Moment generating functions and Normalized implied volatilities: unification and extension via Fukasawa's pricing formula
MMoment generating functions and Normalizedimplied volatilities: unification and extension viaFukasawa’s pricing formula
Stefano De Marco , Claude Martini Abstract
We extend the model-free formula of [Fukasawa 2012] for E [Ψ( X T )], where X T = log S T /F is the log-price of an asset, to functions Ψ of exponential growth. The resulting integral representation is written interms of normalized implied volatilities. Just as Fukasawa’s work provides rigourous ground for Chriss andMorokoff’s (1999) model-free formula for the log-contract (related to the Variance swap implied variance),we prove an expression for the moment generating function E [ e pX T ] on its analyticity domain, thatencompasses (and extends) Matytsin’s formula [Matytsin 2000] for the characteristic function E [ e iηX T ]and Bergomi’s formula [Bergomi 2016] for E [ e pX T ], p ∈ [0 , d ( p, · ) = p d + (1 − p ) d when p lies outside [0 , Ecole Polytechnique, CMAP, Universit´e Paris-Saclay, France. [email protected] Zeliade Systems, 56 rue Jean-Jacques Rousseau, Paris, France.
We thank Masaaki Fukasawa and Jim Gatheral for stimulating discussions. We are much indebted with M. Fukasawa for raisingour interest in this topic and for profound insights.This PDF was obtained from an export from
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126 The invertibility of k (cid:55)→ f ( p, k ) f ( p, · ) is surjective for every p in an interval larger than ( − p − ( β − ) , p + ( β + )) . . . . . . . . . . 136.2 Some results on the monotonicity of f ( p, · ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 θϕ (1 + | ρ | ) = 4 and checking Assumption 2.1 (ii) on v . . . . . . . . . . . . 177.3 Checking Assumption 2.1 (i) on v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.4 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 A.1 Proof of Theorem 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Introduction
We consider an asset price S T at some fixed date T >
0. We denote P the pricing (T-forward) measure, sothat the price of a call option with maturity T and strike K is given by B (0 , T ) E P [( S T − K ) + ], where B (0 , T )denotes the current price of the zero-coupon bond with maturity T . We work with dimensionless quantities:we denote k = log( K/F ) the forward log-strike, where F = E P [ S T ] denotes the forward price for maturity T ,and v ( k ) = √ T σ BS ( T, k ) the dimensionless (or “total”) implied volatility. Recall that v ( k ) is defined for all k ∈ R by the equation Call BS ( k, v ( k )) = E (cid:2)(cid:0) S T F − e k (cid:1) + (cid:3) , where Call BS ( k, v ) = N ( d ( k, v )) − e k N ( d ( k, v )), N ( · ) is the standard normal cdf, and d i ( k, v ) = − kv + ( − i ) v .It is well-known that any C (or convex) payoff ϕ ( S T ) with linear growth can be statically replicated with astrip of call and put options, so that its price can be written via Carr–Madan’s formula B (0 , T ) E [ ϕ ( S T )] = B (0 , T ) ϕ ( F ) + (cid:82) F ϕ (cid:48)(cid:48) ( K ) P ( K ) dK + (cid:82) ∞ F ϕ (cid:48)(cid:48) ( K ) C ( K ) dK , where P ( K ) and C ( K ) denote put and call pricesfor the maturity T , see e.g. [7, Eq (11.1)]. On the other hand, if the law of S T under P is absolutely continuouswith respect to the Lebesgue measure on [0 , ∞ ), using the well-known Breeden-Litzenberger relations, onecan write E [ ϕ ( S T )] = (cid:90) ∞ ϕ ( K ) d dK E [( S T − K ) + ] dK = F (cid:90) R ϕ ( K ) d dK Call BS ( k, v ( k )) | k =log( K/F ) dK. (1)Applying the chain rule to the rightmost integrand in (1) leads to an integral formula containing Black-ScholesGreeks with respect to strike and volatility, and the derivatives of the implied volatility smile v ( · ) up to ordertwo. A stream of literature [2,6,7,12] studies the possibility of re-expressing Equation (1) in such a way thatthe derivatives of the implied volatility do not appear any more on the right hand side. This is a relevantfeature in practice, because observed market data is (in any case) discrete. Such investigations requiredthe introduction of the concept of normalizing transformation of the implied volatility smile, introduced byChriss and Morokoff [3] and Matytsin [12] and formalized in the seminal work of Fukasawa [6], that we recallbelow.One of the most important examples in this field is the following formula for the implied variance of the logcontract , see Chriss and Morokoff [3] or Gatheral [7]: E P (cid:20) − T log (cid:18) S T F (cid:19)(cid:21) = 1 T (cid:90) R v ( g ( z )) φ ( z ) dz. (2)In (2), φ is the standard normal density, and g : R → R is the inverse of the function (called secondnormalizing transformation ) f ( k ) := − d ( k, v ( k )) = kv ( k ) + v ( k )2 . Similarly, the first normalizing transformation (used later on) is given by f ( k ) := − d ( k, v ( k )) = kv ( k ) − v ( k )2 .Apart from its appealing compactness, the formula (2) is amenable for numerical approximations, notably inview of the use of Gauss-Hermite quadrature – see the discussions in [6, Remark 4.9] and Bergomi [2, p. 143].Other examples of similar formulas include: other derivatives such as the S ln S contract (related to theGamma Swap, see again [6] and Section 3 of this paper), and a formula for the characteristic function of X T = log( S T /F ) due to Matytsin [12], see below. The important property that the map f : R → R is which coincides with the fair strike of the Variance Swap, under the assumption that ( S t ) t ≤ T follows a diffusion process.When T = 30 days and S is the S&P500 stock index, the left hand side of (2) defines the (theoretical value) of VIX at t = 0. May 4, 2017 ctually invertible for any arbitrage-free implied volatility v ( · ), implicitly assumed in the aforementionedworks, was first proven by Fukasawa [6]. Matytsin’s formula for characteristic functions and Bergomi’s formula for E [( S T /F ) p ].Denote v ( z ) = v ( g ( z )). Assuming that v is differentiable, Matytsin [12] gives the following formula forthe characteristic function of X T E (cid:2) e iηX T (cid:3) = (cid:90) R e − iηv ( z ) ( v ( z ) − z ) (1 − iηv (cid:48) ( z )) φ ( z ) dz, η ∈ R . (3)The proof of (3) is (only) sketched in Matytsin’s slides. Building on this work, Bergomi [2, Section 4.3.1]derives a formula for the moments of S T /F of order p ∈ [0 , E (cid:20)(cid:18) S T F (cid:19) p (cid:21) = (cid:90) R e p ( p − v p ( z ) φ ( z ) dz, p ∈ [0 , , (4)where the “ p -normalized” implied volatility v p ( · ) is defined in the following way: consider the convex inter-polation f ( p, k ) = pf ( k ) + (1 − p ) f ( k ) = kv ( k ) + (cid:18) − p (cid:19) v ( k )2of the two normalizing transformations f and f . We know from Fukasawa [6] that the two maps k (cid:55)→ f ( k )and k (cid:55)→ f ( k ) are strictly increasing from R to R : therefore, so is k (cid:55)→ f ( p, k ), for every p ∈ [0 , g ( p, · ) be the inverse of f ( p, · ) on R : v p ( · ) is defined by v p ( z ) = v ( g ( p, z )) , for all z ∈ R (5)(hence, with reference to Fukasawa’s notation, we have v = v , v = v ). Note we have the followingnice interpretation of (4): in the Black-Scholes model, where S T = F e σW T − σ T is a geometric Brownianmotion with constant volatility parameter σ = v √ T , one has E (cid:2)(cid:0) S T F (cid:1) p (cid:3) = e p ( p − v = (cid:82) R e p ( p − v φ ( z ) dz .Therefore, we can see Equation (4) as an extension of the pricing formula for power payoffs, from theBlack-Scholes world to models with non-constant implied volatility.The formulas (3) and (4) are the starting point of this work. As mentioned above, Bergomi [2] derives(4) from (3). Here, we will follow a different route: our starting point is the work of Fukasawa. We firstextend the formula for expectations of functions of X T with polynomial growth given in [6, Theorem 4.6]to exponential functions – carrying out in details the plan addressed in [6, Remark 4.8]. This providesa formula for the generalized characteristic function p ∈ C (cid:55)→ E [ e pX T ] on its analyticity domain, writtendirectly in terms of the implied volatility smile. Matytsin’s (3) and Bergomi’s (4) formulas are embedded inthis representation as special cases (along with a dual version of the first, and an extension to the complexplane of the second). By taking real values of p , this formula allows to numerically evaluate the (finite)risk-neutral moments of the underlying asset price from the market smile – therefore identifying model-freequantities that can be used as targets in the calibration of a parametric model.As addressed in [6, Remark 4.8], it is natural, when evaluating expectations of the form E (cid:2)(cid:0) S T F (cid:1) p (cid:3) , to exploitLee’s moment formulas [10] relating the critical moments of S T to the asymptotic slopes of the impliedvolatility for large and small strikes. We stress that our approach here goes the other way round: we provean integral representation for E (cid:2)(cid:0) S T F (cid:1) p (cid:3) without making use of Lee’s result. Then, as a by-product, we candeduce sharp bounds on the exponents that appear in the moment formulas. May 4, 2017 he example of the SSVI parameterisation . Gatheral and Jacquier [8] propose the following parame-terisation for total implied variance (the square of the dimensionless implied volatility v ): v ( k ) = θ T (cid:16) ρϕ ( θ T ) k + (cid:112) ( ϕ ( θ T ) k + ρ ) + 1 − ρ (cid:17) , (6)where θ T > ϕ : (0 , ∞ ) → R + and ρ ∈ ( − , k and every maturity T . In the present setting, we are interestedin parameterisations of a single arbitrage-free smile for a fixed maturity T : for simplicity, we will thereforedrop the index T from the notation, and denote θ = θ T and ϕ = ϕ ( θ ). Important no-arbitrage properties ofthe SSVI model will be recalled in Section 7. Our standing assumptions on the implied volatility are the following:
Assumption 2.1. (i) v is twice differentiable on R and v ( k ) > for all k ∈ R .(ii) lim k →−∞ d ( k, v ( k )) = lim k →−∞ (cid:16) − kv ( k ) − v ( k )2 (cid:17) = + ∞ Denote µ the law of S T /F under P . It is classical that the second differentiability of v ( · ) is equivalent tothe existence of a density with respect to Lebesgue measure for the restriction of µ to (0 , ∞ ). Moreover,the strict positivity of v ( · ) is equivalent to the two conditions inf { supp( µ ) } = 0 and sup { supp( µ ) } = ∞ ,see [13]. In its turn, Assumption 2.1 (ii) is equivalent to µ ( { } ) = P ( S T = 0) = 0. (In general, we havelim k →−∞ d ( k, v ( k )) = − N − ( P ( S T = 0)), see [14], [5].) Remark 2.2.
