Short dated option pricing under rough volatility

Transcription

1 Department of Mathematics, Imperial College London Mathematics of Quantitative Finance Oberwolfach, February 2017 Based on joint works with P. Harms, A. Jacquier, C. Lacombe and with C. Bayer, P. Friz, A. Gulisashvili and B. Stemper

2 Implied volatility Asset price process: S t = e Xt ) t 0, with X 0 = 0. Black-Scholes-Merton BSM) framework: C BS τ, k, σ) := E 0 e Xτ e k) = N d+) + ek N d ), d ± := k σ τ ± 1 2 σ τ. Spot implied volatility σ τ k): the unique non-negative) solution to C observed τ, k) = C BS τ, k, σ τ k)). Implied volatility: unit-free measure of option prices.

3 Implied volatility Asset price process: S t = e Xt ) t 0, with X 0 = 0. Black-Scholes-Merton BSM) framework: C BS τ, k, σ) := E 0 e Xτ e k) = N d+) + ek N d ), d ± := k σ τ ± 1 2 σ τ. Spot implied volatility σ τ k): the unique non-negative) solution to C observed τ, k) = C BS τ, k, σ τ k)). Implied volatility: unit-free measure of option prices. Implied volatility is not available in closed form generally. Its asymptotic behaviour is available via small/large k, τ) approximations.

4 Literature Implied volatility σ τ k)) asymptotics as k, τ 0 or τ : Hagan-Kumar-Lesniewski-Woodward 2003/2015), Ob lój 2008): small-maturity for the SABR model. Berestycki-Busca-Florent 2004): small-τ using PDE methods for diffusions. Henry-Labordère 2009): small-τ asymptotics using differential geometry. Forde et al.2012), Jacquier et al.2012): small/ large τ using large deviations. Lee 2003), Benaim-Friz 2009), Gulisashvili ), Caravenna-Corbetta 2016), De Marco-Jacquier-Hillairet 2013): k. Laurence-Gatheral-Hsu-Ouyang-Wang 2012): small-τ in local volatility models. Mijatović-Tankov 2012): small-τ for jump models. Bompis-Gobet 2015): asymptotic expansions in the presence of both local and stochastic volatility using Malliavin calculus. Fouque et al ): perturbation techniques for slow and fast mean-reverting stochastic volatility models.

5 Related works: Deuschel-Friz-Jacquier-Violante CPAM 2014), De Marco-Friz 2014): small-noise expansions using Laplace method on Wiener space Ben Arous-Bismut approach). Baudoin-Ouyang 2015): small-noise expansions in a fully) fractional setting Gatheral-Jaisson-Rosenbaum 2014), and Bayer-Gatheral-Friz 2015), Forde-Zhang 2015): large deviations in a fractional stochastic volatility setting Fukasawa 2011,2015), Alós-León-Vives 2007) small-time fractional) skew Guennoun-Jacquier-Roome 2015), El Euch-Rosenbaum 2016) fractional Heston.

6 Classical case: H i = 1 ) case: Deuschel-Friz-Jacquier-Violante 2011) 2 dx ε t = bε, X ε t )dt + ε m i=1 σ i X ε t )dw i t, X ε 0 = xε 0 Rd

7 Classical case: H i = 1 ) case: Deuschel-Friz-Jacquier-Violante 2011) 2 dx ε t = bε, X ε t )dt + ε m i=1 σ i X ε t )dw i t, X ε 0 = xε 0 Rd Fractional case: H i = H 1, 1)) case: Baudoin-Ouyang 2015) 4 m dx ε t = bε, Xε t )dt + ε σ i X ε t )dw H ) i t, i=1 Xε 0 = xε 0 Rd

8 Classical case: H i = 1 ) case: Deuschel-Friz-Jacquier-Violante 2011) 2 dx ε t = bε, X ε t )dt + ε m i=1 σ i X ε t )dw i t, X ε 0 = xε 0 Rd Fractional case: H i = H 1, 1)) case: Baudoin-Ouyang 2015) 4 m dx ε t = bε, Xε t )dt + ε σ i X ε t )dw H ) i t, i=1 Xε 0 = xε 0 Rd Our main interest: H 1 = 1 2, H

