The Log Moment formula for implied volatility
aa r X i v : . [ q -f i n . P R ] J a n THE LOG MOMENT FORMULA FOR IMPLIED VOLATILITY
VIMAL RAVAL AND ANTOINE JACQUIER
Dedicated to the memory of Mark H.A Davis
Abstract.
We revisit the foundational Moment Formula proved by Roger Lee fifteen years ago. We showthat when the underlying stock price martingale admits finite log-moments E [ | log S t | q ] for some positive q , thearbitrage-free growth in the left wing of the implied volatility smile is less constrained than Lee’s bound. Theresult is rationalised by a market trading discretely monitored variance swaps wherein the payoff is a function ofsquared log-returns, and requires no assumption for the underlying martingale to admit any negative moment.In this respect, the result can derived from a model-independent setup. As a byproduct, we relax the momentassumptions on the stock price to provide a new proof of the notorious Gatheral-Fukasawa formula expressingvariance swaps in terms of the implied volatility. Introduction
Implied volatility is at the very core of Quantitative Finance and is the day-to-day gadget that traders observeand manipulate. The increasing complexity of stochastic models we have witnessed over the past thirty years isa testimony to its importance and subtlety. One key issue is the absence of closed-form expression for the latter,leaving it to the sometimes capricious moods of numerical analysis. Among the plethora of research in thisdirection, carried out both by academics and by practitioners, model-free results, with minimum assumptions,are scarce. Roger Lee’s Moment Formula [25] was a groundbreaking result and its importance cannot beunderstated: it provides a direct link between the slope of the smile in the wings and the moments of thedistribution of the underlying asset price. It serves not only to infer directly observed information about theimplied volatility smile into constraints on model parameters but also to provide arbitrage-free solutions to theextrapolation problem (how to evaluate options for strikes outside the observed range). Recent refinementshave led to a deeper understanding of the information contained in the implied volatility smile, determiningwhether the probability of default of the underlying could be inferred [16] or the potential lack of martingalityof the latter [23]. These have complemented the otherwise exhaustive literature on the asymptotic behaviourof stochastic models in Finance, a thorough review of which can be consulted in [19].Asymptotic methods have both supporters and enemies, the former trying to expand the abundance oftechniques to every possible model, while the latter sometimes dismiss the usefulness of these results. The truthas often lies somewhere in the middle but asymptotic results nevertheless provide useful information about thequalities and pitfalls of models with regards to real-life practices. With this in mind, we revisit Lee’s formulawhen presented with some underlying stock price, the prices of finitely many co-maturing European Call and Putoptions as well as a variance swap with the same maturity. In [14] conditions were stated under which a given set
Date : January 21, 2021.2010
Mathematics Subject Classification.
Key words and phrases. moment formula, implied volatility, variance swaps.The authors would like to thank Masaaki Fukasawa for insightful comments, in particular leading to Section 3.4. of European Call and Put prices all maturing at the same time T is consistent with absence of arbitrage, whichis shown to be equivalent to the existence of a market model ; a filtered proability space carrying an adapted,integrable process ( S t ) t ∈ [0 ,T ] , with S equal to the time-0 stock price, in which the discounted stock price processis a martingale and the discounted expectations of the Put option payoffs recover the observed time-zero values.In [15] robust model-free conditions are provided, when the process ( S t ) t ∈ [0 ,T ] admits continuous sample paths,for a set of European Put option prices and continuously monitored variance swap price to be consistent withabsence of arbitrage. Our approach here is to make no assumption on the dynamics of the stock-price priceprocess, and to instead infer limiting behaviour of the left-wing given merely the information that the marginaldistribution of S T admits finite log moments, E [ | log S T | q ] for finite positive moments q , which is motivateddirectly from the market since it trades a (discretely or continuously monitored) variance swap. Further, itis feasible for market models to exist that do not admit any negative moment for the stock price and Lee’sMoment formula (on S T rather than log( S T )) implies that the left wing of the smile has slope precisely equalto two. Our newly formulated Log-Moment formula allows us to provide higher-order term in this asymptoticbehaviour, fully characterised by the moments of the log-stock price.We provide a precise formulation of the problem and a thorough review of Roger Lee’s Moment formula inSection 2 before stating and proving the new Log Moment formula in Section 3. As a byproduct, we revisitthe Fukasawa-Gatheral formula expressing variance swaps in terms of the implied volatility and provide a newproof with relaxed assumptions and further show how this improves Fukasawa’s representation [20] of optionprices in terms of implied volatility. We highlight in Section 4 a few stochastic models, both with continuouspaths and with jumps, used in Finance for which our formula refines Lee’s standards.2. Problem Formulation and background
