A General Moment Formula

A General Moment Formula Vladimir Lucic Date: October 7, 2024 Abstract In this work we provide a generalization and unification of several moment formulæ: the Lee Moment Formula [15], the Log-Moment Formula [22], and the modified Piterbarg conjecture [11]. We approach the problem via investigating the asymptotic behaviour of the normalizing volatility transforms rather than the implied volatility itself. Our derivations are elementary and do not rely on the Regular Variation Theory. Keywords implied volatility, Lee moment formula, normalizing volatility transforms 1 Introduction In this section we informally outline the main contributions of this paper, deferring the full rigor to the sections to follow. Since its publication in 2004, the Lee Moment Formula has been embraced by academics and practitioners alike, making it one of the results that “every quant should know”. Over the years the result of Lee has been extended and refined in a number of ways, notably by the work of [3], [11], and, more recently, [22]. The Lee Moment Formula has also become indispensable in studying parametric volatility surfaces (e.g. [10], [12], [16]), and is often used in establishing various model-free properties of implied volatility (e.g. [26], [6]). Finally, extrapolating implied volatility beyond observable strikes is a ubiquitous task for a practitioner, where the Lee Moment Formula inevitably comes into play. For that reason, the formula is commonly found in the literature on applied quantitative finance (e.g. [9], [13], [14]). For some fixed time horizon T > 0 consider an asset price ST with a finite first moment FT := E[ST ] < ∞. In this work we revisit the moment problem Department of Mathematics Imperial College London E-mail: vlucic@ic.ac.uk 2 Vladimir Lucic for ST considered by Lee in [15], and connect it to the asymptotic behavior of the normalizing volatility transforms f1/2 : k 7→ −d1/2 (k, σ(k)), d1/2 (k, a) = − k a ± , a 2 introduced in [8] (here σ(·) stands for the Black-Scholes total implied volatility as a function of log-moneyness k = ln(K/FT )). This change of the viewpoint allows us to arrive at a general moment formula in Theorem 1 which covers and extends all previously developed moment farmulæ. We start by noting that the large-strike Lee Moment Formula from [15] which connects the upper critical moment    p̃ := sup p : E ST1+p < ∞ to the asymptotics of the right-wing implied volatility via σ 2 (k) = ψ(p̃), k k→∞  p x2 + x − x , ψ(x) := 2 − 4 lim sup (1) x ≥ 0, can be written more succinctly as p̃ = lim inf k→∞ f12 (k) . 2k This is a direct consequence of the identity   2 σ 2 (k) f1 (k) ≡ψ k 2k (2) valid for sufficiently large k, and the fact that ψ(·) is continuous and decreasing. f 2 (k) Thus, in particular, ST has moments of all orders iff lim inf k→∞ 12k = ∞, f 2 (k) and ST has no finite moments of order greater than one iff lim inf k→∞ 12k = 0. In the case of ST having moments of all orders, the Lee Moment Formula yields σ(k) lim sup √ = 0, (3) k k→∞ without giving precise information about the asymptotic behaviour of σ(·) at infinity. To address this case, in [18] it was conjectured that if we choose a positive and increasing w : (0, ∞) → (0, ∞) such that
w ek = ∞, (4) lim k→∞ k we shall obtain the limit lim sup k→∞ σ(k) q w ek k
=√ 1 , 2p̃w A General Moment Formula where 3 n o   p̃w := sup p ≥ 0 : E epw(ST ) < ∞ . In view of (3) this extends the formula of Lee. In [11] it was shown that for this conjecture to hold, p̃w has to be replaced with # ) ( "Z ST p̂w := sup p ≥ 0 : E 0 epw(u) du < ∞ . In addition to establishing such modified Piterbarg conjecture, [11] also gives conditions ensuring p̃w = p̂w for w(·) satisfying (4). In our main result, Theorem 1, we show that for an arbitrary non-decreasing w(·) we have f 2 (k) lim inf 1 k = p̂R ≡ p̂w , (5) k→∞ 2w(e ) for the right tail, while for the left tail the analogous formula is lim inf k→−∞ f22 (k) = p̂L , 2w(e|k| ) (6) where, for the nontrivial case of zero mass at the origin, P(ST = 0) = 0, # ) ( " Z 1/ST pw(u) e du < ∞ . p̂L := sup p ≥ 0 : E ST 0 This immediately recovers the Lee Moment Formulæ by setting w(x) ≡ ln(x). Similarly, Corollary p3 gives the modified Piterbarg Conjecture: from (4) and the identity σ(k) = f12 (k) + 2k − f1 (k) valid for large k, we obtain p √ σ(k) w(ek ) 1 2 q 2 =√ . = lim sup q lim sup f12 (k) f1 (k) k 2p̂ k