Short-term at-the-money asymptotics under stochastic volatility models
Omar El Euch Masaaki Fukasawa Jim Gatheral Mathieu Rosenbaum
q-fin.CP math.PR 91G99

   i-Josephson Junction as Topological  Superconductor 
    Jiangping Hu
    December 30, 2023
=======================================================


A small-time Edgeworth expansion of the density
 of an asset price is given
 under a general stochastic volatility model,
 from which asymptotic expansions of put option prices and 
 at-the-money implied volatilities follow.
 A limit theorem for at-the-money implied volatility skew and
 curvature
is also given as a corollary. The rough Bergomi model is treated as an example.
 



§ INTRODUCTION

Stochastic volatility models are extensions of the Black-Scholes model
that explain a number of empirical evidences.
The Heston and SABR models among them are popular in financial
practice owing to (semi-)analytic (approximation) formulas for the vanilla option
prices or the option-implied volatilities.
For a practical guide on stochastic volatility modeling,
we refer to <cit.>.
Recently, attracting much attention is a class of stochastic volatility
models where the volatility is driven by a fractional Brownian motion
with the Hurst parameter smaller than 1/2.
This is due to their consistency to a power law for the term structure of
the implied volatility skew that has been empirically recognized;
see <cit.>.
The small Hurst parameter
implies in particular  that the volatility path is rougher than a Brownian
motion and so, this class is often referred as the rough
volatility models.
Since the models do not admit of explicit expressions for option prices
or implied volatilities, the above mentioned consistency has been
discussed through asymptotic analyses.

The aim of this paper is to provide a general framework under which the
short-term asymptotics of the at-the-money implied volatility is
studied.
 Here, by short-term asymptotics, we mean the asymptotics
with time-to-maturity θ→ 0. 
By at-the-money, we mean a regime of log-moneyness k=
O(√(θ)).
The framework is for a general continuous stochastic volatility model.
The rough Bergomi model introduced by <cit.> is treated as an example.
The asymptotic expansion of the at-the-money implied volatility is given
up to the second-order.


The first-order expansion was already given in
<cit.> by a different method.
 For the SABR model, Osajima <cit.> gave the second-order 
expansion based on the
Watanabe-Yoshida theory; see e.g., <cit.>.
 For a Markov stochastic volatility model with jumps,
Medvedev and Scaillet <cit.> derived 
the expansion  by a formal computation.
For the Markov diffusion case, 
Pagliarani and Pascucci <cit.> proved the validity of the Taylor
 expansion.
An expansion of the  at-the-money implied volatility 
skew is derived under a Lévy jump model with Markov
stochastic volatility
 by Figueroa-López and Ólafsson <cit.>.
Beside these results for the at-the-money regime, considering
near-the-money, that is, a moderate deviation regime,
Friz et al. <cit.> derived the asymptotic skew and curvature of the
implied volatility 
by assuming the asymptotic behavior of the density function of the
underlying asset price.
 Recently, 
Bayer et al. <cit.> has extended the moderate deviation analysis
 to a rough volatility model.


In this paper, we introduce a novel approach based on the
conditional Gaussianity
of a  continuous stochastic volatility model to prove the validity of a second
order density expansion, from which 
follow expansions of the option prices and the implied volatility as well as the asymptotic skew and curvature formula.
 In contrast to <cit.>, we do not rely on the Malliavin
 calculus, which enables us to treat effectively the rough volatility models.
In contrast to the elementary method of <cit.>, our approach can
be extended to higher-order expansions without any
 additional theoretical difficulty.
We choose the square root of the forward variance, that is, the fair strike of a variance swap, as the leading term of our asymptotic expansion, while a recent work <cit.> studies the difference between the implied volatility and the fair strike of a volatility swap in terms of the Malliavin derivatives.
 
The paper is organized as follows. In Section 2, we describe assumptions and general results.
In Section 3, we give the proofs of the general results.
In Section 4, we treat regular stochastic volatility models.
In Section 5, we show that the rough Bergomi model fits into the framework as well
and
compute the coefficients of the expansion for this particular model.

 

 

§ FRAMEWORK

 

 §.§ Assumptions

Let (Ω,ℱ,Q) be a 
probability space equipped with a filtration
{ℱ_t ; t ≥ 0} satisfying the usual assumptions.
A log price process Z is assumed to follow

    dZ_t = rdt - 1/2 v_t dt
     + √(v)_tdB_t,

 where r∈ℝ stands for an interest rate and v is a
 positive continuous process adapted to
 a smaller filtration {𝒢_t; t ≥ 0}, of which the square root is called the volatility
 of Z. The Brownian motion B is decomposed as
 
    dB_t = ρ_t dW_t +
     √(1-ρ^2_t)dW^'_t,

 where W^' is an {ℱ_t}-Brownian motion independent of
 𝒢_t for all t≥ 0,
 W is a {𝒢_t}-Brownian motion and ρ is a progressively
 measurable processes with respect to {𝒢_t} and taking
 values in [-1,1].
A typical situation for stochastic volatility models, including the Heston, SABR and rough Bergomi models,  is that (W,W^') is a two dimensional
{ℱ_t}-Brownian motion and {𝒢_t } is the
filtration generated by W, that is,

    𝒢_t = 𝒩∨σ(W_s; s ≤ t),

where 𝒩 is the null sets of ℱ.
Denote by ·_p the L^p norm under  Q.
Our key assumption is the following:
for any p > 0, 

    sup_θ∈ (0,1)1/θ∫_0^θ v_t dt_p < ∞,  sup_θ∈ (0,1){1/θ∫_0^θ v_t(1-ρ_t^2) dt }^-1_p
    < ∞.

This is satisfied by standard stochastic volatility models (with correlation parameter |ρ| < 1) but not by local volatility models that correspond to
 ρ≡ 1.




An arbitrage-free price p(K,θ) of a put option at time 0 with strike K>0 and
maturity θ>0 is given by

    p(K,θ) = e^-rθE[(K- exp(Z_θ))_+] =
    e^-rθ∫_0^K Q(log x ≥ Z_θ ) dx.

The forward variance curve v_0(t) at time 0 is defined by
v_0(t) = E[v_t].
Changing variable as

    x = Fexp(ζσ_0(θ) ),   
    F = exp(rθ + Z_0),

where

    σ_0(θ) = √(∫_0^θ v_0(t)dt),

we have

    p(Fe^zσ_0(θ),θ)/Fσ_0(θ)
    =e^-rθ∫_-∞^z Q( ζ≥ X_θ) e^σ_0(θ)ζdζ,

where

    X_θ = -1/2σ_0(θ)⟨ M
    ⟩_θ + 1/σ_0(θ) M_θ,   
    M_θ = ∫_0^θ√(v_t)dB_t,  ⟨ M ⟩_θ = ∫_0^θ v_t dt.

