On the harmonic mean representation of the implied volatility
OOn the harmonic mean representation of the implied volatility
Stefano De Marco ∗ July 8, 2020
Abstract
It is well know that, in the short maturity limit, the implied volatility approaches theintegral harmonic mean of the local volatility with respect to log-strike, see [Berestyckiet al., Asymptotics and calibration of local volatility models, Quantitative Finance, 2,2002]. This paper is dedicated to a complementary model-free result: an arbitrage-free implied volatility in fact is the harmonic mean of a positive function for any fixedmaturity. We investigate the latter function, which is tightly linked to Fukasawa’sinvertible map f / [Fukasawa, The normalizing transformation of the implied volatilitysmile, Mathematical Finance, 22, 2012], and its relation with the local volatility surface.It turns out that the log-strike transformation z = f / ( k ) defines a new coordinatesystem in which the short-dated implied volatility approaches the arithmetic (as opposedto harmonic) mean of the local volatility. As an illustration, we consider the case of theSSVI parameterization: in this setting, we obtain an explicit formula for the volatilityswap from options on realized variance. A classical result in the literature on the implied volatility surface, usually referred to asBerestycki, Busca and Florent’s (BBF) formula [4], states that the implied volatility σ BS generated by a local volatility model d S t = ( r − q ) S t d t + σ loc ( t, S t ) S t d W t converges to the harmonic mean of the local volatility in the small maturity limit: lim T → σ BS ( T, k ) = 1 k (cid:82) k yσ loc (0 ,S e y ) ∀ k, (1) ∗ Centre de Mathématiques Appliquées (CMAP), CNRS, Ecole Polytechnique, Institut Polytechnique deParis, France. [email protected]
This research has benefited from the financial sup-port of the
Chaire Risques Financiers (Fondation du Risque) and the
Chaire Stress Test, RISK Managementand Financial Steering (Fondation de l’Ecole Polytechnique). I would like to thank Claude Martini forstimulating discussions. a r X i v : . [ q -f i n . P R ] J u l here k = log KF T → log KS denotes log-forward moneyness. For the asymptotic result (1) tohold, the local volatility surface needs to have a well-behaved limit σ loc (0 , · ) when time tendsto zero; see Berestycki et al. [4, Assumption (7)] for the precise conditions. The limit (1)can also be obtained using small-time large deviations theory; in this setting, the function (cid:0)(cid:82) k yσ loc (0 ,S e y ) (cid:1) stems from the finite-dimensional rate function of the process log S t S .In this work, we show that a representation analogous to (1) actually holds for every fixed maturity and in a model-free setting, that is: as soon as σ BS is a arbitrage-free impliedvolatility surface, σ BS ( T, k ) = 1 k (cid:82) k y Σ( T,y ) ∀ k, ∀ T, (2)for some positive function Σ( T, k ) (which cannot be interpreted as the local volatility any-more). We inspect the representation (2) from the point of view of static no-arbitrage condi-tions, and investigate the link of the function Σ with the local volatility surface associated to σ BS via Dupire’s formula: as expected, the two functions can be identified in the small timelimit (but are different otherwise).The key element in the (simple) proof of (2) is Fukasawa’s seminal work [11] on the strictmonotonicity of the time-dependent Black-Scholes maps k (cid:55)→ d ( k ) = − k √ T σ BS ( T,k ) − √ T σ BS ( T,k )2 and k (cid:55)→ d ( k ) = − k √ T σ BS ( T,k ) + √ T σ BS ( T,k )2 . Since the function Σ is linked to the interpolatedmap f / ( k ) = − ( d ( k )+ d ( k )) , we explore the consequences of the log-strike transformation z = f / ( k ) (referred to as “normalizing transformation” in [11]) on the geometry of theimplied volatility surface. It turns out that such a change of variable transforms the harmonicmean representation (2) into an arithmetic mean representation σ / ( T, z ) = 1 z (cid:90) z Σ / ( T, y )d y ∀ z, ∀ T, where σ / and Σ / are, respectively, the implied volatility and its harmonic mean counter-part in the new coordinate system, that is σ / ( T, f / ( k )) = σ BS ( T, k ) and Σ / ( T, f / ( k )) =Σ( T, k ) , for all k .As an application we show that, under some reasonable conditions on the volatility surface σ BS one starts from, the short-time limit of the function Σ / can also be identified with theshort-time limit of Dupire’s local volatility, showing that BBF asymptotic formula (1) isreplaced by an arithmetic mean formula in the new coordinate system. We refer to Theorem4.5 for precise statements. Notation and basic definitions . We denote c BS : ( k, v ) ∈ R × [0 , ∞ ) → R + :=[0 , ∞ ) the normalized Black-Scholes call price with forward log-moneyness k and total impliedvolatility parameter v = √ τ σ : c BS ( k, v ) = (cid:26) N ( d ( k, v )) − e k N ( d ( k, v )) if v > − e k ) + if v = 0 (3)where N is the standard Gaussian cdf N ( x ) = (cid:82) x −∞ φ ( y )d y , φ ( y ) = √ π e − y , and d i ( k, v ) = − kv − (1 − i ) v , i ∈ { , } , for every v > . 2 The harmonic mean representation of the implied volatil-ity
Let us recall the following
Definition 2.1 (Arbitrage-free implied volatility, fixed maturity) . Let time to maturity T befixed. We say that a function v : R → R + is a total implied volatility free of static arbitrageif the function K ∈ (0 , ∞ ) (cid:55)→ C ( K ) := c BS (cid:18) log KF , v (cid:16) log KF (cid:17)(cid:19) (4) is convex and satisfies lim K →∞ C ( K ) = 0 (for some, hence for any, F > ).If v is an arbitrage-free total implied volatility for time to maturity T , we denote σ BS ( k ) = √ T v ( k ) the related implied volatility tout court . It is well know that the conditions on the function C in Definition 2.1 (which are usuallyreferred to as no–butterfly arbitrage conditions) are equivalent to the existence of a pricingmeasure: if v satisfies Definition 2.1, there exists a non-negative random variable X with E [ X ] = 1 such that c BS ( k, v ( k )) = E (cid:2) ( X − e k ) + (cid:3) , ∀ k ∈ R . (5)Arbitrage-free implied volatilities can fail to be everywhere differentiable and can vanish onsome interval; the regularity and the support of v can of course be linked with the regularityand the support of the law of the random variable X in (5), see [19, Lemma 5.2]. We restrictour analysis to total implied volatilities that are strictly positive and differentiable: Assumption 2.2.