k→∞ k→∞ w + + w(ek ) 2w(ek ) 2w(ek ) Finally, in Corollary 4 we obtain the recent Log-Moment Formula of [22] lim inf k→−∞    f22 (k) = sup p ≥ 0 : E | ln(ST /FT )|p < ∞ , 2 ln(|k|) This is a consequence of a more general log-moment formula that we give in Lemma 4, which also covers the right tail of the implied volatility. We emphasize that the right-tail version of the Log-Moment Formula cannot be obtained from the Piterbarg Conjecture, as (4) does not hold. In fact, it sharpens the Lee Moment Formula in a way precisely complementary to (3), addressing the case where no moments larger than one exist. We also note that Lemma 4 is not a direct corollary of our main result, as it also gives a simplified representation for p̂ (not involving the integral), analogous to the original Piterbarg conjecture. Our derivation are elementary and, in particular, we do not rely on the Regular Variation Theory (as in [11]). We have two main ingredients in our proofs. 4 Vladimir Lucic In Lemma 1 we show that in order for Φ(ST ) to be integrable for some Φ(·) with a non-decreasing derivative ϕ(·), we must have limK→∞ ϕ(K) Call(K) = 0, in which case Z ∞   Call(K) dϕ(K). E Φ(ST ) = ϕ(0) + 0 This in turns provides a direct link between the tail behavior of the call price function and the existence of the asset moments. Our second result, Lemma 2, links the tail behavior of the call price and the first normalizing volatility transform f1 (·), namely we obtain   q
ln f1 (k) −2 ln Call FT ek = f1 (k) + O , k → ∞. f1 (k) This result is important, as the link between the tail behavior of implied volatility and that of the normalizing volatility transforms can be readily obtained from (2). Thus, starting from the existence of the asset moments we ultimately obtain the asymptotic properties of the implied volatility. In the last section we provide applications of some of the previously developed formulæ. We focus on a concrete case which cannot be addressed with the known moment results. We also demonstrate on an example from [22] a way to establish existence of the genuine limit in the Log-Moment Formula. 2 Preliminaries In this section we fix notation and summarize some results from option pricing theory that will be needed in the sequel. To streamline the exposition, with slight loss of generality we will consider the case of zero discount rates. In the probabilistic context the financial term “asset” will refer to a nonnegative random variable. Our analysis will pertain to a single expiry T > 0, which we fix thereafter. In the practice of options pricing one observes the prevailing market Put and Call values, Put(K), Call(K), written on an asset ST , with strike K > 0 and expiry T , together with the asset forward value FT := E[ST ] < ∞. The total (or normalized) implied volatility slice σ(·) is linked to the Vanilla option prices (premia) via the Black-Scholes formulæ: with the “log-spot” XT ≡ ln(ST /FT ) and “log-moneyness” k ≡ ln(K/FT ), K, FT > 0 we have   Put(K)/K = E (1 − e−k eXT )+ = N (−d2 (k)) − e−k N (−d1 (k)) =: pBS (k), (7a)  XT  k k Call(K)/FT = E (e − e )+ = N (d1 (k)) − e N (d2 (k)) =: cBS (k), (7b) where N (·) stands for the standard Normal CDF and d1/2 (k) = − σ(k) k ± , σ(k) 2 k ∈ R. To avoid inessential complications, in what follows we shall restrict ourselves to the case of (strictly) positive total implied volatility. A General Moment Formula 5 Definition 1 A volatility slice σ : R → (0, ∞) is free of strike arbitrage if there exists an integrable asset ST with E[ST ] > 0, such that (7) holds. Following [8], for a volatility slice σ : R → (0, ∞) we define the normalizing volatility transforms (NVTs): f1 (k) := −d1 (k) = σ(k) k − , σ(k) 2 f2 (k) := −d2 (k) = k σ(k) + , σ(k) 2 k ∈ R. Remark 1 A necessary condition for a function K 7→ Call(K), K ≥ 0, to be a Call price function of an integrable asset is that it is free of large strike arbitrage, namely lim Call(K) = 0, K→∞ which is a consequence of the Lebesgue Dominated Convergence Theorem. Remark 2 The absence of the large strike arbitrage (see Remark 1) implies limk→∞ f√1 (k) = ∞ for the corresponding f1 (·): Theorem 5.5
of√[23] shows √ 2k − σ(k) 2k/σ(k) + limk→∞ 2k − σ(k) = ∞, thus limk→∞ f1 (k) = √
1 /2 ≥ limk→∞ 2k−σ(k) = ∞. Note also that limk→∞ f2 (k) = ∞, which is a √ consequence of the AM-GM inequality f2 (k) ≥ 2k, k ≥ 0, holds irrespective of the arbitrage considerations.