Based on this expression, the asymptotic behavior of put option prices is studied through the asymptotic distribution of X_θ.
From the martingale central limit theorem[
The martingale central limit theorem for one-dimensional continuous
local martingales
is proved as follows. Let M^n be a continuous local
martingale with ⟨ M^n ⟩_1 → 1 in probability. 
By the Dambis-Dubins-Schwarz theorem, M^n = W^n_⟨ M^n
⟩ for a Brownian motion W^n. Since (W^n,⟨ M^n
⟩_1) → (W,1) in law, by the continuous mapping theorem, we conclude
M^n_1 → W_1 in law.], 
it is not difficult to see that  X_θ converges in law to
the standard normal distribution
as θ→ 0.
To determine higher-order asymptotic distribution,  
we assume the following structure:
there exists a family of random vectors

    { (M^(0)_θ, M^(1)_θ, M^(2)_θ,
    M^(3)_θ) ; θ∈ (0,1) }

such that 


  * the law of M^(0)_θ is standard normal for all θ
>0,

  * 
    sup_θ∈ (0,1)M^(i)_θ_p < ∞,   i=1,2,3

for all p > 0 and 

  * for some H ∈ (0,1/2] and ϵ∈ (0,H),

    lim_θ→ 0θ^-2H - 2ϵM_θ/σ_0(θ) - M^(0)_θ - θ^H
    M^(1)_θ - θ^2H M^(2)_θ_1+ϵ
    =0, 
       lim_θ→ 0θ^- H-2 ϵ⟨ M ⟩_θ/σ_0(θ)^2 - 1 - θ^H M^(3)_θ_1+ϵ
    =0.


Further, we assume the existence of the derivatives

    a^(i)_θ(x) = d/dx{E[M^(i)_θ |
    M^(0)_θ=x]ϕ(x)},   i= 1,2,3, 
       b_θ(x) = d^2/dx^2{ E[M^(1)_θ |
    M^(0)_θ=x] ϕ(x) }
       c_θ(x) = d^2/dx^2{ E[|M^(1)_θ|^2 |
    M^(0)_θ=x] ϕ(x) }

in the Schwartz space (i.e., the space of the rapidly decreasing smooth
functions), where ϕ is the standard normal density. 

As will be discussed in Section 4, regular stochastic volatility models satisfy these assumptions with H = 1/2, where (<ref>) is a consequence of the Itô-Taylor expansion.
In Section 5, we see that the rough Bergomi model, where the volatility is driven by a fractional Brownian motion, 
satisfies these assumptions with H being the Hurst parameter of the fractional Brownian motion.



 §.§ General results

The fundamental result in this paper is the following.

 
   The law of  X_θ admits a density  p_θ,  and
   for any α∈ℕ∪{0}, 
   
    sup_x ∈ℝ
       (1+x^2)^α|p_θ(x) - q_θ(x)| = o(θ^2H)

   as θ→ 0, where
  
    q_θ(x) =     ϕ(x + σ_0(θ)/2) - θ^H (a^(1)_θ(x + σ_0(θ)/2)-  σ_0(θ)/2 a^(3)_θ(x + σ_0(θ)/2)) 
        -
      θ^2H( a^(2)_θ(x) -  1/2c_θ(x) ).


The proof is given in Section 3.2. In order to derive a neat asymptotic expansion of the put option prices, we introduce an additional assumption which is satisfied by the models in Sections 4 and 5.


Suppose we have (<ref>) with q_θ of the form
  
    q_θ(x)   =     ϕ(x + σ_0(θ)/2) { 1 + 
     κ_3(θ) (H_3 (x + σ_0(θ)/2)- σ_0(θ) H_2 (x + σ_0(θ)/2))θ^H 
    }
        + ϕ(x) (κ_4(θ)  H_4(x) +
      κ_3(θ)^2/2H_6(x)) θ^2H

with bounded functions κ_3(θ)  and κ_4(θ) of θ, 
 where H_k is the kth Hermite polynomial:
  
    H_1(x) = x,  H_2(x) = x^2-1,   
      H_3(x) = x^3-3x,   H_4(x) = x^4 - 6x^2 + 3,  …

  Then, for any z_0 ∈ℝ,
  
    p(Fe^σ_0(θ)z,θ)/Fe^-rθσ_0(θ)    = 
    1/σ_0(θ)(Φ(z + σ_0(θ)/2)e^σ_0(θ)z-Φ(z-σ_0(θ)/2) ) 
        + κ_3(θ) ϕ(z+σ_0(θ)/2) 
      H_1(z+σ_0(θ)/2)e^σ_0(θ)zθ^H 
        
       
     + ϕ(z)(κ_4(θ)  H_2(z) +
      κ_3(θ)^2/2H_4(z)) θ^2H+ o(θ^2H)

  uniformly in z ≤ z_0.
  
The proof is given in Section 3.3. Under the same assumption,  an asymptotic expansion of the Black-Scholes implied volatility follows.
  Denote by p_BS(K,θ,σ) the put option price
  with strike price K and maturity θ under the Black-Scholes
  model with volatility parameter σ > 0.
  Given a put option price p(K,θ), K = Fe^k,
the Black-Scholes implied volatility σ_BS(k,θ) is defined through
  
    p_BS(K,θ,σ_BS(k,θ)) = p(K,θ).

The at-the-money implied volatility skew and curvature
are defined respectively as the first and the second derivatives in k of the
Black-Scholes implied volatility   at k = 0.
   The skew behavior is especially important in order to argue the
   consistency of a  model to the empirically observed power law.

   
Suppose we have (<ref>) with q_θ of the form (<ref>).
Then, 
   for any z ∈ℝ,
  
    σ_BS(√(θ)z,θ) 
        =
      κ_2{
      1 + κ_3 (
       z/κ_2 + κ_2√(θ)/2) θ^H
      + (  3 κ_3^2/2-κ_4 + 
      (κ_4 - 3κ_3^2) z^2/κ_2^2) θ^2H}
      + o(θ^2H),

   where κ_2 = κ_2(θ) = σ_0(θ)/√(θ),
  κ_3 = κ_3(θ) and κ_4 = κ_4(θ).
 
    
  
Suppose we have (<ref>) with q_θ of the form (<ref>).
Then, 
      
    ∂_k σ_BS(0,θ)
          = κ_3(θ)θ^H-1/2 + o(θ^2H-1/2), 
       ∂_k^2 σ_BS(0,θ)
    	    = 2κ_4(θ) - 3κ_3(θ)^2
      /κ_2(θ)θ^2H-1 + o(θ^2H-1).

      
The proofs are given in Sections 3.4 and 3.5 respectively.




Remark: 
The above asymptotic estimates are not uniform in H; the assumed stochastic expansion (<ref>) is not uniform in H and so, there seems no hope to have uniformity. We would have uniformity in H ∈ [H_0, 1/2] for some H_0 > 0 if we could strengthen the condition (<ref>) to uniform convergence on [H_0,1/2]. It seems impossible to argue the uniformity in H ∈ (0,1/2] because Lemma 3.2 below requires  some ϵ-δ argument depending on H.



§ PROOFS



 §.§ Characteristic function expansion

Here we give an asymptotic expansion of the characteristic function of
X_θ. Let

    Y_θ = M^(0)_θ + θ^H M^(1)_θ +
    θ^2HM^(2)_θ
    - σ_0(θ)/2( 1 + θ^H M^(3)_θ).

 
 Let H ∈ (0,1/2] and
ϵ∈ (0,H) be constants under
 which (<ref>) holds. Then, 
 for any α∈ℕ∪{0},

    sup_|u|≤θ^-ϵ |E[X_θ^α e^iuX_θ]
    - E[Y_θ^α e^iuY_θ]| = o(θ^2H + ϵ).