9 Rough volatility models Short-term data suggests a time decay of the ATM skew proportional to τ θ, with θ 0, 1/2) while classical stochastic volatility models generate a constant short-maturity skew. Gatheral-Jaisson-Rosenbaum and Bayer-Gatheral-Friz 2014,1015) proposed a fractional volatility model: ds t = S tσ tdz t + µ tdt), σ t = exp Y t), 1) where dy t = µdw H t by t m)dt, for µ, b > 0, m R for a Bm Z and a fbm motion W H with Hurst parameter H. Time series of the Oxford-Man SPX realised variance as well as implied volatility smiles of the SPX suggest that H 0, 1/2): short-memory volatility. Main drawback: loss of Markovianity H 1/2) rules out PDE techniques, and Monte Carlo is computationally intensive. One way out is an efficient Hybrid scheme of Bennedsen, Lunde and Pakkanen 2015).

10 Rough volatility models: our setting dx t = b 1 Y t)dt + σ 1 Y t)dw t dy t = b 2 Y t)dt + σ 2 dw H t. 2) σ 2 > 0 extendible to bounded, elliptic) and H 0, 1), particular interest in H < 1/2. Forde-Zhang 16: b 1 b 2 0, and σ 1 C α, α 0, 1] second part of the talk); Fractional Stein-Stein: b 1 y) y 2 /2, σ 1 y) y, b 2 y) a by first part). Gatheral-Jaisson-Rosenbaum 14: b 1 y) e 2y /2, σ 1 y) e y, b 2 y) a by.

11 Rough volatility models: our setting dx t = b 1 Y t)dt + σ 1 Y t)dw t dy t = b 2 Y t)dt + σ 2 dw H t. 2) σ 2 > 0 extendible to bounded, elliptic) and H 0, 1), particular interest in H < 1/2. Forde-Zhang 16: b 1 b 2 0, and σ 1 C α, α 0, 1] second part of the talk); Fractional Stein-Stein: b 1 y) y 2 /2, σ 1 y) y, b 2 y) a by first part). Gatheral-Jaisson-Rosenbaum 14: b 1 y) e 2y /2, σ 1 y) e y, b 2 y) a by. To introduce correlation we consider B and B independent, and set W = ρ B + ρb and t Wt H = Kt, s)db s 0 where K the Volterra kernel of the standard) fbm W H.

12 Rough volatility models: our setting dx t = b 1 Y t)dt + σ 1 Y t)dw t dy t = b 2 Y t)dt + σ 2 dw H t. 2) σ 2 > 0 extendible to bounded, elliptic) and H 0, 1), particular interest in H < 1/2. Forde-Zhang 16: b 1 b 2 0, and σ 1 C α, α 0, 1] second part of the talk); Fractional Stein-Stein: b 1 y) y 2 /2, σ 1 y) y, b 2 y) a by first part). Gatheral-Jaisson-Rosenbaum 14: b 1 y) e 2y /2, σ 1 y) e y, b 2 y) a by. To introduce correlation we consider B and B independent, and set W = ρ B + ρb and t Wt H = Kt, s)db s 0 where K the Volterra kernel of the standard) fbm W H. An intuitive remark: Mandelbrot-van Ness representation for fbm: At time zero, the volatility process in 2) has already accumulated some randomness. Hot-start of the process Y with A. Jacquier and C. Lacombe).

13 1 Introduction 2 3 Refined expansions and moderate regimes A non-markovian extention of Osajima s energy expansion Implied volatility asymptotics

14 Varadhan-type asymptotics Recall the Black-Scholes density expansion: heat-kernel asymptotics f BS t, x) t 1/2 exp 1 x ) ) 2, as t 0. 2t σ In the homogenous fractional) case: H 1 = H 2 =... = H d = H Baudoin-Ouyang) dx t = bx t)dt + Extended) Varadhan formula m σ i X t)dw H ) i t, X 0 = x 0 R d 3) i=1 ) f X t, x) cst t H exp d2 x 0, x) 2t 2H, as t 0.