Remark 3 For the limiting behaviour at −∞ we have limk→−∞ N f2 (k) = P(ST = 0) (e.g. Lemma 3.10. of [7]). dP̂ Remark 4 If P(ST = 0) = 0, we can change the measure via dP = ST so that Ŝt := 1/St , t ∈ [0, T ], is a martingale under P̂. The total implied volatility of ŜT (under P̂) is then σ− (k) = σ(−k), k ∈ R, so after putting fi (k; σ), i = 1, 2, to signify the dependence of the NVTs on the underlying implied volatility, we get f1/2 (k; σ) = −f2/1 (−k; σ− ), k ∈ R (see, for instance, [5] or Section 4 in [15]). Recall that the Mills ratio is defined as R(x) := N (−x)/n(x), x ∈ R, where n(·) stands for the standard Normal PDF. Inserting this expression into the Black-Scholes formulæ (7), and using FT n(f1 (k)) = Kn(f2 (k)), k ≡ ln(K/FT ), k ∈ R, gives

Put(K) = Kφ −f2 (k), σ(k) , Call(K) = FT φ f1 (k), σ(k) , (8) where
φ(x, y) := n(x) R(x) − R(x + y) , x ∈ R, y > 0. (9) 6 Vladimir Lucic 3 Price Asymptotics Under Absence of Large-Strike Arbitrage The two technical lemmas presented in this section play a central role in establishing our main result. Lemma 1 Let K 7→ Call(K) be a call price function corresponding to the asset STR, E[ST ] = 1, and let ϕ : [0, ∞) → [0, ∞) be non-decreasing. Put x Φ(x) := 0 ϕ(u) du, x ≥ 0. Then (i)   E Φ(ST ) = ϕ(0) + Z ∞ Call(K) dϕ(K), (10) 0 where both sides are allowed to be infinite. (ii) If Φ(ST ) is integrable, then lim ϕ(K) Call(K) = 0. (11) K→∞ Proof By the Fubini’s theorem for every S ≥ 0 we have the Carr-Madan spanning representation Z ∞ [S − K]+ dϕ(K), Φ(S) = ϕ(0)S + 0 whence the first part of the result follows by the Tonelli’s theorem. For the second part, first assume ϕ(0) > 0, so that Φ(·) is increasing and Φ−1 (·) exists. Then Z ∞ Z ∞  
P ST ≥ Φ−1 (x) dx P (Φ(ST ) ≥ x) dx = E Φ(ST ) = 0 Z ∞ Z0 ∞ ∂K Call(K) ϕ(K) dK, (12) P(ST ≥ K) dΦ(K) = − = 0 0 so the integration-by-parts gives   E Φ(ST ) = ϕ(0) − lim ϕ(K) Call(K) + K→∞ Z ∞ Call(K) dϕ(K). 0 Therefore, from (10) we conclude that (11) holds due to integrability of Φ(ST ). If ϕ(0) = 0, applying the above part of the proof to ϕ(K) + 1, K ≥ 0 and using limK→∞ Call(K) = 0 (see Remark 1) gives (11). Remark 5 In Corollary 2.2 of [15] it was shown that if E[STp+1 ] < ∞, then Call(K) = O(K −p ) as K → ∞. Setting Φ(K) = K p+1 , K ≥ 0 in Lemma 1(ii) yields an improved estimate Call(K) = o(K −p ) as K → ∞. Remark 6 The estimate of Remark 5 can also be obtained combining Lemma 1(i) with the following classical result dueR to Pringsheim (e.g. Problem II.113 in ∞ [19]): if f (·) is monotone on [1, ∞) and 1 xp−1 f (x) dx < ∞, then limx→∞ xp f (x) = p+1 0. We take Φ(K) = K , K ≥ 0 in (10) and consider f (x) = Call(x), x ≥ 0. A General Moment Formula 7 Put FT = 1. From (8) and (9) we have
cBS (k) = n f1 (k) h(k), where k ∈ R,

h(k) := R f1 (k) − R f2 (k) , (13a) k ∈ R. Assume the absence of large strike arbitrage, so that limk→∞ fi (k) = ∞, i = 1, 2 (see Remark 2). In view of the fact that R(k) is decreasing to zero as k → ∞, there exists k0 > 0 such that f1 (k) > 0, 0 ≤ h(k) ≤ 1, k ≥ k0 . (13b) Using the Gordon’s bounds (e.g. [17]) 1 t < R(t) < , t2 + 1 t t > 0, from (13b) we obtain where
h(k) ≥ G f1 (k), f2 (k) , G(x, y) := xy − x2 − 1 , y(x2 + 1) k ≥ k0 , (13c) x, y > 0. Thus, with