Proof: 
Since |e^ix-1|≤ |x|, we have

    |E[X_θ^α e^iuX_θ]
    - E[Y_θ^α e^iuY_θ]|
       ≤ |E[(X^α_θ -Y^α_θ )e^iu X_θ]|
    + |E[Y^α_θ e^iu Y_θ(e^iu(X_θ-Y_θ)-1)]|
       ≤ E[|X_θ^α - Y_θ^α|] + u
    E[|Y_θ|^α|X_θ-Y_θ|]

By (<ref>) and (<ref>) respectively, X_θ and Y_θ 
have moments of any order. Therefore by the Hölder inequality,

    E[|Y_θ|^α|X_θ-Y_θ|] ≤ C_1(α,ϵ)
    X_θ - Y_θ_1+ϵ

for a constant C_1(α,ϵ) > 0.
Since X^α_θ - Y^α_θ =
(X_θ-Y_θ)∑_β=0^α-1 (-1)^β
X_θ^α-1-βY_θ^β,
the Hölder inequality gives also 

    E[|X_θ^α - Y_θ^α|] ≤ 
    C_2(α,ϵ)X_θ - Y_θ_1+ϵ

 for a constant C_2(α,ϵ) > 0.
 Since σ_0(θ) = O(θ^1/2), we have 
X_θ - Y_θ_1+ϵ = o(θ^2H+2ϵ) 
by (<ref>),
from which the result follows. ////

 
Let H and ϵ be as in Lemme <ref>.
Then,  for any δ∈ [0,(H-ϵ)/3),
  
    sup_|u| ≤θ^-δ| E[Y_θ^α e^iuY_θ]
     - E[e^iuM^(0)_θ((M^(0)_θ)^α + A(α,u,M^(0)_θ) +
      B(α,u,M^(0)_θ))] |= o(θ^2H + ϵ),

where
 
    A_θ(α,u,x) =    (iux^α + α
      x^α-1)( E[Y_θ| M^(0)_θ = x]-x),
    
       B_θ(α,u,x) =     ( -u^2/2x^α + iu α x^α-1 +
      α(α-1)/2x^α-2) 
       ×( θ^2HE[|M^(1)_θ|^2 | M^(0)_θ = x] 
    -σ_0(θ)θ^HE[M^(1)_θ | M^(0)_θ = x] +
    σ_0(θ)^2/4).

  
  Proof: 
  This follows from the fact that
  
    | e^ix-1 - ix + x^2/2| ≤|x|^3/6

  for all x ∈ℝ. 
 Indeed, this implies that
  
    sup_|u| ≤θ^-δ| E[Y_θ^α e^iuY_θ]
     - E[Y_θ^α e^iuM^(0)_θ(1 + iu(Y_θ -
    					       M^(0)_θ) -
    					       u^2/2(Y_θ - M^(0)_θ)^2)]
     |= o(θ^2H + ϵ).

Expand Y^α_θ = (M^(0)_θ)^α + α
(M^(0)_θ)^α-1(Y_θ - M^(0)_θ) + … and
take the conditional expectation given M^(0)_θ to obtain
the result.
////

  
  Define q̅_θ(x) by
  
    q̅_θ(x) =     ϕ(x) - θ^H a^(1)_θ(x) -
      θ^2Ha^(2)_θ(x) - σ_0(θ)/2(xϕ(x) -
      θ^Ha^(3)_θ(x)) 
       + θ^2H/2c_θ(x)
    - θ^H  σ_0(θ)/2 b_θ(x) + σ_0(θ)^2/8(x^2-1)ϕ(x),

where a^(i)_θ, b_θ  and c_θ are defined by (<ref>).
  Then,
  
    ∫_ℝ e^iux x^αq̅_θ(x) dx
      = E[e^iuM^(0)_θ((M^(0)_θ)^α + A(α,u,M^(0)_θ) +
      B(α,u,M^(0)_θ))].

  
  Proof: 
Since the density of M^(0)_θ is ϕ by the assumption, 
this simply follows from integration by parts. ////
  

 §.§ Density expansion

  Here we derive an asymptotic expansion of the density of X_θ.
   
There exists a density of X_θ and
  for any α, j ∈ℕ∪{0},
  
    sup_θ∈ (0,1)∫ |u|^j |E[X_θ^α e^iuX_θ]| du < ∞

  
  Proof: 
  Note that the distribution of X_θ is Gaussian conditionally on
  𝒢_θ,
with conditional mean 

    U_θ:=  -1/2σ_0(θ)⟨ M
    ⟩_θ + 1/σ_0(θ)∫_0^θ√(v_t)ρ_tdW_t

and conditional variance

    V_θ :=  1/σ_0(θ)^2∫_0^θ v_t(1-ρ_t^2)dt.

Therefore, for any bounded continuous function f, we have

    E[f(X_θ)]
    = E[E[f(X_θ)|𝒢_θ]]
    = E[ ∫ f(x)ϕ(x,U_θ,V_θ) dx],

where ϕ(·,u,v) is the density of the normal distribution with
mean u and variance v. This means that  
X_θ 
admits a density 

    p_θ(x) = E[ϕ(x,U_θ,V_θ)].

Furthermore, the density function is in the Schwartz space 𝒮
and each Schwartz semi-norm is uniformly bounded  in θ  by (<ref>).
  Therefore,
  
    sup_θ∈ (0,1)∫ |u|^j |E[X_θ^α e^iuX_θ]| du
          = sup_θ∈ (0,1)∫| ∫ u^jx^α e^iuxp_θ(x)dx |
       du
       
       = sup_θ∈ (0,1)∫|
       ∫ e^iux∂_x^j(x^α p_θ (x)) dx
       | du < ∞

   since the Fourier transform is a continuous linear mapping from 𝒮 to 𝒮.
   ////



Proof of Theorem <ref>: 
   As seen in the proof of Lemma <ref>, the density
   p_θ exists in the Schwartz space.
Note that for a function f in the Schwartz space,
by Taylor's theorem,

    | f(x+a) - f(x) -f^'(x)a - f^''(x)a^2/2|
        ≤a^3/2sup_|b|≤ |a|
     |f^'''(x+b)|
    
       ≤a^3/2sup_|b|≤ |a| 1/(1 + (x+b)^2)^αsup_y ∈ℝ
     (1+y^2)^α |f^'''(y)|

and so,

    sup_x ∈ℝ
    (1+x^2)^α| f(x+a) - f(x) -f^'(x)a - f^''(x)a^2/2|
     = O(a^3).

This gives
  
    sup_x ∈ℝ (1+x^2)^α
       |q_θ(x) - q̅_θ(x)|
       = O(θ^1+H) = o(θ^2H),

where q̅_θ is given by (<ref>).
By the Fourier identity,
   
    (1+x^2)^α
       (p_θ(x) - q̅_θ(x))
       = 1/2π∫∫ e^iuy (1+y^2)^α(p_θ(y)-q̅_θ(y))dy e^-i uxdu

   Combining the lemmas in the previous section, taking δ∈ (0,
   min{ϵ, (H-ϵ)/3}),
   we have
   
    ∫_|u|≤θ^-δ|
    ∫ e^iuy (1+y^2)^α(p_θ(y)-q̅_θ(y))dy
       | du = o(θ^2H).