15 Varadhan-type asymptotics Recall the Black-Scholes density expansion: heat-kernel asymptotics f BS t, x) t 1/2 exp 1 x ) ) 2, as t 0. 2t σ In the homogenous fractional) case: H 1 = H 2 =... = H d = H Baudoin-Ouyang) dx t = bx t)dt + Extended) Varadhan formula m σ i X t)dw H ) i t, X 0 = x 0 R d 3) i=1 ) f X t, x) cst t H exp d2 x 0, x) 2t 2H, as t 0. What if Hurst parameters are different H 1 H 2?

16 Rescalings Key to asymptotic expansions for H 1 H 2 : Rescalings Recall the considered processes dx t = b 1 Y t)dt + σ 1 Y t)dw t dy t = b 2 Y t)dt + σ 2 dw H t.

17 Rescalings Key to asymptotic expansions for H 1 H 2 : Rescalings Recall the considered processes Define appropriate rescalings dx ε t dy ε t dx t = b 1 Y t)dt + σ 1 Y t)dw t dy t = b 2 Y t)dt + σ 2 dw H t. = b 1 ε κ 1, Y ε t )dt + εβ σ 1 Y ε t )dwt = b 2 ε κ 2, Y ε t )dt + ε β σ 2 dw H t.

18 Rescalings dx ε t = b 1 ε κ 1, Y ε t )dt + ε β σ 1 Y ε t )dw t dy ε t = b 2 ε κ 2, Y ε t )dt + ε β σ 2 dw H t. 4) Fractional Stein-Stein: Consider X 0, Y 0 ) = 0, y 0 ) and dx t = Y t 2 H dt + YtdWt, dyt = a byt)dt + cdwt 2. 5)

19 Rescalings dx ε t = b 1 ε κ 1, Y ε t )dt + ε β σ 1 Y ε t )dw t dy ε t = b 2 ε κ 2, Y ε t )dt + ε β σ 2 dw H t. 4) Fractional Stein-Stein: Consider X 0, Y 0 ) = 0, y 0 ) and dx t = Y t 2 H dt + YtdWt, dyt = a byt)dt + cdwt 2. 5) Rescaling 1 short-time): Xt ε, Y t ε) := ε2h 1 X ε 2 t, Y ε 2 t ) 4) with κ 1 = 2H + 1, κ 2 = 2 and β = 2H, x0 ε, y 0 ε) = 0, y 0): dx ε t 2H+1 Y ε t = ε )2 dt + ε 2H Yt ε 2 dwt, dy ε t = ε 2 a by ε t )dt + ε2h cdw H t. 6)

20 Rescalings dx ε t = b 1 ε κ 1, Y ε t )dt + ε β σ 1 Y ε t )dw t dy ε t = b 2 ε κ 2, Y ε t )dt + ε β σ 2 dw H t. 4) Fractional Stein-Stein: Consider X 0, Y 0 ) = 0, y 0 ) and dx t = Y t 2 H dt + YtdWt, dyt = a byt)dt + cdwt 2. 5) Rescaling 1 short-time): Xt ε, Y t ε) := ε2h 1 X ε 2 t, Y ε 2 t ) 4) with κ 1 = 2H + 1, κ 2 = 2 and β = 2H, x0 ε, y 0 ε) = 0, y 0): dx ε t 2H+1 Y ε t = ε )2 dt + ε 2H Yt ε 2 dwt, dy ε t = ε 2 a by ε t )dt + ε2h cdw H t. Rescaling 2 tails): X ε t, Y ε t ) := ε2h X t, ε H Y t) 4) with κ 1 = 0, κ 2 = β = H, x ε 0, y ε 0 ) = 0, εh y 0 ): dx ε t = Y t ε)2 dt + ε H Yt ε 2 dwt, dy ε t 6) = aε H + by ε t )dt + εh cdw H t. 7)

21 Theorem Harms-H-Jacquier) Consider an SDE of the form 4). Then the density of XT ε admits an expansion f εt, x) = exp Λx) ε 2β + Λ x) X ) ) T ε β ε minκ 1,β) c 0 + Oε δκ1,β) ), as ε 0, where and d X ) t = [ x b 1 0, φ k 0 t + x σ 1 { } 1 Λx) = inf 2 k 2 H H, k K x x0 0,y 0 0 = 1 2 k 0 2 H H, φ h 0 t ) k ] 0 t) Xtdt+ ε β b 1 0, φ k 0 t ) dt, X0 = ε β x ε 0 ε=0, where φ k 0 denotes the ODE solution of the same SDE 4) replacing ε β dw by k 0 and x0 ε by x0 0.

