Small-maturity asymptotics for the at-the-money implied volatility slope in Lévy models
SSMALL-MATURITY ASYMPTOTICS FOR THEAT-THE-MONEY IMPLIED VOLATILITY SLOPE IN L ´EVYMODELS
STEFAN GERHOLD, I. CETIN G ¨UL ¨UM, AND ARPAD PINTER
Abstract.
We consider the at-the-money strike derivative of implied volatil-ity as the maturity tends to zero. Our main results quantify the behavior ofthe slope for infinite activity exponential L´evy models including a Browniancomponent. As auxiliary results, we obtain asymptotic expansions of short ma-turity at-the-money digital call options, using Mellin transform asymptotics.Finally, we discuss when the at-the-money slope is consistent with the steep-ness of the smile wings, as given by Lee’s moment formula. Introduction
Recent years have seen an explosion of the literature on asymptotics of op-tion prices and implied volatilities (see, e.g., [4, 24] for many references). Suchresults are of practical relevance for fast model calibration, qualitative modelassessment, and parametrization design. The small-time behavior of the level ofimplied volatility in L´evy models (and generalizations) has been investigated ingreat detail [7, 17, 18, 19, 33, 38]. We, on the other hand, focus on the at-the-money slope of implied volatility, i.e., the strike derivative, and investigate itsbehavior as maturity becomes small. For diffusion models, there typically existsa limiting smile as the maturity tends to zero, and the limit slope is just theslope of this limit smile (e.g., for the Heston model, this follows from [14, Sec-tion 5]). Our focus is, however, on exponential L´evy models. There is no limitsmile here that one could differentiate, as the implied volatility blows up off-the-money [38]. In fact, this is a desirable feature, since in this way L´evy models arebetter suited to capture the steep short maturity smiles observed in the market.But it also implies that the limiting slope cannot be deduced directly from thebehavior of implied volatility itself, and requires a separate analysis. (Note thata limiting smile does exist if maturity and log-moneyness tend to zero jointly inan appropriate way [32].)
Date : October 1, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Implied volatility, L´evy process, digital option, asymptotics, Mellintransform.
JEL Classification:
G13.We thank Jos´e Fajardo, Peter Friz, Friedrich Hubalek, Andreas Kyprianou, and MykhayloShkolnikov for helpful discussions, and gratefully acknowledge financial support from the Aus-trian Science Fund (FWF) under grant P 24880-N25. Special thanks are due to the anonymousreferees for their comprehensive and very helpful remarks, some of which led us to alter thecore of the paper. a r X i v : . [ q -f i n . P R ] M a y S. GERHOLD, I. C. G ¨UL ¨UM, AND A. PINTER
It turns out that the presence of a Brownian component has a decisive influence:Without it, the ATM (at-the-money) slope explodes (under mild conditions). Theblowup is of order T − / for many models, but may also be slower (CGMY modelwith Y ∈ (1 , with a Brownian component, though. We provide a result (Corollary 6 in Section 5)that translates the asymptotic behavior of the moment generating function tothat of the ATM slope. When applied to concrete models, we see that theslope may converge to a finite limit (Normal Inverse Gaussian, Meixner, CGMYmodels), or explode at a rate slower than T − / (generalized tempered stablemodel; this kind of behavior seems to be the most realistic one, see [5]). Notethat several studies [1, 2, 9] highlight the importance of a Brownian componentwhen fitting to historical data or option prices. In particular, in many pure jumpL´evy models ATM implied volatility converges to zero as T ↓ β > − .To obtain these results, we investigate the asymptotics of at-the-money digitalcalls; their relation to the implied volatility slope is well known. While, for L´evyprocesses X , the small-time behavior of the transition probabilities P [ X T ≥ x ](in finance terms, digital call prices) has been well studied for x (cid:54) = X (see,e.g., [20] and the references therein), not so much is known for x = X . Still,first order asymptotics of P [ X T ≥ X ] are available, and this suffices if there isno Brownian component. If the L´evy process has a Brownian component, thenit is well known that lim T → P [ X T ≥ X ] = . In this case, it turns out that thesecond order term of P [ X T ≥ X ] is required to obtain slope asymptotics. Forthis, we use a novel approach involving the Mellin transform (w.r.t. time) of thetransition probability (Sections 4 and 5). We believe that this method is of wideapplicability to other problems involving time asymptotics of L´evy processes, andhope to elaborate on it in future work.Finally, we consider the question whether a positive at-the-money slope re-quires the right smile wing to be the steeper one, and vice versa. Wing steepnessrefers to large-strike asymptotics here. It turns out that this is indeed the case forseveral of the infinite activity models we consider. This results in a qualitativelimitation on the smile shape that these models can produce.One of the few other works dealing with small-time L´evy slope asymptoticsis the comprehensive recent paper by Andersen and Lipton [4]. Besides manyother problems on various models and asymptotic regimes, they study the small-maturity ATM digital price and volatility slope for the tempered stable model MPLIED VOLATILITY SLOPE FOR L´EVY MODELS 3 (Propositions 8.4 and 8.5 in [4]). This includes the CGMY model as a special case(see Example 10 for details). Their proof method is entirely different from ours,exploiting the explicit form of the characteristic function of the tempered stablemodel. Using mainly the dominated convergence theorem, they also analyzethe convexity. We, on the other hand, assume a certain asymptotic behavior ofthe characteristic function, and use its explicit expression only when calculatingconcrete examples. Our approach covers, e.g., the ATM slope of the generalizedtempered stable, NIG, and Meixner models without additional effort.The recent preprint [21] is also closely related to our work. There, the Brow-nian component is generalized to stochastic volatility. On the other hand, theassumptions on the L´evy measure exclude, e.g., the NIG and Meixner models.Section 6 has additional comments on how our results compare to those of [4]and [21]. Al`os et al. [3] also study the small time implied volatility slope understochastic volatility and jumps, but the latter are assumed to have finite activity,which is not our focus. Results on the large time slope can be found in [23]; seealso [25], p. 63f. 2.