   On the other hand,
   
    ∫_|u| ≥θ^-δ| ∫ e^iuy
       (1+y^2)^α p_θ(y) dy  | du
          ≤θ^j δ∫_|u| ≥θ^-δ |u|^j| E[(1+X_θ^2)^α
       e^iuX_θ]|du
       
        = O(θ^jδ)

   for any j ∈ℕ by Lemma <ref>.
   The remainder
   
    ∫_|u| ≥θ^-δ| ∫ e^iuy
       (1+y^2)^αq̅_θ(y) dy  | du

  is handled in the same manner. ////

    

 §.§ Put option price  expansion

    Here we consider put option prices.
    Denote by p_θ the density of X_θ as before and 
   consider a normalized  put option price

    p(Fe^σ_0(θ)z,θ)/Fσ_0(θ)
    =e^-rθ∫_-∞^z∫_-∞^ζ p_θ(x)
      dx
      e^σ_0(θ)ζdζ.


Let q_θ(x), θ>0 be a family of functions on ℝ (not necessarily the one given by (<ref>)). If

    sup_x ∈ℝ (1+x^2)^α |p_θ(x)-q_θ(x)| =
    o(θ^β)

for some α > 5/4 and β >0,
then for any z_0 ∈ℝ,

    p(Fe^σ_0(θ)z,θ)/Fσ_0(θ)
    =e^-rθ∫_-∞^z∫_-∞^ζ q_θ(x)
    dx e^σ_0(θ)ζdζ + o(θ^β)

uniformly in z ≤ z_0.

Proof: 
By the Cauchy-Schwarz inequality,

    e^-rθ∫_-∞^z∫_-∞^ζ |p_θ(x) - q_θ(x)|
    dz e^σ_0(θ)ζdζ
       ≤
    e^-rθ∫_-∞^z √(∫_-∞^ζdx/(1+x^2)^2α-1)√(∫_-∞^ζ (1+x^2)^2α-1|p_θ(x)-q_θ(x)|^2
    dz ) e^σ_0(θ)ζdζ
       ≤√(π) e^-rθ + σ_0(θ)zsup_x ∈ℝ (1+x^2)^α |p_θ(x)-q_θ(x)|
    ∫_-∞^z √(∫_-∞^ζdx/(1+x^2)^2α-1)dζ,

which is o(θ^β) if α > 5/4. ////



Proof  of Theorem <ref>: 
  This is a direct consequence of the previous lemma. For example,

    d/dz{e^-σ_0(θ)zd/dz{1/σ_0(θ)(Φ(z + σ_0(θ)/2)e^σ_0(θ)z-Φ(z-σ_0(θ)/2) )
    }}
    = ϕ(z + σ_0(θ)/2).

The derivative of H_k(z)ϕ(z) is -H_k+1(z)ϕ(z).
Recall also σ_0(θ) = O(√(θ)).
 ////
  
  

 §.§ Implied volatility expansion

  Here we prove an expansion formula for the Black-Scholes implied
  volatility. 



Proof  of Theorem <ref>: 
Step 1).
Fix z ∈ℝ.  Note that

    P_θ(σ):=  p_BS(Fe^√(θ)z,θ,σ) /Fe^-rθ√(θ)= 
    1/√(θ)(Φ(z/σ + σ√(θ)/2)e^√(θ)z-Φ(z/σ-σ√(θ)/2) )

and that

    P_θ : [0,∞] →[
     (e^√(θ)z-1)_+/√(θ),  e^√(θ)z/√(θ)]

is a strictly increasing function.
From (<ref>) and Proposition <ref>, we have
  
    p(Fe^√(θ)z,θ) /Fe^-rθ√(θ)
      =        P_θ(κ_2)
    + κ_2 κ_3ϕ(z/κ_2 + κ_2√(θ)/2)
    H_1(z/κ_2 + κ_2√(θ)/2)e^√(θ)zθ^H
        + κ_2 ϕ(z/κ_2)(κ_4H_2(z/κ_2) + κ_3^2/2H_4(z/κ_2) )θ^2H
      + o(θ^2H) 
    
     =      P_θ(κ_2) + O(θ^H).

Therefore

    σ_BS(√(θ)z,θ) =
    P_θ^-1(P_θ(κ_2) + O(θ^H)).

By (<ref>), κ_2 is bounded in θ, say, by L > 0.
The function P_θ converges as θ→ 0 to 

    P_0(σ) := z Φ(z/σ) + σϕ(z/σ)

pointwise, and by Dini's theorem, this convergence is uniform on [0,L].
Since the limit function P_0 is strictly increasing,
the inverse functions P^-1_θ converges to P_0^-1.
Again by Dini's theorem, this convergence is uniform and in particular,
P_θ^-1 are equicontinuous.
Thus we conclude σ_BS(√(θ)z,θ)  - κ_2 → 0 as θ→ 0. Then, 
write  σ_BS(√(θ)z,θ) = κ_2 + β(θ)
and substitute this to the equation 
P_θ(σ_BS(√(θ)z,θ)) = P_θ(κ_2) + O(θ^H). The Taylor expansion gives
β(θ) = O(θ^H).



Step 2).
From (<ref>) we have
  
    P_θ(σ)  = 
        σ F_1(z/σ) +
        σ^2 √(θ)/2
        F_2(z/σ) +
         σ^3 θ/6
        F_3(z/σ) +
        o(θ),

   where
   
    F_1(x) = xΦ(x) + ϕ(x),    F_2(x) = x^2 Φ(x) + xϕ(x),   
          F_3(x) = x^3 Φ(x) + (x^2-1/4)ϕ(x).

    Using that
    
    ∂_σ{σ F_1 (z/σ) 
        } 
        =  ϕ(z/σ),

    we have
 
    κ_2 F_1(z/κ_2) +
        κ_2^2 √(θ)/2  F_2(z/κ_2) + 
      κ_2ϕ(z/κ_2)
      κ_3H_1(z/κ_2)e^√(θ)zθ^H
        = σ_BS(√(θ)z,θ)
      F_1(z/σ_BS(√(θ)z,θ)) +
        σ_BS(√(θ)z,θ)^2 √(θ)/2
        F_2(z/σ_BS(√(θ)z,θ))
        + O(θ^2H)
        
        =
        κ_2 F_1(z/κ_2) +
        κ_2^2 √(θ)/2
        F_2(z/κ_2) +
        ϕ(z/κ_2)(σ_BS(√(θ)z,θ)-κ_2)
        + O(θ^2H),

  from which we conclude
  σ_BS(√(θ)z,θ) =
  κ_2 + κ_3 z e^√(θ)zθ^H + O(θ^2H).