We assume v ∈ C ( R ) and v ( k ) > , for every k ∈ R . Denote f ( k ) := − d ( k, v ( k )) = kv ( k ) + v ( k )2 ,f ( k ) := − d ( k, v ( k )) = kv ( k ) − v ( k )2 . (6)Fukasawa [11] proved the following result. Theorem 2.3 (Fukasawa [11]) . If v is an arbitrage-free total implied volatility satisfyingAssumption 2.2, then ddk f i ( k ) > ∀ k ∈ R , i ∈ { , } . The strict monotonicity of the maps f and f can be exploited to rigorously justify someremarkable model-free pricing formulas for European claims such as the log-contract, see[7, 11, 8] and section 5.1 below, and can also be used as a partial characterization of thestatic no-arbitrage condition on v , see Remark 2.5 below and the work carried out in [18].The current section is devoted to the following result:3 heorem 2.4. Let v be an arbitrage-free total implied volatility satisfying Assumption 2.2.Then, there exists a unique strictly positive function h ∈ C ( R ) such that v is the harmonicmean of h : v ( k ) = 1 k (cid:82) k h ( y ) d y ∀ k (cid:54) = 0 , (7) and v (0) = h (0) .Proof. Let f / ( k ) = kv ( k ) , so that f / ∈ C ( R ) . Since, by definition of f and f , f / ( k ) = 12 ( f ( k ) + f ( k )) , it follows from Theorem 2.3 that f (cid:48) / ( k ) > for every k ∈ R . Set h := f (cid:48) / : the function h isstrictly positive, continuous, and such that kv ( k ) = f / ( k ) = f / (0) + (cid:82) k h ( y ) d y = (cid:82) k h ( y ) d y .Then by construction, v satisfies equation (7) for every k (cid:54) = 0 . Taking the limit as k → in(7) and using the continuity of h , we obtain that v and h coincide at k = 0 .The uniqueness of h follows from (7): taking derivatives on both sides of (cid:82) k h ( y ) d y = kv ( k ) ,we get h ( k ) = dd k kv ( k ) = 1 v ( k ) (cid:18) − k v (cid:48) ( k ) v ( k ) (cid:19) ∀ k ∈ R . (8)which identifies uniquely the function h .In view of Theorem 2.4, it seems reasonable to wonder what is the class of functionshaving an harmonic mean representation as (7). Actually, every strictly positive function v ∈ C ( R ) admits the representation (7) for a uniquely determined function h ∈ C ( R ) :simply define h from (8). The important part of the statement of Thm 2.4 is the positivity of h : if we start from any function v , h will not be positive in general. In this respect, Theorem2.4 provides a necessary condition for arbitrage freeness of a total implied volatility v , butthis condition is unfortunately not sufficient, as we discuss below. Remark 2.5.
Consider a strictly positive function v : we can always define the maps f and f as in (6) .i) If we assume v ∈ C ( R ) , then a computation involving the derivatives of the Black-Scholes call price (3) yields d d K c BS (cid:18) log KF , v (cid:16) log KF (cid:17)(cid:19) = 1 F K φ ( f ( k )) (cid:16) v (cid:48)(cid:48) ( k ) + v ( k ) f (cid:48) ( k ) f (cid:48) ( k ) (cid:17)(cid:12)(cid:12)(cid:12) k =log KF (9) (see for example [11] for the derivation of the expression on right hand side), whichshows that the call price function K (cid:55)→ C ( K ) in (4) is convex if and only if v (cid:48)(cid:48) + v f (cid:48) f (cid:48) ≥ . (10) If Theorem 2.4 were “if and only if”, we could generate implied volatilities parameterizations by simplytaking harmonic means of positive functions. n particular, we see that the strict monotonicity of f and f in Theorem 2.3 is anecessary condition for the convexity of C ( · ) , but not a sufficient condition. ii) We can further define the function h from equation (8) , so that v and h satisfy (7) byconstruction. Since h = ( f + f ) (cid:48) , h is positive if and only if the sum f + f is astrictly increasing function. The latter condition is weaker that the strict monotonicityof f and f separately, which, as seen in the previous bullet point, is itself a necessarybut not sufficient condition for no arbitrage. We might also wonder whether the no butterfly-arbitrage condition (10) simplifies whenrephrased in terms of the function h . In other words: is it easier to generate arbitrage-free implied volatilities via (7) by looking for appropriate functions h , than trying to lookfor arbitrage-free parameterizations of v directly? The condition (10)can be rewritten moreexplicitly as v (cid:48)(cid:48) − v ( v (cid:48) ) + v (cid:0) − k v (cid:48) v (cid:1) ≥ , see [14]. Injecting the expression of h from (8), weobtain a new condition involving the three functions h , h (cid:48) and (cid:82) · h (instead of v , v (cid:48) and v (cid:48)(cid:48) ).The resulting expression does not seem particularly insightful to us (concretely: it does notlook more tractable than condition (10) itself), and is therefore not reported here. Corollary 2.6.