Digital call prices
We denote the underlying by S = e X , normalized to S = 1, and the pricingmeasure by P . W.l.o.g. the interest rate is set to zero, and so S is a P -martingale.Suppose that the log-underlying X = ( X t ) t ≥ is a L´evy process with characteristictriplet ( b, σ , ν ) and X = 0. The moment generating function (mgf) of X T is M ( z, T ) = E [ e zX T ] = exp ( T ψ ( z )) , where(2.1) ψ ( z ) = σ z + bz + (cid:90) R ( e zx − − zx ) ν ( dx ) . This representation is valid if the L´evy process has a finite first moment, whichwe of course assume, as even S t = e X t should be integrable. If, in addition, X has paths of finite variation, then (cid:82) R | x | ν ( dx ) < ∞ , and ψ ( z ) = σ z + b z + (cid:90) R ( e zx − ν ( dx ) , where the drift b is defined by b = b − (cid:90) R x ν ( dx ) . The following theorem collects some results about the small-time behavior of P [ X T ≥ S = e X is a martingale, and so P [ X T ≥ S. GERHOLD, I. C. G ¨UL ¨UM, AND A. PINTER condition from [35] is used:( H - α ) The L´evy measure ν has a density g ( x ) / | x | α , where g is a non-negativemeasurable function admitting left and right limits at zero: c + := lim x ↓ g ( x ) , c − := lim x ↑ g ( x ) , with c + + c − > . Theorem 1.
Let X be a L´evy process with characteristic triplet ( b, σ , ν ) and X = 0 . (i) If X has finite variation, and b (cid:54) = 0 , then lim T ↓ P [ X T ≥
0] = (cid:40) , b > , b < . (ii) If σ > , then lim T ↓ P [ X T ≥
0] = . (iii) If X is a L´evy jump diffusion, i.e., it has finite activity jumps and σ > ,then P [ X T ≥
0] = 12 + b σ √ π √ T + O ( T ) , T ↓ . (iv) Suppose that σ = 0 and that ( H - α ) holds for some α ∈ [1 , . If α = 1 ,we additionally assume c − = c + =: c and (cid:82) x − | g ( x ) − g ( − x ) | dx < ∞ .Then lim T ↓ P [ X T ≥
0] = (cid:40) + π arctan b ∗ πc if α = 1 , + απ arctan (cid:0) β tan (cid:0) απ (cid:1)(cid:1) if α (cid:54) = 1 , where b ∗ = b − (cid:82) ∞ ( g ( x ) − g ( − x )) /x dx and β = ( c + − c − ) / ( c + + c − ) . (v) If e X is a martingale and the L´evy measure satisfies ν ( dx ) = e − x/ ν ( dx ) ,where ν is a symmetric measure, then P [ X T ≥
0] = Φ( − σ imp (1 , T ) √ T / , where Φ denotes the standard Gaussian cdf.Proof. (i) We have P [ X T ≥
0] = P [ T − X T ≥ T − X T converges a.s. to b ,by Theorem 43.20 in [36].(ii) If σ >
0, then T − / X T converges in distribution to a centered Gaussianrandom variable with variance σ (see [36]). For further CLT-type results in thisvein, see [13, 27].(iii) Conditioning on the first jump time τ , which has an exponential distribu-tion, we find P [ X T ≥
0] = P [ X T ≥ | τ ≤ T ] · P [ τ ≤ T ] + P [ X T ≥ | τ > T ] · P [ τ > T ]= O ( T ) + P [ σW T + b T ≥ O ( T ))= P [ σW T + b T ≥
0] + O ( T )= Φ( b √ T /σ ) + O ( T ) . (2.2) MPLIED VOLATILITY SLOPE FOR L´EVY MODELS 5
Now apply the expansion(2.3) Φ( x ) = 12 + x √ π + O ( x ) , x → . (iv) By Proposition 1 in [35], the rescaled process X ε,αt := ε − X ε α t converges inlaw to a strictly α -stable process X ∗ ,αt as ε ↓
0. Thereforelim T ↓ P [ X T ≥
0] = lim ε ↓ P [ ε − X ε α ≥
0] = P [ X ∗ ,α ≥ , and it suffices to evaluate the latter probability. For α = 1, X ∗ , has a Cauchydistribution with characteristic exponentlog E [exp( iuX ∗ , )] = ib ∗ u − πc | u | , hence P [ X ∗ , ≥
0] = π arctan b ∗ πc . (Our b ∗ is denoted γ ∗ in [35].)If 1 < α <
2, then X ∗ ,α has a strictly stable distribution with characteristicexponent log E [exp( iuX ∗ ,α )] = −| du | α (cid:16) − iβ sgn( u ) tan (cid:0) απ (cid:1)(cid:17) , where d α ± = − Γ( − α ) cos (cid:0) απ (cid:1) c ± ≥ , d α = d α + + d α − , β = d α + − d α − d α ∈ ( − , . The desired expression for P [ X ∗ ,α ≥
0] then follows from [11]. See [17] for furtherrelated references.(v) Under this assumption, the market model is symmetric in the sense of [15,16]. The statement is Theorem 3.1 in [15]. (cid:3)
The variance gamma model and the CGMY model with 0 < Y < σ = 0), and so part (i) ofTheorem 1 is applicable. Part (iii) is applicable, clearly, to the well-known jumpdiffusion models by Merton and Kou. In Section 6, we will discuss two examplesfor part (iv) (NIG and Meixner).3. Implied Volatility Slope and Digital Options with SmallMaturity
The (Black-Scholes) implied volatility is the volatility that makes the Black-Scholes call price equal the call price with underlying S : C BS ( K, T, σ imp ( K, T )) = C ( K, T ) := E [( S T − K ) + ] . Since no explicit expression is known for σ imp ( K, T ) (see [26]), many authorshave investigated approximations (see, e.g., the references in the introduction).The following relation between implied volatility slope and digital calls is wellknown [25]; we give a proof for completeness. (Note that absolute continuityof S T holds in all L´evy models of interest, see Theorem 27.4 in [36], and will beassumed throughout.) S. GERHOLD, I. C. G ¨UL ¨UM, AND A. PINTER
Lemma 2.
Suppose that the law of S T is absolutely continuous for each T > ,and that (3.1) lim T ↓ C ( K, T ) = ( S − K ) + , K > . Then, for T ↓ , (3.2) ∂ K σ imp ( K, T ) | K =1 ∼ (cid:114) πT (cid:18) − P [ S T ≥ − σ imp (1 , T ) √ T √ π + O (cid:0)(cid:0) σ imp (1 , T ) √ T (cid:1) (cid:1)(cid:19) . Proof.