22 Notations H: absolutely continuous paths [0, T ] R 2 starting at 0 such that ḣ 2 <. H H H := K H H and k := K H h, where K H denotes the Volterra kernel. For fixed x 0, y 0 ) R 2, φ k is the unique) ODE solution to ) ) ) φ k t = b 1 0, φ k t dt + σ 1 φ h t dk 1 t + σ 2 φ h t dk 2 t, φk 0 = x0 0, y 0 0 ). Denote ψ k := Π 1 φ k its projection on to the first coordinate X. K a := { k H H : ψt k = a R} by Hörmander condition ). { } Λa) := inf 1 2 k 2 H : k Ka.

23 Corollary short-time asymptotics in Stein-Stein) Corollary: Varadhan-type asymptotics dy t = a by t)dt + cdw H t In the fractional Stein-Stein model X t, Y t) with X 0 = 0, Y 0 = y 0 > 0 the density of X t satisfies in a neighbourhood of x 0, y 0 ) the following asymptotic expansion as t 0 f X t, x) = exp Λx) ) ) 1 t 2H t H 2π + OtδH,H+1/2) ) where { } 1 Λx) = inf 2 k 2 H H, k Kx x 0,y 0.

24 Corollary short-time asymptotics in Stein-Stein) Corollary: Varadhan-type asymptotics dy t = a by t)dt + cdw H t In the fractional Stein-Stein model X t, Y t) with X 0 = 0, Y 0 = y 0 > 0 the density of X t satisfies in a neighbourhood of x 0, y 0 ) the following asymptotic expansion as t 0 f X t, x) = exp Λx) ) ) 1 t 2H t H 2π + OtδH,H+1/2) ) where { } 1 Λx) = inf 2 k 2 H H, k Kx x 0,y 0. Proof: Take T = 1, ε 2 = t and consider Xt ε, Y t ε) := ε2h 1 X ε 2 t, Y ε 2 t ) with X0 ε = 0, Y 0 ε = y 0 > 0. Short-time scaling: dx ε t 2H+1 Y ε t = ε )2 dt + ε 2H Yt ε 2 dwt, dy ε t = ε 2 a + by ε t )dt + ε2h cdw H t, 6) Note that the drift vanishes in the limit ε 0 and x ε 0 = x 0 = 0. X t, Ŷt) 0, so that there is no 1/εβ = 1/t β/2 term in the exponential.

25 Corollary tail expansion in Stein-Stein) Corollary: tail asymptotics dy t = a + by t)dt + cdw H t Consider the fractional Stein-Stein model with X 0 = 0, Y 0 = y 0 > 0. Then as x, f X T, x) = exp c 1 x + c 2 x 1/2) 1 x 1/2 c 0 + O x 1/2)) where c 1 := Λ1), c 2 := X T Λ 1). Note that the expression on the RHS is independent of the Hurst-parameter!

26 Corollary tail expansion in Stein-Stein) Corollary: tail asymptotics dy t = a + by t)dt + cdw H t Consider the fractional Stein-Stein model with X 0 = 0, Y 0 = y 0 > 0. Then as x, f X T, x) = exp c 1 x + c 2 x 1/2) 1 x 1/2 c 0 + O x 1/2)) where c 1 := Λ1), c 2 := X T Λ 1). Note that the expression on the RHS is independent of the Hurst-parameter! Proof: Consider X ε T, Y ε T ) := ε2h X T, ε H Y T ) with X ε 0 = ε2h X 0 and Y ε 0 = εh Y 0. dx ε t = Y t ε)2 dt + ε H Yt ε 2 dwt, dy ε t = aε H + by ε t )dt + εh cdw H t, 7) Note that XT ε = ε 2H X T. PXT ε y) = PX T y/ε 2H ), f X T, y/ε 2H ) = ε 2H f εt, y). ) Take y = 1, that is x := ε 2H. By the theorem, f εt, 1) exp Λ1) ε H ) ε, hence H f X T, x) exp Λ1) ε +... ε H = exp Λ1)x +...) 2H 1 x 1/2.