Under the assumptions and notation of Theorem 2.4,(i) v and h coincide at the critical points of v : h ( k ) = v ( k ) ⇐⇒ k = 0 or v (cid:48) ( k ) = 0 . (ii) The functions v and h satisfy the “1/2-skew rule” v (cid:48) (0) = 12 h (cid:48) (0) . (11) Proof. (i) follows immediately from (8). (ii) simply states that the / -derivative rule (11)always holds at k = 0 for a function h and its harmonic mean v , as it can be checked bydirect computation of the limit lim k → v ( k ) − v (0) k using (7). Remark 2.7 (An upper bound) . Since the arithmetic mean exceeds the harmonic mean, weobtain that the upper bound v ( k ) ≤ M ( k ) := (cid:40) k (cid:82) k h ( y )d y ∀ k (cid:54) = 0 ,v (0) k = 0 . holds for any arbitrage-free implied volatility v . In Figure 1, as an illustration we plot two examples of implied volatility smiles and theirrelated functions h . The SVI parameterization, introduced by Gatheral in 2004 [13], is definedby w ( k ) = a + b (cid:0) ρ ( k − m ) + (cid:112) ( k − m ) + σ (cid:1) , where w ( k ) = v ( k ) denotes implied total For example, the strict monotonicity of f and f together with convexity of v would be a sufficientcondition. Unfortunately, this condition would be very restrictive in practice, since implied volatility smilescalibrated to market data are often not convex. h inTheorem 2.4. Left: SVI parameters are a = 0 . , b = 0 . , ρ = − . , m = 0 . , σ = 0 . . Thisset of parameters generate an arbitrage-free smile, as it can be checked using the proceduredescribed in [18]. Right: SSVI parameters are θ = 0 . , ρ = 0 . , ϕ = 3 . This set of parameterssatisfies condition (28) and therefore generates an arbitrage-free smile.variance. SSVI, see (27), is sub-family of SVI based on three parameters ( θ, ϕ, ρ ) instead offive; related no-arbitrage conditions were analyzed in [15]. A more detailed analysis of theSSVI framework and related applications will be carried out in section 5. In Figure 1, theSVI in the left pane has the typical negative-skew shape observed in equity markets, whilethe positive-skew SSVI in the right pane reproduces a typical pattern observed for optionson realized variance or options on the VIX index. f / The function f / ( k ) = ( f ( k ) + f ( k )) we have encountered in Theorem 2.4 is a specialexample of log-strike transformation: it belongs to the family of maps f p ( k ) = p f ( k ) + (1 − p ) f ( k ) = kv ( k ) + (cid:16) − p (cid:17) v ( k ) , p ∈ [0 , . Owing to Theorem 2.3, the functions f p are strictly increasing. When p (cid:54) = 0 , they are alsosurjective: Lemma 3.1.
For every p ∈ (0 , , Im( f p ) := f p ( R ) = R , so that f p is bijective from R onto R . In particular, so is f / .Proof. The no arbitrage condition lim K →∞ C ( K ) = 0 for the call price (4) is equivalent to lim inf k →∞ f ( k ) = ∞ , see [20, Theorem 2.9]. Since f ( k ) = − (cid:16) | k | v ( k ) + v ( k ) (cid:17) ≤ − (cid:112) | k | for k < by the arithmetic-geometric inequality, we have lim inf k →−∞ f ( k ) = −∞ , hence Im( f ) = R . Since f ( k ) > for k > and f is decreasing, we have lim inf k →±∞ (cid:0) p f ( k ) +(1 − p ) f ( k ) (cid:1) = ±∞ , too, for every p ∈ (0 , .6 emark 3.2. Lemma 3.1 is not true in general for p = 0 . It holds that lim k →−∞ f ( k ) = N − ( P ( X = 0)) , see [21, 10], so that Im( f ) fails to coincide with the whole R when P ( X =0) (cid:54) = 0 . Following Fukasawa [11], we can see the map f p as a transformation of the log-strikevariable, from k to z = f p ( k ) . Denoting g p the inverse transformation from Im( f p ) = R to R , g p ( z ) = f − p ( z ) , z ∈ R , the so-called p -normalized implied volatilities v p are defined by v p ( z ) = v ( g p ( z )) , z ∈ R . Inparticular, the “one-half normalized” implied volatility is v / ( z ) = v (cid:0) g / ( z ) (cid:1) . The functions v p allow to write particularly compact and elegant model-free pricing formulasfor European claims, see [11, Theorem 4.6] and [8, Theorem 2.7] for a general treatment:celebrated examples are Chriss and Morokoff’s formula [7] for the log-contract and Bergomi’sformula [5, Section 4.3.1] for the moments of X of order p ∈ [0 , . See Section 5.1 for moredetails, and for a detailed study of the SSVI case. Proposition 3.3.
The log-strike transformation k → z = f / ( k ) maps the harmonic meanrepresentation v ( k ) = 1 k (cid:82) k h ( y ) d y into the arithmetic mean representation v / ( z ) = 1 z (cid:90) z h / ( y )d y ∀ z (cid:54) = 0 , (12) where h / ( y ) := h ( g / ( y )) .Proof. Let k ∈ R . We have k = g / ( z ) if and only if f / ( k ) = z . Since f / ( k ) = kv ( k ) , wededuce z = g / ( z ) v / ( z ) , or yet z v / ( z ) = g / ( z ) . Since g / (0) = 0 and g (cid:48) / ( z ) = 1 f (cid:48) / ( g / ( z )) = h ( g / ( z )) = h / ( z ) , we obtain z v / ( z ) = (cid:82) z h / ( y )d y , which proves (12). Put-Call duality . When P ( X = 0) (equivalently: when lim k →−∞ f ( k ) = −∞ , seeRemark 3.2), the put-call symmetry relation in the Black-Scholes model implies that themirrored function ˆ v ( k ) = v ( − k ) is still an arbitrage-free total implied volatility, associated toa pricing model ˆ X via equation (5). The distribution of the dual model ˆ X can be related tothat of X via a change of measure; see [6] and section 3 in [8] for more details. As notice in[8], it is straightforward to check how the maps f p and the p -normalized implied volatilitieschange under the duality transformation v (cid:55)→ ˆ v : ˆ f p ( k ) = − f − p ( − k ) , ˆ v p ( k ) = v − p ( − k ) , p ∈ [0 , . h is also mirrored under the duality transformation. Indeed, denote ˆ h theunique function associated to ˆ v in the harmonic mean representation (7): we have h ( k ) = dd x x ˆ v ( x ) = − dd x − xv ( − x ) = − dd x f / ( − x ) = f (cid:48) / ( − x ) = 1 h ( − k ) , so that ˆ h ( k ) = h ( − k ) , for all k ∈ R . In this section, we consider a total implied volatility surface v : ( T, k ) ∈ [0 , ∞ ) × R (cid:55)→ v ( T, k ) ,where T denotes the option’s time to maturity and k = log KF T the corresponding forwardlog-moneyness. When arbitrage-free, such a function always satisfies v (0 , · ) ≡ . In thissection, we assume that v is strictly positive for strictly positive maturities and that thesurface is smooth: Assumption 4.1.