g(k) := G f1 (k), f2 (k) , k ≥ k0 , (13d)
cBS (k) ≥ n f1 (k) g(k), k ≥ k0 . (13e) from (13a) and (13c) we obtain The following technical result enables us to establish a link between the tail behavior of the call price function and the first normalizing volatility transform f1 (·). Lemma 2 Suppose that σ(·) is a volatility slice free of large strike arbitrage. Then for g(·) given in (13d) we have (i) g(·) is positive for sufficiently large k. (ii)
1 = o f13 (k) , g(k) k → ∞. 8 Vladimir Lucic Proof Note that g(k) = f1 (k)σ(k) − 1
, f2 (k) f12 (k) + 1 k ≥ k0 . Fix an arbitrary ϵ > 0. The absence of large strike arbitrage implies that there exists k1 > 0 such that f1 (k) ≥ 1ϵ , k ≥ k1 . For k ≥ k1 such that σ(k) > ϵ we have g(k) =  1 f1 (k) − σ(k) f1 (k) − 1ϵ   ≥

=: g1 (k). f1 (k) f1 (k) f12 (k) + 1 + 1 f12 (k) + 1 σ(k) + 1 ϵ
Similarly, for k ≥ max k0 , ϵ2 /2 + 1 =: k2 such that σ(k) ≤ ϵ we have g(k) = k − 12 ϵ2 − 1 k − 21 σ 2 (k) − 1
2
≥

=: g2 (k), f1 (k) + σ(k) f1 (k) + 1 f1 (k) + ϵ f12 (k) + 1
and therefore, g(k) ≥ min g1 (k), g2 (k) , k ≥ max(k1 , k2 ). Thus, as limk→∞ f13 (k)gi (k) = ∞, i = 1, 2, we have limk→∞ f13 (k)g(k) = ∞. Since limk→∞ f1 (k) = ∞, both parts of the result follow. As a consequence of Lemma 2, we have the following asymptotics of the call price on its “natural scale” f1 (·). Corollary 1 Suppose that σ(·) is a volatility slice free of large strike arbitrage. Then for g(·) given in (13d) for sufficiently large k we have 0≤ p −2 ln g(k) + ln(2π) √ −2 ln cBS (k) − f1 (k) ≤ , 2f1 (k) (14) and p −2 ln cBS (k) = f1 (k) + O  ln f1 (k) f1 (k)  , k → ∞. (15) Proof From Lemma 2(i) and (13) for sufficiently large k we have 0≤ p − ln g(k) + 12 ln(2π) f1 (k) −2 ln g(k) + ln(2π) √ − ln cBS (k) − √ ≤ p ≤ , f1 (k) 2 2f1 (k) − ln cBS (k) + √2 which gives
(14). For (15), note that from Lemma 2(ii) we have − ln g(k) = O ln f1 (k) , k → ∞, whence the result follows. A General Moment Formula 9 4 Unifying the Moment Formulæ and Piterbarg Conjecture The main result of this section is the following theorem, generalizing the moment formulæ and the Piterbarg Conjecture. Theorem 1 Let ST be an asset with E[ST ] = 1. Suppose that w : [0, ∞) → [−∞, ∞) is non-decreasing with limK→∞ w(K) > 0, and put Z K epw(u) du, K ≥ 0, p ≥ 0. Φp (K) := 0 Then with p̂R := sup {p ≥ 0 : E [Φp (ST )] < ∞} , we have lim inf k→∞
z(k) := w ek , k ∈ R, (16) − ln cBS (k) = p̂R . z(k) (17) f12 (k) = p̂R . 2z(k) (18) and lim inf k→∞ Proof First note that the equivalence of (17) and (18) follows from Lemma 2, so we focus on establishing (17). cBS (k) . We first show p∗ ≥ p̂R . If p̂R = 0 the Set p∗ := lim inf k→∞ − lnz(k) statement is trivially true, so suppose p̂R > 0. Then for an arbitrary p ∈ (0, p̂R ) from Lemma 1(ii) we obtain limk→∞ cBS (k)epz(k) = 0, so   − ln cBS (k) ∞ = lim (− ln cBS (k) − pz(k)) = lim z(k) −p . k→∞ k→∞ z(k) cBS (k) For this to hold we must have lim inf k→∞ − lnz(k) ≥ p, thus p∗ ≥ p̂R . ∗ To show p ≤ p̂R and complete the proof, note that if p∗ = 0 the statement is trivially true, so assume p∗ > 0. For an arbitrary p ∈ (0, p∗ ) we have BS (k) =: a, so from (13a) and (13b) there exists k ′ ∈ R 0 > lim supk→∞ p + ln cz(k) such that cBS (k)e pz(k) 1 =√ e 2π  p+ ln cBS (k) z(k)  z(k) h(x) ≤ eaz(k)/2 , k ≥ k′ . Thus, as 0 ≤ cBS (k) ≤ 1, k ∈ R, from Lemma 1(i) we obtain E[Φp (ST )] = p Z ∞ −∞ cBS (k)epz(k) dz(k) ≤ p Z k′ epz(k) dz(k)+p −∞ Z ∞ k′ eaz(k)/2 dz(k) < ∞, ∗ giving p ≤ p̂R . The asymptotic behaviour of the left wing is deduced using the well-known duality argument (e.g. [5]). 