 Step 3).  Using that
      
    ∂_σ^2 {σ F_1 (z/σ) 
        } 
        =   z^2/σ^3ϕ(z/σ),   ∂_σ{σ^2 F_2 (z/σ) 
        } 
        =  z ϕ(z/σ),

    we obtain
      
    κ_2ϕ   (z/κ_2 + κ_2 √(θ)/2)  (
      κ_3H_1(z/κ_2 + κ_2 √(θ)/2)e^√(θ)zθ^H
     + (κ_4H_2(z/κ_2) +
          κ_3^2/2H_4(z/κ_2)
          )θ^2H)  
    
     =     p(Fe^√(θ)z,θ) /Fe^-rθ√(θ)
     -  P_θ(κ_2) + o(θ^2H) 
    
    =    
    P_θ(σ_BS(√(θ)z,z))
     -  P_θ(κ_2) + o(θ^2H) 
    
    =   ∂_σ{σ F_1 (z/σ) 
        }|_σ = κ_2  (σ_BS(√(θ)z,θ)-κ_2) + 1/2∂_σ^2 {σ F_1 (z/σ) 
        }|_σ = κ_2 (σ_BS(√(θ)z,θ)-κ_2)^2
        +  √(θ)/2∂_σ{σ^2 F_2 (z/σ) 
        }|_σ=κ_2 (σ_BS(√(θ)z,θ)-κ_2) + o(θ^2H)
    
          =    ϕ(z/κ_2)(σ_BS(√(θ)z,θ)-κ_2)
        	  + √(θ)/2
    	  z
          ϕ(z/κ_2)(σ_BS(√(θ)z,θ)-κ_2)
          
        + z^2/2κ_2^3ϕ(z/κ_2)(σ_BS(√(θ)z,θ)-κ_2)^2
      + o(θ^2H)

from Theorem <ref> and Step 2.
The left hand side is further expanded as

    κ_2 ϕ(z/κ_2) {κ_3 H_1(z/κ_2)e^√(θ)zθ^H -    κ_3 H_2(z/κ_2) κ_2 /2θ^H+1/2
        +  (κ_4H_2(z/κ_2) +
          κ_3^2/2H_4(z/κ_2)
          )θ^2H}  + o(θ^2H).

Denote γ(θ) = σ_BS(√(θ)z,θ) -
  κ_2 - κ_3 z e^√(θ)zθ^H and substitute this to obtain

    γ(θ) =    
    - κ_3 H_2(z/κ_2) κ_2^2 /2θ^H+1/2 +  κ_2 (κ_4H_2(z/κ_2) +
          κ_3^2/2H_4(z/κ_2)
          )θ^2H
       -κ_3 /2 z^2 θ^H+1/2-
    κ_3^2/2κ_2^3 z^4θ^2H + o(θ^2H) 
    
    =    (κ_2^2/2-z^2)κ_3θ^H+1/2
    +κ_2
     ((κ_4-3κ_3^2)z^2/κ_2^2 + 3/2κ_3^2-κ_4) θ^2H + o(θ^2H),

  from which we conclude the result. ////

  

 §.§ Asymptotics for at-the-money skew and curvature

  Here we prove Theorem <ref>.



Proof of Theorem <ref>: 
  It is known (see e.g., Fukasawa <cit.>) that
  
    ∂_k σ_BS(k,θ)
      = Q(k ≥σ_0(θ)X_θ ) -
      Φ(f_2(k,θ))/√(θ)ϕ(f_2(k,θ)), 
       ∂_k^2 σ_BS(k,θ)
      = p_θ(k /σ_0(θ))/σ_0(θ)√(θ)ϕ(f_2(k,θ)) -
      σ_BS(k,θ) ∂_kf_1(k,θ)
         ∂_kf_2 (k,θ),

  where
  
    f_1(k,θ) = k/√(θ)σ_BS(k,θ)
      - √(θ)σ_BS(k,θ)/2,    f_2(k,θ) = k/√(θ)σ_BS(k,θ)
      + √(θ)σ_BS(k,θ)/2.

Since the condition of Theorem <ref> is met, 
we have
  
    Q(0 ≥ X_θ)
      = Φ(σ_0(θ)/2) + κ_3(θ)ϕ(σ_0(θ)/2) θ^H + o(θ^2H).

  On the other hand, by Theorem <ref>,
  
    f_2(0,θ) = √(θ)/2κ_2(θ) +
      O(θ^2H+1/2)

  and so,
  
    Φ(f_2(0, θ))
      = Φ(σ_0(θ)/2) +
      O(θ^2H+1/2),
      
       ϕ(f_2(0,θ)) = ϕ(0) - ϕ(0)θ/8κ_2(θ)^2 +
      O(θ^2H+1).

Then, it follows from (<ref>) that
     
    ∂_k σ_BS(0,θ)
          = κ_3(θ)θ^H-1/2 + o(θ^2H-1/2).
 
 Further, under the condition, we have
 
    p_θ(0)
     = ϕ(σ_0(θ)/2){1 -κ_3(θ)/2σ_0(θ)θ^H
      + (3κ_4(θ) - 15 κ_3(θ)^2/2)θ^2H}
      + o(θ^2H).

    On the other hand, by Theorem <ref> and (<ref>),
    
    σ_BS(0,θ)    ∂_kf_1(0,θ) ∂_kf_2(0,θ) 
     =    1/σ_BS(0,θ) θ + O(θ^2H)
       
      =    1/κ_2(θ) θ(1 -1/2κ_2(θ) κ_3(θ)θ^H+1/2
      -  (3/2κ_3(θ)^2-κ_4(θ))θ^2H)
        + o(θ^2H-1).

    Then, it follows from (<ref>) that
    
    ∂_k^2 σ_BS(0,θ) =
        2κ_4(θ) - 6 κ_3(θ)^2/κ_2(θ)θ^2H-1
        + o(θ^2H-1),

    which completes the
    proof. ////




§ REGULAR STOCHASTIC VOLATILITY MODELS

Here we briefly discuss that regular stochastic
volatility models satisfy all the assumptions in Section 2.1 with H = 1/2.
Consider the volatility process  v_t = v(X_t), where
 X is a Markov process
satisfying a stochastic differential equation

    dX_t = b(X_t) dt + c(X_t)dW_t

and v is a smooth positive function defined
on the state space of X.
Let ρ∈ (-1,1) be a constant and {𝒢_t} be the augmented
filtration generated by W.
We assume (<ref>), which is satisfied in the usual cases including the
log-normal SABR and Heston models. 
Denote by L the generator of X.
Put f = √(v), g = f^' c and h = v^' c. 
Then, by Itô's formula,
we have

    M_θ = f(X_0)B_θ + ∫_0^θ∫_0^t g(X_s)dW_sdB_t + 
        ∫_0^θ∫_0^t Lf(X_s)dsdB_t,
       ⟨ M ⟩_θ
      = v(X_0) θ +
      ∫_0^θ∫_0^t h(X_s)dW_sdt + 
        ∫_0^θ∫_0^t Lv(X_s)dsdt.

Let B̅^θ_t = θ^-1/2B_θ t, W̅^θ_t =
θ^-1/2W_θ t and X^θ_t = X_θ t.
Then

    M_θ/√(θ)
      = f(X_0)B̅^θ_1 + √(θ)∫_0^1
      ∫_0^ug(X^θ_v)dW̅^θ_v
      dB̅^θ_u +
       θ∫_0^1
      ∫_0^uLf(X^θ_v)dv
      dB̅^θ_u,
       ⟨ M ⟩_θ/θ
      = v(X_0) + √(θ)∫_0^1
      ∫_0^uh(X^θ_v)dW̅^θ_v
      du +
       θ∫_0^1
      ∫_0^uLv(X^θ_v)dv
      du.