We consider a time horizon [0 , T ], with
T > (cid:0) Ω , F , ( F ) t ∈ [0 ,T ] , Q (cid:1) satisfyingthe usual assumptions and carrying two adapted processes ( S t ) t ∈ [0 ,T ] , the asset price, and ( B t ) t ∈ [0 ,T ] , startingfrom B = 1, where B T represents the value at time T of £ F t the t -forward price of S from time 0, thus F = S is the observed spot price. Dividendsmay be paid by the asset S , but we do not make any assumptions about these. The process ( S t ) t ∈ [0 ,T ] is assumedto be a strictly positive Q -semimartingale. We finally assume the existence of a zero-coupon-bond maturingat T with face-value £
1, traded with price P T = E [ B − T ] so that F T = E [ S T ], where all expectations are takenunder Q . We consider the setup where ( S t ) t ∈ [0 ,T ] can be traded with no transaction costs and where the interestrate for borrowing and lending is the same, however not necessarily deterministic. Here, the probability Q thusplays the role of the T -forward measure. By no-arbitrage arguments, a vanilla Put option with strike K isworth P ( K ) := P T E [( K − S T ) + ], for K ≥
0. Using normalised units for the stock-price S T := S T /F T andlog-moneyness x := log( K/F T ), the normalised price of the Put option is denoted by P ( x ) := P ( K ) P T F T = E h (e x − S T ) + i . Recall now the Black-Scholes formula for the European Put option:(2.1) P BS ( x, σ ) = e x Φ[ − d ( x, σ )] − Φ[ − d ( x, σ ) − σ ] , where Φ denotes the Gaussian cumulative distribution function and(2.2) d ( x, σ ) := − xσ − σ , HE LOG MOMENT FORMULA FOR IMPLIED VOLATILITY 3 which is nothing else than the usual d or d − from the Black-Scholes formula. Since the maturity T is fixedthroughout the whole paper, we work with normalised volatility σ rather than the classical σ √ T notation. Thishas the clear advantage of avoiding cluttered statements. Definition 2.1.
For any log-moneyness x ∈ R , the implied volatility I( x ) ∈ [0 , ∞ ) is the unique non-negativesolution to P ( x ) = P BS ( x, I( x )).The implied volatility I( x ) is well defined whenever P ( x ) ∈ [(e x − + , e x ], which holds since the stock priceis a true martingale. Our starting point is the following initial bound for the implied volatility [25, Lemma 3.3]: Lemma 2.2.
For any β > , there exists x ∗ < such that I( x ) < p β | x | for all x < x ∗ . For β = 2 , the sameholds if and only if Q [ S T = 0] < . In our setup, S T is strictly positive almost surely and therefore I( x ) = O ( p | x | ) as x tends to −∞ . Whenthe number of finite inverse power moments for the stock price is known, the small-strike Moment Formula dueto Lee [25] refines the above result: Theorem 2.3 (Lee’s Left Moment Formula) . Let p := sup (cid:8) p > E (cid:2) S − pT (cid:3) < ∞ (cid:9) and β L := lim sup x ↓−∞ I( x ) | x | . Then β L ∈ [0 , , p = β L + β L − , with := ∞ . Equivalently, β L = 2 − p p + p − p ) , equal to for p = ∞ . This theorem was one of the first model-free result about the relationship between the distribution of thestock price and the behaviour of the implied volatility. The lim sup in Lee’s result was further strengthened toa genuine limit by Benaim and Friz [3, 4], albeit with additional assumptions. It is really a cornerstone in theimplied volatility modelling literature and has provided academics and practitioners robust consistency checksfor extrapolation of the smile. Lee also proved a symmetric right-wing formula, but we omit its presentationas we shall not require it here. This left-wing behaviour of the smile left two unresolved issues however: if S T has a strictly positive mass at the origin, then Lee’s expression is not able to distinguish it from a mass-lessdistribution with fat tails; this was tackled in [16]. The second issue is that in fact no information about themoments of S T is really available in the market, and the so-called Power options [7] are rarely traded. However,variance swaps are traded on the market and it is thus a natural question to check if Lee’s celebrated MomentFormula could be refined to take into account these highly liquid derivatives.3. Variance swaps and the Log-Moment formula
Characterisation of variance swaps.
