The Smile in Stochastic Volatility Models

Transcription

1 Global Markes Quaniaive Research Modeling and Managing Financial Risks Conference, Paris January 13h, 2011 Join work wih Lorenzo Bergomi

2 Ouline Moivaion Expansion of he smile a order 2 in vol of vol Firs example: a family of Heson-like models Second example: he Bergomi model wih 2 facors on he variance curve Numerical experimens Shor-erm and long-erm asympoics of he smile Rederiving he link beween skew and skewness of log-reurns Conclusion

3 Moivaion Consider he following general dynamics for a diffusive sochasic volailiy model: X = ln S dx = 1 2 ξ d + p ξ dw 1, X 0 = x (1) dξ u = λ(, u, ξ ) dw, ξ u 0 = y u ξ (ξ u, u): insananeous forward variance curve from onwards. For a given mauriy u, ξ u is a drifless process whose iniial value is read on he marke prices of variance swap conracs: ξ0 u = d 2 `ˆσ du u u, where ˆσ u is he implied variance swap volailiy for mauriy u. λ = (λ 1,..., λ d ): volailiy of forward insananeous variances. W = `W 1,..., W d is a d-dimensional Brownian moion. The firs componen of he Brownian moion, W 1, drives he spo dynamics. No dividend. Zero raes and repos (for he sake of simpliciy)

4 No closed-form formula is available for he price of vanilla opions in Model (1). In a few paricular cases of firs generaion sochasic volailiy models, like he Heson model, some approximaions of he price of vanilla opions have been suggesed in he lieraure. Here, we aim a finding a general approximaion of he smile of implied volailiy which does no depend on a paricular specificaion of he model, i.e., on a paricular choice of λ. We will derive general asympoic expansion of he smile, for small volailiy of volailiy, a second order. We inroduce a scaling facor ω for he volailiy of insananeous forward variance: λ ωλ. Abusing language, we will speak of ω as he vol of vol. X and ξ hen depend on ω: X X ω and ξ ξ ω. Our derivaion relies on he fac ha in Model (1), he volailiy of he asse is an auonomous sochasic process, meaning ha i incorporaes no local volailiy componen, and ha λ does no depend on he asse value.

5 Expansion of he price Expansion of he price of a vanilla opion Consider he vanilla opion delivering g(xt ω ) a ime T. Is price is a funcion P ω of (, X ω, ξ,ω ). We wrie P ω (, x, y): he variable y (y u, u T ) is a curve. P ω solves he PDE ( + L ω ) P ω = 0 wih erminal condiion P ω (T, x, y) = g(x), where L ω = L 0 + ωl 1 + ω 2 L 2 wih L 0 = 1 2 y x y 2 x L 1 = L 2 = 1 2 du µ(, u, y) 2 xy u du du ν(, u, u, y) 2 y u y u µ(, u, y) = p y λ 1(, u, y) = E [dxdξu ξ = y] = d h i dx E dξ ν(, u, u, y) = λ i(, u, y)λ i(, u u dξ u ξ = y, y) = d i=1 h i ds E S dξ u ξ = y d

6 Expansion of he price The perurbaion equaions Assume ha P ω = P 0 + ωp 1 + ω 2 P 2 + ω 3 P = ` + L 0 + ωl 1 + ω 2 L 2 `P0 + ωp 1 + ω 2 P 2 + ω 3 P 3 + = ( + L 0) P 0 + ω (( + L 0) P 1 + L 1P 0) +ω 2 (( + L 0) P 2 + L 1P 1 + L 2P 0) +ω 3 (( + L 0) P 3 + L 1P 2 + L 2P 1) + We need o solve he following equaions: ( + L 0) P 0 = 0, P 0(T, x, y) = g(x) ( + L 0) P 1 + L 1P 0 = 0, P 1(T, x, y) = 0 ( + L 0) P n + L 1P n 1 + L 2P n 2 = 0, P n(t, x, y) = 0, n 2

