E Gobet

A Suleiman
email: ali.suleiman9@gmail.com

New approximations in local volatility models

Keywords: Local volatility model, European option, asymptotic expansion, CEV model

 where stochastic expansions combined with Malliavin calculus were performed to obtain approximation formulas based on the local volatility At The Money. Here, we derive alternative expansions involving the local volatility at strike. Averaging both expansions give even more accurate results. Approximations of the implied volatility are provided as well.

Introduction

Framework

We consider a linear Brownian motion (W t ) 0≤t≤T defined on a filtered probability space (Ω, F T , (F t ) 0≤t≤T , P) where T > 0 is a fixed terminal time. Here, (F t ) 0≤t≤T is the completion of the natural filtration of W . This is used to model the dynamics of a risky asset S (e.g. a stock or an index), which price process is (S t ) 0≤t≤T . We are mainly interested in valuing European-style financial contracts written on S, exercised at maturity T , which related payoff is of the form Φ(S T ). We especially pay attention to vanilla options, i.e. Φ(S) = (S -K) + (call options) and Φ(S) = (K -S) + (put options).

We consider the standard framework of complete market (see for instance [START_REF] Musiela | Martingale methods in financial modelling[END_REF]), and more specifically, we assume that 2. the risky asset pays a continuous dividend (q t ) 0≤t≤T , which is deterministic and bounded; 3. S follows a local volatility model, which dynamics is defined by the solution of the following stochastic differential equation: dS t S t = (r tq t )dt + σ(t, S t )dW t .

(1.1)

We denote the compound factor by

C t = exp t 0
(r sq s )ds .

(1.2) Thus, we have

S t = C t exp(X t ), (1.3)
X t = log(S 0 ) + t 0 σ(s, S s )dW s -1 2 t 0 σ 2 (s, S s )ds.

(1.4) Note that the above dynamics are directly given under the risk-neutral measure, since we only focus on pricing formulas. Then, the option price at time 0 is given by E(e - R T 0 rsds Φ(S T )). Of course, due to the general form of the local volatility function σ(t, S), it is hopeless to derive exact closed formulas for such option prices. The aim of this work is to obtain accurate approximations.

Literature background

The interest in local volatility models probably dates back to the work by Dupire [START_REF] Dupire | Pricing with a smile[END_REF] among others, who shows that such models are able to fit all call and put option prices at a given observation date (the calibration date). However, except in a few cases, analytical pricing formulas are available (for instance, in the CEV model σ(t, S) = νS β-1 , see [START_REF] Schroder | Computing the constant elasticity of variance option pricing formula[END_REF]). As alternative numerical methods, one could use a PDE approach but to achieve real-time pricing and calibration routines, it is better to search for approximative formulas, quicker to evaluate. Hagan etal. [START_REF] Hagan | Equivalent Black volatilities[END_REF] use singular perturbation techniques to obtain an implied volatility expansion, in the case of separable volatility σ(t, S) = α(t)A(S). Henry-Labordère [START_REF] Henry-Labordère | Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing[END_REF] transfers heat kernel expansions on price expansions. To tackle the case of non-separable volatility, Piterbarg [START_REF] Piterbarg | Stochastic volatility model with time-dependent skew[END_REF] suggests the use of parameter averaging for some choices of σ(t, S). A different approach has been developed in [START_REF] Benhamou | Smart expansion and fast calibration for jump diffusion[END_REF]: first a model proxy is chosen, then a smart expansion around this proxy is performed, involving Malliavin calculus to determine explicitly the expansion terms. This approach appears to be quite flexible since it naturally handles time-dependent coefficients and various modeling situations including so far jumps, discrete dividends or stochastic interest rates. More precisely, applications to local volatility model including jumps have been developed in [START_REF] Benhamou | Smart expansion and fast calibration for jump diffusion[END_REF] and deeply investigated along further directions in [START_REF] Benhamou | Expansion formulas for European options in a local volatility model[END_REF]. Allowing the interest rates to be stochastic is achieved in [START_REF] Benhamou | Analytical formulas for local volatility model with stochastic rates[END_REF], while [START_REF] Benhamou | Time dependent Heston model[END_REF] considers the case of time-dependent Heston model. In [START_REF] Etore | Stochastic expansion for the pricing of call options with discrete dividends[END_REF], the authors investigate the case of assets paying discrete dividends. Within this approach, we are able to prove explicit error estimates that depend on σ and its derivatives, on the maturity and on the payoff. It helps to better understand the roles of each parameter. In addition, the regularity of the payoff is crucial in order to design the expansion and to establish error estimates. These features are extensively discussed in [START_REF] Benhamou | Expansion formulas for European options in a local volatility model[END_REF] and [START_REF] Benhamou | Analytical formulas for local volatility model with stochastic rates[END_REF].