Recall that f ( k ) ≥ √ k for every k > from the arithmetic mean-geometric mean inequality,therefore we always have lim k →∞ f ( k ) = ∞ . Analogously, f ( k ) = −| k | v ( k ) − v ( k )2 ≤ − (cid:112) | k | for every k < ,hence lim k →−∞ f ( k ) = −∞ . The limit of f as k → + ∞ is related to the arbitrage freeness of v ( k ) : indeed, lim k → + ∞ f ( k ) = + ∞ is equivalent to the no-arbitrage condition lim k → + ∞ Call BS ( k, v ( k )) = 0 . Therefore. f always maps R onto R . Assumption 2.1 (ii) ensures that we have lim k →−∞ f ( k ) = −∞ , so that f issurjective, too. Fukasawa [6] proves the following result:
Theorem 2.3 (Theorem 4.6 in [6]) . Let Ψ be an absolutely continuous function with derivative Ψ (cid:48) of poly-nomial growth, and assume that there exists q > such that E [ S − qT ] < ∞ . Then, E (cid:20) Ψ (cid:18) log S T F (cid:19)(cid:21) = (cid:90) + ∞−∞ (cid:2) Ψ( g ( z )) − Ψ (cid:48) ( g ( z )) + Ψ (cid:48) ( g ( z )) e − g ( z ) (cid:3) φ ( z ) dz. (7) Analogously, if there exists q > such that E [ S qT ] < ∞ , then E (cid:2) S T F Ψ (cid:0) log S T F (cid:1)(cid:3) = (cid:82) + ∞−∞ (cid:2) Ψ( g ( z )) +Ψ (cid:48) ( g ( z )) − Ψ (cid:48) ( g ( z )) e g ( z ) (cid:3) φ ( z ) dz . Equation (7) can be proven starting from (1) and then applying judicious integration by parts, and change ofvariables by means of the transformations f and f . Important steps in this proof, see [6], are (i) ensuring theintegrability of the involved integrands, and (ii) checking that the boundary terms arising from integration May 4, 2017 y parts do give zero contribution (point (ii) amounts to showing that lim k →±∞ Ψ( k ) v (cid:48) ( k ) φ ( f ( k )) = 0, seeLemma A.2 in the Appendix).Setting Ψ( k ) = e pk in (7) would give us a formula for the moment of S T /F of order p . Unless p = 0, sucha function Ψ falls outside the class of functions covered by Theorem 2.3. We therefore proceed to extendTheorem 2.3 to this setting. Define the two functions p + ( β ) = 12 (cid:18) β + β (cid:19) , p − ( β ) = 12 (cid:18) β + β − (cid:19) , β ∈ (0 , , (8)and set p ± ( β ) = + ∞ if β = 0. It is easy to see that p + ( β ) ≥ p − ( β ) ≥
0, for all β ∈ [0 , p ± ( · ) arise from certain integrability conditions of the integrand in (7)when Ψ( k ) = e pk , as we will explain more precisely below. Now, if we denote p ∗ = sup { p > E [( S T ) p ] < ∞} , q ∗ = sup { q > E [( S T ) − q ] < ∞} , (9)the right, resp. left, critical exponent of S T (note we define q ∗ to be positive), then Roger Lee’s momentformula [10] states that: p ∗ = p + ( β + ) , q ∗ = p − ( β − ) (10)where (cid:112) β + = lim sup k →∞ v ( k ) √ k , (cid:112) β − = lim sup k →−∞ v ( k ) (cid:112) | k | . (11)As pointed out in the introduction, in the present work we do not make use of Lee’s result in order toextend Equation (7) to functions Ψ with exponential growth. In other words, we do not use the information p ∗ = p + ( β + ) or q ∗ = p − ( β − ) at any point. Example 2.4.
For the SSVI parameterisation (6) we have β ± (SSVI) = θϕ (1 ± ρ )2 . (12) As we will recall in section 7.1 below, a necessary condition for no arbitrage is θϕ (1 ± ρ ) ≤ , so that ≤ β ± (SSVI) ≤ . The quantitative link between (7) and (10) can be highlighted with some simple calculations: assume(only within this paragraph) that the first equation in (11) hold as a limit, and that β + / ∈ { , } . Then, fromthe definition of f and f , f ( k ) ∼ (cid:32) (cid:112) β + + (cid:112) β + (cid:33) √ k, f ( k ) ∼ (cid:32) (cid:112) β + − (cid:112) β + (cid:33) √ k as k → ∞ . (13)Using the definition f i ( g i ( z )) = z of the maps g i , it is easy to see that Eq (13) implies a quadratic behavior May 4, 2017 f g and g for large z : g ( z ) ∼ z (cid:16) √ β + + √ β + (cid:17) = z p + ( β + ) , g ( z ) ∼ z (cid:16) √ β + − √ β + (cid:17) = z p − ( β + ) as z → ∞ . (14)Now, a formal application of (7) to Ψ( k ) = e pk leads to E (cid:2)(cid:0) S T F (cid:1) p (cid:3) = (cid:82) R (cid:2) pe ( p − g ( z ) + (1 − p ) e pg ( z ) (cid:3) φ ( z ) dz .Applying (14), a straightforward calculation yields l ( z ) := e ( p − g ( z ) φ ( z ) = exp (cid:16) p − p + ( β + ) p − ( β + ) z (1 + o (1)) (cid:17) as z → ∞ ,l ( z ) := e pg ( z ) φ ( z ) = exp (cid:16) p − p + ( β + ) p + ( β + ) z (1 + o (1)) (cid:17) as z → ∞ . The last estimates above show that p = p + ( β + ) is precisely the threshold between Gaussian integrability(when p < p + ( β + )) or, on the contrary, exploding behavior (when p > p + ( β + )) for the two functions l and l inside the integral (for positive z ). The analogous argument allows to show that p = − p − ( β − ) isthe threshold between integrability and explosion of the integrand for negative z . Some more work (whichwe perform in the proof of Theorem 2.7 below) is required to study the asymptotic behavior of the linearcombination p l ( · ) + (1 − p ) l ( · ), and to deal with the remaining cases β ± ∈ { , } . Definition 2.5.
We say that a function
Ψ : R → R has exponential growth of order p for some p ∈ R if thefunction k (cid:55)→ e − pk Ψ( k ) is bounded. Lemma 2.6.
Let p ∈ R such that − p − ( β − ) < p < p + ( β + ) where p ± ( · ) are defined in (8) , and let Ψ be an absolutely continuous function such that Ψ and Ψ (cid:48) haveexponential growth of order p . Then, the functions z (cid:55)→ Ψ( g ( z )) φ ( z ) , z (cid:55)→ Ψ (cid:48) ( g ( z )) φ ( z ) , and z (cid:55)→ Ψ (cid:48) ( g ( z )) e − g ( z ) φ ( z ) (15) are integrable on R . With a view on Lemma 2.6, note that the bound f ( k ) ≥ √ k for k ≥ f ( k ) ≤ − (cid:112) | k | for k ≤ g ( z ) ≤ z z large enough (resp. g ( z ) ≥ − z for z small enough). (16)Since g ( z ) is eventually positive for large z , the estimate (16) follows from z = f ( g ( z )) ≥ (cid:112) g ( z ) (resp. z = f ( g ( z )) ≤ − (cid:112) | g ( z ) | for z sufficiently small). Therefore, for every p ∈ (0 , pg ( z ) → z →−∞ −∞ , e pg ( z ) φ ( z ) ≤ e ( p − z for z large enough e ( p − g ( z ) φ ( z ) ≤ e − p z for z small enough , ( p − g ( z ) → z →∞ −∞ . (17)For every function F ( · ) with exponential growth of order p , we have | F ( g ( z )) φ ( z ) | ≤ const. × e pg ( z ) φ ( z )and | F ( g ( z )) e − g ( z ) φ ( z ) | ≤ const. × e ( p − g ( z ) φ ( z ). Consequently, the estimates in (17) show that the May 4, 2017 unctions in (15) are always integrable for every p ∈ (0 ,
1) = ( p − (2) , p + (2)) and for every arbitrage-free smile v (regardless of the values of β ± ). Theorem 2.7.
Let p ∈ R such that − p − ( β − ) < p < p + ( β + ) (18) where p ± ( · ) are defined in (8) and β ± in (11) . Let Ψ be an absolutely continuous function such that Ψ and Ψ (cid:48) have exponential growth of order p . Then, (cid:90) + ∞−∞ (cid:2) Ψ( g ( z )) − Ψ (cid:48) ( g ( z )) + Ψ (cid:48) ( g ( z )) e − g ( z ) (cid:3) φ ( z ) dz = E (cid:20) Ψ (cid:18) log S T F (cid:19)(cid:21) . (19) In particular, for all − p − ( β − ) < p < p + ( β + ) , L ( p ) := (cid:90) + ∞−∞ (cid:2) pe ( p − g ( z ) + (1 − p ) e pg ( z ) (cid:3) φ ( z ) dz = E (cid:20)(cid:18) S T F (cid:19) p (cid:21) =: M ( p ) < ∞ . (20)The proofs of Lemma 2.6 and Theorem 2.7 are given in Appendix A. The formula (20) for the moments of S T was mentioned in the introduction of [6], the extension of Thm 2.3 to functions with exponential growthbeing implicitly assumed therein.We stress once again that our proof of Theorem 2.7 does not make use of Lee’s result [10]. As animmediate consequence of Theorem 2.7, we have the following bounds: Corollary 2.8.