By the implicit function theorem, the implied volatility slope has therepresentation ∂ K σ imp ( K, T ) = ∂ K C ( K, T ) − ∂ K C BS ( K, T, σ imp ( K, T )) ∂ σ C BS ( K, T, σ imp ( K, T )) . Since the law of S T is absolutely continuous, the call price C ( K, T ) is continuouslydifferentiable w.r.t. K , and ∂ K C ( K, T ) = − P [ S T ≥ K ]. Inserting the explicitformulas for the Black-Scholes Vega and digital price, and specializing to theATM case K = S = 1, we get ∂ K σ imp ( K, T ) | K =1 = Φ( − σ imp (1 , T ) √ T / − P [ S T ≥ √ T ϕ ( σ imp (1 , T ) √ T / , where Φ and ϕ denote the standard Gaussian cdf and density, respectively. ByProposition 4.1 in [34], our assumption (3.1) implies that the annualized impliedvolatility σ imp (1 , T ) √ T tends to zero as T ↓
0. (The second assumption usedin [34] are the no-arbitrage bounds ( S − K ) + ≤ C ( K, T ) ≤ S , for K, T >
0, butthese are satisfied here because our call prices are generated by the martingale S .)Using the expansion (2.3) and ϕ ( x ) = √ π + O ( x ), we thus obtain (3.2). (cid:3) The asymptotic relation (3.2) is, of course, consistent with the small-moneynessexpansion presented in [12], where (cid:112) π/T (cid:0) − P [ S T ≥ K ] (cid:1) appears as secondorder term (i.e., first derivative) of implied volatility.Lemma 2 shows that, in order to obtain first order asymptotics for the at-the-money (ATM) slope, we need first order asymptotics for the ATM digital callprice P [ S T ≥ S = 1.) For models where lim T ↓ P [ S T ≥
1] = ,we need the second order term of the digital call as well, and the first order termof σ imp (1 , T ) √ T . The limiting value 1 / P [ S T ≥
1] may convergeto 1 / Y ∈ (1 , X is such that S = e X isa martingale with S = 1. Proposition 3.
Suppose that the L´evy process X has finite variation (and thus,necessarily, that σ = 0 ), and that b (cid:54) = 0 . Then the ATM implied volatility slope MPLIED VOLATILITY SLOPE FOR L´EVY MODELS 7 satisfies ∂ K σ imp ( K, T ) | K =1 ∼ − (cid:112) π/ b ) · T − / , T ↓ . Note that T − / is the fastest possible growth order for the slope, in any model(see Lee [30]).If X is a L´evy jump diffusion with σ >
0, then by part (iii) of Theorem 1,(3.2), and the fact that σ imp → σ (implied volatility converges to spot volatility),we obtain the finite limit(3.3) lim T ↓ ∂ K σ imp ( K, T ) | K =1 = − b σ − σ . (It is understood that the substitution K = 1 is to be performed before the limit T ↓ b , fixed by the condition E [exp( X )] = 1,depends on them. Moreover, (3.3) is consistent with the formal calculation ofthe variance slope lim T ↓ ∂ K σ ( K, T ) | K =1 = − b − σ on p. 61f in [25]. In fact (3.3) is well known for jump diffusions, see [3, 39].4. General remarks on Mellin transform asymptotics
As mentioned after Lemma 2, we need the second order term for the ATMdigital call if we want to find the limiting slope in L´evy models with a Browniancomponent. While this is easy for finite activity models (see the end of thepreceding section), it is more difficult in the case of infinite activity jumps. Wewill find this second order term using Mellin transform asymptotics. For furtherdetails and references on this technique, see e.g. [22]. The Mellin transform of afunction H , locally integrable on (0 , ∞ ), is defined by( M H )( s ) = (cid:90) ∞ T s − H ( T ) dT. Under appropriate growth conditions on H at zero and infinity, this integral de-fines an analytic function in an open vertical strip of the complex plane. Thefunction H can be recovered from its transform by Mellin inversion (see for-mula (7) in [22]):(4.1) H ( T ) = 12 πi (cid:90) κ + i ∞ κ − i ∞ ( M H )( s ) T − s ds, where κ is a real number in the strip of analyticity of M H . For the validityof (4.1), it suffices that H is continuous and that y (cid:55)→ ( M H )( κ + iy ) is integrable.Denote by s ∈ R the real part of the left boundary of the strip of analyticity.A typical situation in applications is that M H has a pole at s , and admits ameromorphic extension to a left half-plane, with further poles at s > s > s >. . . Suppose also that the meromorphic continuation satisfies growth estimatesat ± i ∞ which allow to shift the integration path in (4.1) to the left. We then S. GERHOLD, I. C. G ¨UL ¨UM, AND A. PINTER collect the contribution of each pole by the residue theorem, and arrive at anexpansion (see formula (8) in [22]) H ( T ) = Res s = s ( M H )( s ) T − s + Res s = s ( M H )( s ) T − s + . . . Thus, the basic principle is that singularities s i of the transform are mapped toterms T − s i in the asymptotic expansion of H at zero. Simple poles of M H yieldpowers of T , whereas double poles produce an additional logarithmic factor log T ,as seen from the expansion T − s = T − s i (1 − (log T )( s − s i ) + O (( s − s i ) )).5. Main results: digital call prices and slope asymptotics
The mgf M ( z, T ) of X T is analytic in a strip z − < Re( z ) < z + , given by thecritical moments(5.1) z + = sup { z ∈ R : E [ e zX T ] < ∞} and(5.2) z − = inf { z ∈ R : E [ e zX T ] < ∞} . Since X is a L´evy process, the critical moments do not depend on T . We willobtain asymptotic information on the transition probabilities (i.e., digital callprices) from the Fourier representation [29] P [ S T ≥
1] = P [ X T ≥ iπ (cid:90) a + i ∞ a − i ∞ M ( z, T ) z dz = 1 π Re (cid:90) ∞ M ( a + iy, T ) a + iy dy, (5.3)where the real part of the vertical integration contour satisfies a ∈ (0 , ⊆ ( z − , z + ), and convergence of the integral is assumed throughout. We are goingto analyze the asymptotic behavior of this integral, for T ↓
0, by computing itsMellin transform. Asymptotics of the probability (digital price) P [ X T ≥
0] arethen evident from (5.3). The linearity of log M as a function of T enables us toevaluate the Mellin transform in semi-explicit form. Lemma 4.
