → ∞ , and lim n →∞ Z S f  s, t ′ n  µ ( ds ) = lim sup t → 0 Z S f ( s, t ) µ ( ds ) . (B.44) So, (C2) implies h ( s ) + g ( s, t ) − f ( s, t ) ≥ 0, and applying F atou’s lemma again, Z S  h ( s ) − ¯ f ( s )  µ ( ds ) = Z S lim inf t → 0 ( h ( s ) + g ( s, t ) − f ( s, t )) µ ( ds ) ≤ Z S lim inf n →∞  h ( s ) + g  s, t ′ n  − f  s, t ′ n  µ ( ds ) ≤ lim inf n →∞ Z S  h ( s ) + g  s, t ′ n  − f  s, t ′ n  µ ( ds ) = Z S h ( s ) µ ( ds ) + lim inf n →∞  − Z S f  s, t ′ n  µ ( ds )  = Z S h ( s ) µ ( ds ) − lim sup n →∞ Z S f ( s, t ′ n ) µ ( ds ) = Z S h ( s ) µ ( ds ) − lim sup t → 0 Z S f ( s, t ) µ ( ds ) . Note that the limsup in the last line is finite due to (C2) and the integrabilit y of h and g ( · , t ). Canceling the h term, we obtain lim sup t → 0 Z S f ( s, t ) µ ( ds ) ≤ Z S ¯ f ( s ) µ ( ds ) . (B.45) Com bining ( B.43 ) and ( B.45 ), w e ha ve Z S ¯ f ( s ) µ ( ds ) ≤ lim inf t → 0 Z S f ( s, t ) µ ( ds ) ≤ lim sup t → 0 Z S f ( s, t ) µ ( ds ) ≤ Z S ¯ f ( s ) µ ( ds ) , whic h pro ves the result. Pr o of of The or em 3.4 . W e mak e use of the fact that lim t → 0 E ∗  exp  iu X t √ t   = exp  − 1 2 σ 2 u 2  , (B.46) and lim t → 0 P ∗  X t √ t ≥ x  = P ∗ ( σ W 1 ≥ x ) . (B.47) Con tinuing, 1 √ t E ( S t − S 0 ) + = 1 √ t Z ∞ 0 e − x P ∗ ( X t ≥ x ) dx = Z ∞ 0 e − √ tu P ∗  X t ≥ u √ t  du. (B.48) Note that e − √ tu P ∗  X t ≥ u √ t  ≤ P ∗  σ W t ≥ √ tu 2  + P ∗  L t ≥ √ tu 2  = P ∗  σ W 1 ≥ u 2  + P ∗  L t ≥ √ tu 2  . (B.49) W e next use Lemma B.5 with f ( u, t ) = e − √ tu P ∗  X t ≥ u √ t  , ¯ f ( u ) = P ∗ ( σ W 1 ≥ u ) , h ( u ) = P ∗ ( σ W 1 ≥ u/ 2) , 26 and g ( u, t ) = P ∗  L t ≥ √ tu/ 2  . It is easy to see that conditions (C1) – (C2) of the lemma are satisfied. W e need to sho w that (C3) and (C4) also hold. It is not too difficult to see that (C3) holds from ( B.46 ) and the indep endence of W and L under P ∗ . W e no w sho w that (C4) is satisfied as well. Once done, w e immediately ha ve the result b y applying Lemma B.5 to ( B.48 ). First note Z ∞ 0 P ∗  L t ≥ √ tu 2  du = Z ∞ 0 P ∗  L t ≥ √ tu 2   1 { √ tu 8 ≥ t ¯ µ ∗ } + 1 { √ tu 8 1 and δ > 0 such that α ± δ ∈ (1 , 2) and let M 0 > 0 be suc h that the P otter bounds hold for ℓ on [ M 0 , ∞ ) . W e ha v e D 1 ( t ) = Z ∞ 0 P ∗  L t ≥ √ tu 2   1 { M 0 ≤ 8 √ tu ≤ 1 t ¯ µ ∗ } + 1 { 8 √ tu M 0 √ t . W e pro ceed b y estimating (and using κ as a general p ositiv e coefficient that can change from line to line) D 11 ( t ) ≤ Z ∞ 0  (1 + C e 2 ) tγ ∗  √ tu 8  ∨ 1  1 { M 0 ≤ 8 √ tu ≤ 1 t ¯ µ ∗ } du = Z ∞ 0 κt  √ tu 8  − α ℓ  8 √ tu  ∨ 1 ! 1 { M 0 ≤ 8 √ tu ≤ 1 t ¯ µ ∗ } du = Z ∞ 0  κt 1 − α/ 2 u − α ℓ  8 √ tu  ∨ 1  1 { M 0 ≤ 8 √ tu ≤ 1 t ¯ µ ∗ } du. W e ha ve  κt 1 − α/ 2 u − α ℓ  8 √ tu  ∨ 1  1 { M 0 ≤ 8 √ tu ≤ 1 t ¯ µ ∗ } =   κt 1 − α/ 2 u − α ℓ  8 √ tu  ℓ  8 √ t  ℓ  8 √ t  ∨ 1   1 { M 0 ≤ 8 √ tu ≤ 1 t ¯ µ ∗ } ≤ κt 1 − α/ 2 ℓ  8 √ t  A max  u − α − δ , u − α + δ  ∨ 1 . (B.50) No w, letting β t = 8 / √ t , t ( 1 − α 2 ) ℓ  8 √ t  = 8 2 − α  8 √ t  2 ( α 2 − 1 ) ℓ ( β t ) = 8 2 − α β α − 2 t ℓ ( β t ) → 0 , since α − 2 < 0 and β t → ∞ as t → 0. Thus, there exists t 0 suc h that t (1 − α/ 2) ℓ (8 / √ t ) ≤ 1 for all 0 ≤ t < t 0 . So, ( B.50 ) is bounded b y κ max  u − α − δ , u − α + δ  ∨ 1 ∈ L 1 ([0 , ∞ )) . 