Let p + ( β + ) , p − ( β − ) be defined by (8) and p ∗ , q ∗ by (9) . Then p ∗ ≥ p + ( β + ) and q ∗ ≥ p − ( β − ) . In particular, p ∗ = p + ( β + ) if p + ( β + ) = ∞ and q ∗ = p − ( β − ) if p − ( β − ) = ∞ .Proof. From Equation (20), we have M ( p ) < ∞ for all p ∈ ( − p − ( β − ) , p + ( β + )). The claim then follows fromthe definition of p ∗ and q ∗ . Remark 2.9.
Our proof of Theorem 2.7 in Appendix A is obtained essentially by rerunning the explicitcomputations linking the LHS to the RHS in (19) , showing that i) all the required integrability conditionsare met, and ii) that integration by parts produce zero boundary terms. Another approach to the proof of (20) would go as follows: apply Fukasawa’s result (Theorem 2.3 above) to the functions Ψ n ( x ) = (cid:0) pxn (cid:1) n ,which have polynomial growth for every n . Under the assumption q ∗ > , this yields E [Ψ n ( X T )] = (cid:82) (cid:2)(cid:0) pg ( z ) n (cid:1) n − p (cid:0) pg ( z ) n (cid:1) n − + p (cid:0) pg ( z ) n (cid:1) n − e − g ( z ) (cid:3) φ ( z ) dz . By monotone convergence, the left hand side E [Ψ n ( X T )] converges to E [ e pX T ] for every p ∈ ( − q ∗ , p ∗ ) . On the other hand, if p ∈ ( − p − ( β − ) , p + ( β + )) theRHS converges to (cid:82) (cid:2) pe ( p − g ( z ) +(1 − p ) e pg ( z ) (cid:3) φ ( z ) dz by dominated convergence, thanks to Lemma 2.6. Thisargument proves Eq (20) for every p such that max( − p − ( β − ) , − q ∗ ) < p < min( p + ( β + ) , p ∗ ) ; some additionalwork then allows to prove the inequalities p ∗ ≥ p + ( β + ) and q ∗ ≥ q − ( β − ) (we use a similar argument inAppendix A), and the claimed result then follows for every p in (18) .Following our (alternative) proof in Appendix A, we get rid of the limitation q ∗ > (or p ∗ > ) from Theorem4.6 in [6]. This is related to the growth condition we consider on the function Ψ , which is “one-sided”: while,in Definition 2.5, Ψ is allowed to grow faster than a polynome for large arguments (if, say, p > ), on theother side Ψ( x ) has to go to zero for x → −∞ (as opposed to the polynomial growth condition in [6, Theorem4.6], which allows Ψ to diverge on both sides). May 4, 2017 emark 2.10 (About the proof of the converse inequalities for p ∗ , p + and q ∗ , p − , using Theorem 2.7) . Thefunction p (cid:55)→ M ( p ) = E (cid:2)(cid:0) S T F (cid:1) p (cid:3) in Theorem 2.7 is the bilateral Laplace transform of a probability measureon R (the law of X T ). Its positive and negative abscissas of convergence are, respectively, p ∗ and − q ∗ definedin (9) ; moreover, M has a unique holomorphic extension to the strip D ∗ = { p ∈ C : − q ∗ < Re( p ) < p ∗ } .Assume p ∗ > p + ( β + ) : then, M would be a holomorphic extension of the function p (cid:55)→ L ( p ) , defined on ( − p − ( β − ) , p + ( β + )) , to D ∗ . If we prove that this is not possible – that is, that L cannot be extended analyticallyto any neighbourhood of p + ( β + ) – then we would have shown by contradiction that p ∗ ≤ p + ( β + ) , therefore p ∗ = p + ( β + ) by Lemma 2.8. The symmetric argument of course applies to the inequality q ∗ ≤ p − ( β − ) .It is not difficult to prove, using similar arguments to the proof of Theorem 2.7, that the function L (asdefined by the left hand side of Eq (20) ) is infinite for every p / ∈ [ − p − ( β − ) , p + ( β + )] if the implied volatilityslope has a limit. More precisely, we can show: • Assume that lim k →∞ v ( k ) √ k exists. Then, L ( p ) = + ∞ for every p > p + ( β + ) . • Assume that lim k →−∞ v ( k ) √ | k | exists. Then, L ( p ) = + ∞ for every p < − p − ( β − ) .Note we are not (yet) showing here that L does not admit a holomorphic extension to a neighbourhood of p + ( β + ) or − p − ( β − ) . Let us just observe, for the moment, that the proof of this statement (that we leave forfuture work) might not be so straightforward. We can see L as a linear combination of Laplace transformsof positive functions, L ( p ) = p L ( p ) + (1 − p ) L ( p ) . Since the coefficients p and − p have opposite signs as soon as p > or p < , even if each Laplace transform L i cannot be extended analytically above p + ( β + ) (or below − p − ( β − ) ), the behavior of their linear combination is more subtle to study. We can extend Theorem 2.7 to the following
Proposition 2.11.
The identity E (cid:2) e pX T (cid:3) = (cid:90) + ∞−∞ (cid:2) pe g ( z )( p − + (1 − p ) e pg ( z ) (cid:3) φ ( z ) dz (21) holds for all p ∈ C with − p − ( β − ) < Re( p ) < p + ( β + ) .Proof. By Lemma 2.6, Equation (20) defines a holomorphic function L on D = { p ∈ C : − p − ( β − ) < Re( p )
0. Then, the distributions of M under P and M under ˆ P are the same. The equality between the outermost terms in the above chain of equalities rewrites: C BS ( K, F, V ) =
KF P BS (cid:16) F K , F, V (cid:17) (23)where C BS ( K, F, V ) (resp. P BS ( K, F, V )) denotes the price of the Black-Scholes call (resp. put) option withstrike K , forward value F , and total implied volatility V = σ √ T . Now, by the definition of the impliedvolatility we have E ˆ P [( ˆ S − K ) + ] = C BS ( K, F, ˆ V ( K )) and E P (cid:2)(cid:0) F K − S (cid:1) + (cid:3) = P BS (cid:0) F K , F, V (cid:0) F K (cid:1)(cid:1) . Applying(22) and (23), we get C BS ( K, F, ˆ V ( K )) = C BS (cid:16) K, F, V (cid:16) F K (cid:17)(cid:17) , therefore ˆ V ( K ) = V (cid:0) F K (cid:1) for all K >
0. In terms of v ( k ) = V ( F e k ) (resp. ˆ v ( k ) = ˆ V ( F e k )), this readsˆ v ( k ) = v ( − k ) , for all k ∈ R . (24)We immediately have the following: Lemma 3.1.
Let f , f (resp. ˆ f , ˆ f ) the first and second normalizing transformations associated to S (resp. to ˆ S = F S ). Let f ( p, k ) = pf ( k ) + (1 − p ) f ( k ) , p ∈ R , and g ( p, · ) the inverse function of f ( p, · ) for p ∈ [0 , (and likewise for ˆ f ( p, · ) and ˆ g ( p, · ) ). Then ˆ f ( p, k ) = − f (1 − p, − k ) for all k ∈ R , p ∈ R . and ˆ g ( p, z ) = − g (1 − p, − z ) for all z ∈ R , p ∈ [0 , . In particular ˆ f ( k ) = − f ( − k ) , ˆ g ( z ) = − g ( − z )ˆ f ( k ) = − f ( − k ) , ˆ g ( z ) = − g ( − z ) . (25) Proof.
The first bullet point follows immediately from (24) and the definition of the normalizing transforma-tions: ˆ f ( p, k ) = k ˆ v ( k ) + (cid:0) − p (cid:1) ˆ v ( k )2 = − − kv ( − k ) − (cid:0) − (1 − p ) (cid:1) v ( − k )2 = − f (1 − p, − k ), and the claim follows.The second bullet point is a direct consequence of the first: ˆ f ( p, ˆ g ( p, z )) = z rewrites − f (1 − p, − ˆ g ( p, z )) = z ,therefore − ˆ g ( p, z ) = g (1 − p, − z ). Equations (25) are special cases for p = 1 and p = 0 (recall that, with ournotation, f (0 , · ) = f ( · )). Proposition 3.2.
Let v p (resp. ˆ v p ) the p -normalized implied volatility associated to S (resp. ˆ S ), as definedin (5) . Then, ˆ v p ( z ) = v − p ( − z ) , for all z ∈ R . (26) In particular, ˆ v ( z ) = v ( − z ) and ˆ v ( z ) = v ( − z ) . May 4, 2017 roof.
Applying (24) and Lemma 3.1, we have ˆ v p ( z ) = ˆ v (ˆ g ( p, z )) = v ( − ˆ g ( p, z ))) = v ( g (1 − p, − z ))) = v − p ( − z ). The last claim is a special case of (26), recalling that, with our notation, v = v , v = v .As an example, using formula (2) for the price of the log contract and Proposition 3.2, we obtain the followingformula for the implied variance of the S ln S contract (corresponding to the de-annualized fair strike of theGamma Variance Swap in a diffusion model):2 E P (cid:20) S T F log (cid:18) S T F (cid:19)(cid:21) = 2 E ˆ P (cid:34) − log (cid:18) ˆ S T F (cid:19)(cid:35) = (cid:90) R ˆ v ( z ) φ ( z ) dz = (cid:90) R v ( − z ) φ ( z ) dz = (cid:90) R v ( z ) φ ( z ) dz, which is formula (1.2) in [6]. Remark 3.3.