The function ( T, k ) (cid:55)→ v ( T, k ) is C , ((0 , ∞ ) × R ) and v ( T, k ) > forevery k ∈ R and T > . In particular, Assumption 2.2 holds for every fixed T . According to Theorem 2.4, there existsa unique strictly positive function h defined over (0 , ∞ ) × R satisfying v ( T, k ) = 1 k (cid:82) k h ( T,y ) d y ∀ k (cid:54) = 0 , ∀ T > . (13)Assuming a market with constant interest rate r and repo (or dividend) rate q , so that F T = S e ( r − q ) T , Dupire’s formula for the local volatility σ Dup reads σ Dup ( T, k ) = 2 dd T (cid:16) e − rT F T c BS (cid:16) log KF T , v (cid:0) T, log KF T (cid:1)(cid:17)(cid:17) e − rT F T K d K c BS (cid:16) log KF T , v (cid:0) T, log KF T (cid:1)(cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K = F T e k = 2 ∂ T v ( T, k ) (cid:16) v (cid:48)(cid:48) − v ( v (cid:48) ) + v (cid:0) − k v (cid:48) v (cid:1) (cid:17) ( T, k ) ∀ T > , k ∈ R , (14)where we denote v (cid:48) ( T, k ) := ∂ k v ( T, k ) , v (cid:48)(cid:48) ( T, k ) := ∂ kk v ( T, k ) . The derivation of the expres-sion of σ Dup in terms of the total implied volatility and its space-time derivatives can befound for example in Lee [17]. It has already been noted, see again [17], that the right-most summand in the denominator of (14) contains precisely the squared derivative of thefunction kv ( T,k ) with respect to k , since v (cid:0) − k v (cid:48) v (cid:1) = (cid:0) dd k kv (cid:1) = h , see (8). This remarkprovides a quick way to infer Berestycki et al.’s asymptotic formula (1) from Dupire’s equa-tion (14). Following [17], we can re-express (14) in terms of the standard implied volatility σ BS ( T, k ) = √ T v ( T, k ) : since v (cid:48) = √ T σ (cid:48) BS , v (cid:48)(cid:48) = √ T σ (cid:48)(cid:48) BS and ∂ T v = σ BS √ T + √ T ∂ T σ BS , weobtain σ Dup ( T, k ) = σ BS ( T, k ) + 2
T ∂ T σ BS ( T, k ) (cid:16) T σ (cid:48)(cid:48) BS − T σ BS ( σ (cid:48) BS ) + σ BS (cid:16) − k σ (cid:48) BS σ BS (cid:17) (cid:17) ( T, k ) . (15)8ormally taking T = 0 inside (15), one obtains an ODE for the function σ BS | T =0 , namely σ BS (0 ,k ) (cid:0) − k σ (cid:48) BS(0 ,k ) σ BS(0 ,k ) (cid:1) = σ Dup (0 , k ) . Now, it is straightforward to see that the function k (cid:82) k σ Dup(0 ,y ) d y solves this differential equation. One could therefore conjecture that the latter is the limit of σ BS as T → ; Berestycki et al.’s [4] show that this is actually the case, under the assumptionthat the local volatility is uniformly continuous, bounded and bounded away from zero on [0 , T ] × R for some T > .If we take T to be fixed (but not equal to zero), dividing both sides of (15) by σ BS ( T, k ) (which is possible under Assumption 4.1) and rearranging terms, we get σ Dup ( T, k ) (cid:18) dd k kσ BS ( T, k ) (cid:19) = (cid:20) T (cid:18) ∂ T σ BS σ BS − σ (cid:48)(cid:48) BS σ BS σ (cid:19) + 14 T ( σ (cid:48) BS ) σ (cid:21) ( T, k ) . (16)Combining with Theorem 2.4, we can formulate an equation relating the harmonic meanfunction h to the local volatility. Proposition 4.2.
Let v be a total implied volatility surface satisfying Assumption 4.1, anddenote σ BS ( T, k ) = √ T v ( T, k ) the corresponding implied volatility. Denote h : (0 , ∞ ) × R → (0 , ∞ ) the function in the harmonic mean representation (13) . For every ( T, k ) such that ∂ T v ( T, k ) > , we have Σ( T, k ) := 1 √ T h ( T, k ) = σ Dup ( T, k ) (cid:112) T a ( T, k ) + T b ( T, k ) , (17) where a ( T, k ) = (cid:18) ∂ T σ BS σ BS − σ (cid:48)(cid:48) BS σ BS σ (cid:19) ( T, k ) , b ( T, k ) = 14 σ (cid:48) BS ( T, k ) σ Dup ( T, k ) . Since σ BS = √ T v , note that the function Σ defined in (17) appears in the harmonic meanrepresentation of the standard Black-Scholes implied volatility: we have σ BS ( T, k ) = 1 k (cid:90) k T, y ) d y (18)for every T > and k (cid:54) = 0 . Proof.
It follows from (14) that σ Dup ( T, k ) > if and only if ∂ T v ( T, k ) > for the samepoint ( T, k ) . For such ( T, k ) , we can divide both sides of (16) by σ Dup ( T, k ) , obtaining that (cid:0) dd k kσ BS ( T,k ) (cid:1) − is equal to the square of the right hand side of (17). According to Theorem2.4, dd k kσ BS ( T,k ) is positive and equal to √ Th ( T,k ) , which concludes the proof of (17).We note in passing that equation (16) does not allow to infer that the the function dd k kσ BS ( T,k ) is positive; this information is provided by Theorem 2.4.The functions a and b in Proposition 4.2 depend on the implied volatility σ BS and itsspace-time derivatives: in this respect, Proposition 4.2 does provide an explicit link betweenthe local volatility σ Dup and the functions h and Σ (in the sense: it does not allow to evaluate h or Σ explicitly from the knowledge of the function σ Dup ). Nevertheless, we can interpretequation (17) when the maturity T is small: if the functions a and b remain bounded as T → σ BS and its derivatives have non-trivial limits as T → ), equation (17)takes the form Σ = 1 √ T h = σ Dup (cid:0) O ( T ) (cid:1) , where the O ( T ) correction term is precisely − T a ( T, k ) . Remark 4.3.