7 Expansion of he price L 0 is he infiniesimal generaor associaed o X 0, he unperurbed diffusion for which ω = 0. I is he sandard one-dimensional Black-Scholes operaor wih deerminisic volailiy y a ime. Each P n is soluion o he radiional one-dimensional diffusion equaion wih a source erm H n = L 1P n 1 + L 2P n 2: ( + L 0) P n + H n = 0 Feynmann-Kac heorem 0,,x P 0(, x, y) = E ˆg `XT,» P n(, x, y) = E H n(s, Xs 0,,x, y)ds, n 1 where X 0,,x is he unperurbed process where ω = 0, saring a log-spo x a ime : dxs 0,,x = 1 2 ys ds + y s dws 1, X 0,,x = x

8 Expansion of he price The price a order 0 P 0 is jus he Black-Scholes price wih ime-dependen volailiy y :» P 0(, x, y) = E g x + ys dws 1 1 «y s ds = P BS x, 2 where «y s ds» P BS(x, v) = E g x + vg 12 «v, G N (0, 1) (2) v = R T y s ds is he oal variance of X 0 inegraed from o T. P 0(, x, y) depends on he curve y (y s, s T ) only hrough v. P BS is soluion o he PDE vp BS = 1 2 ` 2 x x PBS, P BS(x, 0) = g(x) (3) Links he vega and gamma of a vanilla opion in he unperurbed sae.

9 Expansion of he price The price a order 0 An imporan observaion: Because L 0 incorporaes no local volailiy, L 0 and x commue so ( + L 0) xp p 0 = x p ( + L 0) P 0 = 0. xp p BS X 0, R T y s ds p xp 0(, X 0, y) is a maringale for all ineger p. Equaion (3) hen shows ha for all inegers m, n, v m x n P BS X 0, R T y s ds is a maringale. This is crucial in he compuaions of P 1 and P 2.

10 Expansion of he price The price a order 1 Le us define he inegraed spo-variance covariance funcion C Xξ (y): h i Z ds T E s C Xξ S s dξs u ξ s = y (y) = ds du µ(s, u, y) = ds du s s ds We hen have» P 1(, x, y) = E L 1P 0(s, Xs 0,,x, y)ds» = E» = E ds = ds ds s s = C Xξ (y) 2 xvp BS x, s du µ(s, u, y) y u xp BS Xs 0,,x, du µ(s, u, y) xvp 2 BS Xs 0,,x, s» du µ(s, u, y)e xvp 2 BS Xs 0,,x, «y r dr s s «y r dr «y r dr ««y r dr

11 Expansion of he price The price a order 2 A similar resul holds for he second order correcion: C ξξ (y) = C µ (y) = C ξξ P 2 = P L 2P P L 1P 1 2 P L 2P 0 2 (, x, y) = 1 2 Cξξ (y) 2 vp BS x, P L 1P 1 2 = P L 1P 1 2,0 + P L 1P 1 2,1 P L 1P 1 2,0 (, x, y) = 1 2 CXξ (y) 2 2 x 2 vp BS x, P L 1P 1 2,0 (, x, y) = C µ (y) 2 x vp BS x, ds ds du s s s du ν(s, u, u, y) = du µ(s, u, y) y u C Xξ s (y) ds s du «y r dr «y r dr (y): inegraed variance-variance covariance funcion s du «y r dr h E dξ u s dξ u ds i ξ s = y

12 Expansion of he implied volailiy Expansion of he implied volailiy I AT M T = We wrie C Xξ = C Xξ 0 (y), C ξξ = C ξξ 0 (y) and Cµ = C µ 0 (y). In he general diffusive sochasic volailiy model (1), a second order in he vol of vol ω, he implied volailiy for mauriy T and S T = κ T = srike K is quadraic in L = ln I ω (T, K) = I AT M T Coefficiens are r v T + CXξ 4 vt ω + 1 C Xξ 2v 3/2 T ω v 7/2 T K S 0 : ««K + S T ln + κ T ln 2 + O(ω S 0 KS0 3 ) (4) 32v 5/2 T 12C Xξ2 C ξξ v (v + 4) + 4C µ v (v 4) ω 2 4C µ v 3C Xξ2 ω 2 8v 5/2 T 4C µ v + C ξξ v 6C Xξ2 ω 2 v = R T 0 yr dr = inegraed variance.