Suppose that S = e X is a martingale, and that σ > . Then, for any a ∈ (0 , , the Mellin transform of the function (5.4) H ( T ) := (cid:90) ∞ e T ψ ( a + iy ) a + iy dy, T > , is given by (5.5) ( M H )( s ) = Γ( s ) F ( s ) , < Re( s ) < , where (5.6) F ( s ) = (cid:90) ∞ ( − ψ ( a + iy )) − s a + iy dy, < Re( s ) < . Moreover, | ( M H )( s ) | decays exponentially, if Re( s ) ∈ (0 , ) is fixed and | Im( s ) | →∞ . MPLIED VOLATILITY SLOPE FOR L´EVY MODELS 9
See the appendix for the proof of Lemma 4. With the Mellin transform inhand, we now proceed to convert an expansion of the mgf at i ∞ to an expansionof P [ X T ≥
0] for T ↓
0. The following result covers, e.g., the NIG and Meixnermodels, and the generalized tempered stable model, all with σ >
0. See Section 6for details.
Theorem 5.
Suppose that S = e X is a martingale, and that σ > . Assumefurther that there are constants a ∈ (0 , , c ∈ C , ν ∈ [1 , and ε > such thatthe Laplace exponent satisfies (5.7) ψ ( z ) = 12 σ z + cz ν + O ( z ν − ε ) , Re( z ) = a, Im( z ) → ∞ . Then the ATM digital call price satisfies (5.8) P [ X T ≥
0] = 12 + C ˜ ν T ˜ ν + o ( T ˜ ν ) , T ↓ , where C ˜ ν = ˜ ν π (cid:0) σ (cid:1) ˜ ν − Im( e − iπ ˜ ν c )Γ( − ˜ ν ) with ˜ ν = (2 − ν ) / ∈ (0 , ] . For ν = 1 ,this simplifies to P [ X T ≥
0] = 12 + Re( c ) σ √ π √ T + o ( √ T ) , T ↓ . Together with Lemma 2, this theorem implies the following corollary, which isour main result on the implied volatility slope as T ↓ Corollary 6.
Under the assumptions of Theorem , the ATM implied volatilityslope behaves as follows: (i) If ν = 1 , then lim T ↓ ∂ K σ imp ( K, T ) | K =1 = − Re( c ) σ − σ , with c from (5.7) . (ii) If < ν < and C ˜ ν (cid:54) = 0 , then ∂ K σ imp ( K, T ) | K =1 ∼ −√ πC ˜ ν T ˜ ν − / , T ↓ . Proof of Theorem . From (5.3) and (5.4) we know that(5.9) P [ X T ≥
0] = 1 π Re H ( T ) . We now express H ( T ) by the Mellin inversion formula (4.1), with κ ∈ (0 , ).This is justified by Lemma 4, which yields the exponential decay of the transform M H along vertical rays. (Continuity of H , which is also needed for the inversetransform, is clear.) Therefore, we have(5.10) H ( T ) = 12 πi (cid:90) / i ∞ / − i ∞ Γ( s ) F ( s ) T − s ds, T ≥ . As outlined in Section 4, we now show that Γ( s ) F ( s ) has a meromorphic contin-uation, then shift the integration path in (5.10) to the left, and collect residues.It is well known that Γ is meromorphic with poles at the non-positive integers, so it suffices to discuss the continuation of F , defined in (5.6). As in the proof ofLemma 4, we put h ( y ) := − ψ ( a + iy ), y ≥
0. To prove exponential decay of thedesired meromorphic continuation, it is convenient to split the integral: F ( s ) = (cid:90) y h ( y ) − s a + iy dy + (cid:90) ∞ y h ( y ) − s a + iy dy (5.11) =: A ( s ) + ˜ F ( s ) , < Re( s ) < . The constant y ≥ A is analyticin the half-plane Re( s ) < , and so ˜ F captures all poles of F in that half-plane.By (5.7), the function h has the expansion (with a possibly decreased ε , to beprecise)(5.12) h ( y ) = σ y + ˜ cy ν + O ( y ν − ε ) , y → ∞ , where ˜ c := (cid:40) − ci ν ν > , − ( c + σ a ) i ν = 1 . The reason why F (or ˜ F ) is not analytic at s = 0 is that the second integralin (5.11) fails to converge for y large. We thus subtract the following convergence-inducing integral from ˜ F :˜ G ( s ) := (cid:90) ∞ y ( σ y ) − s a + iy dy = − πi ( a σ ) − s e iπs sin 2 πs − (cid:90) y ( σ y ) − s a + iy dy (5.13) =: G ( s ) + A ( s ) . Note that G is meromorphic, and that A is analytic for Re( s ) < . From theexpansion(5.14) h ( y ) − s = ( σ y ) − s − csσ (cid:18) σ (cid:19) − s y ν − s − + O ( y ν − s ) − − ε ) , y → ∞ , for s fixed, we see that the function(5.15) ˜ F ( s ) := (cid:90) ∞ y a + iy (cid:0) h ( y ) − s − ( σ y ) − s (cid:1) dy is analytic for − ˜ ν < Re( s ) < , and, clearly, for 0 < Re( s ) < we have(5.16) ˜ F ( s ) = ˜ F ( s ) + ˜ G ( s ) . We have thus established the meromorphic continuation of ˜ F to the strip − ˜ ν < Re( s ) < . To continue ˜ F even further, we look at the second term in (5.14) and MPLIED VOLATILITY SLOPE FOR L´EVY MODELS 11 define˜ G ( s ) := − csσ (cid:18) σ (cid:19) − s (cid:90) ∞ y y ν − s − a + iy dy = − cπσ (cid:18) σ (cid:19) − s sa ν − s − e (2 s − ν +3) πi/ sin π ( ν − s ) + 2˜ csσ (cid:18) σ (cid:19) − s (cid:90) y y ν − s − a + iy dy =: G ( s ) + A ( s )and the compensated function˜ F ( s ) := (cid:90) ∞ y a + iy (cid:32) h ( y ) − s − ( σ y ) − s + 2˜ csσ (cid:18) σ (cid:19) − s y ν − s − (cid:33) dy. By (5.14), the function ˜ F is analytic for Re( s ) ∈ ( − ˜ ν − ε/ , ( ν − / F ( s ) = ˜ F ( s ) + ˜ G ( s ) , − ˜ ν < Re( s ) < ν − , and so the meromorphic continuation of ˜ F to the region − ˜ ν − ε/ < Re( s ) < is established.In order to shift the integration path in (5.10) to the left, we have to ensurethat the integral converges. This is the content of Lemma 7 below, which alsoyields the existence of an appropriate y ≥