27 From density to implied volatility: small-time Recall the Black-Scholes density expansion: f BS t, x) t 1/2 exp 1 x ) ) 2, as t 0, for any x R. 2t σ The corollary Varadhan-type asymptotics) implies that in the fractional Stein-Stein model ) f X t, x) cst t H exp d2 x 0, y 0 ; x) 2t 2H, as t 0.

28 From density to implied volatility: small-time Recall the Black-Scholes density expansion: f BS t, x) t 1/2 exp 1 x ) ) 2, as t 0, for any x R. 2t σ The corollary Varadhan-type asymptotics) implies that in the fractional Stein-Stein model ) f X t, x) cst t H exp d2 x 0, y 0 ; x) 2t 2H, as t 0. Matching the leading-orders gives σ BS t, x) x dx 0, y 0 ; x) th 1/2 as t 0. Skew explodes with rate H 1/2 in the short end whenever H < 1/2.

29 From density to implied volatility: tails Recall the Black-Scholes density expansion: f BS t, x) exp x2 2σ 2 t x ) 4 as x, for any t > 0. Our theorem corollary) says that in the fractional Stein-Stein model 5), we have f X t, x) cst x 1/2 exp c 1 x + c 2 x ), as x.

30 From density to implied volatility: tails Recall the Black-Scholes density expansion: f BS t, x) exp x2 2σ 2 t x ) 4 as x, for any t > 0. Our theorem corollary) says that in the fractional Stein-Stein model 5), we have f X t, x) cst x 1/2 exp c 1 x + c 2 x ), as x. Matching the leading-orders gives x 2 c 1 x + c 2 x 2σ 2 t x 4, and we recover Roger Lee s formula independently of the Hurst exponent in 5).

31 Proof of Theorem 1 dx t = ε 2H Y 2 t dt + ε 2H Y tdw t, dy t = ε 2H dw H t, with the same initial condition X 0 = Y 0 = 0. [ Density: f εt, x) = exp Λx) ε 4H + Λ x) X ] ) T ε 2H ε 2H c 0 + Oε 2H ).

32 Proof of Theorem 1 dx t = ε 2H Y 2 t dt + ε 2H Y tdw t, dy t = ε 2H dw H t, with the same initial condition X 0 = Y 0 = 0. [ Density: f εt, x) = exp Λx) ε 4H + Λ x) X ] ) T ε 2H ε 2H c 0 + Oε 2H ). Proof: Take x R and a C -bounded function F such that F x) = 0. f εt, x)e F x)/ε4h = 1 { [ 2πε 2H E exp iζ, 0) R X ε T x, 0) ε 2H ) F X T ε ) ]} ε 4H dζ. Choose F such that F ) + Λ x0 ) has a non-degenerate minimum at z. This implies that k F φ k T x 0, y 0 )) k 2 H H has a non-degenerate minimum at k 0 H H. For instance F z) = λ z x 2 [Λ x0,y 0 z) Λ x0,y 0 x)] with λ > 0).

33 Proof of Theorem 1 Replace ε 2H db B := W, W H )) in the SDE by ε 2H dw + k 0. Call the corresponding Girsanov-transformed process Z t ε = X t ε, Ỹ t ε): d X t ε = ε 2H+1 1 Ỹt 2 dt + 2 Ỹ t ε ε 2H dw t + k 0 ) 1 ), Girsanov factor dỹt = ε2h dw H t + k 0 ) 2. G = exp 1 T ε 2H ψk 0 ) tdb t 1 ) 0 2ε 4H k 0 2 H H. Therefore f x, T )e F x)/4ε4h = 1 2πε 2H R = 1 2πε 2H R where [ E e ε2h iζ X T x) ε 4H F X ] T ) G dζ [ E e )] dζ T ) = ε 2H iζ X T x) ε 4H F X T ) ε 2H ψγ) tdb t ε 4H γ 2 H H.