Nevertheless, regarding the results in [START_REF] Benhamou | Expansion formulas for European options in a local volatility model[END_REF], one could legitimately formulate the criticism that we use the local volatility only At The Money (ATM in short) when we take the model proxy as Black-Scholes model and when we compute the expansions. For arbitrary payoffs, this is natural, but for call/put options, this may be strange since the spot and strike variables play somewhat symmetric roles.

Here, we correct this drawback by providing new expansion formulas based on the local volatility at strike (and we even mix the expansions). This article is organized as follows: in the next paragraphs, we define the assumptions and notations used throughout the paper. Then, in the next section, main results are stated. The main proofs are postponed in Appendix. Numerical experiments are presented in Section 1.3.

Standing assumptions for the approximations

Throughout the paper, we assume the following:

• Assumption (E). The function σ is bounded and positive (σ inf = inf (t,x)∈[0,T ]×R + σ(t, S) > 0). We denote by c E ≥ 1 the smallest constant such that

sup (t,S)∈[0,T ]×R + σ(t, S) ≤ c E inf (t,S)∈[0,T ]×R + σ(t, S).
• Assumption (R). The function σ is seven-times continuously differentiable in the S-variable and

M 1 = max 1≤i≤7 sup (t,x)∈[0,T ]×R ∂ i ∂x i [σ(t, exp(x))] < ∞, (1.5)
M 0 = max M 1 , sup (t,S)∈[0,T ]×R + σ(t, S) < ∞. (1.6)
The assumption (R) is used in at least two respects: it allows for differentiating coefficients to obtain an expansion formula; it is used to derive error estimates.

The assumption (E) is an ellipticity-type condition that enables us to handle the error analysis for non-smooth payoffs Φ (such as call/put options). This is the standard framework developed in [START_REF] Benhamou | Smart expansion and fast calibration for jump diffusion[END_REF]. Note that for deterministic volatility functions, one has M 1 = 0.

Definitions and other notations

In the representation of the expansion formulas, we repeatedly use the following integral operator.

Definition 1 (Integral Operator). The integral operator ω T is defined as follows: for any integrable function l, we set

ω(l) T t = T t l u du for t ∈ [0, T ]. Similarly, for integrable functions (l 1 , l 2 ), we put for t ∈ [0, T ] ω(l 1 , l 2 ) T t = ω(l 1 ω(l 2 ) T . ) T t = T t l 1,r T r l 2,s ds)dr.
The n-times iteration is defined analogously: for any integrable functions

(l 1 , • • • , l n ), we set ω(l 1 , • • • , l n ) T t = ω(l 1 ω(l 2 , • • • , l n ) T . ) T t for t ∈ [0, T ].
We also use a short notation for Greeks.

Definition 2 (Greeks). Let Z be a random variable and let h be a payoff function. We define the i th Greek for the variable Z by the quantity (if it has a meaning):

Greek h i (Z) = ∂ i E[h(Z + x)] ∂x i | x=0 .
Definition 3 (Black-Scholes formula and related Greeks). Using usual notation, the Black-Scholes formula for call option and constant parameters (σ, r, q) writes Call BS (t, S; T, K; σ, r, q

) = Se -q(T -t) N (d 1 ) -Ke -r(T -t) N (d 2 ),
where

N (d) = d -∞
e -u 2 /2 √ 2π du and

d 1 = d 1 (t, S; T, K; σ, r, q) = 1 σ √ T -t log Se -q(T -t) Ke -r(T -t) + 1 2 σ √ T -t, d 2 = d 2 (t, S; T, K; σ, r, q) = d 1 -σ √ T -t.
For time dependent coefficients (σ s , r s , q s ) 0≤s≤T , the call price formula is deduced from the Black-Scholes formula by replacing the arguments σ 2 , r and q by their time-average on the interval [t, T ]. The resulting formula is denoted by Call BS (t, S; T, K; (σ s ) s , (r s ) s , (q s ) s ).