13 Expansion of he implied volailiy Commens ATM implied volailiy: r IT AT M v = T + CXξ 4 vt ω v 5/2 12C Xξ2 C ξξ v (v + 4) + 4C µ v (v 4) ω 2 T ATM implied volailiy = variance swap volailiy + spread. A firs order, spread = CXξ 4 vt ω. Typically, on he equiy marke, C Xξ < 0: he ATM implied volailiy lies below he variance swap volailiy. When spo reurns and forward variances are uncorrelaed, C Xξ = C µ = 0 so ha r IT AT M v = T Cξξ (v + 4) 32v 3/2 T ω2 Because C ξξ 0, he ATM implied volailiy lies again below he variance swap volailiy. The higher he volailiy of variances, he smaller he ATM implied volailiy.

14 Expansion of he implied volailiy Commens (coninued) ATM skew: S T = CXξ 2v 3/2 ω + 1 `4C T 8v 5/2 µ v 3C Xξ2 ω 2 T The ATM skew S T is of order ω. I has he sign of C Xξ. S T vanishes when spo reurns and forward variances are uncorrelaed, even a second order. The skew is produced by he spo-variance correlaion. If µ 0, S T 0 a firs and a second order. A firs order in ω, he ATM skew has he same sign as he difference beween he ATM implied volailiy and he variance swap volailiy. ATM convexiy: κ T = `4C µ v + C ξξ v 6C Xξ2 ω 2 1 8v 7/2 T The convexiy κ T is of order ω 2. I is an increasing funcion of C ξξ and of C µ, and a decreasing funcion of he absolue value of he spo-variance covariance CXξ. If spo and variances are uncorrelaed, κ T = Cξξ 8v 5/2 T ω2 0 The convexiy is posiive, and produced by variance-variance correlaion.

15 Expansion of he implied volailiy Anoher derivaion which says a he level of operaors Recall ha he price P ω of he vanilla opion is soluion o ( + L ω ) P ω = 0 wih L ω = L 0, + ωl 1, + ω 2 L 2,, and erminal condiion P ω (T, x, y) = g(x). The price can be expressed in erms of he semigroup (U ω s, 0 s T ) aached o he family of differenial operaors L ω : P ω (, ) = U ω T g. The semigroup is defined by U ω s = lim n (1 δlω 0 ) (1 δl ω 1 ) `1 δl ω n 1, s δ =, i = s+iδ n I saisfies Ur ω = UrsU ω s ω for 0 r s T, hence he noaion R : exp s Lω τ dτ :, where :: denoes ime ordering. We can direcly expand U ω s in powers of ω. This is he usual ime-dependen perurbaion echnique in quanum mechanics. U 0 s is called he free propagaor.

16 Expansion of he implied volailiy Consider he general siuaion where a differenial operaor L is perurbed by anoher operaor H : L ε = L + εh From he definiion of he semigroup, Us ε = U (0) s + εu (1) s + ε 2 U (2) s + wih U (1) s = U (2) s = Z s Z s dτ U 0 sτ H τ U 0 τ dτ 1 Z P ω = P 0 + ωp 1 + ω 2 P 2 +, wih P 1 = P 2 = dτ U 0 τ L 1,τ U 0 τt g τ 1 dτ 2 U 0 sτ 1 H τ1 U 0 τ 1 τ 2 H 2 U 0 τ 2 dτ Uτ 0 L 2,τ UτT 0 g + dτ 1 dτ 2 Uτ 0 1 L 1,τ1 Uτ 0 1 τ 2 L 1,τ2 Uτ 0 2 T g τ 1 We recover he expressions of P 1 and P 2. Quicker.

17 Firs example: a Heson-like model dx ω = 1 2 V ω d + p V ω dw 1, X0 ω = x (5), V0 ω = v dv ω = k (V ω v ) d + ω (V ω ) ϕ ρdw 1 + p 1 ρ 2 dw 2 The insananeous forward variance reads ξ u,ω and has dynamics = E [Vu ω V ω ] = v + (V ω dξ u,ω = ωe k(u ),ω `ξ ϕ ρdw 1 v ) e k(u ) + p 1 ρ 2 dw 2 The iniial erm-srucure of insananeous forward variances is y u ξ u 0 = v + (v v ) e ku Like in all classic firs generaion sochasic volailiy models, his erm-srucure is deermined by he model parameers, and he curren value of he insananeous volailiy.