34 Proof of Theorem 1 Replace ε 2H db B := W, W H )) in the SDE by ε 2H dw + k 0. Call the corresponding Girsanov-transformed process Z t ε = X t ε, Ỹ t ε): d X t ε = ε 2H+1 1 Ỹt 2 dt + 2 Ỹ t ε ε 2H dw t + k 0 ) 1 ), Girsanov factor dỹt = ε2h dw H t + k 0 ) 2. G = exp 1 T ε 2H ψk 0 ) tdb t 1 ) 0 2ε 4H k 0 2 H H. By a stochastic Taylor expansion of Z t ε ) exp F X ε t ε 4H = exp = X ε t, Ỹ ε t ) for ε2h 0, [ 1 T )] F ε 4H x) ε 2H ψk 0 ) tdb t ε 2H XT Λ x 0 x) + Oε 4H ) 0

35 Proof of Theorem 1 Replace ε 2H db B := W, W H )) in the SDE by ε 2H dw + k 0. Call the corresponding Girsanov-transformed process Z t ε = X t ε, Ỹ t ε): d X t ε = ε 2H+1 1 Ỹt 2 dt + 2 Ỹ t ε ε 2H dw t + k 0 ) 1 ), Girsanov factor dỹt = ε2h dw H t + k 0 ) 2. G = exp 1 T ε 2H ψk 0 ) tdb t 1 ) 0 2ε 4H k 0 2 H H. By a stochastic Taylor expansion of Z t ε ) exp F X ε t ε 4H = exp = X ε t, Ỹ ε t ) for ε2h 0, [ F x) ε 4H + 1 T ε 2H ψk 0 ) tdb t + 1 ] X 0 ε 2H T Λ x 0 x) + O1).

36 Proof of Theorem 1 Replace ε 2H db B := W, W H )) in the SDE by ε 2H dw + k 0. Call the corresponding Girsanov-transformed process Z t ε = X t ε, Ỹ t ε): where d X t ε = ε 2H+1 1 Ỹt 2 dt + 2 Ỹ t ε ε 2H dw t + k 0 ) 1 ), dỹt = ε2h dw H t + k 0 ) 2. G = exp 1 T ε 2H ψk 0 ) tdb t 1 ) 0 2ε 4H k 0 2 H H. f x, T )e F x)/4ε4h = 1 [ 2πε 2H E gz ε )e ε 4H F X ] T ) G dζ R = 1 [ 2πε 2H E gz ε )e ε2h R e 1 ε 4H F x)+ 1 ] 2 k 0 HH ) 1 exp ε 2H X T Λ x x) 0 dζ R ) = ε 4H F x) ε 2H T T ε 2H ψγ) tdb t ε 4H γ 2 H H. 0 ) ψk 0 ) tdb t ε 2H XT Λ x 0 x)

37 Refined expansions and moderate regimes A non-markovian extention of Osajima s energy expansion Implied volatility asymptotics 1 Introduction 2 3 Refined expansions and moderate regimes A non-markovian extention of Osajima s energy expansion Implied volatility asymptotics

38 Refined expansions and moderate regimes A non-markovian extention of Osajima s energy expansion Implied volatility asymptotics Direct call price expansion: For x 0 the option price satisfies cε 1 2H x, t) : = E [expx t) expε 1 2H x)) +] = E [expz t) expε 1 2H x)) + G ] I x)g 1 e ε 2H where Z is the controlled process around the optimal path k and G I x) = e ε 4H is the Girsanov factor for the optimal path, and g 1 a Gaussian random variable. Then cε 1 2H x, t) = exp [ Jε, x) := E exp I x) ε 4H ) exp xε 1 2H) Jε, x), where Û := Ẑ 1 ε x and I x) ε 2H ) Û expε 1 2H Û) 1) exp ] I ) x)r 2 1Û 0. Expansion uniformly in x) via epansion of the energy I directly Osajima).