10 Vladimir Lucic Theorem 2 Let ST be an integrable asset with E[ST ] = 1. With Φp (·) defined in Theorem 1 with limK→∞ w(K) = ∞, and put Then with we have   KΦp (1/K) Φ̂p (K) := 1,   ∞, K>0 K = 0 and p = 0 K = 0 and p > 0. i o n h p̂L := sup p ≥ 0 : E Φ̂p (ST ) < ∞ , lim inf k→−∞ f22 (k) = p̂L . 2z(|k|) (19) (20) Proof If P(ST = 0) > 0, from Remark 3 it follows that (20) is trivially satisfied, the both sides being zero. If P(ST = 0) = 0, we change the measure and use the relationship between the NVTs of ST and 1/ST , as outlined in Remark 4. The following results are now immediate. Corollary 2 (Lee Moment Formulæ, [15]) Let ST be an integrable asset, and put    p̃R := sup p ≥ 0 : E ST1+p < ∞ , Then lim inf    p̃L := sup p ≥ 0 : E ST−p < ∞ . 2 f1/2 (k) k→±∞ 2|k| = p̃R/L , or, equivalently, lim sup k→±∞ σ 2 (k) = βR/L , |k| where βU := ψ(p̃U ), U ∈ {L, R}, with ψ(x) := 2 − 4 √
x2 + x − x , x ≥ 0. Proof Setting w(x) ≡ ln x in Theorem 1 and Theorem 2 gives the first equality, and the second one then follows from (2). Corollary 3 (Piterbarg Conjecture, [11]) Let ST be an integrable asset with E[ST ] = 1. Suppose w(·) is a positive, increasing function on (0, ∞) such that limk→∞ w(ek )/k = ∞. Then for p̂ given in (16) we have lim sup k→∞ σ(k) p w(ek ) 1 =√ , k 2p̂ where the both sides are allowed to be infinite. A General Moment Formula 11 Proof Note that lim sup k→∞ σ(k) p w(ek ) = lim sup q k k→∞ √ k w(ek ) + 2 f12 (k) 2w(ek ) + q f12 (k) 2w(ek ) . Since by assumption limk→∞ k/w(ek ) = 0 and by Theorem 1 lim inf k→∞ p̂, the result follows. f12 (k) 2w(ek ) = Lemma 3 Suppose ϕ : [0, ∞) 7→ [0, ∞) is non-decreasing and limu→∞ ϕ(u) = ∞. Further suppose that ϕ(·) differentiable on [u0 , ∞) for some u0 ≥ 0, with dϕ(u) ≤ ϕ(u), u ≥ u0 . (21) du Then for every nonnegative random variable U we have ( "Z # ) U   p  p sup p ≥ 0 : E U ϕ (U ) < ∞} = sup p ≥ 0 : E ϕ (u) du < ∞ . u 0 Proof Note that from (21) for p ≥ 0 we have Z K Z K dϕ(u) p p p du, uϕp−1 (u) ϕ (u) du + p Kϕ (K) = u0 ϕ (u0 ) + du u0 u0 Z K ϕp (u) du, K ≥ u0 . ≤ u0 ϕp (u0 ) + (1 + p) K ≥ u0 , u0 On the other hand, since ϕ(·) is non-decreasing, for p ≥ 0 we also have RK Kϕp (K) ≥ 0 ϕp (u) du, K ≥ 0, which combined with the previous inequality yields the desired result. The following result extends the Log-Moment Formula of [22]. Lemma 4 Let θ : [0, ∞) → [0, ∞) be a non-decreasing function with limx→∞ θ(x) = ∞, and such that dθ(x) dx ≤ 1 for sufficiently large x. Then for an integrable asset ST with E[ST ] = 1 we have the following: (i) lim inf k→∞ f12 (k) = p̃+ , 2θ(ln k) (22) where    p̃+ := sup p ≥ 0 : E ST θp (| ln ST |) < ∞ . (ii) lim inf k→−∞ where f22 (k) = p̃− , 2θ(ln |k|)    p̃− := sup p ≥ 0 : E θp (| ln ST |) < ∞ . (23) (24) 12 Vladimir Lucic Proof For the first part of the proof we w.l.o.g. can consider the case P(ST = 0) = 0, since ST and ST I{ST >0} + I{ST =0} have the same p̃+ and the limit on the LHS in (22).