As another application of the duality (25) , we can deduce the invertibility of the first normal-izing transformation from the invertibility of the second (or vice-versa). Precisely, we have the following: themap f (resp. f ) is strictly increasing for every arbitrage-free smile v ( · ) if and only if f (resp. f ) is such. Remark 3.4.
The formula (20) for the moments of S T is invariant under the duality transformations p → − p , g ( z ) → − g ( − z ) and g ( z ) → − g ( − z ) . This is consistent with Lemma 3.1: indeed, note that M ( p ) = E P (cid:20)(cid:18) S T F (cid:19) p (cid:21) = E P (cid:34) S T F (cid:18) S T F (cid:19) p − (cid:35) = E ˆ P (cid:34)(cid:18) ˆ S T F (cid:19) − p (cid:35) =: ˆ M (1 − p ) . Equation (20) yields, for every p ∈ ( − p − ( β − ) , p + ( β + )) , ˆ M (1 − p ) = ˆ L (1 − p ) := (cid:90) R (cid:104) (1 − p ) e − p ˆ g ( z ) + pe (1 − p )ˆ g ( z ) (cid:105) φ ( z ) dz. Applying the identities ˆ g ( z ) = − g ( − z ) and ˆ g ( z ) = − g ( − z ) in Lemma 3.1, it is immediate to check thatthe rightmost term in the above equation coincides with L ( p ) , therefore with M ( p ) as expected. The formula (21) is written in terms of the two normalizing transformations g and g . We can recast it intoan equivalent formula containing only the normalized implied volatility v ( · ) and its derivative (or yet, a dualformula in terms of v ( · )). The original formula (3) of Matytsin [12] is a special case of this representation.Recall the definition v = v ◦ g and the following relation between v and g : by definition, we have g ( z ) = k ⇔ f ( k ) = z ⇔ kv ( k ) + v ( k )2 = z ⇔ k = zv ( k ) − v ( k ) , which yields g ( z ) = zv ( z ) − v ( z ) , for all z ∈ R . (27)Analogously, g ( z ) = zv ( z ) + v ( z ) , for all z ∈ R . (28) Proposition 4.1.
For every p ∈ C with Re( p ) ∈ ( − p − ( β − ) , p + ( β + )) , E (cid:2) e pX T (cid:3) = (cid:90) R e p ( zv ( z ) − v ( z ) )[1 − pv (cid:48) ( z )] φ ( z ) dz. (29) May 4, 2017 roof.
Starting from (21), we have E (cid:20)(cid:18) S T F (cid:19) p (cid:21) = (cid:90) + ∞−∞ [ pe ( p − g ( z ) + (1 − p ) e pg ( z ) ] φ ( z ) dz = (cid:90) + ∞−∞ (cid:110) pe ( p − k φ ( f ( k )) f (cid:48) ( k ) + (1 − p ) e pk φ ( f ( z )) f (cid:48) ( k ) (cid:111) dk = (cid:90) + ∞−∞ e pk φ ( f ( k ))[ pf (cid:48) ( k ) + (1 − p ) f (cid:48) ( k )] dk (30)where we used the identity e − k φ ( f ( k )) = φ ( f ( k )). By definition of f and f , we have f (cid:48) ( k ) = f (cid:48) ( k ) − v (cid:48) ( k ),which yields e pk φ ( f ( k )) [ pf (cid:48) ( k ) + (1 − p ) f (cid:48) ( k )] = e pk φ ( f ( k ))[ f (cid:48) ( k ) − pv (cid:48) ( k )] = e pk φ ( f ( k ))[1 − pv (cid:48) ( f ( k ))] f (cid:48) ( k ) (31)for ddk v ( k ) = ddk v ( f ( k )) = v (cid:48) ( f ( k )) f (cid:48) ( k ). Plugging (31) into (30), we obtain E (cid:20)(cid:18) S T F (cid:19) p (cid:21) = (cid:90) + ∞−∞ e pk φ ( f ( k ))[1 − pv (cid:48) ( f ( k ))] f (cid:48) ( k ) dk = (cid:90) + ∞−∞ e pg ( z ) φ ( z )[1 − pv (cid:48) ( z )] dz from which the claim follows using (27).We have the following dual formula: Proposition 4.2.
For every p ∈ C with Re( p ) ∈ ( − p − ( β − ) , p + ( β + )) , E (cid:2) e pX T (cid:3) = (cid:90) R e ( p − ( zv ( z )+ v ( z ) )[1 + (1 − p ) v (cid:48) ( z )] φ ( z ) dz. (32) Proof.
Follows by mimicking the proof of Prop 4.1, now applying the identities f (cid:48) ( k ) = f (cid:48) ( k ) + v (cid:48) ( k ) and(28). Remark 4.3.
The two formulas (29) and (32) are the dual of each other: that is, they are related via thetransformations p ↔ − p , v ( z ) ↔ v ( − z ) . In fact, instead of proving Equation (32) , we could infer itfrom (29) , applying the duality results of Section 3. This goes as follows: use Equation (29) to compute ˆ M (1 − p ) = E ˆ P (cid:2)(cid:0) ˆ S T F (cid:1) − p (cid:3) in terms of ˆ v ( · ) . Applying the identities ˆ M (1 − p ) = M ( p ) from Remark 3.4 and ˆ v ( z ) = v ( − z ) from Proposition 3.2, we obtain (32) . Corollary 4.4.
Taking p = iη for η ∈ R in (29) , we obtain Matytsin’s formula [12] E (cid:2) e iηX T (cid:3) = (cid:90) e iη ( zv ( z ) − v ( z ) )[1 − iηv (cid:48) ( z )] φ ( z ) dz. Taking p = iη for η ∈ R in (32) , we obtain the dual formula E (cid:2) e iηX T (cid:3) = (cid:90) e ( iη − ( zv ( z )+ v ( z ) )[1 + (1 − iη ) v (cid:48) ( z )] φ ( z ) dz. As an exercise, we can reobtain Chriss and Morokoff’s formula [3] for E [ X T ] = E (cid:2) log S T F (cid:3) by computing i ddη E (cid:2) e iηX T (cid:3)(cid:12)(cid:12) η =0 . May 4, 2017
Extension of Bergomi’s formula to p ∈ C , Re( p ) ∈ [0 , In this section, we follow Bergomi’s idea [2] of interpolating the two transformations f and f . When p is real-valued, this procedure allows to turn Equation (20) into (yet) another formula for E [( S T /F ) p ], nowwritten in terms of the p -normalized implied volatility v p ( · ) : the result is the compact and elegant formula(4) (which is also self-dual, cf. Remark 5.3 below). Starting from Proposition 2.11, we can now extendBergomi’s formula to the complex plane.Note that we can extend the definition of the function f ( p, · ) to p ∈ C in the obvious way, setting f ( p, k ) = pf ( k ) + (1 − p ) f ( k ) for every k ∈ R . This gives f ( p, k ) = f (Re( p ) , k ) + i Im( p )( f ( k ) − f ( k )) = f (Re( p ) , k ) − i Im( p ) v ( k )= f (Re( p ) , k ) − i Im( p ) v Re( p ) ( f (Re( p ) , k )) , (33)so that the map f ( p, · ) is one-to-one from R onto the following curve in the complex plane: γ p : R → C , γ p : a (cid:55)→ a − i Im( p ) v Re( p ) ( a ) . (34)Consequently, for every p with Re( p ) ∈ [0 , f ( p, · ) from γ p to R by g ( p, · ) : z ∈ γ p (cid:55)→ f ( p, · ) − ( z ) = g (Re( p ) , Re( z )) , where, with a slight abuse of notation, we still denote γ p the support { γ p ( a ) : a ∈ R } of the curve. Finally,we can extend the definition of the p -normalized implied volatility v p to complex p :for every z ∈ γ p , v p ( z ) := v ( g ( p, z ))so that v p ( z ) = v Re( p ) (Re( z )). Theorem 5.1.
For every p in the strip { Re( p ) ∈ [0 , } , E (cid:2) e pX T (cid:3) = (cid:90) γ p e i Im( p ) g ( p,z ) e Re( p )(Re( p ) − v p ( z ) φ (Re( z )) dz, (35) where g ( p, z ) = zv p ( z ) − (cid:0) − Re( p ) (cid:1) v p ( z ) and the curve γ p is defined in (34) .Proof. In the proof of Prop. 4.1, we have shown that E (cid:104)(cid:0) S T F (cid:1) p (cid:105) = (cid:82) + ∞−∞ e pk φ ( f ( k ))[ pf (cid:48) ( k ) + (1 − p ) f (cid:48) ( k )] dk .Therefore, using (33), E (cid:20)(cid:18) S T F (cid:19) p (cid:21) = (cid:90) + ∞−∞ e pk φ ( f ( k )) ∂ k f ( p, k ) dk = (cid:90) + ∞−∞ e pk φ (cid:16) f (Re( p ) , k ) + Re( p ) v ( k ) (cid:17)(cid:104) − i Im( p ) (cid:16) v Re( p ) (cid:17) (cid:48) ( f (Re( p ) , k )) (cid:105) ∂ k f (Re( p ) , k ) dk = (cid:90) + ∞−∞ e pg (Re( p ) ,a ) φ (cid:16) a + Re( p ) v Re( p ) ( a ) (cid:17) γ (cid:48) p ( a ) da = (cid:90) γ p e pg ( p,z ) φ (cid:0) Re( z ) + Re( p ) v p ( z ) (cid:1) dz, (36) May 4, 2017 here we have used the identity f ( k ) = kv ( k ) + v ( k ) = kv ( k ) + (cid:0) − a (cid:1) v ( k ) + av ( k ) = f ( a, k ) + av ( k ) inthe second line. Now, Eqs (27) and (28) are easily generalized to g ( a, z ) = zv a ( z ) − (cid:0) − a (cid:1) v a ( z ) , whichyields g ( p, z ) = Re( z ) v p ( z ) + Re( p ) v p ( z ) − v p ( z ) . Plugging this last identity inside (36), after somestraightforward simplifications, we obtain (35). Remark 5.2.