For t < T and σ > 0, the function (S, K) → Call BS (t, S; T, K; σ, r, q) is smooth and its sensitivities ∂ i ∂S i Call BS (t, S; T, K; σ, r, q) and ∂ i ∂K i Call BS (t, S; T, K; σ, r, q) are given explicitly in Proposition 1 (see Appendix C), for i = 1, . . . , 6. They will be used in our expansion formulas (see Theorems 2 and 3).

Expansion formulas

In this section, we give several expansion formulas, with a second and third order accuracy. The general principle for deriving such approximations is to choose a relevant proxy and to expand the quantities of interest around this proxy. First, we recall the general results from [START_REF] Benhamou | Smart expansion and fast calibration for jump diffusion[END_REF], where the proxy is obtained by freezing the local volatility at the initial spot value (ATM). Second we apply these expansions to call options. Third, using the Dupire forward PDE satisfied by the call price as a function of maturity and strike, we propose a new proxy where the volatility is frozen at the strike value K (instead of S 0 ). We then derive new second and third approximation formulas. Finally, some expansions of implied volatility are provided.

A general result

We first state two expansion results in a quite general form, so that we can apply it later to various situations. Let (Y t ) 0≤t≤T the solution of • the function a is seven-times continuously differentiable in the y-variable and

dY t = - 1 2 a 2 (
M Y,1 = max 1≤i≤7 sup (t,y)∈[0,T ]×R ∂ i y a(t, y) < ∞, (1.8)
M Y,0 = max M Y,1 , sup (t,y)∈[0,T ]×R a(t, y) < ∞.
(1.9)

• the function h : R → R is a.e. differentiable. In addition, h and h ′ have at most an exponential growth:

|h(x)| + |h ′ (x)| ≤ c h e c h |x| for any x, for a constant c h .

Define

• the Gaussian process (Y P t ) 0≤t≤T by

Y P t = Y 0 - 1 2 t 0 a 2 (s, Y 0 )ds + t 0 a(s, Y 0 )dW s ;
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• a(t) := a(t, Y 0 ), a (1) (t) := ∂ 1 y a(t, Y 0 ) and a (2) (t) := ∂ 2 y a(t, Y 0 ); • the expansion coefficients computed using the function a(t, .) at Y 0 : c 1,T = ω(a 2 , aa (1) ) T 0 , c 2,T = ω(a 2 , (a (1) ) 2 ) T 0 , c 3,T = ω(a 2 , aa (2) ) T 0 , c 4,T = ω(a 2 , a 2 , (a (1) ) 2 ) T 0 , c 5,T = ω(a 2 , a 2 , aa (2) ) T 0 , c 6,T = ω(a 2 , aa (1) , aa (1) ) T 0 , c 7,T = ω(a 2 , a 2 , aa (1) , aa (1) ) T 0 , c 8,T = ω(a 2 , aa (1) , a 2 , aa (1) ) T 0 . Then, the following expansion formulas hold. a) Second order approximation. One has

E[h(Y T )] = E[h(Y P T )] + c 1,T 1 2 Greek h 1 (Y P T ) - 3 2 Greek h 2 (Y P T ) + Greek h 3 (Y P T ) + Error 2 , (1.10)
where

|Error 2 | ≤ C sup v∈[0,1] h (1) (vY T + (1 -v)Y P T ) 2 ( M Y,0 a inf )M Y,1 M 2 Y,0 T 3/2
and the constant C depends (in an increasing way) only on the upper bounds of the model parameters, on c E and on the maturity.

b) Third order approximation. One has

E[h(Y T )] = E[h(Y P T )] + 6 i=1 η i,T Greek h i (Y P T ) + Error 3 , (1.11)
where

η 1,T = c 1,T 2 - c 2,T 2 - c 3,T 2 - c 4,T 4 - c 5,T 4 - c 6,T 2 , η 2,T = - 3c 1,T 2 + c 2,T 2 + c 3,T 2 + 5c 4,T 4 + 5c 5,T 4 + 7c 6,T 2 + c 7,T 2 + c 8,T 4 , η 3,T =c 1,T -2c 4,T -2c 5,T -6c 6,T -3c 7,T - 3c 8,T 2 , η 4,T =c 4,T + c 5,T + 3c 6,T + 13c 7,T 2 + 13c 8,T 4 , η 5,T = -6c 7,T -3c 8,T , η 6,T =2c 7,T + c 8,T , and
|Error 3 | ≤ C sup v∈[0,1] h (1) (vY T + (1 -v)Y P T ) 2 ( M Y,0 a inf ) 2 M Y,1 M 3 Y,0 T 2 .
As before, the constant C depends (in an increasing way) only on the upper bounds of the model parameters, on c E and on the maturity.