Putting w(K) := ln θ (ln K)+ , K ≥ 0 in Theorem 1 yields lim inf k→∞ where ( p̂R := sup p ≥ 0 : E f12 (k) = p̂R , 2θ(ln k) "Z ST 0 (25) # ) θp (ln u)+ du < ∞ , which by Lemma 3 gives
 
 p̂R = sup p ≥ 0 : E ST θp (ln ST )+ < ∞ . (26) It remains to establish p̃+ = p̂R . We introduce P̂ via ddPP̂ = ST , and note that, since ŜT := 1/ST is P̂-integrable, for every p ≥ 0 there exists cp > 0 
 
 such that cp Ê θp (ln ŜT )+ ≤ Ê[ŜT ] = 1. Thus, as E ST θp (ln ST )− =
  p Ê θ (ln ŜT )+ , we have 
 
 
 E ST θp (ln ST )+ ≤ E ST θp | ln ST | ≤ E ST θp (ln ST )+ + 1/cp , p ≥ 0, which gives p̃+ = p̂R . For the second part of the result, note that if P(ST = 0) > 0 from Remark 3 it follows that (23) is trivially satisfied (the both sides being zero). If P(ST = 0) = 0, the second part follows by symmetry, after applying the first part of the result to ŜT and P̂. Corollary 4 (The Log-Moment Formula, [22]) For an integrable asset ST with E[ST ] = 1 we have lim inf k→−∞    f22 (k) = sup p ≥ 0 : E | ln ST |p < ∞ . 2 ln |k| Proof Set θ(x) = x, x ≥ 0, in Lemma 4(ii). Remark 7 It is instructive to try to apply the argument of Lemma 3 to the w(·) featured in the Piterbarg Conjecture, for which (see (4)) w(K) = ∞. (27) ln K RK Applying integration-by-parts to Φp (K) = 0 epw(u) du, K ≥ 0, p ≥ 0, gives Z K Z K dw(u) pw(u) pw(u) e du = Kepw(K) , K ≥ 1. u e du + p du 0 0 lim K→∞ Thus, if (27) holds, the L’Hôpital’s rule yields limu→∞ u dw(u) = ∞, and the du second integral dominates the LHS. For that reason the argument of Lemma 3 cannot be applied. A General Moment Formula 13 Remark 8 The following variant of the classic Cramér moment result (e.g. Theorem 3.1 in [20] or Theorem 1.1.1 in [21]) is in the same spirit as the moment formulæ discussed here: if X is a random variable whose CDF F (·) is supported by the whole real line, and δ := lim inf x→∞ − ln T (x) , ln x T (x) := 1 − F (x) + F (−x), x > 0, then E[|X|p ] is finite for p ∈ [0, δ) and infinite for p > δ. An alternative proof of this statement can be obtained as follows. As in (12) we obtain Z ∞ Z ∞  p p E |X| = P(|X| ≥ k) dk = p P(|X| ≥ k)k p−1 dk, p > 0. 0 0 Thus, when E |X| < ∞, as in Remark 6, we deduce limk→∞ P(|X| ≥ k)k p = 0, and the rest of the proof follows the steps of Theorem 1.   p 5 Applications In this section we apply the moments formula to obtain asymptotic expansions of implied volatility smile. To this end, we first need to connect the implied volatility asymptotics to those of the normalizing volatility transforms, which is done next. 5.1 Implied volatility asymptotics In this section we apply the moments formula to obtain asymptotic expansions of implied volatility smile. To this end, we first need to connect the implied volatility asymptotics to those of the normalizing volatility transforms, which is done next. 5.2 Implied volatility asymptotics We focus on the cases where we have z(|k|) = o(k), k → ±∞ in Theorem 1 and Theorem 2, as this relates to the examples we will present in the next section. Considering the right tail first, we assume that a nontrivial limit in (18) exists, namely f 2 (k) ∈ (0, ∞). (28) p̂R = lim 1 k→∞ 2z(k) This implies p̂R z(k) f12 (k) = +o 2k k  z(k) k  , (29) 14 Vladimir Lucic so from σ(k) = √ 2k r f 2 (k) f1 (k) − √ 1+ 1 2k 2k ! , k > 0, √ using the expansion 1 + x = 1 + x2 + o(x) we obtain r    2  f1 (k) z(k) p̂R z(k) f 2 (k) f 2 (k) =1+ . 1+ 1 =1+ 1 +o +o 2k 4k 2k 2k k Putting together the last three equalities and using z(·) = o(k) gives p  p √ z(k) σ(k) = 2k − 2p̂R z(k) + p̂R √ + o z(k) , k → ∞. 