Variance swaps are highly liquid traded derivatives on the Equitymarket. One can describe them as a standard swap, where, over the time horizon [0 , T ] the floating leg is equalto the (annualised) realised variance V dT as measured by V dT := 252 T n X i =1 log (cid:18) S t i S t i − (cid:19) , in which 0 = t < t < · · · < t n = T is an equidistant partition with t i − t i − = iT /n corresponding to oneday for some positive integer n . The superscript d here refers to the fact that this definition corresponds to theso-called discretely monitored variance swap. The early advances on the hedging and pricing of the varianceswap by Neuberger [29], Dupire [12], Carr and Madan [8] and Demeterfi, Derman, Kamal and Zou [18] led VIMAL RAVAL AND ANTOINE JACQUIER to the instrument being used extensively by traders to express views on future realised variance and hedgingvolatility risk. These advances hinged on assuming (i) the stock price process is a continuous semi-martingalewith strictly positive values, (ii) the realised variance is continuously monitored and measured by the quadraticvariation h log S i T , and (iii) Call or Put options maturing at time T are traded for all strikes K ∈ R + . Itˆo’sformula for continuous semi-martingales applied to − log S T , then gives h log S i T = Z T d h S i t S t = − (cid:18) S T S (cid:19) + 2 Z T d S t S t . The variance swap is replicated by holding a contract paying − log( S T /S ) and dynamic trading in the under-lying stock. Now, the log payoff − log S T is redundant, since − log (cid:18) S T S (cid:19) = S T − S S + Z S ( K − S T ) + K d K + Z ∞ S ( S T − K ) + K d K, i.e. it is hedged by a static position in the underlying asset, the continuum of Call and Put options, and cash.The variance swap payoff in this setup is therefore fully replicated, with no assumptions on the dynamics of theprice process S , except for continuity. It thus follows that the variance swap-rate is the forward cost of the fullhedging portfolio. When interest rates are zero and dividends are not paid by the underlying asset, this is2 Z S P ( K ) K d K + 2 Z ∞ S C ( K ) K d K, provided both integrals are finite, with P ( K ) and C ( K ) the prices of Put and Call options with strike K . Thesubtle impact of jumps on the prices of variance swaps was treated thoroughly by Broadie and Jain [5]. In boththe discretly monitored and the continuously monitored case, the moments of the underlying stock price arenot at play, but rather the moments of its logarithm, thus creating the need to refine Lee’s formula to this case.3.2. The Log-Moment formula.
Our main result is the following Log-Moment Formula:
Theorem 3.1.
Let q := sup { q ≥ E [ | log S T | q ] < ∞} be finite. Then lim inf x ↓−∞ d ( x, I( x )) p | x | = √ q . It is clear that q does not provide information about the right tail of the distribution of S T . Since S is amartingale, its first moment is finite and therefore, for any q ≥
0, there exists some constant c q ≥ E (cid:2) | log S T | q { S T ≥ c q } (cid:3) ≤ E (cid:2) | S T | { S T ≥ c q } (cid:3) ≤ E [ S T ] , which is finite, since S T is strictly positive almost surely. The following corollary is immediate but shows theimmediate consequences of the Log-Moment Formula on the behaviour of the implied volatility in the left wing. Corollary 3.2.
In the setting of Theorem 3.1, at least along a subsequence, we have, as x ↓ −∞ , I( x ) = p q log( | x | ) − x − p q log( | x | ) , = p | x | − p q log | x | + q log( | x | ) p | x | + O (cid:16) | x | − / (cid:17) , Benaim and Friz [3, 4] refined Lee’s result, with additional assumptions, from a lim inf / lim sup statementto a genuine limit. One could investigate how this might apply here, but we defer it to a future analysis inorder not to clutter our main result with extra technical assumptions. An interesting feature however is theform of the small-strike implied volatility expansion in Corollary 3.2. The slope equal to 2 of the total implied HE LOG MOMENT FORMULA FOR IMPLIED VOLATILITY 5 variance I is trivial from Lee’s result (Theorem 2.3) since q finite implies p = 0 (no negative moment of thestock price exists). Lee’s formulation however does not provide further details. In the case of a strictly positivemass at the origin, De Marco, Hillairet and Jacquier [16, Theorem 3.6] proved thatI( x ) = p | x | + c + ϕ ( x ) , as x ↓ −∞ , where the constant c is related to the mass at zero and the function ψ tends to zero as x tends to infinity, which,while capturing the slope 2, is markedly different from our new formula here. Before being able to prove thetheorem, we need two lemmas providing bounds on prices of Put options and on the implied volatility. Lemma 3.3.
Let q ≥ be such that E [ | log S T | q ] is finite. Then for all x < ( q − { q< } , P BS ( x, I( x )) ≤ e x | x | − q E [ | log S T | q ] . Proof.