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As explained in [START_REF] Benhamou | Expansion formulas for European options in a local volatility model[END_REF], the approximation order is related to the power m in the error upper bounds M Y,1 M m Y,0 ( √ T ) m+1 . The smaller the volatility (M Y,0 → 0) or its variations (M Y,1 → 0) or the maturity (T → 0), the more accurate the approximations. Since the proxy is Gaussian, the computation of E[h(Y P T )] and Greek h i (Y P T ) can be performed in closed forms for usual functions h (such as call/put payoffs), or by using efficient numerical integration techniques in other cases.

An interesting property of these expansion formulas is that they are exact for h

(x) = e x (indeed E[h(Y T )] = E[h(Y P T )] = Greek h i (Y P T ) = e Y0
, and the sum of expansion coefficients is equal to 0). In particular, when further applied to the local volatility model (1.1), this implies that the call/put parity will be preserved within these approximations.

When the function (t, x) → a(t, x) is piecewise constant w.r.t. the time variable, the coefficients (c i,T ) 1≤i≤8 can be quickly and simultaneously computed for different maturities T , using recursion (see [BGM09, Proposition 4.1]). In other situations, numerical integration is likely needed.

Application to expansion formulas for call price

We go back to the local volatility model (1.1) and to the evaluation of call options. In view of (1.4), the call price at time 0 is equal to

Call(T, K) = E(e - R T 0 rsds (S T -K) + ) = E(h(X T )) where h(x) = e - R T 0 rsds (C T e x -K) + .
In order to apply previous expansion results, it remains to identify the function a(•) in the dynamics of dX t = a(t, X t )dW t -1 2 a 2 (t, X t )dt. Comparing with (1.4), it follows that a(t, x) = σ(t, C t exp(x)).

Owing to the assumptions (R) and (E) on σ, one can apply Theorem 1 to Y = X and to h(x) = e - R T 0 rsds (C T e x -K) + , in order to obtain expansion formulas for call prices in local volatility models. The next step consists in transforming the Greeks in the X-variable in the (usual) Greeks in the Svariable, and in expressing the coefficients c i,T using the derivatives of σ. These computations are detailed in Appendix A. We obtain the following Theorem 2 (Second and third order approximations for call options, based on the ATM local volatility).

Assume (E) and (R). Set σ t := σ(t, C t S 0 ), σ (1) t := ∂ S σ(t, C t S 0 ), σ (2) t := ∂ 2 S σ(t, C t S 0 ) and α 1,T = ω(σ 2 , S 0 Cσσ (1) ) T 0 , α 2,T = ω(σ 2 , (S 0 Cσ (1) ) 2 ) T 0 , α 3,T = ω(σ 2 , S 2 0 C 2 σσ (2) + S 0 Cσσ (1) ) T 0 , α 4,T = ω(σ 2 , σ 2 , (S 0 Cσ (1) ) 2 ) T 0 , α 5,T = ω(σ 2 , σ 2 , S 2 0 C 2 σσ (2) + S 0 Cσσ (1) ) T 0 , α 6,T = ω(σ 2 , S 0 Cσσ (1) , S 0 Cσσ (1) ) T 0 , α 7,T = ω(σ 2 , σ 2 , S 0 Cσσ (1) , S 0 Cσσ (1) ) T 0 , α 8,T = ω(σ 2 , S 0 Cσσ (1) , σ 2 , S 0 Cσσ (1) ) T 0 .
a) Second order approximation. One has