Theorem 5.1 shows that Matytsin’s formula (3) and Bergomi’s (4) can be written as lineintegrals of the same function on different curves in the complex plane. Bergomi’s formula is obtainedsetting
Im( p ) = 0 in (35) : in this case, γ p = R . Matytsin’s formula is obtained (again) setting Re( p ) = 0 in (35) : in this case, γ p = { a − i Im( p ) v ( a ) : a ∈ R } (recall that v ( · ) corresponds to Fukasawa’s v ( · ) ). Re-injecting the expression of γ (cid:48) p , formula (35) can of course be written as the following (less compact)integral on the real-line E (cid:2) e pX T (cid:3) = (cid:90) R e i Im( p ) g (Re( p ) ,z ) e Re( p )(Re( p ) − v Re( p ) ( z ) (cid:104) − i Im( p ) ddz v Re( p ) ( z ) (cid:105) φ ( z ) dz, (37)which does not require the notion of transformation g ( a, z ) and volatility v a ( z ) for complex-valued a and z . Remark 5.3.
Just as the Equation (20) we started from, the formulas (35) and (37) are also self-dual: theyare invariant under the transformations p ↔ − p, v p ( z ) ↔ v − p ( − z ) . The restriction Re( p ) ∈ [0 ,
1] in Theorem 5.1 is imposed by the invertibility requirement for the interpo-lated transformation f (Re( p ) , · ). In the next section, we investigate the invertibility of this map for values ofRe( p ) that lie outside [0 ,
1] . We will see that (apart from the trivial Black-Scholes case) there are examplesof smiles for which the map f ( a, · ) remains strictly monotone on some interval (and possibly on the whole R ) for all a > a < k (cid:55)→ f ( p, k ) We know (from the result of Fukasawa [6]) that the interpolated transformation k (cid:55)→ f ( p, k ) = pf ( k ) + (1 − p ) f ( k ) = kv ( k ) + (cid:0) − p (cid:1) v ( k )2 associated to any arbitrage-free implied volatility v ( · ) is strictly increasing if p is in [0 , σ , the map k (cid:55)→ f ( p, k ) = kσ √ T + (cid:16) − p (cid:17) σ √ T is (linear, hence) trivially invertible, for every p ∈ R . It is natural, then,to wonder what happens in general – that is, if there are other cases where f ( p, · ) is invertible when p liesoutside [0 , p for which this happens. f ( p, · ) is surjective for every p in an interval larger than ( − p − ( β − ) , p + ( β + )) A reasonable guess would seem to conjecture that k (cid:55)→ f ( p, k ) is invertible when p is between the criticalexponents, p ∈ ( − p − ( β − ) , p + ( β + )). The following proposition shows that f ( p, · ) is actually surjective forevery p in an interval that is larger than ( − p − ( β − ) , p + ( β + )). Proposition 6.1.
Recall that β ± = lim sup k →±∞ v ( k ) √ | k | . Define ˜ p + = 1 β + + 12 , ˜ p − = 1 β − − , May 4, 2017 here ˜ p ± = ∞ if β ± = 0 . We have ˜ p + ≥ p + ( β + ) resp. ˜ p − ≥ p − ( β − ) , where the inequalities are strict if β + (cid:54) = 2 , resp. β − (cid:54) = 2 . Moreover- if p < ˜ p + , then lim k →∞ f ( p, k ) = + ∞ .- if p > − ˜ p − , then lim k →−∞ f ( p, k ) = −∞ .In particular, if p ∈ ( − ˜ p − , ˜ p + ) , the map f ( p, · ) is surjective on R .Proof. The comparison between ˜ p + and p + ( β + ) (resp. ˜ p − and p − ( β − )) is immediate to check. Let us, then,prove the statements about the limits of f ( p, · ).Assume p < ˜ p + . First note that, if p ≤
0, we have f ( p, k ) = kv ( k ) + ( − p ) v ( k ) ≥ kv ( k ) → ∞ as k → ∞ .Then, assume p >
0. If β + = 2, there is nothing more to prove, because in this case ˜ p + = 1. If β + ∈ [0 , β + < β < f ( p, k ) = f ( k ) − pv ( k ) ≥ (cid:20) √ β + √ β − p (cid:112) β (cid:21) √ k, ∀ k > k β , (38)for some k β >
0. The condition p < ˜ p + entails √ β + + √ β + − p (cid:112) β + >
0. Therefore, when β is sufficientlyclose to β + , the coefficient in front of √ k in (38) is positive, so that lim k →∞ f ( p, k ) = + ∞ .The symmetric argument holds when p > − ˜ p − . Let us provide the details: First, if p ≤
1, then f ( p, k ) ≤ kv ( k ) → −∞ as k → −∞ . Then, assume p <
1. Once again, if β − = 2, there is nothing more to prove,because in this case ˜ p − = 0. If β − ∈ [0 , β − < δ < f ( p, k ) = f ( k ) + (1 − p ) v ( k ) ≤ (cid:34) − (cid:18) √ δ + √ δ (cid:19) + (1 − p ) √ δ (cid:35) (cid:112) | k | , ∀ k < k δ , for some k δ <
0. The condition p > − ˜ p − entails − √ δ − √ δ + (1 − p ) √ δ <
0. Therefore, when δ is sufficientlyclose to β − , the coefficient in front of (cid:112) | k | is negative, so that lim k →−∞ f ( p, k ) = −∞ . f ( p, · ) In this section we show that there exist (non trivial, and practically interesting) situations where the map f ( p, · ) is strictly monotone for values of p that lie outside [0 , Lemma 6.2.
On the set { k : v (cid:48) ( k ) ≤ } , we have ∂ k f ( p, k ) > , for every p > .On the set { k : v (cid:48) ( k ) ≥ } , we have ∂ k f ( p, k ) > , for every p < . The first case in Lemma 6.2 is relevant in particular in Equity markets, where smiles on stock indices areoften monotonically decreasing on the observed interval of strikes (in particular for larger maturities).
Proof.
We first focus on the case p >
1. Using the identities f (cid:48) ( k ) = 1 v ( k ) − kv ( k ) v (cid:48) ( k ) − v (cid:48) ( k )2 = 1 v ( k ) (1 − v (cid:48) ( k ) f ( k )) f (cid:48) ( k ) = 1 v ( k ) − kv ( k ) v (cid:48) ( k ) + v (cid:48) ( k )2 = 1 v ( k ) (1 − v (cid:48) ( k ) f ( k )) + v (cid:48) ( k ) (39) May 4, 2017 e have ∂ k f ( p, k ) = pf (cid:48) ( k ) + (1 − p ) f (cid:48) ( k ) = 1 v ( k ) [ p (1 − v (cid:48) ( k ) f ( k )) + (1 − p )(1 − v (cid:48) ( k ) f ( k ))] + (1 − p ) v (cid:48) ( k )= 1 v ( k ) [1 − v (cid:48) ( k ) f ( k )] + (1 − p ) v (cid:48) ( k ) . We know from [6, Lemma 2.6] that 1 − v (cid:48) ( k ) f ( k ) > k ∈ R . The conclusion follows.The case p < f ( p, k ) = − ˆ f ( q, − k ), therefore ∂ k f ( p, k ) = ∂ k ˆ f ( q, − k ), where q = 1 − p is larger than 1. In the first part of the proof, we have shown that ∂ k ˆ f ( q, l ) > { l : ˆ v (cid:48) ( l ) ≤ } . Since ˆ v ( k ) = v ( − k ) from Eq (24), we have { l : ˆ v (cid:48) ( l ) ≤ } = −{ k : v (cid:48) ( k ) ≥ } ,and the second claim follows. Proposition 6.3. i ) If the implied volatility k (cid:55)→ v ( k ) is decreasing, then ˜ p + = p + ( β + ) = ∞ and thefunction f ( p, · ) is invertible from R to R , for all p ≥ . ii ) If the implied volatility k (cid:55)→ v ( k ) is increasing, then ˜ p − = p − ( β − ) = ∞ and the function f ( p, · ) isinvertible from R to R , for all p ≤ . Proposition 6.3 allows us to extend the definition of the normalized implied volatility v p ( · ) in Eq (5) to every p ≥ v ( · ) (resp. to every p ≤ Corollary 6.4.
The extended Bergomi’s formula (5.1) holds for all { p : Re( p ) ≥ } in case i) of Proposition6.3, resp. { p : Re( p ) ≤ } in case ii).Proof of Proposition 6.3. Let us consider the first case ( v ( · ) is increasing). We have to prove the statementfor p >
1, for we already know that f ( p, · ) is invertible for p ∈ [0 , f ( p, · )is strictly monotone for every p >
1. Since, by assumption, v ( k ) has a finite limit for k → ∞ , we havelim k →∞ v ( k ) √ k = β + = 0, therefore ˜ p + = p + ( β + ) = ∞ . It then follows from Proposition 6.1 that f ( p, · ) issurjective on R for every p >
1, and the claim is proved. The second case ( v ( · ) is decreasing) is proven inthe same way, using Lemma 6.2 and Proposition 6.1 with ˜ p − = ∞ . A class of smiles for which f ( p, · ) is not invertible when p / ∈ [ − ˜ p − , ˜ p + ]. We now exhibit a largeclass of implied volatilities (with finite critical moments, hence finite coefficients ˜ p ± ) for which the function f ( p, · ) fails to be both monotone and surjective when p is outside the interval ( − ˜ p − , ˜ p + ). Consider thefollowing assumption (H) v (cid:48) ( k ) ∼ ± (cid:112) β ± (cid:112) | k | , k → ±∞ , where β ± (cid:54) = 0 . By de L’Hopital’s rule, (H) implies v ( k ) ∼ (cid:112) β ± | k | as k → ±∞ , therefore we have (11). Note that the SSVIparameterisation (6) with ϕ > f ( p, k ), it is straightforward to check that assumption (H) implies f ( p, k ) ∼ (cid:112) β + (˜ p + − p ) √ k as k → ∞ , f ( p, k ) ∼ − (cid:112) β − (˜ p − + p ) (cid:112) | k | as k → −∞ . May 4, 2017 he limits of f ( p, k ) for k → ±∞ are then easily assessed: the case p ∈ ( − ˜ p − , ˜ p + ) is already considered inProposition 6.1; if p lies outside the closure of this interval, then both limits have the same sign: p > ˜ p + ⇒ lim k →±∞ f ( p, k ) = −∞ ,p < − ˜ p − ⇒ lim k →±∞ f ( p, k ) = ∞ . (40)In particular we see that, being continuous, the map f ( p, · ) cannot be surjective on R when p / ∈ [ − ˜ p − , ˜ p + ]. Proposition 6.5.