The case q = 0 is a consequence of no-arbitrage bounds for the Put option. Now consider q >
0. Forease of exposition only, we work in the moneyness unit k = e x . The map k
7→ | log( k ) | q is strictly convex on K := (0 , e q − { q< } + 11 { q ≥ } ). Let now v q ( k ) denote the solution the equation to(3.1) k = v q ( k ) (cid:18) − q log v q ( k ) (cid:19) , for k ∈ K , such that lim k ↓ v q ( k ) = 0. This equation van be solved explicitly as v q ( k ) = exp (cid:8) W − (cid:0) − qk e − q (cid:1) + q (cid:9) . Recall [13] that the Lambert W function is such that for z ∈ R , W ( z ) solves W ( z )e W ( z ) = z , which is multi-valued for − e − < z ≤
0. The W − branch is the one that satisfies lim z ↑ W − ( z ) = −∞ . Then, for u ∈ [0 , k ],the curves k q v q ( k ) | log v q ( k ) | − q | log u | q and k ( k − u ) + are equal and have the same gradient at u = v q ( k ).Since k ∈ K , strict convexity and positivity of | log( u ) | q over (0 , k ) imply on taking expectations that(3.2) P BS (log( k ) , I(log( k ))) ≤ q v q ( k ) | log v q ( k ) | − q E [ | log S T | q ] . By construction 0 < v q ( k ) < k ≤
1, hence | log v q ( k ) | − q < | log( k ) | − q . In particular using log v q ( k ) < q v q ( k ) | log v q ( k ) | < k . Combining these, a larger bound (than in (3.2)) for Put prices is given by(3.3) P BS (log( k ) , I(log( k ))) ≤ k | log( k ) | − q E [ | log S T | q ] . (cid:3) Remark 3.4.
In the limit x ↓ −∞ (or k ↓ k ↓ q v q ( k ) | log v q ( k ) | − q k | log k | − q = 1 . To see this, first recall that lim k ↓ v q ( k ) = 0, then from (3.1),lim k ↓ k q v q ( k ) | log v q ( k ) | = 1 . Further, taking logarithm of both sides of (3.1) it follows that lim k ↓ k )log v q ( k ) = 1, which implies (3.4). Lemma 3.5.
Let q ≥ such that E [ | log S T | q ] is finite. Then for any p ∈ [0 , q ) , there exists x p < such that I( x ) < p − x + 2 p log( | x | ) − p p log( | x | ) , for all x < x p . VIMAL RAVAL AND ANTOINE JACQUIER
Proof.
The case q = 0 is clear from Lemma 2.2. Let q >
0. Note that when the implied volatility is of the formI( x ) = √ f ( x ) − g ( x )) , for x ∈ R , where f, g : R → R satisfy f ( x ) − g ( x ) = − x , the corresponding price of the Put option (2.1) is given by P BS ( x, I( x )) = e x Φ (cid:16) −√ g ( x ) (cid:17) − Φ (cid:16) −√ f ( x ) (cid:17) . In our case, the two functions are given by f ( x ) = p p log( | x | ) − x and g ( x ) = p p log( | x | ). With φ denotingthe Gaussian density, the asymptotic relationship(3.5) lim z ↑∞ z Φ( − z ) φ ( z ) = 1 , holds trivially by L’Hˆopital’s rule and thereforelim x ↓−∞ Φ (cid:0) −√ f ( x ) (cid:1) e x Φ (cid:0) −√ g ( x ) (cid:1) = 0 , which implies lim x ↓−∞ P BS (cid:0) x, √ f ( x ) − g ( x )) (cid:1) e x Φ (cid:0) −√ g ( x ) (cid:1) = 1 . We can then deducelim x ↓−∞ e x | x | − q P BS (cid:0) x, √ f ( x ) − g ( x )) (cid:1) = lim x ↓−∞ e x | x | − q e x Φ (cid:0) −√ g ( x ) (cid:1) = lim x ↓−∞ | x | − q Φ (cid:0) −√ g ( x ) (cid:1) = lim x ↓−∞ −√ g ( x ) | x | − q φ (cid:0) −√ g ( x ) (cid:1) = lim x ↓−∞ √ πg ( x ) | x | − q e − g ( x ) = lim x ↓−∞ √ πg ( x ) | x | p − q = ( , if p < q, ∞ , if p ≥ q, (3.6)and the lemma follows from Lemma 3.3 and the monotonicity of P BS ( · , · ) in its second argument. (cid:3) Before stating the proof of Theorem 3.1, recall the following lemma, which will be used repeatedly:
Lemma 3.6.