Call(T, K) = Call BS (0, S 0 ; T, K) (1.12) +α 1,T 3 2 S 2 0 ∂ 2 S Call BS (0, S 0 ; T, K) + S 3 0 ∂ 3 S Call BS (0, S 0 ; T, K) + Error 2 , |Error 2 | ≤CS 0 exp - [log(S 0 C T /K)] 2 8|σ| 2 ∞ T ( M 0 σ inf ) M 1 M 2 0 T 3/2
where the Black-Scholes price and greeks are computed using the time dependent parameters (σ t , r t , q t ) 0≤t≤T . b) Third order approximation. One has

Call(T, K) =Call BS (0, S 0 ; T, K) (1.13) + 6 i=2 π i,T S i 0 ∂ i S Call BS (0, S 0 ; T, K) + Error 3 ,
where

π 2,T = 3 2 α 1,T + 1 2 α 2,T + 1 2 α 3,T + 9 4 α 4,T + 9 4 α 5,T + 13 2 α 6,T + 9α 7,T + 9 2 α 8,T , π 3,T = α 1,T + 4α 4,T + 4α 5,T + 12α 6,T + 66α 7,T + 33α 8,T , π 4,T = α 4,T + α 5,T + 3α 6,T + 153 2 α 7,T + 153 4 α 8,T , π 5,T = 24α 7,T + 12α 8,T , π 6,T = 2α 7,T + α 8,T , |Error 3 | ≤ CS 0 exp - [log(S 0 C T /K)] 2 8|σ| 2 ∞ T ( M 0 σ inf ) 2 M 1 M 3 0 T 2 .
In the above expansions, the constant C depends (in an increasing way) only on the upper bounds of the model parameters, on c E and on the maturity.

Note that the local volatility and its derivatives are computed along the ATM forward curve (S 0 C t ) 0≤t≤T .

Other expansions based on the local volatility at strike

In the previous approximation formulas, the ATM local volatility plays a central role. This is quite natural for arbitrary functions h, like in the general form of Theorem 1. But when dealing with call-put options, the local volatility at strike presumably plays a similarly important role. The aim of this paragraph is to derive similar expansion formulas, but using the volatility at strike. To achieve this goal, we follow the Dupire approach [START_REF] Dupire | Pricing with a smile[END_REF], which writes a PDE satisfied by the call price function

(T, K) → Call(T, K) = E(e - R T 0 rsds (S T -K) + ). Indeed,

we know that

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∂Call(T, K) ∂T = -q T Call(T, K) -(r T -q T )K ∂Call(T, K) ∂K + 1 2 σ 2 (T, K)K 2 ∂ 2 Call(T, K) ∂K 2 , Call(0, K) = (S 0 -K) + .
In other words, instead of handling a PDE in the backward variables (t, S) with a call payoff as a terminal condition, we now deal with a PDE in the forward variables (T, K), with a put payoff as an initial condition. This latter has a probabilistic Feynman-Kac representation

Call(T, K) = e - R T 0 q T -t dt E[(S 0 -K T ) + ].
using the following diffusion process (K t ) 0≤t≤T :

dK t K t = -(r T -t -q T -t )dt + σ(T -t, K t )dW t , K 0 = K. (1.14)
Define the process (Y t ) 0≤t≤T as follows:

K t = e - R t 0 (r T -s -q T -s )ds e Yt = C T -t C T e Yt . (1.15)
Then, Y has a dynamics of the form (1.7) with a(t, y

) = σ(T -t, C T -t
C T e y ). Thus, we are in a position to apply the general Theorem 1, to Y and to the function

h(y) = e - R T 0 q T -t dt (S 0 -C0
C T e y ) + . Retransforming the Greeks w.r.t. the Y -variable into usual Greeks w.r.t. K, we obtain the new following expansion formulas (see Appendix B for the proof).

Theorem 3 (Second and third order approximations for call options, based on the local volatility at strike). Assume (E) and (R). (1) , K C σσ (1) ) T 0 , α7,T = ω(σ 2 , σ2 , K C σσ (1) , K C σσ (1) ) T 0 , α8,T = ω(σ 2 , K C σσ (1) , σ2 , K C σσ (1) ) T 0 .