Assume (H) . Then, if p > ˜ p + or p < − ˜ p − , the map f ( p, · ) : R → R is neither monotone,nor surjective.More precisely: If p > ˜ p + (resp. p < − ˜ p − ), there exist k and k such that f ( p, · ) is strictly increasing(resp. decreasing) on ( −∞ , k ) and strictly decreasing (resp. increasing) on ( k, ∞ ) . A numerical example of the situation described in Proposition 6.5 will be given in the next section – seeFigure 2.
Proof of Proposition 6.5.
The statement about the surjectivity of f ( p, · ) has already been proven above(recall (40)); we prove that f ( p, · ) is not monotone.Let us first consider the case p > ˜ p + . The condition v (cid:48) ( k ) ∼ − √ β − √ | k | as k → −∞ implies that v (cid:48) is negativeon the half-line ( −∞ , k ), for some k . It follows from Lemma 6.2 that f ( p, · ) is strictly increasing on ( −∞ , k ).Using the first equation in (39), together with the identity f (cid:48) ( k ) = v ( k ) (1 − v (cid:48) ( k ) f ( k )) + v (cid:48) ( k ), we have ∂ k f ( p, k ) = 1 v ( k ) [ p (1 − v (cid:48) ( k ) f ( k )) + (1 − p )(1 − v (cid:48) ( k ) f ( k ))] = 1 v ( k ) [1 − v (cid:48) ( k )( pf ( k )) + (1 − p ) f ( k ))]= 1 v ( k ) [1 − v (cid:48) ( k ) f (1 − p, k )] . (41)It follows from assumption (H) that f (1 − p, k ) = kv ( k ) + (cid:18) p − (cid:19) v ( k ) ∼ (cid:18) (cid:112) β + + (cid:18) p − (cid:19)(cid:112) β + (cid:19) √ k as k → ∞ , (42)therefore lim k →∞ v (cid:48) ( k ) f (1 − p, k ) = (cid:112) β + (cid:32) (cid:112) β + + (cid:18) p − (cid:19)(cid:112) β + (cid:33) = 12 + (cid:18) p − (cid:19) β + A ( p, β + ) . It is immediate to see that sign( A ( p, β + ) −
1) = sign( p − ˜ p + ). Consequently, if p > ˜ p + , it follows from(41) that ∂ k f ( p, k ) is negative for k large enough, and the claim on the intervals of monotonicity of f ( p, · )isproven.We can now deduce the claim in the case p < − ˜ p − from duality: consider the dual implied volatility ˆ v ( · )defined in Section 3. It follows from (24) and assumption (H) that ˆ v (cid:48) ( k ) ∼ √ β − √ k as k → ∞ and ˆ v (cid:48) ( k ) ∼ − √ β + √ | k | as k → −∞ , so that ˆ β ± = β ∓ (the duality transformation exchanges the right and left slopes of the smile).Denote ˆ p + the coefficient ˜ p + associated to ˆ v , that is: ˆ p + = β + + = β − + = ˜ p − + 1. We know fromLemma 3.1 that f ( p, k ) = − ˆ f (1 − p, − k ). Since p < − ˜ p − , then 1 − p > p − = ˆ p + . In the first part of the May 4, 2017 roof, we have proven that ˆ f (1 − p, k ) is strictly increasing on the half-line ( −∞ , k ) for some k , and strictlydecreasing on ( k, ∞ ) for some k , therefore the claim on the monotonicity of f ( p, · ) follows.In view of Propositions 6.1, 6.3 and 6.5, it seems reasonable to conjecture that the map f ( p, · ) : R → R is invertible if p ∈ ( − ˜ p − , ˜ p + ), and only if p ∈ [ − ˜ p − , ˜ p + ]. Leaving the proof of this statement for future work,we numerically check this fact on an arbitrage-free SSVI parameterisation in the next section. Recall the SSVI parameterisation (6), where, for fixed
T > θ T = θ > , ϕ ( θ T ) = ϕ ≥ , ρ ∈ ( − , . Theorem 4.2 in [8] proves that the implied variance v ( k ) is free of arbitrage (for the given maturity T )if the following conditions are satisfied: Condition 7.1. (1) θϕ (1 + | ρ | ) < θϕ (1 + | ρ | ) ≤ . Moreover, [8, Lemma 4.2] shows that the condition θϕ (1 + | ρ | ) ≤ v are (as onecould expect) satisfied. We also further discuss the limiting case θϕ (1 + | ρ | ) = 4. θϕ (1 + | ρ | ) = 4 and checking Assumption 2.1 (ii) on v We show that the case ρ ≥ θϕ (1 + | ρ | ) = 4 is not arbitrage-free. Therefore, if ρ ≥
0, Condition 7.1(1)(with strict inequality) is also necessary. When ρ <
0, we show that the case θϕ (1 + | ρ | ) = 4 is ruled out byour Assumption 2.1 (ii) of zero mass at K = 0.Assume θϕ (1 + | ρ | ) = 4. We separate the two cases ρ ≥ ρ <
0: assume first that ρ ≥
0. It is easy tosee that lim k →∞ v ( k ) k = θϕ ρ = 2. Then we can compute, for k > v ( k ) − k = k (cid:20) v ( k ) k − (cid:21) = k (cid:34) θ (cid:32) k + ρϕ + (cid:114) ϕ + 2 ϕρk + 1 k (cid:33) − (cid:35) = k (cid:20) θ (cid:18) k + ρϕ + ϕ + ρk + O (cid:16) k (cid:17)(cid:19) − (cid:21) = k (cid:20) θ k + O (cid:16) k (cid:17)(cid:21) = θ ρ ) + O (cid:16) k (cid:17) → θ ρ ) > , as k → ∞ . The limit above contradicts the following property from [10, Lemma 3.1]: if v is an arbitrage-free smile,there exists k such that v ( k ) − k < k > k . This property was subsequently improved to lim k →∞ ( v ( k ) − k ) = −∞ by Rogers and Tehranchi [13]. May 4, 2017 ow assume ρ <
0. The analogous computation for negative k gives v ( k ) − | k | = θ − ρ ) + O (cid:16) | k | (cid:17) → θ − ρ, as k → −∞ . In general, a positive value of lim k →−∞ ( v ( k ) − | k | ) is not in contradiction with no-arbitrage: indeed, weknow from [5, Propositions 2.4 and 2.5] that an arbitrage-free implied volatility satisfieslim k →−∞ (cid:0) v ( k ) − (cid:112) | k | (cid:1) = N − ( P ( S T = 0)) , where the right hand side is worth −∞ if P ( S T = 0) (see also [4, Thm 3.6]). As discussed in the Introduction,our Assumption 2.1 (ii) on the coefficient d ( k, v ( k )) is equivalent to P ( S T = 0) = 0, therefore v ( k ) − (cid:112) | k | →−∞ as k → −∞ . Consequently, v ( k ) − | k | = (cid:0) v ( k ) − (cid:112) | k | (cid:1)(cid:0) v ( k ) + (cid:112) | k | (cid:1) tends to −∞ as well, and thecase θϕ (1 + | ρ | ) = 4 and ρ ≤ θϕ (1 + | ρ | ) <
4, we have v ( k ) − | k | ∼ − a | k | as k → −∞ with a = (cid:0) − θϕ (1 + | ρ | ) (cid:1) >
0, therefore Assumption 2.1 (ii) is satisfiedwhen Condition 7.1(1) is in force. v Denote w ( k ) = v SSVI ( k ) . We can compute w (cid:48) ( k ) = θϕ (cid:32) ρ + ϕk + ρ (cid:112) ( ϕk + ρ ) + 1 − ρ (cid:33) , w (cid:48)(cid:48) ( k ) = θϕ − ρ (( ϕk + ρ ) + 1 − ρ ) / . (43)Note that the second equation shows that w ( · ) is a convex function. If ϕ = 0, w ( · ) is identically equal to θ > ϕ >
0. Fromthe first equation in (43), w (cid:48) ( k ) = 0 if and only if k = − ρϕ =: k min . At this point, we have w ( k ) ≥ w ( k min ) = θ (cid:16) − ρ + (cid:112) ρ + 1 (cid:17) , ∀ k ∈ R . A straightforward computation shows that the function of ρ in the RHS above is strictly positive, for any ρ ∈ ( − , v ( k ) = (cid:112) w ( k ) > , ∀ k ∈ R .Moreover, the argument of the square-root function in (6) being lower bounded by 1 − ρ >
0, we have w ∈ C ( R ) (actually, w ∈ C ∞ ( R )). Since w ( k ) ≥ w ( k min ) >
0, we also have that k (cid:55)→ v ( k ) = (cid:112) w ( k ) is C .Overall, Assumption 2.1 (i) is also satisfied. It is immediate to check that the set of parameters θ = (0 . = 0 . , ρ = − . , ϕ = 1 .