0, to be used in the definition of ˜ F in (5.11). By the residue theorem, we obtain(5.17) H ( T ) = Res s =0 ( M H )( s ) T − s + Res s = − ˜ ν ( M H )( s ) T − s + 12 πi (cid:90) κ + i ∞ κ − i ∞ ( M H )( s ) T − s ds, T ≥ , where κ = − ˜ ν − ε/
4, and M H now of course denotes the meromorphic con-tinuation of the Mellin transform. We then compute the residues. Accordingto (5.11) and (5.16), the continuation of M H in a neighborhood of s = 0 is givenby Γ( s )( A ( s ) + ˜ F ( s ) + ˜ G ( s )). Therefore,Res s =0 ( M H )( s ) T − s = A (0) + ˜ F (0) + A (0) + Res s =0 Γ( s ) G ( s ) T − s = Res s =0 Γ( s ) G ( s ) T − s (5.18) = π + i ( γ − log( aσ/ √
2) + log T ) , where γ is Euler’s constant. Note that A (0) = − A (0) and ˜ F (0) = 0 by defi-nition. The remaining residue (5.18) is straightforward to compute from (5.13)(with a computer algebra system, e.g.) and has real part π . Notice that thelogarithmic term log T , resulting from the double pole at zero (see the end ofSection 4), appears only in the imaginary part. Recalling (5.9), we see that thefirst term on the right-hand side of (5.17) thus yields the first term of (5.8). Similarly, we compute for ν > s = − ˜ ν ( M H )( s ) T − s = Res s = − ˜ ν Γ( s ) G ( s ) T − s = Γ( − ˜ ν )2 π (cid:34) csσ (cid:18) σ (cid:19) − s πa ν − s − e (2 s − ν +3) πi/ T − s (cid:35) s = − ˜ ν . In the case ν = 1, the function G also has a pole at − ˜ ν = − , and we obtainRes s = − ˜ ν ( M H )( s ) T − s = Res s = − / Γ( s )( G ( s ) + G ( s )) T − s = (cid:114) π (cid:18) i ˜ cσ − aσ (cid:19) √ T .
A straightforward computation shows that the stated formula for C ˜ ν is correctin both cases. The integral on the right-hand side of (5.17) is clearly O ( T − κ ) = o ( T ˜ ν ), and so the proof is complete. (cid:3) Lemma 7.
There is y ≥ such that the meromorphic continuation of M H constructed in the proof of Theorem , which depends on y via the definition of ˜ F in (5.11) , decays exponentially as | Im( s ) | → ∞ . Lemma 7 is proved in the appendix.6.
Examples
We now apply our main results (Theorem 5 and Corollary 6) to several concretemodels.
Example 8.
The NIG (Normal Inverse Gaussian) model has Laplace exponent ψ ( z ) = σ z + µz + δ ( (cid:112) ˆ α − β − (cid:112) ˆ α − ( β + z ) ) , where δ > , ˆ α > max { β + 1 , − β } . (The notation ˆ α should avoid confusion with α from Theorem .) Since S is a martingale, we must have µ = − σ + δ ( (cid:112) ˆ α − ( β + 1) − (cid:112) ˆ α − β ) . The relation between µ and b from (2.1) is µ + βδ/ (cid:112) ˆ α − β = b , as seen fromthe derivative of the Laplace exponent ψ at z = 0 . The L´evy density is ν ( dx ) dx = δ ˆ απ | x | e βx K (cid:0) ˆ α | x | (cid:1) , where K is the modified Bessel function of second order and index 1.First assume σ = 0 . Since K ( x ) ∼ /x for x ↓ , condition ( H - α ) is satisfiedwith α = 1 , with c + = c − = δ/π . The integrability condition in part (iv) ofTheorem is easily checked, and we conclude lim T ↓ P [ X T ≥
0] = 12 + 1 π arctan (cid:16) µδ (cid:17) , σ = 0 . Note that b ∗ = µ = b − δ ˆ απ (cid:82) ∞ K ( ˆ αx )( e βx − e − βx ) dx . By Lemma , the impliedvolatility slope of the NIG model thus satisfies ∂ K σ imp ( K, T ) | K =1 ∼ − (cid:112) /π arctan( µ/δ ) · T − / , T ↓ , σ = 0 , µ (cid:54) = 0 . MPLIED VOLATILITY SLOPE FOR L´EVY MODELS 13 - - - - - Figure 1.
The volatility smile, as a function of log-strike, of theNIG model with parameters σ = 0 . α = 4 . β = − . δ = 0 . T = 0 . T =0 .
01 (right panel). The parameters were calibrated to S&P 500call prices from Appendix A of [8]. The dashed line is the slopeapproximation (6.1). We did the calibration and the plots withMathematica, using the Fourier representation of the call price.
Now assume that σ > . Since (cid:112) ˆ α − ( β + z ) = − iz + O (1) as Im( z ) → ∞ ,the expansion (5.7) becomes ψ ( z ) = σ z + ( µ + i ) z + O (1) , Re( z ) = a, Im( z ) → ∞ . We can thus apply Theorem to conclude that the ATM digital price satisfies P [ X T ≥
0] = 12 + µσ √ π √ T + o ( √ T ) , T ↓ , σ > . By part (i) of Corollary , the limit of the implied volatility slope is given by lim T ↓ ∂ K σ imp ( K, T ) | K =1 = − µσ − σ δσ ( (cid:112) ˆ α − β − (cid:112) ˆ α − ( β + 1) ) , σ > . (6.1) This limit is positive if and only if β > − . See Figure 1 for a numerical example. Let us stress again that we identify thecorrect sign of the slope, while we find that explicit asymptotics do not approx-imate the value of the slope very accurately. Still, in the right panel of Figure 1we have zoomed in at very short maturity to show that our approximation givesthe asymptotically correct tangent in this example.