For any convex function f : R + → R , the identity f ( x ) = f ( x ) + f ′ ( x )( x − x ) + Z x ( y − x ) + µ (d y ) + Z ∞ x ( x − y ) + µ (d y ) , holds for Lebesgue almost all x, x ∈ R + , where µ = f ′′ in the sense of distributions.Proof of Theorem 3.1. Let ζ := lim inf x ↓−∞ d ( x, I( x )) √ | x | . Suppose by contradiction that ζ < √ q and let q suchthat √ q ∈ ( ζ, √ q ). Then there exists a sequence ( x n ) n ∈ N with x n ↓ −∞ such that for all n , d ( x n , I( x n )) < p q log | x n | . Inverting this yieldsI( x n ) > p − x n + 2 q log | x n | − p q log | x n | , which contradicts Lemma 3.5 since q < q . Assume now that ζ > √ q and let q such that √ q ∈ ( √ q , ζ ). We showthat this implies E [ | log S T | p ] is finite for all p ∈ ( q , q ). Indeed, in this case,I( x ) < p − x + 2 q log( | x | ) − p q log( | x | ) HE LOG MOMENT FORMULA FOR IMPLIED VOLATILITY 7 for all x along a sequence. From (3.6) it follows that there exists x ∗ < x < x ∗ ,(3.7) P BS ( x, I( x )) < e x | x | − q . Now, reverting to moneyness units k = e x , one sees that for p ∈ ( q , q ) and z p = e p − { p< } + 11 { p ≥ } , E [ | log S T | p ] = E (cid:2) | log S T | p { S T 7→ | log x | p is strictly convex on ( −∞ , z p ), the second line above follows from Lemma 3.6 applied to theconvex function x 7→ | log x | p { x 7→ | log x | p in the interval considered and (3.7). The final equalityholds since q > q and the second expectation on the first line is finite. (cid:3) Refinement of the Fukasawa-Gatheral formula. In his volatility Bible [21], Gatheral derived an ele-gant formula expressing the log contract directly in terms of the implied volatility. This has obvious appeal astraders can plug in their favourite implied volatility smile (parametric or not) and obtain the fair value of a vari-ance swap. Earlier versions of this formula, albeit with more sketchy proofs, were proposed by Matytsin [28] andChriss and Morokoff [11]. A fully thorough derivation though has only recently been provided by Fukasawa [20](see also [27] for interesting connectiong with absence of arbitrage) who not only proved the key ingredient, thedecreasing property of the map k d ( k, ˙), but extended the formula to more general payoff contract. In allthese proofs, the main assumption is the existence of moments E [ S εT ] for some ε > 0. We show hereafter thatthis additional condition is in fact not required. Following [20], let(3.8) f ( x ) := − d ( x, I( x )) = x I( x ) + I( x )2 , and note that, as proved by Fukasawa [20], the inverse function f ← is well defined. This yields the following: Theorem 3.7. If E [ | log( S T ) | ] is finite (namely q ≥ ), then − E [log( S T )] = Z R I( f ← ( z )) φ ( z )d z. Proof. Note first that, by [20, Theorem 2.8] the map x d ( x, · ) is decreasing. By (2.1) and the Put-Call parity,a Call option with log-moneyness x = log( K/F T ) is worth C BS ( x, σ ) = Φ[ d ( x, σ ) + σ ] − e x Φ[ d ( x, σ )] . By Lemma 3.6, with c (e x ) := C BS ( x, I( x )) and p (e x ) := P BS ( x, I( x )), we can write L := E [ − log( S T )] = Z −∞ p (e x )e − x d x + Z ∞ c (e x )e − x d x = (cid:2) − p (e x ) e − x (cid:3) −∞ + (cid:2) − c (e x ) e − x (cid:3) ∞ + Z −∞ p ′ (e x )d x + Z ∞ c ′ (e x )d x = Z −∞ p ′ (e x )d x + Z ∞ c ′ (e x )d x. VIMAL RAVAL AND ANTOINE JACQUIER The boundary terms vanish because c (1) = p (1) by Put-Call parity, because c ( · ) tends to zero for large strikesand by Lemma 3.3 since p ( x ) ≤ e x | x | − E [ | log S T | ] for x < x ↓−∞ p (e x )e − x = 0. Now, p ′ (e x )e x = dd x P BS ( x, I( x )) and c ′ (e x )e x = dd x C BS ( x, I( x )) . Hence, with δ ( x ) := d ( x, I( x )), p ′ (e x ) =Φ[ − δ ( x )] − φ ( δ ( x )) δ ′ ( x ) + e − x φ ( − δ ( x ) − I( x ))[ δ ′ ( x ) + I ′ ( x )] c ′ (e x ) =e − x φ ( δ ( x ) + I( x ))[ δ ′ ( x ) + I ′ ( x )] − Φ[ δ ( x )] − φ ( δ ( x )) δ ′ ( x ) . (3.9)Since the Gaussian density φ satisfies φ ( a + b ) = φ ( a − b )e − ab for any a, b ∈ R , thene − x φ ( δ ( x ) + I( x )) = e − x φ (cid:18) − x I( x ) + I( x )2 (cid:19) = φ ( δ ( x )) , and hence the system (3.9) simplifies, by symmetry of φ , to p ′ (e x ) = Φ[ − δ ( x )] + φ ( δ ( x ))I ′ ( x ) and c ′ (e x ) = φ ( δ ( x ))I ′ ( x ) − Φ[ δ ( x )] . Therefore L = Z −∞ Φ[ − δ ( x )]d x − Z ∞ Φ[ δ ( x )]d x + Z R φ ( δ ( x ))I ′ ( x )d x = [ x Φ[ − δ ( x )]] −∞ − [ x Φ[ δ ( x )]] ∞ + Z R xφ ( δ ( x )) δ ′ ( x )d x + Z R φ ( δ ( x ))I ′ ( x )d x. For the boundary terms, observe first from the log-moment formula, Theorem 3.1, that q ≥ δ ( x ) ≥ p | x | eventually for x < 0, and so exp (cid:8) − δ ( x ) (cid:9) ≤ | x | − . Combining with the identity (3.5) one seeslim x ↓−∞ x Φ[ − δ ( x )] = 0. Now, Lemma [25, Lemma 3.1] (the right-tail analogue of Lemma 2.2), implies thetrivial bound I( x ) ≤ √ x for x > δ ( x ) = − (cid:18) x I( x ) + I( x )2 (cid:19) ≤ − x I( x ) ≤ − √ x , which diverges to −∞ as x tends to infinity. Therefore, for x large enough,0 ≤ xφ ( − δ ( x )) − δ ( x ) = 1 √ π x exp (cid:8) − δ ( x ) (cid:9) − δ ( x ) ≤ √ π x exp (cid:8) − x (cid:9) x I( x ) = I( x ) exp (cid:8) − x (cid:9) √ π ≤ √ x exp (cid:8) − x (cid:9) √ π , which tends to zero as x tends to infinity. The limit (3.5) thus implies lim x ↑∞ x Φ[ δ ( x )] = 0 and therefore L = Z R xφ ( δ ( x )) δ ′ ( x )d x + Z R φ ( δ ( x ))I ′ ( x )d x = Z R xφ ( δ ( x )) δ ′ ( x )d x + [I( x ) φ ( δ ( x ))] R + Z R φ ( δ ( x )) δ ( x ) δ ′ ( x )I( x )d x = Z R φ ( δ ( x )) δ ′ ( x )[ x + I( x ) δ ( x )]d x = − Z R φ ( δ ( x )) δ ′ ( x ) I ( x )2 d x, where the boundary terms cancel as above and by Lemma 2.2, and applying (2.2) for δ ( x ). Substituting z = δ ( x ),using the symmetry of φ , the proposition follows from the limits lim x →±∞ δ ( x ) = ∓∞ . (cid:3) HE LOG MOMENT FORMULA FOR IMPLIED VOLATILITY 9 Pricing formulae for European options. In [20], Fukasawa not only proved a version of Theorem 3.7(with more restrictive assumptions), but also extended it to options with payoffs of the form Ψ(log( S T )) for anytwice differentiable function Ψ with derivative of at most polynomial growth. More precisely, he derived [20,Theorem 4.4] an integral form for E [Ψ(log( S T ))] assuming either that E [ S pT ] exists for some p > E [ S − qT ] exists for some q > 0. The former case is not affected by our setup and we instead provide a refinementof the latter case when no such q exists but instead log-moments are available. This in fact extends the scopeof Theorem 3.7 above. Recall that the function f is defined in (3.8) and let P q denote the set of functions withat most polynomial growth of order q at −∞ . Theorem 3.8. Assume that q := sup { q ≥ E [ | log S T | q ] < ∞} belongs to [1 , ∞ ) . • For any twice differentiable function Ψ ∈ P q with q ∈ [0 , q ] , E [Ψ(log( S T ))] = Z R ( Ψ( f ← ( z )) − Ψ ′ ( f ← ( z )) " f ← ( z ) + I ( f ← ( z )) φ ( z )d z + Z R Ψ ′′ ( x )I( x ) φ ( f ( x ))d x. • For any absolutely continuous function Ψ ∈ P q with q ∈ [0 , q ] , E [Ψ(log( S T ))] = Z R n Ψ( f ← ( z )) − Ψ ′ ( f ← ( z )) + Ψ ′ ( h ( z ))e − h ( z ) o φ ( z )d z, where h is the inverse function of the map x f ( x ) − I( x ) . Remark 3.9. With Ψ( x ) ≡ x , then Ψ ∈ P and Ψ ′ ∈ P , proving Theorem 3.7. Proof. The proof of this theorem follows that of [20, Theorem 4.4], or indeed that of Theorem 3.7 above. Thesteps are analogous, but one has to pay special attention to the boundary terms arising from the differentintegrations by parts involved. In our setting, the two terms that need special care are(3.10) lim x ↓−∞ Ψ ′ ( x )I( x ) φ ( f ( x )) and lim x ↓−∞ Ψ( x ) | I ′ ( x ) | φ ( f ( x )) , which we need to send