40 (44)satisfies Condition (7.1). Figure 1 shows the resulting SSVI implied volatility smile and the two correspondingtransformations f and f , on a large interval ( k min , k max ). May 4, 2017 igure 1: Left: SSVI parameterisation (6) of the implied volatility v ( · ) with the arbitrage-free parametersin (44). Right: the induced transformations f , f .Recall that β ± (SSVI) = θϕ ± ρ . In Figure 2, we compute the values of the coefficients ˜ p ± in Proposition6.1, and plot the function k (cid:55)→ f ( p, k ) for different values of p . As predicted by Proposition 6.5, one can see(and we did check on the numerical values) that f ( p, · ) is not increasing anymore for large (resp. small) k when p is larger than ˜ p + (resp. p is smaller than − ˜ p − ). On the contrary, f ( p, · ) does appear to be strictlyincreasing (at least on the considered interval of log-strikes) for p within the interval ( − ˜ p − , ˜ p + ). Having derived formulas for the extended characteristic function of the log-price, prices of European optionson S T can (also) be recovered with standard transform-based methods. As a consistency check, we show thatthe formula P ( K ) = KN ( f ( k )) − F N ( f ( k )) for a put option (the Black-Scholes formula) can be restoredfrom Theorem 2.7. We assume B (0 , T ) = 1 and F = 1 for simplicity. We apply the following inversiontheorem (see e.g. [11]): Theorem 8.1.
Denote ϕ T ( u ) = E [ e iuX T ] the characteristic function of the log-price. Then, for every K > ,the price of a put option with strike K and maturity T is given by: P ( K ) = R α + 12 π (cid:90) + ∞−∞ K − α − iu +1 ( α + iu )( α − iu ) ϕ T ( u − iα ) du, where R α = K − if < α < p ∗ K if < α < if − q ∗ < α < . May 4, 2017 igure 2: Plot of the function k (cid:55)→ f ( p, k ) induced by the SSVI parameterisation (6) with the parameters in(44), for different values of p around the thresholds ˜ p + (left pane) and − ˜ p − (right pane) from Proposition6.1.Applying Theorem 8.1 and Proposition 2.11 we get, choosing α ∈ (0 ,
1) and using i ( u − iα ) = iu + α : P ( K ) = K + 12 π (cid:90) + ∞−∞ K − α − iu +1 ( α + iu )( α − iu ) (cid:90) + ∞−∞ (cid:104) ( α + iu ) e g ( z )( α + iu − + (1 − α − iu ) e ( α + iu ) g ( z ) (cid:105) φ ( z ) dz du. Using Fubini’s Theorem, P ( K ) = K + 12 π (cid:90) + ∞−∞ φ ( z ) dz (cid:90) + ∞−∞ K − α − iu +1 ( α + iu )( α − iu ) (cid:104) ( α + iu ) e g ( z )( α + iu − + (1 − α − iu ) e ( α + iu ) g ( z ) (cid:105) du or yet, after simplification: P ( K ) = K + 12 π (cid:90) + ∞−∞ φ ( z ) dz (cid:90) + ∞−∞ (cid:20) K − α − iu +1 e g ( z )( α + iu − α − iu − K − α − iu +1 e ( α + iu ) g ( z ) α + iu (cid:21) du. (45)Set k = log K, a ( z ) = k − g ( z ) , b = k − g ( z ). Then P ( K ) = K + 12 iπ (cid:90) + ∞−∞ φ ( z ) (cid:2) I ( a ( z ) , α − − KI ( b ( z ) , α ) (cid:3) dz (46)where I ( c, d ) = (cid:90) d + i ∞ d − i ∞ e − cω ω dω. We show now:
Lemma 8.2. If c < and d < , then I ( c, d ) = 0 .If c < and d > , then I ( c, d ) = 2 iπ . May 4, 2017 f c > and d < , then I ( c, d ) = − iπ .If c > and d > , then I ( c, d ) = 0 .Proof. Consider first the case c >
0, and the rectangle defined by the real coordinates d and d + R andthe imaginary coordinates − R , R wih positive R and R . By Cauchy residue formula, the integral of thefunction ω → e − cω ω over this (clockwise) rectangle contour is equal to 0 if d >
0, or to the residue at the pole0, which is 1, times − iπ because we go clockwise, if the pole 0 is inside the rectangle, i.e. if d <
0. Now take R to ∞ : since c is positive, the integral on the right segment goes to zero, and the integral on the bottomand top segments are absolutely convergent. By Lebesgue dominated convergence theorem, those integralsgo to zero as R goes to infinity. Since the remaining integral is exactly I ( c, d ), we have proven the last twostatements. The proof of the case c < d and d − R instead, and running anticlockwise on the rectangle contour.Since α ∈ (0 , d = α − <
0, sothat 12 iπ (cid:90) + ∞−∞ φ ( z ) I ( a ( z ) , α −
1) = − (cid:90) + ∞−∞ φ ( z )1 { a ( z ) > } dz = − (cid:90) g ( z )
0, so that the contribution is − K iπ (cid:90) + ∞−∞ φ ( z ) I ( b ( z ) , α ) dz = − K (cid:90) + ∞−∞ φ ( z )1 { b ( z ) < } dz = − K (cid:90) g ( z ) >k φ ( z ) dz = − K + KN ( f ( k )) . Summing up, we have recovered the Black-Scholes formula for the put option: P ( K ) = KN ( f ( k )) − N ( f ( k )) = Put BS ( k, v ( k )) . A Appendix
Proof of Lemma 2.6
We focus on the case p >
0. As explained above, the statement of the Lemma istrue for every p ∈ (0 ,
1) = ( − p − (2) , p + (2)). Therefore, we can limit ourselves to β + ∈ [0 , p > β > β + , there exists k β > v ( k ) < √ βk for all k > k β . We claim that this impliesFor every β + < β < , f ( k ) ≥ (cid:18) √ β + √ β (cid:19) √ k, ∀ k > k β . (47)It follows from (47) that z = f ( g ( z )) ≥ (cid:18) √ β + √ β (cid:19) (cid:112) g ( z ) , for z large enough,which entails g ( z ) ≤ z (cid:16) √ β + √ β (cid:17) = z p + ( β ) . On the other hand, using f ( z ) = f ( z ) − v ( z ), we obtain May 4, 2017 ( k ) ≥ (cid:16) √ β − √ β (cid:17) √ k for all k > k β , therefore g ( z ) ≤ z p − ( β ) for z large enough. For such z , we have e ( p − g ( z ) φ ( z ) ≤ exp (cid:18) z p − p − ( β ) − z (cid:19) = exp (cid:18) z p − p + ( β )2 p − ( β ) (cid:19) and e pg ( z ) φ ( z ) ≤ exp (cid:18) z p p + ( β ) − z (cid:19) = exp (cid:18) z p − p + ( β )2 p + ( β ) (cid:19) . Recall that p ± ( β ) > β <
2. Since p − p + ( β + ) < β sufficiently close to β + , we have p − p + ( β ) <
0, too. Therefore, using the last two estimates above, we obtain that the functionsin (15) are integrable at + ∞ . Using the fact that g ( z ) , g ( z ) → −∞ as z → −∞ and p ≥
1, we obtain thatthe functions in (15) are also integrable at −∞ .The case p < Proof of (47) : We proceed along the lines of [5, Lemma 2.6] (note that we are referring here to an ArXivpreprint: this lemma was not reported in the published version of the article). For every a > k > k β ,we have f ( k ) = kv ( k ) + v ( k )2 = kv ( k ) + av ( k )2 − ( a − v ( k )2 ≥ √ ak − ( a − √ βk a = 2 /β (which in fact provides the optimal lower bound in (48)). A.1 Proof of Theorem 2.7
We first prove a weaker version of Theorem 2.7:
Proposition A.1.
Assume max( − p − ( β − ) , − q ∗ ) < p < min( p + ( β + ) , p ∗ ) . (49) Then, Equation (19) holds for every absolutely continuous function Ψ such that Ψ and Ψ (cid:48) have exponentialgrowth of order p . In order to prove Proposition A.1, we need the following intermediate result.
Lemma A.2.
Assume that p satisfies (49) , and let Ψ be a function with exponential growth of order p .Then, the function k (cid:55)→ Ψ( k ) v (cid:48) ( k ) φ ( f ( k )) is integrable on R , and satisfies lim k →±∞ Ψ( k ) v (cid:48) ( k ) φ ( f ( k )) = 0 . In the proof of Lemma A.2, we will make use of the identity φ ( f ( k )) v (cid:48) ( k ) = N ( − f ( k )) − P ( X T > k ) , ∀ k ∈ R , (50)which can easily be derived from Black-Scholes formula and the definition of v ( k ), Call BS ( k, v ( k )) = E (cid:2)(cid:0) S T F − e k (cid:1) + (cid:3) . Using the identity N ( − f ) = 1 − N ( f ), we also have the equivalent formulation φ ( f ( k )) v (cid:48) ( k ) = − N ( f ( k )) + P ( X T ≤ k ) , k ∈ R . (51) May 4, 2017 emark A.3.