Set Ct = C T -t C T , σt := σ(T -t, Ct K), σ (1)
t := ∂ S σ(T - t, Ct K), σ(2) t = ∂ 2 S 2 σ(T -t, Ct K) and α1,T = ω(σ 2 , K C σσ (1) ) T 0 , α2,T = ω(σ 2 , (K C σ(1) ) 2 ) T 0 , α3,T = ω(σ 2 , K 2 C2 σσ (2) + K C σσ (1) ) T 0 , α4,T = ω(σ 2 , σ2 , (K C σ(1) ) 2 ) T 0 , α5,T = ω(σ 2 , σ2 , K 2 C2 σσ (2) + K C σσ (1) ) T 0 , α 6,T = ω(σ 2 , K C σσ
a) Second order approximation. One has

Call(T, K) =Call BS (0, S 0 ; T, K) (1.16) +α 1,T 3 2 K 2 ∂ 2 K Call BS (0, S 0 ; T, K) + K 3 ∂ 3 K Call BS (0, S 0 ; T, K) + Error 2 , |Error 2 | ≤CK exp - [log(S 0 C T /K)] 2 8|σ| 2 ∞ T ( M 0 σ inf ) M 1 M 2 0 T 3/2
where the Black-Scholes price and greeks are computed using the time dependent parameters (σ t , r t , q t ) 0≤t≤T . b) Third order approximation. One has Call(T, K) =Call BS (0, S 0 ; T, K) (1.17)

+ 6 i=2 πi,T K i ∂ i K Call BS (0, S 0 ; T, K) +
|Error 3 | ≤ CK exp - [log(S 0 C T /K)] 2 8|σ| 2 ∞ T ( M 0 σ inf ) 2 M 1 M 3 0 T 2 .
In the above expansions, the constant C depends (in an increasing way) only on the upper bounds of the model parameters, on c E and on the maturity.

Expansion formulas for implied volatility

Interestingly, the previous expansions of call price can be turned into expansions of implied volatility σ I (0, S 0 ; T, K) defined by Call(T, K) = Call BS (0, S 0 ; T, K; σ I (0, S 0 ; T, K), (r t ) 0≤t≤T , (q t ) 0≤t≤T ). To achieve this, we use nice relations between Greeks (see below and Proposition 1), omitting to indicate all the parameters whenever unambiguous:

Vega = ∂ ∂σ Call BS (0, S; T, K) = Se -qT N ′ (d 1 ) √ T = Ke -rT N ′ (d 2 ) √ T , S 2 Γ S = S 2 ∂ 2 ∂S 2 Call BS (0, S; T, K) = Se -qT N ′ (d 1 ) σ √ T = Vega σT , S 3 Speed S = S 3 ∂ 3 ∂S 3 Call BS (0, S; T, K) = -S 2 Γ S ( d 1 σ √ T + 1) = - Vega σT ( d 1 σ √ T + 1), K 2 Γ K = ∂ 2 ∂K 2 Call BS (0, S; T, K) = Ke -rT N ′ (d 2 ) σ √ T = Vega σT , K 3 Speed K = K 3 ∂ 3 ∂K 3 Call BS (0, S; T, K) = -K 2 Γ K (1 - d 2 σ √ T ) = - Vega σT (1 - d 2 σ √ T ).

Now, consider the second order expansion formula based on the ATM local volatility: it becomes

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Call(T, K) =Call BS (0, S 0 ; T, K) -Vega α 1,T √ T ( T 0 σ 2 s ds) 3/2 log S 0 C T K + Error 2 .
This reads as an expansion of the implied volatility. The derivation is similar for the second order expansion formula based on the local volatility at strike. This proves the following Theorem 4 (Second order approximations on implied volatilities).

Assume (E) and (R). Using the notations of Theorems 2 and 3, we have

σ I (0, S 0 ; T, K) = 1 T T 0 σ 2 s ds 1 2 - α 1,T T 1 2 ( T 0 σ 2 s ds) 3 2 log S 0 C T K (1.18) + Error I 2 , σ I (0, S 0 ; T, K) = 1 T T 0 σ2 s ds 1 2 + α1,T T 1 2 ( T 0 σ2 s ds) 3 2 log S 0 C T K (1.19) + Error I 2 .
Note that in the first case (1.18), the local volatility is computed ATM, while in the second one (1.19), it is computed at strike. In addition to these direct implied volatility approximations, one can upper bound the residual terms Error I 2 and Error I 2 , simply applying the error estimates from Theorems 2 and 3. We do not give the details of this derivation. As it can be expected, the error estimates depend on the ratio log(S0C T /K) |σ|∞ √ T , but actually, they are locally uniform w.r.t. this ratio. More precisely, for any ξ > 0, there is a constant C ξ which depends (in an increasing way) on ξ, on the upper bounds of the model parameters, on c E , on the maturity and on the ratio M 0 /σ inf such that for any S 0 and