Example 9.
The Laplace exponent of the Meixner model is ψ ( z ) = σ z + µz + 2 ˆ d log cos(ˆ b/ ( − ˆ aiz − i ˆ b ) , where ˆ d > , ˆ b ∈ ( − π, π ) , and < ˆ a < π − ˆ b . (We follow the notation ofSchoutens [37] , except that we write µ instead of m , and ˆ a, ˆ b, ˆ d instead of a, b, d .) The L´evy density is ν ( dx ) dx = ˆ d exp(ˆ bx/ ˆ a ) x sinh( πx/ ˆ a ) . We can proceed analogously to Example . For σ = 0 we again apply part (iv) ofTheorem , with α = 1 , where now c + = c − = ˆ d ˆ a/π . Consequently, lim T ↓ P [ X T ≥
0] = 12 + 1 π arctan (cid:18) µ ˆ a ˆ d (cid:19) , σ = 0 , and ∂ K σ imp ( K, T ) | K =1 ∼ − (cid:112) /π arctan (cid:18) µ ˆ a ˆ d (cid:19) · T − / , T ↓ , σ = 0 , µ (cid:54) = 0 . Now assume σ > . The expansion of the Laplace exponent is ψ ( z ) = σ z + ( µ + ˆ a ˆ di ) z + O (1) , Re( z ) = a, Im( z ) → ∞ . By Theorem , the ATM digital price in the Meixner model thus satisfies P [ X T ≥
0] = 12 + µσ √ π √ T + o ( √ T ) , T ↓ . The limit of the implied volatility slope is given by lim T ↓ ∂ K σ imp ( K, T ) | K =1 = − µσ − σ
2= 2 ˆ dσ log (cid:32) cos(ˆ b/ ( − (ˆ a + ˆ b ) i ) (cid:33) , σ > . Example 10.
The Laplace exponent of the CGMY model is (6.2) ψ ( z ) = σ z + µz + C Γ( − Y )(( M − z ) Y − M Y + ( G + z ) Y − G Y ) , where we assume C > , G > , M > , < Y < , and Y (cid:54) = 1 .The case σ = 0 and Y ∈ (0 , need not be discussed, as it is a special caseof Proposition 8.5 in [4] . Our Proposition could also be applied, as the CGMYprocess has finite variation in this case.If σ = 0 and Y ∈ (1 , , then the ATM digital call price converges to , and theslope explodes, of order T / − /Y . This is a special case of Corollary 3.3 in [21] .Note that Proposition 8.5 in [4] is not applicable here, because the constant C M from this proposition vanishes for the CGMY model, and so the leading term ofthe slope is not obtained. Theorem (iv) from our Section is not useful, either;it gives the correct digital call limit price , but does not provide the second orderterm necessary to get slope asymptotics.We now proceed to the case σ > , which is our main focus. The expansionof ψ at i ∞ is ψ ( z ) = σ z + c Y z Y + µz + O ( z Y − ) , Re( z ) = a, Im( z ) → ∞ , MPLIED VOLATILITY SLOPE FOR L´EVY MODELS 15 with the complex constant c Y := C Γ( − Y )(1 + e − iπY ) . First assume < Y < .Then we proceed analogously to the preceding examples, applying Theorem andCorollary . The ATM digital price thus satisfies (6.3) P [ X T ≥
0] = 12 + µσ √ π √ T + o ( √ T ) , T ↓ , and the limit of the implied volatility slope is given by lim T ↓ ∂ K σ imp ( K, T ) | K =1 = − µσ − σ
2= 1 σ C Γ( − Y )(( M − Y − M Y + ( G + 1) Y − G Y ) . (6.4) Now assume < Y < . In principle, Theorem is applicable, with ν = Y ;however, the constant C ˜ ν in (5.8) is zero, and so we do not get the second termof the expansion immediately. What happens is that the Mellin transform of H (see the proof of Theorem ) may have further poles in − < Re( s ) < , butnone of them gives a contribution, since the corresponding residues have zero realpart. Therefore, (6.3) and (6.4) are true also for < Y < . See A. Pinter’sforthcoming PhD thesis for details. Note that (6.3) and (6.4) also follow fromconcurrent work by Figueroa-L´opez and ´Olafsson [21] . For < Y < , theyalso follow from Proposition 8.5 in [4] , but not for < Y < , because then theconstant C M from that proposition vanishes when specializing it to the CGMYmodel. In the following example, we discuss the generalized tempered stable model.The tempered stable model, which is investigated in [4], is obtained by setting α − = α + . Example 11.