It follows from expression (50) that − ≤ v (cid:48) ( k ) φ ( f ( k )) ≤ , in particular this quantity isbounded, so that the condition lim k →±∞ Ψ( k ) v (cid:48) ( k ) φ ( f ( k )) = 0 holds for functions Ψ going to zero at infinity.Proof of Lemma A.2. In what follows, c denotes a positive constant that can change from line to line, butdoes not depend on k nor on any other parameter. Let Ψ be of exponential growth of order p . • Using the boundedness of v (cid:48) ( k ) φ ( f ( k )) from Eq (50), we have | Ψ( k ) v (cid:48) ( k ) φ ( f ( k )) | ≤ c e pk . Therefore,if p >
0, lim k →−∞ Ψ( k ) v (cid:48) ( k ) φ ( f ( k )) = 0 and this function is integrable in a neighborhood of −∞ . Onthe other hand, using again Eq (50), the bound f ( k ) ≥ √ k for k >
0, and the bound on Mill’s ratio N ( − f ) ≤ φ ( − f ) f for f >
0, we have | Ψ( k ) v (cid:48) ( k ) φ ( f ( k )) | = | Ψ( k ) N ( − f ( k )) − Ψ( k ) P ( X T > k ) |≤ c √ k e pk φ ( √ k ) + c e pk P ( X T > k ) ≤ c √ k e k ( p − + c e pk P ( X T > k ) , ∀ k > . For the second term, note that E [ e X T ] = E (cid:2) S T F (cid:3) = 1 entails P ( X T > k ) = O ( e − αk ) as k → ∞ , for every α < p <
1, lim k →∞ Ψ( k ) v (cid:48) ( k ) φ ( f ( k )) = 0and that this function is integrable in a neighborhood of + ∞ . Overall, the conclusion of Lemma A.2 is truefor every p ∈ (0 ,
1) = ( p − (2) , p + (2)) and every function Ψ of exponential growth of order p (regardless of thevalues of β ± ). • According to the first bullet point, we can limit ourselves to β + ∈ [0 , p is in the interval(49). It follows from Eq (50) and estimate (47) that, for every β ∈ ( β + , | Ψ( k ) v (cid:48) ( k ) φ ( f ( k )) | ≤ | Ψ( k ) |√ k φ (cid:18)(cid:18) √ β + √ β (cid:19) √ k (cid:19) + | Ψ( k ) | P ( X T > k )= c √ k exp (cid:18) pk − (cid:18) β + β (cid:19) k (cid:19) + c e pk P ( X T > k )= c √ k exp ( k ( p − p + ( β ))) + c e pk P ( X T > k ) , ∀ k > k β . By assumption, p − p + ( β + ) <
0. Choosing β sufficiently close to β + , we have p − p + ( β ) <
0, too (in theparticular case β + = 0, we can make p − p + ( β ) arbitrarily small by taking β > p ∗ satisfies p ∗ = sup { α > P ( X T > k ) = O ( e − kα ) as k → ∞} , see again [9, Lemma 4.4]. Therefore, for every α < p ∗ , e pk P ( X T > k ) = O ( e k ( p − α ) ) as k → ∞ . Taking p < α < p ∗ , we can conclude that lim k →∞ Ψ( k ) v (cid:48) ( k ) φ ( f ( k )) = 0 and that k (cid:55)→ Ψ( k ) v (cid:48) ( k ) φ ( f ( k )) is integrable in a neighborhood of + ∞ . • The analogous argument holds for the left side behavior of Ψ( k ) v (cid:48) ( k ) φ ( f ( k )): let us provide the detailsfor completeness. From the first bullet point, we can assume β − ∈ [0 , δ > β − ,there exists k δ < v ( k ) < (cid:112) δ | k | for all k < k δ . It follows thatFor every β − < δ < , f ( k ) ≤ − (cid:32) √ δ − √ δ (cid:33) (cid:112) | k | =: − r δ (cid:112) | k | , ∀ k < k δ . (52) May 4, 2017 n order to prove (52), note that for every a > k < k δ f ( k ) = − | k | v ( k ) − v ( k )2 = − (cid:18) | k | v ( k ) + av ( k )2 (cid:19) + ( a − v ( k )2 ≤ − (cid:112) a | k | + ( a − (cid:112) δ | k | f ( k ) ≤ − (cid:16) √ δ + √ δ (cid:17) (cid:112) | k | by choosing a = 2 /δ . Then, (52) follows from f ( k ) = f ( k ) + v ( k ).Consequently, if p is in the interval (49), using Eq (51) and estimate (52), for every δ ∈ ( β − ,
2) we have | Ψ( k ) v (cid:48) ( k ) φ ( f ( k )) | ≤ | Ψ( k ) | (cid:112) r δ | k | φ (cid:32)(cid:16) √ δ − √ δ (cid:17) √ k (cid:33) + | Ψ( k ) | P ( X T < k )= c (cid:112) r δ | k | exp (cid:18) pk − (cid:18) δ + δ − (cid:19) | k | (cid:19) + c e pk P ( X T < k )= c (cid:112) r δ | k | exp ( −| k | ( p + p − ( δ ))) + c e pk P ( X T < k ) , ∀ k < k δ . By assumption, p + p − ( β − ) >
0. Choosing δ sufficiently close to β − , we have p + p − ( δ ) >
0, too. For the secondterm in the last line, we use the property q ∗ = sup { α > P ( X T < k ) = O ( e −| k | α ) as k → −∞} [9, Lemma4.4], which entails e pk P ( X T < k ) = O ( e −| k | ( α − p ) ) for every α < q ∗ . Taking p < α < q ∗ , we conclude thatlim k →−∞ Ψ( k ) v (cid:48) ( k ) φ ( f ( k )) = 0 and that k (cid:55)→ Ψ( k ) v (cid:48) ( k ) φ ( f ( k )) is integrable at −∞ .Putting the three bullet points together, we have shown that Lemma A.2 holds for any value of β ± ∈ [0 , Proof of Proposition A.1
We follow the lines of [6, Theorems 4.6 and 4.4]. Denote L Ψ ( p ) the LHS of(19): L Ψ ( p ) = (cid:82) + ∞−∞ [Ψ( g ( z )) − Ψ (cid:48) ( g ( z ))+Ψ (cid:48) ( g ( z )) e − g ( z ) ] φ ( z ) dz . Using the identity φ ( f ( k )) = φ ( f ( k )) e k ,we have (cid:90) Ψ (cid:48) ( g ( z )) e − g ( z ) φ ( z ) dz = (cid:90) Ψ (cid:48) ( k ) e − k φ ( f ( k )) f (cid:48) ( k ) dk = (cid:90) Ψ (cid:48) ( k ) φ ( f ( k ))( f (cid:48) ( k ) − v (cid:48) ( k )) dk = (cid:90) Ψ (cid:48) ( g ( z )) φ ( z ) dz − (cid:90) Ψ (cid:48) ( k ) φ ( f ( k )) v (cid:48) ( k ) dk. (53)Integrating by parts, applying Lemma A.2 and using the identity φ (cid:48) ( f ) = − f φ ( f ), we get − (cid:90) Ψ (cid:48) ( k ) φ ( f ( k )) v (cid:48) ( k ) dk = − [Ψ( k ) φ ( f ( k )) v (cid:48) ( k ))] k →∞ k →−∞ + (cid:90) Ψ( k ) ddk [ φ ( f ( k )) v (cid:48) ( k )] dk = (cid:90) Ψ( k ) ( − f ( k ) f (cid:48) ( k ) v (cid:48) ( k ) + v (cid:48)(cid:48) ( k )) φ ( f ( k )) dk. (54)It follows from Eqs (53) and (54) that L Ψ ( p ) = (cid:82) Ψ( k ) [ f (cid:48) ( k )(1 − f ( k ) v (cid:48) ( k )) + v (cid:48)(cid:48) ( k )] φ ( f ( k )) dk . Now recallthat, under the assumption that v is twice differentiable, the density function of S T at the point K = F e k May 4, 2017 s given by d P ( K ) dK = d dK E [( K − S T ) + ] = d dK P BS (cid:18) K, v (cid:16) ln KF (cid:17)(cid:19) = ddK (cid:18) N ( f ( k )) + ∂ v P BS (cid:18) K, v (cid:16) ln KF (cid:17)(cid:19) v (cid:48) ( k ) 1 K (cid:19) = ddK ( N ( f ( k )) + φ ( f ( k )) v (cid:48) ( k ))= φ ( f ( k )) [ f (cid:48) ( k )(1 − f ( k ) v (cid:48) ( k )) + v (cid:48)(cid:48) ( k )] 1 F e k . (55)where we have used the identities ∂ v P BS ( F e k , v ) = F e k φ ( f ( k )) and, again, φ (cid:48) ( f ) = − f φ ( f ). Applying (55),we obtain L Ψ ( p ) = (cid:90) ∞−∞ Ψ( k ) d P ( F e k ) dK F e k dk = (cid:90) ∞ Ψ (cid:16) log KF (cid:17) d P ( K ) dK dK = E (cid:20) Ψ (cid:16) log S T F (cid:17)(cid:21) therefore Equation (19) is proved under condition (49) on p .We finally have to strengthen Proposition A.1 into Theorem 2.7. If we know that p ∗ ≥ p + ( β + ) and q ∗ ≥ p − ( β − ), this is immediate. Recall anyhow that our intention here is not to make use of Lee’s result [10],therefore these bounds need to be proved.The key ingredient will be the following result from the theory of Laplace transforms. Let ϕ ∈ L loc ( R + )such that ϕ ≥ p ) = (cid:82) ∞ e pt ϕ ( t ) dt and denote abs (Φ) = sup { p ∈ R : Φ( p ) < ∞} the abscissaof convergence of Φ, where inf ∅ = −∞ . Then, Φ defines a holomorphic function on { Re( p ) < abs ( ϕ ) } . Lemma A.4 (Theorem 2.7.1 in [1]) . Let ϕ ∈ L loc ( R + ) such that ϕ ≥ a.e. Assume that −∞ < abs ( ϕ ) < ∞ .Then, Φ cannot be extended to a holomorphic function on a neighbourhood of abs ( ϕ ) . Lemma A.4 allows to prove the bounds we need in order to conclude.
Lemma A.5. p ∗ ≥ p + ( β + ) and q ∗ ≥ p − ( β − ) .Proof. We focus on the first inequality, p ∗ ≥ p + ( β + ). Assume p ∗ < p + ( β + ). We have M ( p ) = M − ( p ) + M + ( p ) := (cid:82) −∞ e px f X ( x ) dx + (cid:82) ∞ e px f X ( x ) dx , where f X ∈ L ( R ) is the density of X T (which exists underAssumption 2.1 (i) on the implied volatility v ). From Proposition A.1, the identity M + ( p ) = L ( p ) − M − ( p )holds on { < Re( p ) < p ∗ } . By definition of p ∗ , abs ( M + ) = p ∗ . On the other hand, M − is holomorphicon the half-plane { Re( p ) > } , and L is holomorphic on the strip { < Re( p ) < p + ( β + ) } by Lemma2.6. In other words, the function L ( · ) − M − ( · ) is a holomorphic extension of M + to the strictly largerstrip { < Re( p ) < p + ( β + ) } , contradicting Lemma A.4. The second inequality q ∗ ≥ p − ( β − ) is provenanalogously.As pointed out above, the proof of Theorem 2.7 is now immediate. Proof of Theorem 2.7
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