It  would follow that

    σ_0(θ)^2/θ
      = E[⟨ M ⟩_θ]/θ
      = v(X_0) + 1/2Lv(X_0) θ + O(θ^3/2),

and so

    σ_0(θ)/√(θ)
      = f(X_0) + 1/4Lv(X_0)/f(X_0)θ + O(θ^3/2)

under a mild regularity condition.
Then, we have (<ref>) with H = 1/2,
M^(0)_θ = B̅^θ_1 and

    M^(1)_θ = g(X_0)/f(X_0)∫_0^1 W̅^θ_u
      dB̅^θ_u, 
         M^(2)_θ =
      -Lv(X_0)/4v(X_0)B̅^θ_1
       + g^'(X_0)c(X_0)/f(X_0)∫_0^1 ∫_0^u 
       W̅^θ_v
       dW̅^θ_v
      dB̅^θ_u +
      Lf(X_0)/f(X_0)∫_0^1 u
      dB̅^θ_u,
      
        M^(3)_θ = 2g(X_0)/f(X_0)∫_0^1
      W̅^θ_u du

again under a mild regularity condition.
By Nualart et al. <cit.> or Appendix A below,

    E[M^(1)_θ|M^(0)_θ = x]
    = g(X_0)/f(X_0)ρ/2H_2(x),   
     E[M^(3)_θ|M^(0)_θ = x]
    = g(X_0)/f(X_0)ρ H_1(x)

and so,

    a^(1)_θ   (x + σ_0(θ)/2) -
    σ_0(θ)/2 a^(3)_θ(x + σ_0(θ)/2)
    
        = - κ_3
    (H_3 (x + σ_0(θ)/2)
    - σ_0(θ)H_2 (x + σ_0(θ)/2)
    )ϕ(x + σ_0(θ)/2)

with

    κ_3  =ρ/2g(X_0)/f(X_0).

Further,

    E[M^(2)_θ|M^(0)_θ = x]
       = -Lv/4v(X_0)x + g^' c/f(X_0) ρ^2/6H_3(x)
    + Lf/2f(X_0) x
    
        = - g^2/4f^2(X_0)x + g^' c/f(X_0) ρ^2/6H_3(x)

and

    E[(M^(1)_θ)^2|M^(0)_θ = x] =
    g^2/f^2(X_0)(ρ^2 (1/2 + H_2(x)+ 1/4H_4(x) ) + (1-ρ^2)(1/2 + 1/3H_2(x))).

Therefore,

    a^(2)_θ(x) - 1/2c_θ(x) = 
    -κ_4 H_4(x)ϕ(x) - κ_3 ^2/2H_6(x)ϕ(x)

with

    κ_4  =  g^' c/f(X_0) ρ^2/6
    + g^2/f^2(X_0)1+ 2ρ^2/6.

Thus we have observed that (<ref>) has the form of (<ref>) and so, all the theorems in Section 2.2 are applied.
In particular, Theorem <ref> proves the Medvedev-Scaillet formula (Proposition 1 of <cit.> that was obtained by a formal computation when f is the identity function).

    

§ THE ROUGH BERGOMI MODEL

    Here we show that the rough Bergomi model proposed by <cit.>
 fits into the framework and compute the expansion terms.
 Let ρ_t = ρ∈ (-1,1) be a constant and 
  
    v_t = v_0(t) exp{η√(2H)∫_0^t (t-s)^H-1/2dW_s -η^2/2 t^2H}.

The deterministic function v_0(t) =E[v_t] is assumed to be continuous.
   
  We have (<ref>) for q_θ given by 
(<ref>) with
  
    κ_3(θ)
      =ρη√(H/2)1/θ^Hσ_0(θ)^3∫_0^θ∫_0^t
      (t-s)^H-1/2√(v_0(s))ds v_0(t)dt,
      
       κ_4(θ) =   (1+2ρ^2)η^2 H/(2H+1)^2(2H+2)
      +  ρ^2η^2 H β(H+3/2,H+3/2)/2(H+1/2)^2,

where β is the beta function.
  
Proof: 
 Since v_t is log-normally distributed, (<ref>) holds by Jensen's inequality. 
The conditions (<ref>) and (<ref>) follow from
Lemma <ref> below. The functions a^(i)_θ and c_θ
are computed in Lemmas <ref>, <ref>, <ref> and <ref> below. The function b_θ is obtained as the 
the derivative of a^(1)_θ.  They are apparently rapidly decreasing smooth functions.
Then, by Theorem <ref>, it suffices to show that q_θ defined by
(<ref>) has the form (<ref>) up to o(θ^2H) with 
κ_3(θ) and κ_4(θ) specified above.
////


 Theorems <ref>, <ref> and  <ref>
 are therefore valid here as well.
When H < 1/2 and
  the forward variance curve is flat (i.e., v_0 is constant),
Theorem <ref> gives a similar formula to 
the Bergomi-Guyon expansion formally derived in <cit.>[ 
 Note however that there is a typo in the second order term in <cit.>.].
In fact, the expansion of O(η) given in <cit.> coincides with
our expansion of O(θ^H) when v_0 is constant.
When v_0 is not constant, or when looking at the second-order terms,
the formulas are not the same; this is not surprising 
because the asymptotics are 
η→ 0 in <cit.>  while θ→ 0 here. 
  Further, when v_0 is constant,  the same formula of O(θ^H) can 
be  obtained by expanding the rate function of the large deviation result of  <cit.> as in <cit.>. To be more precise, note that by Theorem <ref>,

    κ_3(θ) = ρη√(2H)/2(H+1/2)(H+3/2)

when v_0 is constant and Theorem <ref> implies

    σ_BS(√(θ)z,θ) - σ_BS(√(θ)ζ,θ) /√(θ)z - √(θ)ζ = κ_3(θ)θ^H-1/2 + O(θ^2H-1/2)

for z ≠ζ. A weaker assertion, where 
O(θ^2H-1/2) is replaced with o(θ^H-1/2),
 was already shown in <cit.> by a different method.
What is shown in <cit.> via an expansion of the rate function 
 is that this formula up to  o(θ^H-1/2) remains valid even 
if √(θ) z and 
√(θ)ζ are replaced with
θ^β z and 
θ^βζ respectively for β∈ (1/2-H,1/2).









How small θ has to be for reasonable accuracy of our asymptotic formulas should be examined via numerical experiments. Our extensive experiments suggest
 ηθ^H < 1 as a rough criterion[Note that ηθ^H is the standard deviation of log-spot-variance.].  Here we present only a few examples of the volatility surfaces. In Figures <ref> and <ref>, the points are by the Monte Carlo and the curves are by the asymptotic formula given in Theorems <ref>  and <ref>.
The different colors are for different time-to-maturities;
black for θ = .02, red for θ = .05, green for θ=.1,
blue for θ = .2, cyan for θ = .3 and magenta for θ = .6.
Note that the sets of  parameters  in Figure <ref>
 are those calibrated from option data by Bayer et al <cit.>.



In order to prove Lemmas below, we need some preparation. 
Let H_k, k=0,1,… be the Hermite polynomials as before:
  
    H_k(x) = (-1)^k e^x^2/2d^k/dx^k
      e^-x^2/2

  and H_k(x,a) = a^k/2H_k(x/√(a)) for a > 0.
  As is well-known, we have
  
    exp{ux - au^2/2} = ∑_k=0^∞
      H_k(x,a)u^k/k!

  and for any continuous local martingale M and n ∈ℕ,
  
    d L^(n)_t = n L^(n-1)_t dM_t,

  where L^(k) = H_k(M,⟨ M ⟩) for k ∈ℕ.
  See, e.g., Revuz and Yor <cit.>.