18 The volailiy λ(, u, y) of insananeous forward variances depends on he insananeous forward variance curve y = (y s, s T ) only hrough he insananeous spo variance y : As a consequence, C Xξ = ρ k 2X C ξξ = C µ = i=1 0 0 ϕ ds (y s ) ϕ+ 1 2 λ 1(, u, y) = ρ `y ϕ e k(u ) λ 2(, u, y) = p 1 ρ 2 `y ϕ e k(u ) k(t 1 e s) «2 ds du λ i(s, u, y) = 1 s k 2 «ρ 2 k 0 ds (y s ) ϕ+ 1 2 s 0 ds (y s ) 2ϕ 1 e k(t s) 2 du (y u ) ϕ 1 2 e k(u s) k(t 1 e u) This coincides wih Equaions (3.7) o (3.10) in Lewis [5], where J (1) = C Xξ, J (3) = 1 2 Cξξ, and J (4) = C µ

19 Second example: he Bergomi model dx ω = 1 q 2 ξ,ω d + ξ,ω dw S dξ u,ω = ωξ u,ω α θ (1 θ) e k X (u ) dw X + θe k Y (u ) dw Y = ωλ(, u, ξ,ω ) dw d W S, W X = ρ SXd, d W S, W Y = ρ SY d, d W X, W Y = ρ XY d. The normalizing facor α θ = `(1 θ) 2 + 2ρ XY θ (1 θ) + θ 2 1/2 is such ha he very-shor erm variance ξ,ω has log-normal volailiy ω. We pick k X > k Y, θ is a parameer which mixes he shor-erm facor W X and he long-erm facor W Y.

20 Afer a Cholesky ransform, his can be resaed using independen Brownian moions W 1, W 2 and W 3 as follows: W S = W 1 W X = ρ SXW 1 + q 1 ρ 2 SX W 2 W Y = ρ SY W 1 + χ XY q1 ρ 2 SY W 2 + q (1 χ 2 XY ) (1 ρ2 SY )W 3 where χ XY = ρ XY ρ SX ρ SY 1 ρ 2 SX 1 ρ 2 SY ρ SX, ρ SY and ρ XY define a correlaion marix χ XY [ 1, 1]. The volailiy of variance λ = (λ 1, λ 2, λ 3) reads λ 1(, u, y) = y u α θ (1 θ) ρ SXe k X (u ) + θρ SY e k Y (u ) λ 2(, u, y) = y u α θ (1 θ) q «q1 ρ 2SX e k X (u ) + θχ XY 1 ρ 2 SY e k Y (u ) λ 3(, u, y) = y u α θ θ q (1 χ 2 XY ) (1 ρ2 SY )e k Y (u ) We wrie λ i(, u, y) = y u α θ w ixe k X (u ) + w iy e k Y (u )

21 The covariance funcions read C Xξ = C ξξ = C µ = 0 du Z u 0 d p y λ 1(, u, y) Z u = α θ (1 θ) ρ SX du y u d p y e k X (u ) 0 Z u +α θ θρ SY du y u d p y e k Y (u ) 3X i=1 = α 2 θ 0 3X i=1 0 0 Z u + ds s 0 ds du λ i(s, u, y) s 0 0 «2 ds w ix du y u e k X (u s) + w iy s du y s λ 1(s, u, y u 1 ) s 2 y u dr «y r λ1 (r, u, z) z=yu z u s dλ 1(u,, y ) «2 du y u e k Y (u s)