The boundedness of the functions a and b as T → hinges on the boundednessof σ BS , σ (cid:48) BS , σ (cid:48)(cid:48) BS and ∂ T σ BS . When the law of the underlying asset price is specified via astochastic model, the asymptotic behavior of σ BS and of the strike derivatives σ (cid:48) BS and σ (cid:48)(cid:48) BS can be assessed for certain models including stochastic volatility models, see [12] and [1], butboundedness might fail to hold in certain cases, as in rough fractional stochastic volatilitymodels where lim T → σ (cid:48) BS ( T,
0) = ∞ , see [2, 12, 3]. It is nevertheless interesting to noticethat, while the function σ BS and its harmonic mean counterpart Σ always satisfy the 1/2–skewrule d σ BS ( T, k )d k (cid:12)(cid:12)(cid:12)(cid:12) k =0 = 12 dΣ( T, k )d k (cid:12)(cid:12)(cid:12)(cid:12) k =0 ∀ T > (see Corollary 2.6), Proposition 4.2 allows to characterize the possible situations where theimplied volatility σ BS and the local volatility σ Dup do not satisfy the 1/2–skew rule in the shorttime limit : they are precisely the situations where lim T → k Σ( T, k ) | k =0 (cid:54) = lim T → k σ Dup ( T, k ) | k =0 . Since the total implied volatility v ( T, k ) tends to zero for all k as T → , the normalizingtransformations f / ( T, k ) = kv ( T,k ) become trivial for short maturity, in the sense lim T → f / ( T, k ) = + ∞ if k > if k = 0 −∞ if k < . (19)The inverse transformations g / ( T, · ) = f / ( T, · ) − therefore flatten out as T → : we have lim T → g / ( T, z ) = 0 for every z ∈ R . As a consequence, the normalized implied volatility σ / ( T, z ) := √ T v (cid:0) T, g / ( T, z ) (cid:1) tends to a constant: σ / ( T, z ) = σ BS (cid:0) T, g / ( T, z ) (cid:1) → σ BS (0 , as T → , (20)where the last limit holds whenever the implied volatility σ BS ( T, k ) tends to a limiting function σ BS (0 , k ) uniformly in a neighbourhood of k = 0 .Equation (20) shows that the normalized implied volatility σ / is a rather uninterestingobject as T becomes small. We rather expect the time-rescaled function σ / (cid:16) T, z √ T (cid:17) to havea non-trivial limit as T → , see Figure 2.We have already seen in Proposition 3.3 that, for every T , the log-strike transformation z = f / ( T, k ) maps a coordinate system where the implied volatility σ BS is the harmonicmean of a positive function Σ into a system where the new implied volatility σ / is the even if this criterion might be difficult to apply in practice, for it requires to evaluate the asymptoticbehavior of the functions a and b . σ / for the SSVI surface v SSVI (27) with parameters θ T = θ T = (0 . T , ϕ T = ϕ = 4 and ρ T = ρ = − . . The triple ( θ, ϕ, ρ ) satisfies theno-arbitrage condition (28), so that the resulting SSVI surface is arbitrage-free for T ≤ .We used the explicit expression of σ / = √ T v / in Proposition 5.1 to obtain the plots. Left:the normalized implied volatility σ / tends to the constant value √ θ , see (20). Right: thetime-rescaled function σ / (cid:0) T, z √ T (cid:1) has a non-constant limit when T → .arithmetic mean of the transformed function Σ / : dividing both sides of (12) by √ T andusing Σ = h √ T , we have σ / ( T, z ) = 1 z (cid:90) z Σ / ( T, y )d y ∀ T > , where Σ / ( T, y ) := Σ(
T, g / ( T, y )) . We want to investigate the consequence of this fact inthe small-maturity limit, making the link with the local volatility σ Dup explicit. To this end,we also define the normalized local volatility σ Dup , / (cid:0) T, z (cid:1) := σ Dup (cid:0)
T, g / ( T, z ) (cid:1) ∀ T > , z ∈ R . Theorem 4.5 below essentially states that, if the implied volatility surface σ BS has a non-degenerate behavior for small maturity (in the precise sense of Assumption 4.4), then therescaled normalized implied volatility σ / (cid:0) T, z √ T (cid:1) and local volatility σ Dup , / (cid:0) T, z √ T (cid:1) alsohave non-trivial short-maturity limits, and these limits obey an arithmetic mean formula. Assumption 4.4.
The implied volatility σ BS and the local volatility σ Dup have non-triviallimits in the short maturity regime, in the sense:(i) There exists a strictly positive and differentiable function σ BS (0 , · ) such that σ BS ( T, k ) → σ BS (0 , k ) together with ∂ k σ BS ( T, k ) → ∂ k σ BS (0 , k ) as T → , uniformly over k in com-pact sets.(ii) The function k (cid:55)→ kσ BS (0 ,k ) is strictly increasing from R onto R .(iii) The local volatility has a short-maturity limit: σ Dup ( T, k ) → σ Dup (0 , k ) as T → forsome strictly positive function σ Dup (0 , · ) , uniformly over k in compact sets. heorem 4.5 (Arithmetic mean formula for the short-dated implied volatility σ / ) . LetAssumption 4.4 be in force. Then, the rescaled normalized implied volatility σ / (cid:0) T, z √ T (cid:1) andnormalized local volatility σ Dup , / (cid:0) T, z √ T (cid:1) have non-trivial limits as T → : σ / (cid:18) T, z √ T (cid:19) −→ σ / ( z ) , σ Dup , / (cid:18) T, z √ T (cid:19) −→ σ Dup , / ( z ) , (21) where the convergence is uniform over z in compact sets. If, in addition, the two functions ∂ T σ BS ( T, k ) and ∂ kk σ BS ( T, k ) remain bounded as T → , we have the following arithmeticmean formula σ / ( z ) = 1 z (cid:90) z σ Dup , / ( y ) d y , ∀ z ∈ R . (22) Proof.