K satisfying | log(S 0 C T /K)| ≤ ξ|σ| ∞ √ T we have |Error I 2 | + | Error I 2 | ≤ C ξ M 1 M 2 0
T. Thus, inaccuracies may occur for very small or very large strikes, a feature which is confirmed by the further numerical experiments. In view of the above upper bounds, the relative errors on implied volatility are locally of order M 1 M 0 T , justifying the label of second order approximations.

This paves the way for the derivation of a third order expansion of implied volatility, but unfortunately, we have not been able to simplify the computations in order to get a sufficiently nice expression. This will be further investigated.

Applications to time-dependent CEV model

To conclude this section, we specify the results when the volatility has the form

σ(t, S) = ν t S βt-1 , (1.20)
i.e. a CEV-type volatility with a time-dependent level (ν t ) 0≤t≤T and a timedependent skew (β t ) 0≤t≤T . In order to apply Theorems 2 and 3, all what is needed is to give the expressions for the coefficients (α i,T , αi,T ) 1≤i≤6 . First, the proxy volatilities are given by σ t = ν t (C t S 0 ) βt-1 and σt = ν T -t ( Ct K) β T -t -1 , where Ct = C T -t /C T ; then, we have

α 1,T = ω(σ 2 , (β -1)σ 2 ) T 0 , α 2,T = α 3,T = ω(σ 2 , (β -1) 2 σ 2 ) T 0 , α 4,T = α 5,T = ω(σ 2 , σ 2 , (β -1) 2 σ 2 ) T 0 , α 6,T = ω(σ 2 , (β -1)σ 2 , (β -1)σ 2 ) T 0 , α 7,T = ω(σ 2 , σ 2 , (β -1)σ 2 , (β -1)σ 2 ) T 0 , α 8,T = ω(σ 2 , (β -1)σ 2 , σ 2 , (β -1)σ 2 ) T 0 .
The expressions are similar for (α i,T ) 1≤i≤6 , by replacing σ t by σt and (β t -1) by (β T -t -1) in the above formulas. In the case of constant parameters ν t = ν, β t = β and µ = rq, all the previous quantities can be expressed in closed forms (the values of the integral operator ω(.) T 0 are given by iterated integrals of exponential functions). We give them in the simple case µ = 0. By setting σ = νS β-1 0 and σ = νK β-1 , we obtain

α 1,T = (β -1)σ 4 T 2 2 , α 2,T = α 3,T = (β -1) 2 σ 4 T 2 2 , α 4,T = α 5,T = α 6,T = (β -1) 2 σ 6 T 3 6 , α 7,T = α 8,T = (β -1) 2 σ 8 T 4 24 .
Replacing σ by σ gives the values for (α i,T ) 1≤i≤6 .

Numerical results

In the numerical tests we report here, we take r = q = 0 and we consider a CEV model (1.20) for the volatility, with constant parameters ν and β. For additional tests with time-dependent parameters, see [START_REF] Benhamou | Expansion formulas for European options in a local volatility model[END_REF]. We choose S 0 = 1, ν = 25% and we allow β to vary. Actually, we consider two values: β = 0.8 which is not far from the log-normal case, and β = 0.2 which is rather different. We test the accuracy of different approximations, for various maturities (3-6 months, 1-1.5-2-3-5-10 years) and various strikes.

INSERT TABLE 1.1 ABOUT HERE

The range of strikes depends on the maturity: the tested values are reported in Table 1.1. Essentially, the strikes are roughly equal to S 0 exp(ξν √ T ) where ξ is taken as various quantiles of the standard Gaussian law (we take the quantiles 1%-5%-10%-20%-30%-40%-50%-60%-70%-80%-90%-95%-99%): this means that the first and last columns of strikes are associated to very ITM options or very OTM options.

For the sake of completeness, in Table 1.2 and 1.3 we report the implied volatilities related to the (exact) call price in CEV model with constant parameters (our computations are based on the work by Schroder [START_REF] Schroder | Computing the constant elasticity of variance option pricing formula[END_REF]).