The generalized tempered stable process [10] is a generalization ofthe CGMY model, with L´evy density ν ( dx ) dx = C − | x | α − e − λ − | x | ( −∞ , ( x ) + C + | x | α + e − λ + | x | (0 , ∞ ) ( x ) , where α ± < and C ± , λ ± > . For α ± (cid:54)∈ { , } the Laplace exponent of thegeneralized tempered stable process is ψ ( z ) = σ z + µz + Γ( − α + ) C + (cid:16) ( λ + − z ) α + − λ α + + (cid:17) + Γ( − α − ) C − (cid:16) ( λ − + z ) α − − λ α − − (cid:17) . For σ > , α + ∈ (1 , , and α − < α + we have the following expansion: ψ ( z ) = σ z + Γ( − α + ) C + e − iπα + z α + + O ( z max { ,α − } ) , Re( z ) = a, Im( z ) → ∞ . We now apply Theorem with ν = α + , and find that the second order expansionof the ATM digital call is P [ X T ≥
0] = 12 + C ˜ ν T ˜ ν + o ( T ˜ ν ) , T ↓ , with ˜ ν = 1 − α + / ∈ (0 , ) and the real constant C ˜ ν = ˜ ν π (cid:0) σ (cid:1) ˜ ν − Γ( − α + ) C + Im( e − iπ ˜ ν e − iπα + ) (cid:124) (cid:123)(cid:122) (cid:125) =sin( − π (1+ α + / Γ( − ˜ ν ) . By Corollary (ii), the ATM implied volatility slope explodes, but slower than T − / : ∂ K σ imp ( K, T ) | K =1 ∼ −√ πC ˜ ν T ˜ ν − / , T ↓ . Note that these results also follow from the concurrent paper [21] , which treatstempered stable-like models.If σ > and α + < , then part (i) of Corollary is applicable, and formulasanalogous to (6.3) and (6.4) hold. Robustness of Lee’s Moment Formula
As we have already mentioned, our first order slope approximations give limitedaccuracy for the size of the slope, but usually succeed at identifying its sign, i.e.,whether the smile increases or decreases at the money. It is a natural questionwhether this sign gives information on the smile as a whole: If the slope is positive,does it follow that the right wing is steeper than the left one, and vice versa?To deal with this issue, recall Lee’s moment formula [28]. Under the assumptionthat the critical moments z + and z − , defined in (5.1) and (5.2), are finite, Lee’sformula states that(7.1) lim sup k →∞ σ imp ( K, T ) √ k = (cid:114) Ψ( z + − T and(7.2) lim sup k →−∞ σ imp ( K, T ) √− k = (cid:114) Ψ( − z − ) T , where
T > k = log K , and Ψ( x ) := 2 − √ x + x − x ). According toLee’s formula, the slopes of the wings depend on the size of the critical moments.In L´evy models, the critical moments do not depend on T . The compatibilityproperty we seek now becomes:(7.3) lim k →∞ σ imp ( K, T ) √ k > lim k →−∞ σ imp ( K, T ) √− k for all T > ∂ K σ imp ( K, T ) | K =1 > T. That is, the right wing of the smile is steeper than the left wing deep out-of-the-money if and only if the small-maturity at-the-money slope is positive. We nowshow that this is true for several infinite activity L´evy models. By our methods,this can certainly be extended to other infinite activity models. It does not hold,though, for the Merton and Kou jump diffusion models. The parameter rangesin the following theorem are the same as in the examples in Section 6.
MPLIED VOLATILITY SLOPE FOR L´EVY MODELS 17
Theorem 12.
Conditions (7.3) and (7.4) are equivalent for the following models.For the latter three, we assume that σ > or µ (cid:54) = 0 . • Variance gamma with σ = 0 , b (cid:54) = 0 • NIG • Meixner • CGMY
Put differently, these models are not capable (at short maturity) of producinga smile that has, say, its minimum to the left of log K = k = 0, and thus apositive ATM slope, but whose left wing is steeper than the right one. Proof.
The critical moments are clearly finite for all of these models. Moreover,it is well known that the lim sup in (7.1) and (7.2) can typically be replaced by agenuine limit, for instance using the criteria given by Benaim and Friz [6]. Theirconditions on the mgf are easily verified for all our models; in fact Benaim andFriz [6] explicitly treat the variance gamma model with b = 0 and the NIGmodel. We thus have to show that (7.4) is equivalent to Ψ( z + − > Ψ( − z − ).Since Ψ is strictly decreasing on (0 , ∞ ), the latter condition is equivalent to z + − < − z − . It remains to check the equivalence(7.5) z + − < − z − ⇐⇒ (7.4) . The mgf of the variance gamma model is (see [31]) M ( z, T ) = e T b z (1 − θνz − ˆ σ νz ) − T/ν , where ˆ σ, ν > θ ∈ R . Its paths have finite variation, and so Proposition 3shows that (7.4) is equivalent to b <
0. The critical moments are z ± = − νθ ± √ ν ˆ σ + ν θ ν ˆ σ , and we have − z − + 1 − z + = 1 + 2 θ/ ˆ σ . This is positive if and only if b = ν − log(1 − θν − ˆ σ ν ) < , which yields (7.5).As for the other three models, first suppose that σ >
0. The examples inSection 6 show that (7.4) is equivalent to µ < − σ . The critical moments ofthe NIG model are z + = ˆ α − β and z − = − ˆ α − β . Therefore, z + − < − z − ifand only if β > − , and this is indeed equivalent to µ + σ = δ ( (cid:112) ˆ α − ( β + 1) − (cid:112) ˆ α − β ) < . For the Meixner model, we have z ± = ( ± π − ˆ b ) / ˆ a , which yields − z − + 1 − z + =1 + 2ˆ b/ ˆ a . On the other hand, µ + σ = − d log cos(ˆ b/ a + ˆ b ) / , which is negative if and only if cos(ˆ b/ > cos((ˆ a + ˆ b ) / a + 2ˆ b > Finally, in case of the CGMY model, we have µ + σ = − C Γ( − Y ) (cid:0) ( M − Y − M Y + ( G + 1) Y − G Y (cid:1) . Since, for Y ∈ (0 , − Y ) < x (cid:55)→ x Y − ( x + 1) Y is strictlyincreasing on (0 , ∞ ), we see that µ + σ < M − < G . This isthe desired condition, since the explicit expression (6.2) shows that z + = M and z − = − G . The case Y ∈ (1 ,
2) is analogous.It remains to treat the case σ = 0. First, note that the critical moments donot depend on σ . Furthermore, from the examples in Section 6, we see that (7.4)holds if and only if µ <
0. Now observe that adding a Brownian motion σW t to a L´evy model adds − σ to the drift, if the martingale property is to bepreserved. Therefore, the assertion follows from what we have already provedabout σ > (cid:3) Conclusion
Our main result (Corollary 6) translates asymptotics of the log-underlying’smgf to first-order asymptotics for the ATM implied volatility slope. Checking therequirements of Corollary 6 only requires Taylor expansion of the mgf, which hasan explicit expression in all models of practical interest. Higher order expansionscan be obtained by the same proof technique, if desired. They will follow ina relatively straightforward way from higher order expansions of the mgf, bycollecting further residues of the Mellin transform. In future work, we hope toconnect our assumptions on the mgf with properties of the L´evy triplet, whichshould give additional insight on how the slope depends on model characteristics.