Denote g / the inverse of the function k (cid:55)→ kσ BS (0 ,k ) . We claim that g / (cid:18) T, z √ T (cid:19) −→ g / ( z ) as T → , uniformly over z in compact sets . (23)Let us postpone the proof of (23)to Appendix A and focus on the asymptotic behavior of Σ / (cid:16) T, z √ T (cid:17) and σ / (cid:16) T, z √ T (cid:17) . Recall from (18) that T,k ) = dd k kσ BS ( T,k ) = σ BS ( T,k ) (cid:16) − k σ (cid:48) BS ( T,k ) σ BS ( T,k ) (cid:17) for every T > . According to point (ii) in Assumption 4.4, we can define thefunction Σ(0 , k ) from ,k ) = dd k kσ BS (0 ,k ) , so that Σ( T, k ) = σ BS ( T, k ) (cid:16) − k σ (cid:48) BS ( T,k ) σ BS ( T,k ) (cid:17) −→ σ BS (0 , k ) (cid:16) − k σ (cid:48) BS (0 ,k ) σ BS (0 ,k ) (cid:17) = Σ(0 , k ) as T → (24)holds uniformly over k in compact sets, owing to Assumption 4.4 (i). For simplicity, denote k zT = g / (cid:0) T, z √ T (cid:1) and k z = g / ( z ) . For every compact set C ⊂ R , sup z ∈ C | Σ( T, k zT ) − Σ(0 , k z ) | ≤ sup z ∈ C | Σ( T, k zT ) − Σ(0 , k zT ) | + sup z ∈ C | Σ(0 , k zT ) − Σ(0 , k z ) | . The first term on the right hand side tends to zero as T → because, on the one hand, Σ( T, k ) − Σ(0 , k ) → uniformly over k in compacts, and on the other hand, according to(23), sup z ∈ C | k zT | ≤ sup z ∈ C | g / ( z ) | + 1 for T small enough. The second term also tends tozero as T → because k zT → k z uniformly over z ∈ C and Σ(0 , · ) is continuous by assumption.Summing up, we have shown that Σ (cid:18) T, g / (cid:18) T, z √ T (cid:19)(cid:19) = Σ / (cid:18) T, z √ T (cid:19) −→ Σ (cid:0) , g / ( z ) (cid:1) =: Σ / ( z ) as T → , (25)uniformly over z in compact sets. Concerning σ / , it is immediate to see that (25) impliesthat the limit σ / (cid:18) T, z √ T (cid:19) = √ Tz (cid:90) z/ √ T Σ / ( T, y )d y = 1 z (cid:90) z Σ / (cid:18) T, x √ T (cid:19) d x −→ z (cid:90) z Σ / ( x )d x (26)holds as T → uniformly over z in compact sets, too, which proves the first part of (21)with σ / ( z ) = z (cid:82) z Σ / ( x )d x . 12et us move to the normalized local volatility. It follows from (23) and Assumption4.4 (iii) that σ Dup , / (cid:16) T, z √ T (cid:17) = σ Dup (cid:16)
T, g / (cid:16) T, z √ T (cid:17)(cid:17) → σ Dup (cid:0) , g / ( z ) (cid:1) as T → , hencethe second limit in (21) follows with σ Dup , / ( z ) := σ Dup (cid:0) , g / ( z ) (cid:1) . Using the additionalassumption on the boundedness of the functions ∂ T σ BS and ∂ kk σ BS as T → , we can see thatthe functions a and b defined in Proposition 4.2 remain bounded as T → , which impliesthat T a (cid:0)
T, g / (cid:0) T, z √ T (cid:1)(cid:1) + T b (cid:0) T, g / (cid:0) T, z √ T (cid:1)(cid:1) → as T → . Therefore, it follows fromequation (17) that Σ (cid:18) T, g / (cid:18) T, z √ T (cid:19)(cid:19) ∼ σ Dup (cid:18)
T, g / (cid:18) T, z √ T (cid:19)(cid:19) −→ σ Dup , / ( z ) as T → . Comparing with (25), we identify Σ / ( z ) with σ Dup , / ( z ) , hence formula (22) follows from(26). The SSVI parameterisation for total implied variance w = v proposed by Gatheral andJacquier [15] reads w SSVI ( k ) = θ (cid:16) ρϕk + (cid:112) ( ϕk + ρ ) + 1 − ρ (cid:17) . The corresponding totalimplied volatility is v SSVI ( k ) = (cid:114) θ (cid:16) ρϕk + (cid:112) ∆( k ) (cid:17) , ∆( k ) = ( ϕk + ρ ) + 1 − ρ . (27)Equation (27) parametrizes a slice of the implied volatility surface at fixed maturity: theSSVI parameters are θ > , ϕ > and ρ ∈ ( − , . Since min { k ∈ R } w SSVI ( k ) = θ (cid:0) − ρ + (cid:112) ρ + 1 (cid:1) > for every ρ ∈ ( − , , v SSVI satisfies Assumption 2.2 for any triple ( θ, ϕ, ρ ) asabove. Theorem 4.2 in [15] proves that v SSVI is a total implied volatility free of arbitrage (forthe given maturity) if the following sufficient conditions are satisfied: θϕ (1 + | ρ | ) < θϕ (1 + | ρ | ) ≤ . (28)The condition θϕ (1 + | ρ | ) ≤ (including the equality) is known to be necessary, see [15,Lemma 4.2], while the second inequality in (28) is not necessary in general.We can evaluate the function h for SSVI from (8): h SSVI ( k ) = (cid:18) v ( k ) − k v ( k ) w (cid:48) (cid:19) − = (cid:32) v ( k ) − kθ v ( k ) (cid:18) ρϕ + ϕk + ρ (cid:112) ∆( k ) (cid:19)(cid:33) − where v = v SSVI and w = w SSVI . Moreover it turns out that, for the SSVI parameterisation, itis possible to explicitly compute the inverse transformation g / and the normalized impliedvolatility v / . Proposition 5.1.
Assume that the triple ( θ, ϕ, ρ ) satisfies (28) . Then, the inverse transfor-mation g / = f − / for the SSVI parameterisation (27) is given by g / ( z ) = 12 (cid:0) θρϕz + z (cid:112) θ ϕ z + 4 θ (cid:1) , z ∈ R ; nd the / –normalized SSVI implied volatility is v / ( z ) = 12 (cid:0) θρϕz + (cid:112) θ ϕ z + 4 θ (cid:1) , z ∈ R . (29) Remark 5.2.