  Define Ŵ, Ŵ^', B̂ by
  
    Ŵ_t =1/σ_0(θ)∫_0^τ^-1(t)√(v_0(s))dW_s,   Ŵ^'_t =1/σ_0(θ)∫_0^τ^-1(t)√(v_0(s))dW^'_s

  and B̂ = ρŴ + √(1-ρ^2)Ŵ^',
  where
  
    τ(s) = 1/σ_0(θ)^2∫_0^s v_0(t)dt.

  Then, (Ŵ,Ŵ^') is a 2-dimensional Brownian motion under E
  and
for any square-integrable function f,
  
    ∫_0^a f(s)dW_s
      =
      σ_0(θ)∫_0^τ(a)f(τ^-1(t))/√(v_0(τ^-1(t)))dŴ_t.

  Therefore,
  
    M_θ =
      σ_0(θ)∫_0^1
      exp{θ^HF^t_t
      - η^2/4|τ^-1(t)|^2H}dB̂_t

  where
  
    F^t_u =
      η√(H/2)σ_0(θ)/θ^H∫_0^u
      (τ^-1(t)-τ^-1(s))^H-1/2/√(v_0(τ^-1(s)))dŴ_s,
        u ∈ [0,t].

  Let
  
    G^(k)_t = H_k(F^t_t,⟨ F^t ⟩_t).

 Then, we have
  
    M_θ   =
    σ_0(θ)∫_0^1
      exp{-η^2/8|τ^-1(t)|^2H}exp{θ^H F^t_t - θ^2H/2⟨ F^t ⟩_t
      }dB̂_t
        =  σ_0(θ)∫_0^1
      exp{-η^2/8|τ^-1(t)|^2H}∑_k=0^∞G^(k)_tθ^Hk/k!dB̂_t.

    
  We have (<ref>) with
  
    M^(0)_θ = B̂_1, 
        M^(1)_θ = 
      ∫_0^1  h_θ(t)
      G^(1)_t dB̂_t, 
        M^(2)_θ =
      ∫_0^1  {h_θ(t)-1/θ^2H 
      + h_θ(t) 
      G^(2)_t/2}dB̂_t, 
        M^(3)_θ = 2
       ∫_0^1 F^t_t
       dt,

  where
  
    h_θ (t)
      =  exp{-η^2/8|τ^-1(t)|^2H}.

  
  Proof: 
  For M^(i)_θ, i=0,1,2,
  it suffices to show
  
    ∫_0^1 h_θ(t)
      ∑_k=J^∞ G^(k)_t θ^Hk/k!dB̂_t
      _2 = O(θ^HJ)

  for any J ≥ 3. The proof for M^(3)_θ is similar and so omitted.
  It suffices to show
  
    E[∫_0^1 
      | ∑_k=J^∞ G^(k)_t θ^Hk/k!|^2
      dt ] = O(θ^2HJ).

  By the Cauchy-Schwarz inequality,  the left hand side is dominated by
  
    ∑_k=J^∞θ^Hk∑_k=J^∞θ^Hk/(k!)^2∫_0^1
      E[|G^(k)_t|^2]dt

  Let
  
    G^(k)_t,s = H_k(F^t_s,⟨ F^t ⟩_s),   s ∈ [0,t].

  Then, by (<ref>),
  
    E[|G^(k)_t|^2]     =   E[|G^(k)_t,t|^2] 
       = k^2 ∫_0^t
      E[|G^(k-1)_t,s|^2]d⟨ F^t ⟩_s 
       = k^2(k-1)^2
      ∫_0^t ∫_0^s_1
      E[|G^(k-2)_t,s_2|^2]d⟨ F^t ⟩_s_2d⟨ F^t ⟩_s_1
       ≤ (k!)^2 ⟨ F^t ⟩_t^k 
      =
       (k!)^2 (η^2/4|τ^-1(t)|^2H/θ^2H)^k.

  Note that τ^-1(t) ≤τ^-1(1) = θ.
  Therefore, for sufficiently small θ,
 
    ∑_k=J^∞θ^Hk∑_k=J^∞θ^Hk/(k!)^2∫_0^1
      E[|G^(k)_t|^2]dt
      ≤( η^2/4)^J
    θ^2HJ/(1 - θ^H)(1-θ^Hη^2/4),

  which completes the proof. ////

  Now we compute a^(i)_θ,  b_θ and c_θ based on Lemma <ref>.
 The following lemmas follow from the results in Section <ref> by
  straightforward computations.
    
  
    a^(1)_θ(x)    = -H_3(x)ϕ(x)ρη√(H/2)σ_0(θ)/θ^H∫_0^1 h_θ(t)∫_0^t(τ^-1(t) -
      τ^-1(s))^H-1/2/√(v_0(τ^-1(s)))dsdt
       =-H_3(x)ϕ(x)ρη√(H/2)
       ×1/θ^Hσ_0(θ)^3∫_0^θexp{-η^2/8t^2H}∫_0^t
      (t-s)^H-1/2√(v_0(s))ds v_0(t)dt 
       ∼
     -H_3(x)ϕ(x)ρη√(2H)/2(H+1/2)(H+3/2).

  
   
  
    a^(2)_θ(x) = - H_2(x)ϕ(x)∫_0^1 h_θ(t)-1/θ^2Hdt
       
      -H_4(x)ϕ(x)ρ^2 η^2 H/4σ_0(θ)^2/θ^2H∫_0^1 h_θ(t)
      (∫_0^t
      (τ^-1(t)-τ^-1(s))^H-1/2/√(v_0(τ^-1(s)))ds)^2
      dt 
       ∼ H_2(x)ϕ(x)∫_0^1 η^2/8θ^2H|τ^-1(t)|^2Hdt
      - H_4(x)ϕ(x)ρ^2 η^2 H/(2H+1)^2(2H+2).

  
    
  
    a^(3)_θ(x)    =
      -2H_2(x)ϕ(x)ρη√(H/2)σ_0(θ)/θ^H∫_0^1∫_0^t(τ^-1(t) -
      τ^-1(s))^H-1/2/√(v_0(τ^-1(s)))dsdt
      
       ∼ -2H_2(x)ϕ(x)ρη√(2H)/2(H+1/2)(H+3/2).

  
    
  
    c_θ(x) =      H_6(x)ϕ(x) ρ^2
      η^2 H/2σ_0(θ)^2/θ^2H( ∫_0^1 h_θ(t) ∫_0^t
      (τ^-1(t)-τ^-1(s))^H-1/2/√(v_0(τ^-1(s)))dsdt)^2 
        +  H_4(x)ϕ(x)ρ^2
       η^2 H/2σ_0(θ)^2/θ^2H∫_0^1 h_θ(t)^2( ∫_0^t
      (τ^-1(t)-τ^-1(s))^H-1/2/√(v_0(τ^-1(s)))ds)^2dt 
        +  H_4(x)ϕ(x)ρ^2
       η^2 H σ_0(θ)^2/θ^2H∫_0^1 h_θ(t) ∫_0^t
      (τ^-1(t)-τ^-1(s))^H-1/2/√(v_0(τ^-1(s)))ds 
       ×∫_t^1 h_θ(u)
      (τ^-1(u)-τ^-1(t))^H-1/2/√(v_0(τ^-1(t)))du
      dt 
        + H_4(x)ϕ(x)
       η^2 H/2σ_0(θ)^2/θ^2H∫_0^1 h_θ(t)^2( ∫_s^1
      (τ^-1(t)-τ^-1(s))^H-1/2/√(v_0(τ^-1(s)))dt)^2ds 
        +
      H_2(x)ϕ(x)
       η^2 H/2σ_0(θ)^2/θ^2H∫_0^1 h_θ(t)^2 ∫_0^t
      (τ^-1(t)-τ^-1(s))^2H-1/v_0(τ^-1(s))dsdt 
    ∼     H_6(x)ϕ(x)ρ^2
       η^2 H/2(H+1/2)^2(H+3/2)^2
       + H_4(x)ϕ(x) 2(1+ρ^2)η^2 H/(2H+1)^2(2H+2)
         + H_4(x)ϕ(x) ρ^2η^2 H β(H+3/2,H+3/2)/(H+1/2)^2
        + H_2(x)ϕ(x)∫_0^1 η^2/4θ^2H|τ^-1(t)|^2Hdt.