27 Th us, we can apply Leb esgue’s Dominated Conv ergence Theorem to D 11 ( t ) whic h gives D 11 ( t ) → 0, as t → 0, since P ∗  L t ≥ √ tu/ 2  → 0, as t → 0, for u > 0. W e no w consider D 12 ( t ) and estimate D 12 ( t ) = Z ∞ 0 P ∗  L t ≥ √ tu 2  1 { 8 √ tu 0 suc h that B t ≤ √ t for all 0 ≤ t ≤ t 0 . F or 0 ≤ t ≤ t 0 , w e ha v e P ∗  L t ≥ √ tu  ≤ P ∗ ( L t ≥ B t u ) . for ev ery u ≥ 0. T o simplify notation, let F ( t, u ) = P ∗  L t ≥ √ tu/ 2  , G ( t, u ) = P ∗ ( L t ≥ B t u/ 2), and ¯ G ( u ) = P ∗ ( Z ≥ u ). Note that R ∞ 0 G ( t, u ) du → R ∞ 0 ¯ G ( u ) du as t → 0. It is clear that 0 ≤ lim inf t → 0 R ∞ 0 F ( t, u ) du . Now, G ( t, u ) − F ( t, u ) ≥ 0, so w e can apply F atou’s lemma to get Z ∞ 0 ¯ G ( u ) du ≤ lim inf t → 0 Z ∞ 0 ( G ( t, u ) − F ( t, u )) du = lim inf t → 0  Z ∞ 0 G ( t, u ) du − Z ∞ 0 F ( t, u ) du  = Z ∞ 0 ¯ G ( u ) du − lim sup t → 0 Z ∞ 0 F ( t, u ) du. Canceling terms gives lim sup t → 0 R ∞ 0 F ( t, u ) du ≤ 0. Thus, 0 ≤ lim inf t → 0 Z ∞ 0 P ∗  L t ≥ √ tu 2  du ≤ lim sup t → 0 Z ∞ 0 P ∗  L t ≥ √ tu 2  du ≤ 0 , and therefore (C4) is satisfied, pro ving the result. 28 Pr o of of Cor ol lary 3.6 . W e pro ceed exactly as in the pro of of Corollary 3.3 . W e are now comparing ( B.40 ) with ( 3.4 ) multiplied by S 0 . So, S 0 ˆ σ ( t ) √ 2 π √ t ∼ S 0 σ √ t E ∗ ( W 1 ) + , as t → 0. This implies that ˆ σ ( t ) ∼ σ √ 2 π E ∗ ( W 1 ) + , as t → 0. References [1] Leif Andersen and Alexander Lipton, Asymptotics for exp onential L´ evy pr o c esses and their volatility smile: survey and new r esults , Int. J. Theor. Appl. Finance 16 (2013), no. 1, 1350001, 98. [2] Da vid Applebaum, L´ evy pr o cesses and sto chastic calculus , Second, Cambridge Studies in Adv anced Mathematics, vol. 116, Cambridge University Press, 2009. [3] Marco Azzone and P eter T anko v, Short-time implie d volatility of additive normal temp er e d stable mo dels , Annals of Op era- tions Research (2024). Online first. [4] Nic holas H. Bingham, Charles M. Goldie, and Jozef L. T eugels, R e gular variation , Encyclop edia of Mathematics and its Applications, vol. 27, Cam bridge Universit y Press, 1987. [5] Sv etlana I. Bo yarc henko and Sergei Z. Levendorski ˘ ı, Non-Gaussian Merton-Black-Scholes the ory , Adv anced Series on Sta- tistical Science & Applied Probability , vol. 9, W orld Scientific Publishing Co. Inc., River Edge, NJ, 2002. [6] Jean-Christophe Breton, Christian Houdr´ e, and Nicolas Priv ault, Dimension fr e e and infinite varianc e tail estimates on Poisson sp ac e , Acta Appl. Math. 95 (2007), no. 3, 151–203. [7] P eter Carr and Dilip B. Madan, Sadd lep oint metho ds for option pricing , J. Comput. Finance 13 (2009), no. 1, 49–61. [8] Rama Cont and P eter T anko v, Financial mo del ling with jump pr o c esses , Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, 2004. [9] Robert M. Corless, Gaston H. Gonnet, David E. G. Hare, David J. Jeffrey, and Donald E. Knuth, On the Lamb ert W function , Adv ances in Computational Mathematics 5 (1996), no. 4, 329–359. [10] Bruno Dupire, Pricing with a smile , Risk Magazine 7 (1994), 18–20. [11] Omar El Euch, Masaaki F uk asaw a, Jim Gatheral, and Mathieu