22 In he case of a fla iniial erm srucure of variance swaps (y 0 ξ), his reads C Xξ = α θ ξ 3/2 (w 1XJ (k X, T ) + w 1Y J (k Y, T )) C ξξ = α 2 θξ 2 (w 0T + w XI(k X, T ) + w Y I(k Y, T ) + w XXI(2k X, T ) +w Y Y I(2k Y, T ) + w XY I(k X + k Y, T )) C µ = «αθξ A1 + A2 wih I(k, u) = J (k, T ) = K(k, T ) = w 0 = Z u = 1 e ku 0 k Z u du = kt 1 + e kt 0 0 k 2 ds (T s)e k(t s) = 0 0 = 3X i=1 w ix k X 3X w iy w Y = 2 i=1 k Y + w iy k Y w ix k X! 2 X 3, wx = 2 w ix + w iy k Y 1 (1 + kt )e kt w ix i=1 k X k X! 3X w, w XX = ix 2 i=1 k X 2 k 2 + w iy k Y! w Y Y = 3X w iy 2 3X w ix w iy i=1 k Y 2, w XY = 2 i=1 k X k Y

23 and A 1 = w2 1X K(k X, T ) w 2 1Y k X k Y K(k Y, T ) + w X J `k X, T + w Y J `k Y, T w 1X w 1Y k X + k Y k X T 1 e k X T k X 2 + e k Y T 1 e k Y T k Y 2 e 2k X T 1 e k Y T + e 2k Y T 1 e k X T 1 A k X k Y A 2 = w X J `k X, T + w Y Y, T + w XX J `2k X, T + w Y Y J `2k Y, T + w XY J `k X + k Y, T wih w X = w 2 1X k X w X = w 2 1X k X + w 1X w 1Y, w Y = w 2 1Y k Y k Y + w 1X w 1Y, w Y = w2 1Y k Y k Y + w 1X w 1Y k X + w 1X w 1Y k X w XX = w2 1X, w Y Y = w2 1Y, w XY = w 1X w 1Y k X k Y k X w 1X w 1Y k Y

24 Firs order Numerical experimens We pick he Bergomi model wih a fla iniial erm srucure of variance swap prices and θ k X k Y ρ SX ρ SY ρ XY χ XY ξ (0.2) 2 ATM implied volaily 20.2% 20.0% 19.8% 19.6% 19.4% 19.2% 19.0% 18.8% 18.6% 18.4% mauriy in years omega = 20%, order 1 omega = 20%, MC omega = 60%, order 1 omega = 60%, MC omega = 200%, order 1 omega = 200%, MC

25 Firs order Firs order ATM skew 4.5% 4.0% 3.5% 3.0% 2.5% 2.0% 1.5% 1.0% omega = 20%, order 1 omega = 20%, MC omega = 60%, order 1 omega = 60%, MC omega = 200%, order 1 omega = 200%, MC 0.5% 0.0% mauriy in years

26 Firs order Firs order smile, omega = 60% 24% 23% 22% 21% 20% 1Y, order 1 1Y, MC 3Y, order 1 3Y, MC 8Y, order 1 8Y, MC 19% 18% 0% 50% 100% 150% 200% 250% srike as percenage of iniial spo

27 Firs order Firs order smile, omega = 200% 40% 35% 30% 25% 20% 1Y, order 1 1Y, MC 3Y, order 1 3Y, MC 8Y, order 1 8Y, MC 15% 10% 0% 50% 100% 150% 200% 250% srike as percenage of iniial spo

28 Firs order Firs order The ATM skew is very sharply esimaed by he firs order expansion, even for large values of he volailiy of variance ω. The ATM volailiy is well capured by he expansion a firs order in ω only for small values of ω (say, up o 60%). True ATM implied volailiies are below heir firs order approximaes he ATM volailiy is a very concave funcion of ω, around ω = 0. In view of he expression for IT AT M, his means ha, for he se of parameers picked, 12C Xξ2 C ξξ v (v + 4) + 4C µ v (v 4) 0 The global shape of he smile is well capured by he firs order expansion: he rue implied volailiy for srike K is indeed approximaely affine in ln(k/s 0), Bu he level of he smile is well capured only for small valus of ω.