  

  

§ CONDITIONAL EXPECTATIONS OF WIENER-ITÔ INTEGRALS

  Here we collect results on the conditional expectations of  Wiener-Itô integrals that follow from Proposition 3 of Nualart et al <cit.>.
  Let x ∈ℝ and B be a standard Brownian motion (B_0 = 0).
  Let f be a continuous function on
  
    { (s,t) ∈ (0,1)^2 ; s < t }

  with
  
    ∫_0^1 ∫_0^t |f(s,t)|^2dsdt < ∞.

  
  
    E[∫_0^1 ∫_0^t f(s,t)dB_s dt  |   B_1 = x
      ] =
       H_1(x)∫_0^1 ∫_0^t f(s,t)dsdt, 
         E[∫_0^1 ∫_0^t f(s,t)dB_s dB_t   |   B_1 = x] =
       H_2(x)∫_0^1 ∫_0^t f(s,t)dsdt, 
        E[∫_0^1( ∫_0^t f(s,t)dB_s)^2 dB_t   |   B_1 = x]
       =
       H_3(x)∫_0^1 (∫_0^t f(s,t)ds)^2
       dt 
        + H_1(x) ∫_0^1 ∫_0^tf(s,t)^2
       dsdt,
        E[
       ∫_0^1 (∫_s^1 f(s,t)dB_t)^2 ds
          |   B_1 = x ]
       = H_2(x)   ∫_0^1 (∫_s^1 f(s,t)dt)^2 ds
       
        +  ∫_0^1 ∫_s^1f(s,t)^2
       dtds.

  
  
  
    E[
    (∫_0^1∫_0^t f(s,t)dB_s dB_t)^2
       |   B_1 = x ]  - ∫_0^1 ∫_0^t f(s,t)^2dsdt
        = 
    H_4(x) (∫_0^1∫_0^t f(s,t)ds
      dt)^2  +H_2(x)∫_0^1
      ( ∫_0^tf(s,t)ds + ∫_t^1 f(t,u)du)^2
      dt.

  
    
00
 Alos
	 Alòs, E., León, J.A. and Vives, J., On the short-time
	 behavior of the implied volatility for jump-diffusion models
	 with stochastic volatility, Finance Stoch. 11, (2007), 
	 571-589.
AlSh
Alòs, E. and Shiraya, K.,Estimating the Hurst parameter from short
	term volatility swaps, to appear in Finance Stoch..

 BFG
	 Bayer, C., Friz, P.K. and Gatheral, J., Pricing under rough
	 volatility,
	 Quant. Finance 16:6, (2016),  887-904.
 Friz
	 Bayer, C. Friz, P.K.,  Gulisashvili, A., Horvath, B. and
	 Stemper B.,
	 Short-time near-the-money skew in  rough fractional volatility models,
	 to appear in Quant. Finance.

FO
Figueroa-López, J.E. and Ólafsson, S.,
Short-term asymptotics for the implied volatility skew under a
	stochastic volatility model with Lévy jumps,
Finance Stoch. 20 (2016), 973-1020.

 FZ
	 Forde, M. and Zhang, H., Asymptotics for rough stochastic
	 volatility models,
SIAM J. Finan. Math., (2017), 8(1), 114-145. 

FGP
Friz, P., Gerhold, S and Pinter, A., 
Option pricing in the moderate deviations regime,
 Math. Finance, (2018), 28(3), 962-988.

 F2011
	 Fukasawa, M., Asymptotic analysis for stochastic volatility:
	 Martingale expansion, Finance Stoch. 15, (2011),
	 635-654.
 MF
	 Fukasawa, M. The normalizing transformation of the implied
	 volatility smile, Math. Finance 22, (2012) 753-762.
 F2017
	 Fukasawa, M., Short-time at-the-money skew and rough fractional
	 volatility, Quant. Finance. 17:2, (2017), 189-198.
	 GS
	 Garnier, J. and Solna, K., Correction to Black-Scholes formula
		 due to fractional stochastic volatility, SIAM J. Finan. Math., (2017), 8(1) 560-588.
		 
G2006
		 Gatheral, J., The Volatility Surface: A
			 Practioner's Guide. (2006) (John Wiley & Sons
			 Inc: Hoboken, NJ).
GJR
	 Gatheral, J., Jaisson, T. and Rosenbaum, M., Volatility is
	 rough,  Quant. Finance. 18:6, (2018), 933-949.
 GJR2
	 Guennoun, H., Jacquier, A. and Roome, P., Asymptotic behaviour
	 of the fractional Heston model,
SIAM J. Finan. Math., (2018),  9(3), 1017-1045.

 KT
	 Kunitomo, N. and Takahashi, A., On validity of the asymptotic
expansion approach in contingent claim analysis, Ann. Appl.
	 Probab., (2003), 13(3), 914-952.

MS0  Medvedev, A. and Scaillet, O.,
A Simple Calibration Procedure of Stochastic Volatility Models with Jumps
by Short Term Asymptotics,
Research Paper No. 93, September 2003,
FAME - International Center for Financial Asset Management and Engineering.

 MS
	 Medvedev, A. and Scaillet, O., Approximation and calibration of
short-term implied volatilities under jump-diffusion stochastic volatility, The Review of Financial Studies , (2006),  20(2), 427-459.



NUZ
Nualart, D., Üstünel, A. S. and Zakai, M., On the moments of a multiple Wiener-Itô integral and the space induced by the polynomials of the integral. Stochastics 25 (1988), no. 4, 233-240. 

 Osa
	 Osajima, Y., The asymptotic expansion formula of implied volatility for dynamic SABR model and FX hybrid model, 2006, UTMS 2006-29, The University of Tokyo.
 PP
Pagliarani, S. and Pascucci, A.,
The exact Taylor formula of the implied volatility,
Finance Stoch. 21 (2017), 661-718.

RY
	 Revuz, D. and  Yor, M.,
	 Continuous Martingales and Brownian Motion,
	 (1999), Springer-Verlag Berlin Heidelberg.
	 
 Y1992
	 Yoshida, N., Asymptotic expansion for statistics related to
	 small diffusions, J. Jpn. Stat. Soc., (1992) 22, 139-159.
 Y2001
	 Yoshida, N., Malliavin calculus and martingale expansion,
	 Bull. Sci. math. 125, 6-7 (2001) 431-456.