Rosen baum, Short-term at-the-money asymptotics under sto chastic volatility mo dels , SIAM Journal on Financial Mathematics 10 (2019), no. 2, 491–511. [12] William F eller, A n intr o duction to prob ability theory and its applic ations. Vol. II , Second, John Wiley & Sons, Inc., New Y ork, 1971. [13] Jos ´ e E. Figueroa-L´ opez and Martin F orde, The smal l-maturity smile for exp onential L ´ evy mo dels , SIAM Journal on Financial Mathematics 3 (2012), no. 1, 33–65. [14] Jos ´ e E. Figueroa-L´ op ez, Ruoting Gong, and Y uzhao Han, Estimation of temp er e d stable L ´ evy mo dels of infinite variation , Methodology and Computing in Applied Probability 24 (2022), 713–747. [15] Jos ´ e E Figueroa-L´ opez, Ruoting Gong, and Christian Houdr´ e, Smal l-time exp ansions of the distributions, densities, and option pric es of sto chastic volatility mo dels with l´ evy jumps , Sto c hastic Pro cesses and their Applications 122 (2012), no. 4, 1808–1839. [16] Jos ´ e E. Figueroa-L´ op ez, Ruoting Gong, and Christian Houdr´ e, High-or der short-time exp ansions for A TM option pric es of exp onential L´ evy mo dels , Mathematical Finance 24 (2014), no. 4, 664–690. [17] , Third-or der short-time exp ansions for close-to-the-money option pric es under the CGMY mo del , Applied Mathe- matical Finance 24 (2017), no. 6, 547–574. [18] Jos ´ e E. Figueroa-L´ op ez and Sv einn ´ Olafsson, Short-time expansions for close-to-the-money options under a L´ evy jump mo del with sto chastic volatility , Mathematical Finance 26 (2016), no. 3, 620–668. [19] Mic hael Grabchak, Inversions of L´ evy me asur es and the relation b etwe en long and short time b ehavior of L´ evy pr o c esses , J. Theor. Probab. 28 (2015), no. 1, 184–197. [20] Christian Houdr´ e, R emarks on deviation ine qualities for functions of infinitely divisible random ve ctors , Ann. Probab. 30 (2002), no. 3, 1223–1237. [21] Christian Houdr´ e and Philippe Marc hal, On the c onc entr ation of me asur e phenomenon for stable and r elated r andom vectors , Ann. Probab. 32 (2004), no. 2, 1496–1508. [22] Ross A. Maller and David M. Mason, Convergenc e in distribution of L ´ evy pro c esses at smal l times with self-normalization , Acta Sci. Math. (Szeged) 74 (2008), no. 1–2, 315–347. [23] Mark M. Meersc haert and Hans-Peter Scheffler, Limit distributions for sums of indep endent r andom vectors , John Wiley & Sons Inc., New Y ork, 2001. Heavy tails in theory and practice. MR1840531 (2002i:60047) 29 [24] Aleksandar Mijatovi ´ c and Peter T anko v, A new lo ok at short-term implied volatility in asset pric e mo dels with jumps , Mathematical Finance 26 (2016), no. 1, 149–183. [25] Johannes Muhle-Karb e and Marcel Nutz, Small-time asymptotics of option pric es and first absolute moments , Journal of Applied Probability 48 (2011), no. 4, 1003–1020. [26] Ken-iti Sato, L´ evy pr o c esses and infinitely divisible distributions , Cambridge Studies in Adv anced Mathematics, vol. 68, Cambridge University Press, 1999. [27] P eter T ankov, Pricing and he dging in exp onential L´ evy models: r eview of r e c ent results , P aris-Princeton Lectures on Math- ematical Finance 2010, 2011, pp. 319–359. MR2762364 (2012e:91291) 30