Appendix A. Proofs of Lemmas and Proof of Lemma . Since S = e X is a martingale, we have ψ (cid:48) (0) = E [ X ] < ψ (0) = 0 implies that ψ ( a ) < a >
0. In fact,it easily follows from ψ (1) = 0 and the concavity of ψ that all a ∈ (0 ,
1) satisfy ψ ( a ) <
0. Let us fix such an a . FromRe( − ψ ( a + iy )) = − ψ ( a ) + 12 σ y + (cid:90) R e ax (1 − cos( yx )) (cid:124) (cid:123)(cid:122) (cid:125) ≥ ν ( dx )we obtain that the function h ( y ) := − ψ ( a + iy ), y ≥
0, satisfies(A.1) Re h ( y ) > σ y ≥ , y ≥ . For 0 < Re( s ) < define the function g ( T ) = T Re( s ) − (cid:90) ∞ e − T Re( h ( y )) | a + iy | dy, T > . MPLIED VOLATILITY SLOPE FOR L´EVY MODELS 19
Using Fubini’s theorem and substituting T Re( h ( y )) = u , we then calculate forRe( s ) > (cid:90) ∞ g ( T ) dT = (cid:90) ∞ | a + iy | (cid:90) ∞ e − T Re( h ( y )) T Re( s ) − dT dy = (cid:90) ∞ Re( h ( y )) − Re( s ) | a + iy | (cid:18)(cid:90) ∞ e − u u Re( s ) − du (cid:19) dy = Γ(Re( s )) (cid:90) ∞ Re( h ( y )) − Re( s ) | a + iy | dy. From (A.1), we get (cid:90) ∞ Re( h ( y )) − Re( s ) | a + iy | dy ≤ ( σ ) − Re( s ) (cid:90) ∞ y − s ) | a + iy | dy. The restriction Re( s ) < ensures that the last integral is finite and thus the inte-grability of g . Using the dominated convergence theorem and Fubini’s theorem,the Mellin transform of H can now be calculated as (cid:90) ∞ H ( T ) T s − dT = (cid:90) ∞ a + iy (cid:90) ∞ e − T h ( y ) T s − dT dy. The substitution
T h ( y ) = u gives us the result. Note that h ( y ) is in generalnon-real; it is easy to see, though, that Euler’s integralΓ( s ) = (cid:90) ∞ u s − e − u du, Re( s ) > , still represents the gamma function if the integration is performed along anycomplex ray emanating from zero, as long as the ray stays in the right half-plane.The latter holds, since Re( h ( y )) > M H ( s ) =Γ( s ) F ( s ) for large | Im( s ) | . First, note thatIm ψ ( a + iy ) = by + σ ay + (cid:90) R ( e ax sin xy + xy ) ν ( dx )= O ( y ) , y → ∞ , which together with (A.1) yields the existence of an ε > | arg h ( y ) | ≤ π − ε for all y ≥
0. We then estimate, with Re( s ) ∈ (0 , ) fixed, | F ( s ) | ≤ (cid:90) ∞ e − Re( s log h ( y )) | a + iy | dy = (cid:90) ∞ e − Re( s ) log | h ( y ) | +Im( s ) arg h ( y ) | a + iy | dy ≤ e ( π/ − ε ) | Im( s ) | (cid:90) ∞ ( σ y ) − Re( s ) | a + iy | dy. The integral converges, and thus this estimate is good enough, since Stirling’sformula yields | Γ( s ) | = exp (cid:0) − π | Im( s ) | (1 + o (1)) (cid:1) . (cid:3) Proof of Lemma . Recall that, in the proof of Theorem 5, we defined the fol-lowing meromorphic continuation of F ( s ), to the strip − ˜ ν − ε < Re( s ) < : A ( s ) + ˜ G ( s ) + ˜ F ( s ) , − ˜ ν < Re( s ) < ,A ( s ) + ˜ G ( s ) + ˜ G ( s ) + ˜ F ( s ) , − ˜ ν − ε < Re( s ) < ( ν − . As noted at the end of the proof of Lemma 4, Stirling’s formula implies | Γ( s ) | =exp (cid:0) − π | Im( s ) | (1 + o (1)) (cid:1) . By (5.5), it thus suffices to argue that the con-tinuation of F ( s ) is O (exp(( π − ε ) | Im( s ) | )) for some ε >
0. The functions ˜ G and ˜ G are clearly O (1). As for A , defined in (5.11), we have | A ( s ) | ≤ (cid:90) y e − Re( s log h ( y )) | a + iy | dy = (cid:90) y | h ( y ) | − Re( s ) e Im( s ) arg h ( y ) | a + iy | dy. Now note that | h ( y ) | − Re( s ) ≤ (cid:40) ( σ y ) − Re( s ) < Re( s ) < , (max ≤ y ≤ y | h ( y ) | ) − Re( s ) Re( s ) ≤ , and that exp(Im( s ) arg h ( y )) ≤ exp(( π − ε ) | Im( s ) | )for some ε >
0, as argued in the proof of Lemma 4.It remains to establish a bound for ˜ F , defined in (5.15). (The bound for ˜ F is completely analogous, and we omit the details.) In what follows, we assumethat − ˜ ν < Re( s ) < . By (5.12), we have (where the O is uniform w.r.t. s , and y ≥ F ( s ) = (cid:90) ∞ y a + iy (cid:0) ( σ y ) − s (1 + O ( y ν − )) − s − ( σ y ) − s (cid:1) dy = (cid:90) ∞ y a + iy ( σ y ) − s (cid:0) (1 + O ( y ν − )) − s − (cid:1) dy. (A.2)We now choose y such that, for some constant C > (cid:12)(cid:12) log | O ( y ν − ) | (cid:12)(cid:12) ≤ π, (cid:12)(cid:12) arg(1 + O ( y ν − )) (cid:12)(cid:12) ≤ π, (cid:12)(cid:12) log(1 + O ( y ν − )) (cid:12)(cid:12) ≤ C y ν − , hold for all y ≥ y . (By a slight abuse of notation, here O ( y ν − ) of course denotesthe function hiding behind the O ( y ν − ) in (A.2).) For all w ∈ C we have theestimate | e w − | ≤ | w | e | Re( w ) | . MPLIED VOLATILITY SLOPE FOR L´EVY MODELS 21
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