39 Refined expansions and moderate regimes A non-markovian extention of Osajima s energy expansion Implied volatility asymptotics Moderate regimes Moderate Regimes in the sense of Friz-Gerhold-Pinter ) 16) interpolate between out-of-the-money calls with fixed strike log K = k > 0 and at-the-money S0 k = 0 calls: Now k t = ct θ MOTM for 0 < θ < 1 ) and AATM for larger θ) 2 Reflects market reality: options closer expiry strikes closer to the money first observed by Mijatović-Tankov on FX markets The moderate regime MOTM) permits explicit computations for the rate function Λk) in terms of the model parameters Moderate deviations Advantage over large deviations OTM) case where the Λk) often related to geodesic distance problems and not explicitly available. MOTM expansions naturally involve quantities very familiar to practitioners, notably spot implied) volatility, implied volatility skew... In some cases fractional volatility models) the scaling θ permits a fine-tuning to understand the behavior and derivatives of the energy function.

40 Refined expansions and moderate regimes A non-markovian extention of Osajima s energy expansion Implied volatility asymptotics Moderate regimes for rough volatility Rescalings = We tacitly agreed to consider P X t t 1/2 H x ). Now it is only a small step to consider instead for some suitable small θ > 0) ) P X t t 1/2 H+θ x.

41 Refined expansions and moderate regimes A non-markovian extention of Osajima s energy expansion Implied volatility asymptotics Moderate regimes for rough volatility Rescalings = We tacitly agreed to consider P X t t 1/2 H x ). Now it is only a small step to consider instead for some suitable small θ > 0) ) P X t t 1/2 H+θ x. Theorem Bayer-Friz-Gulisashvili-H-Stemper) Consider a moderately out-of-the-money call k t = xt 1/2 H+θ ; θ 0, H) resp. θ 0, 2H ). Then as t 0, the following holds 3 log ck t, t) 1 x 2 2 Λ 0) t 2H 2θ + 1 x 3 6 Λ 0) t 2H 3θ, where we have explicit expressions: Λ 0) = 1 σ 2 0 Here K denotes the Volterra kernel and K, 1 := 1 0 and Λ 0) = ρ 6σ 0 σ0 4 K, 1. t 0 Kt, s)dsdt.

42 Refined expansions and moderate regimes A non-markovian extention of Osajima s energy expansion Implied volatility asymptotics Moderate regimes for rough volatility Corollary MOTM Implied volatility skew) In the moderately out-of-the-money case k t = xt 1/2 H+θ ; θ 0, H); θ 0, 2H )) the 3 implied volatility satisfies the expansion σ impl k t, t) = σ 0 ρ σ 0 σ 0 1 t 0 0 Kt, s) dsdt k t t H 1/2 ) 1 + o1).

43 Refined expansions and moderate regimes A non-markovian extention of Osajima s energy expansion Implied volatility asymptotics Moderate regimes for rough volatility Corollary MOTM Implied volatility skew) In the moderately out-of-the-money case k t = xt 1/2 H+θ ; θ 0, H); θ 0, 2H )) the 3 implied volatility satisfies the expansion σ impl k t, t) = σ 0 ρ σ 0 σ 0 1 t 0 0 Kt, s) dsdt k t t H 1/2 ) 1 + o1). Proof: The statement follows from the Theorem via matching components in the asymptotic expansions and by see Gao-Lee 2014)) using tσimpl 2 kt 2 kt, t) 2 log ck t, t).

44 Refined expansions and moderate regimes A non-markovian extention of Osajima s energy expansion Implied volatility asymptotics Moderate regimes for rough volatility Corollary MOTM Implied volatility skew) In the moderately out-of-the-money case k t = xt 1/2 H+θ ; θ 0, H); θ 0, 2H )) the 3 implied volatility satisfies the expansion σ impl k t, t) = σ 0 ρ σ 0 σ 0 1 t 0 0 Kt, s) dsdt k t t H 1/2 ) 1 + o1). Proof: The statement follows from the Theorem via matching components in the asymptotic expansions and by see Gao-Lee 2014)) using tσimpl 2 kt 2 kt, t) 2 log ck t, t). This formula for the skew is in accord with ones previously derived by Alòs-León-Vives 2007) and Fukasawa 2011, 2016) in different settings.

45 Refined expansions and moderate regimes A non-markovian extention of Osajima s energy expansion Implied volatility asymptotics Thank you!