2k (30) Similarly, by considering (20) we obtain the asymptotics for the left wing as p  p p z(|k|) +o z(|k|) , k → −∞. (31) σ(k) = 2|k| − 2p̂L z(|k|) + p̂L p 2|k| Remark 9 Contrasting this with the Lee Moment Formula, we see that normalizing implied volatility by z(k) = o(k), insteadq of k, does not yield a constant value in the limit, but a perturbation around σ(k) p = 2z(k) s k z(k) : p k − p̂R + o(1), z(k) k → ∞. 5.3 Example In this section we discuss examples related to the moment formula, and present a case in which Theorem 1 yields a result which cannot be obtained by the previously known moment formulæ. Recall from Lee Moment Formula (Corollary 2) that the maximum left tail slope of two σ 2 (k) lim sup =2 (32) |k| k→−∞ is equivalent to    sup p ≥ 0 : E ST−p < ∞ = 0, (33) thus no additional information can be obtained form the asset moments. In order to refine (32), the Log-Moment Formula (Corollary 4) relies on the existence of nontrivial lower critical log-moment    p̃− = sup p ≥ 0 : E | ln ST |p < ∞ < ∞. (34) To produce examples amenable for studying the behavior of the log-moments, in [22] the asset dynamics is taken as St = eµt+Xt −Yt , where Xt and Yt are two independent Lévy processes, µ := −ψ(−i), and ψ(·) is the characteristic A General Moment Formula 15 exponent of Xt −Yt . Here Xt is meant to control the right wing behavior, while Yt controls the left wing behavior. In order for Xt not to impact the limit in (34), any process with finite moments of all orders would do. To produce p̃− ∈ (0, ∞) in (34), Yt was chosen as a special case of the Generalized Inverse Gaussian, namely, the Inverse Gamma Process (see Section 5.3.5 of [25] for a more detailed account on this class of Lévy processes) for which Y1 has the (infinitely divisible) Inverse Gamma distribution fIG (x; α, β) = β α −α−1 −β/x x e , Γ (α) x > 0, (35) where α, β > 0 are the shape and scale parameters respectively, and Γ (·) is the Gamma function. Since ( Z ∞ Γ (α−p) p p < α, βα p p−α−1 −β/x Γ (α) β x e dx = (36) E[Y1 ] = Γ (α) 0 ∞, else, the existence of nontrivial log-moments of ST follows. We note that we can also take Yt to be the compound Poisson process Ŷt = Nt X i=1 Ui , t ≥ 0, (37) where Ui are independent random variables with the common distribution Ui ∼ Y1 , and Nt is an independent Poisson process with intensity λ > 0. The Lévy measure of Ŷt is ν(dx) = λfIG (x; α, β) dx, and, according to Corollary 25.8 of [24], E[Ŷtp ] < ∞ iff E[Y1p ] < ∞, t ≥ 0, p > 0. This construction gives the same critical log-moment of ST , but allows more flexibility, since prescribing the distribution of the increments directly (as in the case of the Generalized Inverse Gaussian Process) imposes the constraint that the distribution be infinitely divisible. We note that the presence of polynomially decaying heavy tails in (35) rendered the Lee Moment Formula ineffective, and the Log-Moment Formula had to come into play. Further tail uncertainty can be modeled via so-called super-heavy tailed distributions, having slowly varying complementary CDFs (in the sense of the Regular Variation theory – for more information about the super-heavy tailed distributions see [1] and references therein). Thus, changing the tail behavior of the complimentary CDF of the jump sizes in (37) from polynomially decaying to logarithmically decaying (super-heavy) Ui ∼ eY1 , according to (36) and Theorem 25.3, Proposition 25.4(ii), (iii) of [24], yields p sup {p ≥ 0 : E [(1 ∨ ln | ln ST |) ] < ∞} = α. (38) Lemma 4(ii) with θ(x) = 1 ∨ ln(x+ ), x ≥ 0, now gives lim inf k→−∞ f22 (k) = α, 2 ln ln |k| (39) 16 Vladimir Lucic so from (31) we obtain, assuming that the limit in (39) exists, s √ σ(k) |k| p − α + o(1), k → −∞. = ln ln |k| 2 ln ln |k| (40) From (39) it follows that the above case cannot be addressed with Corollary 4. 