29 Second order Second order We firs consider he siuaion when spo reurns and forward variances are uncorrelaed. In his case, he ATM skew vanishes, and so does is expansion a second order in ω. We pick θ k X k Y ρ SX ρ SY ρ XY ξ (0.2) 2 a-he-money implied volailiy 20.50% 20.00% 19.50% 19.00% 18.50% 18.00% omega = 60%, order 2 omega = 60%, MC omega = 120%, order 2 omega = 120%, MC omega = 200%, order 2 omega = 200%, MC omega = 400%, order 2 omega = 400%, MC 17.50% mauriy in years

30 Second order Second order smile, omega = 120% 22.0% 21.5% 21.0% 20.5% 20.0% 1Y, order 2 1Y, MC 3Y, order 2 3Y, MC 8Y, order 2 8Y, MC 19.5% 0% 50% 100% 150% 200% 250% 300% srike as percenage of iniial spo

31 Second order Second order smile, omega = 200% 26% 25% 24% 23% 22% 21% 20% 1Y, order 2 1Y, MC 3Y, order 2 3Y, MC 8Y, order 2 8Y, MC 19% 18% 0% 50% 100% 150% 200% 250% 300% srike as percenage of iniial spo

32 Second order Second order smile, omega = 400% 29% 27% 25% 23% 21% 1Y, order 2 1Y, MC 3Y, order 2 3Y, MC 8Y, order 2 8Y, MC 19% 17% 0% 50% 100% 150% 200% 250% 300% srike as percenage of iniial spo

33 Second order Second order The ATM implied volailiy is very sharply esimaed by he second order expansion, even up o ω = 400% and o long mauriies. For T = 15 years, he esimae is less han 15 bps above he rue ATM volailiy. Looking a he whole smile: he second order expansion of he implied volailiy is excellen around he money, bu becomes oo large for srikes far from he money. No surprising, because no arbirage requires ha for very small and very large srikes, he implied volailiy squared I(T, K) 2 grows a mos linearly wih ln(k/s 0) (see Lee [4]), whereas he second order esimae for I(T, K) 2 grows like ln 4 (K/S 0), see (4). This means ha he remainder O(ω 3 ) = R(ω, T, K) is large for large K, for finie ω. Neverheless, even for ω = 400%, a mauriy of 8 years and a deep ou-he-money srike of 250%, he error is only 1.5 poin of volailiy.

34 Second order Second order We now check numerically he accuracy of he second order expansion of he smile in he general case of correlaed spo reurns and variances. ATM implied volailiy, omega = 120% 20.00% 19.90% 19.80% 19.70% 19.60% 19.50% order 1 order 2 MC 19.40% 19.30% 19.20% mauriy in years

35 Second order Second order 20.2% ATM implied volailiy, omega = 200% 20.0% 19.8% 19.6% 19.4% 19.2% 19.0% order 1 order 2 MC 18.8% 18.6% 18.4% mauriy in years

36 Second order Second order smile, omega = 120% 24% 23% 22% 21% 20% 19% 18% 17% 1Y, order 2 1Y, MC 3Y, order 2 3Y, MC 8Y, order 2 8Y, MC 16% 15% 0% 50% 100% 150% 200% 250% srike as percenage of iniial spo

37 Second order Second order smile, omega = 200% 26% 24% 22% 20% 18% 16% 14% 1Y, order 2 1Y, MC 3Y, order 2 3Y, MC 8Y, order 2 8Y, MC 12% 10% 0% 50% 100% 150% 200% 250% srike as percenage of iniial spo

38 Shor-erm asympoics of implied volailiy Shor-erm asympoics of implied volailiy Assume dξ = d + ω(ξ) ϕ db Le ρ SV be he correlaion beween S and insananeous variance V = ξ Heson: ϕ = 1, ρsv = ρ; 2 Bergomi: ϕ = 1, ρ SV = α θ ((1 θ)ρ SX + θρ SY ) Then for shor mauriies qξ `IAT 00 + ρsv M 2ϕ T T ω + T 4ϕ 3 IT AT M `(5 8ϕ) ρ 2 SV 4 ω 2 + O `ω `IAT S T ρsv M 2ϕ 2 T ω + T 4ϕ 3 IT AT M ρ 2 2 SV ϕ 5 «ω 2 + O `ω 3 12 I AT M T κ T `IAT M 4ϕ 5 T ϕ 7 «ρ 2 SV + 1 «ω 2 + O `ω Shor-erm ATM skew does no depend on shor-erm ATM vol iff ϕ = 1 (observed in equiy markes) Shor-erm ATM convexiy does no depend on shor-erm ATM vol iff ϕ = 5 4. And ( ρsv, κt 0) ϕ 5 4