The normalized SSVI volatility v / is asymptotically linear for large arguments z , with lim z ±∞ v / ( z ) | z | = θϕ (1 ± ρ ) , as opposed to the SSVI implied volatility (27) , which isproportional to (cid:112) | k | for large values of | k | .Proof of Proposition 5.1. Denoting v ( k ) = v SSVI for simplicity, we have f / ( k ) = z ⇐⇒ k = v ( k ) z. (30)By the invertibility of f / and Lemma 3.1, we know that the equation on the right hand sideof (30) has a unique solution k for every z ∈ R , which coincides with g / ( z ) . We alreadyknow that g / ( z ) = 0 for z = 0 , hence we assume z (cid:54) = 0 in what follows, which implies k (cid:54) = 0 .By squaring both sides in (30), we obtain that, for every z , the equation k = v ( k ) z hasexactly two solutions k ± (given by g / ( z ) and g / ( − z ) ). Using the SSVI formula (27), theequation k = v ( k ) z is equivalent to (cid:112) ∆( k ) = k θz − − ρϕk . If we square both sides again,passing to the quartic equation ∆( k ) = (cid:16) k θz − − ρϕk (cid:17) , we might add spurious solutions:it turns out that this quartic equation has only two roots, which we can therefore identifywith k ± . Let us work this out: expanding the squares in ∆( k ) = (cid:16) k θz − − ρϕk (cid:17) , aftersome cancellations and rearrangements we obtain θ z k − ϕρθz k + (cid:18) ρϕ − ϕ − θz (cid:19) k = 0 . The special feature of the equation above it that it has no constant term and no linear termin k . Dividing by k , we are left with the quadratic equation θ z k − ϕρθz k + c = 0 , where c = ρϕ − ϕ − θz . The two roots k ± = θz (cid:18) ρϕ ± (cid:114) θz + ϕ (cid:19) are therefore the two solutions of k = v ( k ) z . Since (cid:113) θz + ϕ > ϕ ≥ | ρϕ | , we have k + > and k − < for every z , which allows us to identify g / ( z ) with k + for z > , resp. k − for z < . Overall, we have obtained g / ( z ) = θz (cid:18) ρϕ + sign( z ) (cid:114) θz + ϕ (cid:19) = 12 (cid:18) θρϕz + z (cid:112) θ ϕ z + 4 θ (cid:19) , ∀ z ∈ R . Finally, since (30) implies g / ( z ) = v / ( z ) z , we have v / ( z ) = g / ( z ) z for every z (cid:54) = 0 , fromwhich (29) follows. E (cid:2) √ X (cid:3) When v is arbitrage-free and twice differentiable (as in the case of the SSVI parameterisation(27)), the law of the random variable X in (5) has a absolutely continuous part with density14 X ( K ) = d C ( K )d K K> (the function C being defined in (4)), plus a possible atom at zerowith mass P ( X = 0) = 1 + lim K → ∂ K C ( K ) . According to Remark 3.2, we have P ( X =0) = 0 if and only if lim k →−∞ f ( k ) = −∞ : a sufficient condition for this to hold is β − :=lim sup k →−∞ v ( k ) | k | < , see e.g. [20]. Under such condition, any integrable claim h ( X ) can bepriced as E [ h ( X )] = (cid:90) ∞ h ( K ) d C ( K )d K d K = (cid:90) ∞ h ( K ) d d K c BS ( k, v ( k )) | k =log KF d K . (31)For some payoff functions h , it is possible to convert equation (31) into a particularly compactand elegant formula written in terms of one of the normalized implied volatilities v p , see [11,Theorem 4.6] and [8, Theorem 2.7]. Celebrated examples are Chriss and Morokoff’s formula[7] for the log-contract, E [ − X )] = (cid:82) R v ( z ) φ ( z )d z (recall that φ denotes the standardnormal density), and Bergomi’s formula [5, Section 4.3.1] for the moments of X of order p ∈ [0 , , E [ X p ] = (cid:82) R e p ( p − v p ( z ) φ ( z )d z . In particular, we have E (cid:104) √ X (cid:105) = (cid:90) R e − v / ( z ) φ ( z )d z. (32)Formula (32) is appealing because it only requires to know the values of v / (on, say, a setof quadrature points) and does not require to access the values of the derivatives of v / .When the number of liquid log-moneyness k i and implied volatilities v mkt ( k i ) observed onthe market is sufficiently large, one can discretize the right hand side of (32) on the points z i = f / ( k i ) = k i v mkt ( k i ) , thus obtaining a model-free pricing formula for the claim √ X . On theother hand, when the starting point of pricing operations is to fit market option data with avolatility parameterization (such as SSVI), possibly because market data is relatively scarce,it is of course interesting to be provided with explicit pricing formulas, instead of having torely on numerical integration of (32). When v is given by v SSVI , this is the program we carryout in the rest of this section.When the underlying is the annualized realized variance of an observable asset S , X T = 1 T (cid:88) 0) = (cid:82) R e − A θ y d y √ π = √ A θ .When ρ is different from zero, we did not manage to find an explicit expression for I ( θ, ϕ, ρ ) (neither did WolframAlpha online integrator ), but we can investigate the asymptotic behav-ior of the integral in limiting parameter regimes. When SSVI (27) is calibrated to marketoption data, typical values of ϕ are of order or , see [16], while ρ can approach − (forequity index smiles) or (as in the case of realized variance options, which usually displaya positive implied volatility skew, see [9], just as options on the VIX index). Being an ATMimplied variance, the parameter θ is usually small, often of order − : it seems reasonable,then, to look for an asymptotic approximation of I ( θ, ϕ, ρ ) as θ → . A quick inspectionreveals that the asymptotic behavior of the integral in this regime is governed by the factor e − A θ y : since A θ → ∞ as θ → , the function e − A θ y tends