to zero for a suitable class of functions Ψ.By Theorem 3.1, √ q is the largest value such that for any ε > 0, there exists x ε for which(3.11) d ( x, I( x )) p | x | > √ q − ε =: √ q ε , for all x ≤ x ε . Now, the equation (in σ ) d ( x,σ ) √ | x | = √ q ε admits two roots σ ± = − p q ε log( | x | ) ± p q ε log( | x | ) − x ,so that, for x < x ε , the inequality (3.11) holds if (similarly to Lemma 3.5 in fact)(3.12) I( x ) < − p q ε log( | x | ) + p q ε log( | x | ) − x. Note that when q = 0 and replacing the lim inf by a genuine limit, this reads I( x ) < p | x | for x small enough,which was proved by Lee [25]. This further implies directly that for x < x ε ,(3.13) f ( x ) < − p q ε log( | x | ) . Therefore for any function Ψ : ( −∞ , x ε ] → R ,Ψ ′ ( x )I( x ) φ ( f ( x )) = Ψ ′ ( x )I( x ) √ π exp (cid:26) − f ( x ) (cid:27) ≤ Ψ ′ ( x )I( x ) √ π e − q ε log( | x | ) = Ψ ′ ( x )I( x ) √ π | x | − q ε . From the bound (3.12) on I( x ), this expression tends to zero as x ↓ −∞ if and only if Ψ ′ ∈ P q ′ with q ′ ∈ [0 , q − ].Clearly when q ∈ [0 , ], this cannot tend to zero as I( x ) dominates φ ( f ( x )). This refines the analysis of [20, Lemma 4.2] which assumed the existence of strictly negative moments for the stock price. Now Fukasawashowed [20, Lemma 2.6] that, independently of any moment (or log-moment) assumptions, f ( x )I ′ ( x ) < x ∈ R ; combining this with the new upper bound (3.13), we obtain a new version of [20, Theorem 3.6], namelyI ′ ( x ) > − p q log( | x | ) , for x small enough, so that | I ′ ( x ) | < (2 q log( | x | )) − / and thereforeΨ( x ) | I ′ ( x ) | φ ( f ( x )) = Ψ( x ) | I ′ ( x ) |√ π exp (cid:26) − f ( x ) (cid:27) ≤ Ψ( x ) | I ′ ( x ) |√ π e − q ε log( | x | ) = Ψ( x )2 √ π q | x | − q ε p log( | x | )converges to zero as x ↓ −∞ as soon as Ψ ∈ P q with q ∈ [0 , q ]. This therefore implies that the two limits (3.10)are equal to zero if and only if Ψ ∈ P q for q ∈ [0 , q ]. All the other statements in [20, Lemma 4.3] remain identical,and therefore the proof of Theorem 4.4 follows analogously, the boundary terms cancelling out under our newassumptions, thus proving the first bullet point in the theorem. Indeed, the two conditions are that Ψ ∈ P q for q ∈ [0 , q ] and Ψ ′ ∈ P q ′ with q ′ ∈ [0 , q − ]; the intersection of both is in fact the same as the former. Aclose look at the proof of the second bullet point in [20, Theorem 4.4] shows that only the second limit in (3.10)needs to tend to zero, which, as just discussed, is true as soon as Ψ ∈ P q , and the theorem follows. (cid:3) Examples Corollary 3.2 gives us a recipe to estimate q (whenever it exists) from market data by simple regression of theimplied volatility against the log-moneyness. This also facilitates informed initial guesses for model calibration,with a direct relationship between model parameters and the number of log-moments of the stock price admits.We provide several examples of models where this is feasible.4.1. Exponential L´evy models. In exponential L´evy models the stock-price process is modelled by(4.1) S t = S exp( L t ) , where ( L t ) t ∈ [0 ,T ] is a real-valued L´evy process [30, Chapter 3], namely a c`adl`ag stochastically continuous processwith independent and identically distributed increments starting from L = 0. For any t > 0, the characteristicfunction of the random variable L t satisfies log E (cid:2) e i uL t (cid:3) = ψ ( u ) t, for all u ∈ R , where the characteristic exponent ψ admits the L´evy-Khintchine representation ψ ( u ) = − ξu γu + Z R (cid:0) e i ux − − i ux {| x |≤ } (cid:1) ν (d x ) , with ξ ≥ γ ∈ R and ν a measure on R satisfying ν ( { } ) = 0 and R R (1 ∧ x ) ν (d x ) < ∞ . Sato [30, Theorem 25.3]proved that for any submultiplicative, locally bounded function g , the expectation E [ g ( S