5.4 Existence of the limit It is well known that the limit superior in the Lee Moment Formula cannot in general be replaced by the genuine limit (see Section 2.3 in [4]). To ensure the existence of the limit, the tools of the Regular Variation Theory were introduced in the series of papers [2], [3], [4], with the idea that regularity of the moment generating function p 7→ E[STp ] can be utilized to ensure the 2 (k) existence of limk→±∞ σ |k| . We follow the same approach, aiming to establish the asymptotic relationship (40) for the example from [22] discussed in the previous section (c.f. Remark 9). Other cases where the moment generating function of ln ST is known (e.g. the Finite Moment Log Stable model of Carr and Wu, see Section 4.1.1 in [22]) can be treated similarly. The key result of [2] needed in the sequel is stated below. For the necessary background on the Regular Variation Theory, the reader is referred to [3] and references therein. Theorem 3 (Criterion 2 in [2]) Let F (·) be a bounded, non-decreasing, right-continuous function on R, and let Z ∞ ept dF (t), r∗ := sup{p : M (p) < ∞}. M (p) := −∞ ∗ (n) ∗ If r > 0, M (r − p) ∼ p−ρ l1 (1/p) for some n ∈ N0 , ρ > 0, and slowly varying l1 (·) as p → 0+, then
ln F (x, ∞) ∼ −r∗ x, x → ∞. Let St := exp(µt − Yt ), where Yt is the Generalized Inverse Gaussian process introduced in the previous section, and µ > 0 is chosen so that St is a martingale. Let σ(·) be the implied volatility of ST , and put Ŝt := 1/St , 0 ≤ t ≤ T , ddPP̂ := ST . Setting Φ(K) ≡ K(ln K)p+ , p > 0, in (12) gives h  i Z ∞

p −µT µT lnp K + p lnp−1 K d −Call(K) Ê Φ e ŜT E [YT ] = e = 1 = M0 (p) + pM0 (p − 1), (41) where Call(·) is the call option referencing eµT ŜT , with implied volatility µ σ− (k) := σ µ (−k), σ µ (k) := σ(k − µT ), k ∈ R, and Z ∞ Z ∞


M0 (p) := lnp K d −Call(K) = ept d −cBS et , p > 0. 1 −∞ A General Moment Formula 17 Since 0 ≤ M0 (p − 1) ≤ M0 (p) + Z e 1 e lnp−1 K dK ≤ M0 (p) + , p p > 0, we have M0 (p) + pM  0 (p − 1) = M0 (p)(1 + O(p)) + O(1), as p → 0+. Thus, as (36) implies E YTα−p ∼ Γ (p) ∼ p1 , p → 0+, setting F (t) ≡ −cBS (et ), M (p) ≡ E[YTp ] in Theorem 3 together with (41) gives ln cBS (et ) ∼ −αt, t → ∞, i.e. ln cBS (k) ∼ −α ln k, k → ∞. Therefore, as Corollary 1 implies ln cBS (k) ∼ − f 2 (k;σ µ ) limk→−∞ 22 ln |k| that are given by (40). µ f12 (k;σ− ) , 2 k → ∞, it follows exists and that the asymptotic of the left wing of σ(·) Remark 10 In [2] Theorem 3 is used in the context of establishing the existence of the limit in the Lee Moment Formula. There, F (·) is taken to be the CDF of the underlying asset, and the regularity of the tail behaviour of F (·) secured by Theorem 3 is then linked to the tail behaviour of the option price. In the final step, this was used to yield the existence of the limit in (1). We note that using the argument above gives the existence of the limit without the need to involve the asset CDF: setting Φp (K) ≡ K p+1 in (12) yields Z ∞ Z ∞

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