to zero exponentially fast forevery y (cid:54) = 0 and, in the spirit of Laplace’s method for integral approximation, the asymptoticbehavior is influenced by the second factor f θ ( y ) = e − ρ y √ y + θ only in a neighbourhood of y = 0 . We have to pay attention to the fact that the function f θ ( · ) also depends on θ , butthis does not add substantial difficulties to the analysis. Proposition 5.3 (Volatility swap value when X T models realized variance) . For every ρ ∈ ( − , and ϕ > , the no-arbitrage condition (28) is satisfied for θ small enough. When thetotal implied volatility of the underlying X T is given by SSVI (27) , the following asymptoticformula holds: E (cid:104)(cid:112) X T (cid:105) = e − θ (cid:113) ρ θ ϕ (1 + o (1)) as θ → . (34) Moreover, if ρ = 0 , the above formula is exact: we have E (cid:2) √ X T (cid:3) = e − θ (cid:113) θ ϕ for every couple ( θ, ϕ ) satisfying condition (28) . The proof of Proposition 5.3 is postponed to the Appendix. We test the asymptoticformula (34) numerically under two parameter configurations, one with positive and onewith negative ATM skew; the results are shown in Figures 3 and 4. Formula (34) appears tobe very accurate even for values of θ corresponding to ATM implied volatilities of asin Figure 3 (in the right pane, relative errors remain below for θ = 1 ). Figure 4 displaysa situation that is more typical of equity indices. Overall, we deem that the accuracy of theasymptotic approximation in Proposition 5.3 is more than satisfactory in a wide range ofmarket conditions, including realized variance options or possibly stressed equity markets. θ → (dashedline) with the values of E (cid:2) √ X (cid:3) obtained by quadrature of the right hand side of (33), fordifferent values of θ . A Appendix Proof of (23) . For every T > , define the time–rescaled transformations f / ( T, k ) := √ T f / ( T, k ) = kσ BS ( T, k ) g / ( T, z ) := f / ( T, · ) − ( z ) = g / (cid:16) T, z √ T (cid:17) . Since the functions g / ( T, · ) are strictly increasing from R onto R for every T , it is sufficientto show that the convergence lim T → g / ( T, z ) = g / ( z ) holds pointwise. Thanks to Assump-tion (4.4) (i), we know that f / ( T, k ) tends to f / ( k ) := kσ BS (0 ,k ) uniformly over k in compactsets. We have to transfer the uniform convergence of f / ( T, · ) towards f / ( k ) := kσ BS (0 ,k ) ,granted by Assumption (4.4) (i), to pointwise convergence of the inverse functions g / ( T, · ) .This is a rather standard procedure: fix z ∈ R , and denote for simplicity k zT = g / ( T, z ) .First, it is not difficult to see that k zT remains in a compact set as T → . Then, since f / ( k zT ) = z + (cid:0) f / ( k zT ) − f / ( T, k zT ) (cid:1) −→ z as T → due to the uniform convergence on compact sets of f / ( T, · ) to f / ( · ) , using the continuityof f − / ( · ) we obtain k zT → f − / ( z ) = g / ( z ) as T → , which concludes the proof. Proof of Proposition 5.3 . The proof of Proposition 5.3 is essentially based on thefollowing Lemma. Lemma A.1. For every ρ ∈ ( − , and ϕ > , the following asymptotics holds I ( θ, ϕ, ρ ) ∼ √ A θ = θϕ (cid:113) ρ θ ϕ , as θ → . (35)17igure 4: Same functions as in Figure 3 (left: SSVI implied volatility; right: values of E (cid:2) √ X (cid:3) for different values of θ ), now for a SSVI parameterisation with negative ATM skew. Proof. It is not difficult to see that, for every δ > , (cid:82) | y |≥ δ e − A θ y d y = o ( e − A θ δ ) = o (cid:16) e − δ ϕ θ (cid:17) = o ( θ ) as θ → , so that (cid:90) | y | <δ e − A θ y d y √ π ∼ (cid:90) y ∈ R e − A θ y d y √ π = 1 √ A θ ∼ θϕ . (36)The proof of (35) implements the same idea for the integral I ( θ, ϕ, ρ ) . Let δ > and denote f θ ( y ) = e − ρ y √ y + θ , so that √ π I ( θ, ϕ, ρ ) = (cid:90) R e − A θ y f θ ( y )d y = (cid:90) | y | <δ e − A θ y f θ ( y )d y + (cid:90) | y |≥ δ e − A θ y f θ ( y )d y := I ( θ ) + I ( θ ) . (37)Assume without loss of generality θ < δ . Then, | y | ≥ δ implies (cid:112) y + θ ≤ √ | y | , hence f θ ( y ) ≤ e | ρ | √ y = e c y with c = | ρ | √ . Consequently, for θ small enough (precisely: suchthat A θ − c > ), we have I ( θ ) ≤ (cid:90) | y |≥ δ e − ( A θ − c ) y d y = e − ( A θ − c ) δ (cid:90) y ≥ δ yy e − ( A θ − c )( y − δ ) d y ≤ e − ( A θ − c ) δ δ ( A θ − c ) (cid:104) e − ( A θ − c )( y − δ ) (cid:105) y = δy = ∞ = e c δ δ ( A θ − c ) e − A θ δ = o θ → ( e − A θ δ ) = o θ → (cid:16) e − δ ϕ θ (cid:17) . (38)Let us now estimate I ( θ ) . For | y | < δ , we have f θ ( y ) ≤ e | ρ | δ √ δ + θ ≤ e | ρ | √ δ = e c δ ,hence | f θ ( y ) − | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) y f (cid:48) θ ( z )d z (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) y f θ ( z ) dd z (cid:16) − ρ z √ z + θ (cid:17) d z (cid:12)(cid:12)(cid:12)(cid:12) ≤ e c δ | ρ | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) max( y, y, dd z ( z √ z + θ )d z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e c δ | ρ | δ √ δ + θ ≤ e c δ c δ 18t follows that − e c δ c δ ≤ f θ ( y ) ≤ e c δ c δ for every y with | y | < δ , therefore (1 − e c δ c δ ) (cid:90) | y | <δ e − A θ y d y ≤ I ( θ ) ≤ (1 + e c δ c δ ) (cid:90) | y | <δ e − A θ y d y , hence, in light of (36), (1 − e c δ c δ ) ≤ lim inf θ → I ( θ ) (cid:82) y ∈ R e − A θ y d y ≤ lim sup θ → I ( θ ) (cid:82) y ∈ R e − A θ y d y ≤ (1 + e c δ c δ ) . 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