T )] is finite if and onlyif R R g ( x ) ν (d x ) is finite. In light of Theorem 3.1, we thus consider the function g ( x ) ≡ log( | x | ) q with q ≥ HE LOG MOMENT FORMULA FOR IMPLIED VOLATILITY 11 Finite moment log stable process. The Finite Moment Log Stable (FMLS) model was introduced by Carrand Wu [10] to capture the observed negative skew observed on S&P options. There the driving L´evy process L in (4.1) is α -stable with tail index α ∈ (1 , 2) and skew parameter β = − 1, so that [30, Chapter 3], for any T > • E [ | S T | p ] is finite for all p ≥ • the support of L T is the whole real line; • E [ | log S T | q ] is finite for all q ∈ (0 , α ) and is infinite if q ≥ α .Theorem 3.1 thus applies with q = α ∈ (0 , 2) and E (cid:2) | log( S T ) | (cid:3) is infinite. While the model may capture thefat left tail and thin right tail of the stock price, it is too extreme if a discrete variance swap is traded.4.1.2. Finite moment log mixture model. In (4.1) let L := X − Y for two independent processes X and Y with • q X := sup { q ≥ E [ | X | q ] < ∞} > E (cid:2) e p X X (cid:3) is finite for some p X ≥ • q Y := sup { q ≥ E [ | Y | q ] < ∞} ∈ (0 , q X ) and E (cid:2) e − p Y Y (cid:3) for some p Y ∈ [1 , p X ),so that X and Y respectively influence the right and left tails in the distribution. Before identifying somecandidates for the process X and Y , we note: Lemma 4.1. E (cid:2) e p Y L (cid:3) is finite and q L := sup { q ≥ E [ | L | q ] < ∞} = q Y .Proof. The first statement follows by independence of X and Y , so that the moment generating function of L is simply the product of those of X and Y . Now, it is clear that E | L | q is finite for q < q Y . For q > q Y , observe | Y | q ≤ (cid:12)(cid:12)(cid:12) ( | Y | − | X | ) + + | X | (cid:12)(cid:12)(cid:12) q < q (cid:16)n ( | Y | − | X | ) + o q + | X | q (cid:17) and ( | Y | − | X | ) + ≤ || Y | − | X || ≤ | Y − X | , where this last inequality is due to the reverse triangularinequality. This implies the assertion about q L . (cid:3) Choices for X abund, as any process with finite moments and finite exponential moments of all orders willdo, in particular the Brownian motion, the generalised Inverse Gaussian process, the generalised Hyperbolicprocess [2], the CGMY process [6]. For Y , the choices are scarcer, but the inverse Gaussian process is a validone, whereby Y is a pure-jump L´evy process with density at time 1 equal to f IG ( y ; α, β ) = β α Γ( α ) y − α − e − β/y , for y > , where α, β > · ) is the Gamma function. Jørgensen [24] showed that E [ Y r ] = Γ( α − r )Γ( α ) β r , if r < α, and infinite otherwise . The reciprocal Gamma distribution is a special case of the Generalised Inverse Gaussian (GIG) distributionand hence is infinitely divisible [2]. With this specification, the log-returns have exploding negative momentsbeyond order q L = α (possibly larger than 2) and positive moments of arbitrary order depending on X .4.2. Stochastic volatility models. The final example we are interested in belongs to the class of classicalstochastic volatility models, where S satisfies the following dynamics under the risk-neutral probability measure:d S t = σ δt S t (cid:16) ρ d W t + p − ρ d W ⊥ t (cid:17) , d σ t = b ( σ t )d t + νσ γt d W t , starting from S , σ > 0, where ρ ∈ [ − , δ, γ, ν > b ( · ) is some drift. Lions and Musiela [26] providednecessary (and often sufficient) conditions on the parameters and the drift ensuring that S is a true martingale and that moments of a certain order exist. 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L´evy processes and infinitely divisible distributions. Cambridge University Press, 1999. Email address : [email protected] Department of Mathematics, Imperial College London and the Alan Turing Institute Email address ::