39 Long-erm asympoics of implied volailiy Long-erm asympoics of implied volailiy Assume he erm-srucure of variance swaps volailiies is fla: ξ 0 ξ. Assume ha for large u, µ(, u, y) (u ) α, α > 0. Then a higher order in ω, for long mauriies, S T T α if α < 1 S T T 1 if α > 1 Cf Bergomi, Smile Dynamics 4 [2], for he link wih he skew-sickiness raio. Assume ha for large u and u, ν(, u, u, y) (u ) α (u ) α, α > 0. Also assume ha spos and volailiies are uncorrelaed (µ 0). Then a higher order in ω, for long mauriies, Exponenial decay α > 1. κ T T 2α if α < 1 κ T T 2 if α > 1

40 Remember S T = CXξ 2v 3/2 ω + O `ω 2 T Le us now compue he skewness s T of log-reurns: s T = E ˆX T 3, E [XT 2 XT = XT E [XT ] = ]3/2 We have E ˆX R T 2 T =,ω R T E ˆξ 0 d = 0 ξ 0d and E ˆXT 3 = 3ωC Xξ + O `ω 2 0 q ξ,ω dw 1 A firs order in he vol of vol, he skewness of (he disribuion of) ln (S T /S 0) is hus 3ωC Xξ s T = R 3/2 T 0 ξ 0 d The ATM skew S T simply reads S T = st 6 T + O(ω2 )

41 Conclusion We have considered general second generaion sochasic volailiy models and derived an expansion of he smile of implied volailiy a second order in he volailiy of variance. A his order, he smile is quadraic in L = ln(k/s 0). This expansion shows ha he smile is driven by hree model-dependen quaniies: C Xξ, he inegraed spo-variance covariance funcion, C ξξ, he inegraed variance-variance covariance funcion, C µ, which, like C Xξ, depends only on he insananeous spo-variance covariance, bu in a more complex way. C Xξ drives he ATM implied volailiy and ATM skew, a firs order. When spo reurns are uncorrelaed wih variances, he smile is U-shaped, C Xξ = C µ = 0 and C ξξ drives boh he ATM implied volailiy and ATM convexiy. In he general case where spo reurns are correlaed wih variances, he second order correcion for ATM implied volailiy, he second order correcion for he ATM skew, and he ATM convexiy are all driven by a linear combinaion of `C Xξ 2, C ξξ and C µ.

42 Our derivaion relies on he fac ha in Model (1), he volailiy of he asse is an auonomous sochasic process, meaning ha i incorporaes no local volailiy componen, and ha λ does no depend on he asse value. These hree fundamenal covariance funcions are compued for Heson-like radiional sochasic volailiy models (In paricular we give a new derivaion of A. Lewis resuls [5]). They are also compued for he Bergomi model wih wo facors on he variance curve. In his case, numerical experimens show an excellen agreemen beween he esimae and he rue quaniy a firs order for he ATM skew, and a second order for he ATM implied volailiy, hence for he whole smile, up o ypical values of he volailiy of variance (say, 400% in he equiy marke). We have given shor-erm and long-erm behaviour of he smile. We have also rederived ha he ATM skew is jus he skewness of he disribuion of he log-spo, divided by 6 imes he square roo of mauriy, a firs order in he volailiy of variance.

43 Bergomi L., Smile Dynamics 2, Risk Magazine, pages 67-73, Ocober Bergomi L., Smile Dynamics 4, Risk Magazine, December Backus D., Foresi S., Li K. and Wu L., Accouning for Biases in Black-Scholes, unpublished. Lee R., The momen formula for implied volailiy a exreme srikes, Sanford Universiy and Couran Insiue, Lewis A., Opion valuaion under sochasic volailiy, Finance Press, 2000.