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INSERT TABLES 1.2 and 1.3 ABOUT HERE

We aim at comparing the following different approximations.

1. ImpVol(AppPrice(2,S0)): this is the implied volatility of the second order expansion based on the ATM local volatility (see (1.12) in Theorem 2). 2. AppImpVol(2,S0): this is the second order implied volatility expansion based on the ATM local volatility (see (1.18) in Theorem 4). 3. ImpVol(AppPrice(2,K)): this is the implied volatility of the second order expansion based on the local volatility at strike (see (1.16) in Theorem 3). 4. AppImpVol(2,K): this is the second order implied volatility expansion based on the local volatility at strike (see (1.19) in Theorem 4). 5. ImpVol(AppPrice(3,S0)): this is the implied volatility of the third order expansion based on the ATM local volatility (see (1.13) in Theorem 2). 6. ImpVol(AppPrice(3,K)): this is the implied volatility of the third order expansion based on the local volatility at strike (see (1.17) in Theorem 3). 7. Av.ImpVol(AppPrice(3,.)): this is the average of ImpVol(AppPrice(3,S0))

and ImpVol(AppPrice(3,K)). The interest in this approximation is explained later.

INSERT TABLES 1.4 and 1.5 ABOUT HERE

In Table 1.4 (resp. Table 1.5), we report the errors on implied volatility using the six first aforementioned approximations, for β = 0.8 (resp. β = 0.2). The errors are expressed in bps (basis points): an implied volatility of 25.01% instead of 25% yields 1bp error. For instance, on the first row of Table 1.4, the value -12.3 is associated to the approximation error of ImpVol(AppPrice(2,S0)) for the first strike of maturity T = 3M (i.e. K = 0.70); on the fourth row of Table 1.4, the value -0.9 refers to the approximation error of AppImpVol(2,K) for the second strike of maturity T = 3M (i.e. K = 0.75), and so one. Sometimes (especially for very small and very large strikes), the price approximation is out of the non-arbitrage interval for call options: in this case, one can not define a value for the implied volatility and we report ND in the tabular. For all these results, a medium (or large) error on implied volatility may yield a small (or reasonable) error on prices: this is especially true for ITM or OTM options, for which the Vega is small (see the discussion in [START_REF] Benhamou | Time dependent Heston model[END_REF]).

Influence of β and T . Generally speaking, we observe that for β = 0.8, the errors are smaller compared to β = 0.2: it is not surprising since the lognormal proxy suits better in the first case. This can also be explained by our error estimates, since M 1 is essentially proportional to |β -1|. Errors are increasing w.r.t. T , which is also coherent with our error estimates. Influence of K. For usual values of strike (essentially in the Gaussian quantile range [10%, 90%]), errors are small (or very small, depending on the approximation that is used), usually smaller than 10bps for β = 0.8 up to 10Y maturity, and smaller than 20bps for β = 0.2 up to maturity 5Y. Error approximations on implied volatility are much larger for very ITM or very OTM options. For these situations, it may be a good idea to incorporate known asymptotic on the implied volatility (see for instance [START_REF] Lee | The moment formula for implied volatility at extreme strikes[END_REF]).

Influence of the type of approximation. Regarding the second order approximations, within this model it gives lower bounds on implied volatility (and on price). This systematic underestimation is a drawback of these approximations. Notice that it is usually much better to use the direct approximation on implied volatility (Theorem 4) compared to the implied volatility of the price approximation. However, these implied volatility expansions underestimate the true value as well.

As expected, third order approximations are more accurate than second order ones. The improvement is more significant for β = 0.2. In Figures 1.1 and1.2, we plot the errors on implied volatility for the maturity T = 1.5Y (this choice is unimportant) for both values of β.

INSERT FIGURES 1.1 AND 1.2 ABOUT HERE

We first observe that ImpVol(AppPrice(3,S0)) overestimates the true value for K ≫ S 0 and yields an underestimation for K ≪ S 0 . This is the converse regarding ImpVol(AppPrice(3,K)). On Tables 1.4 and 1.5, we can check that this is generally satisfied for any maturity. Thus, an heuristic rule may be to consider the following confidence interval for the exact implied volatility:

σ I (0, S 0 ; T, K) ∈ ImpVol(AppPrice(3, K)), ImpVol(AppPrice(3, S0)) .
If the width of this interval is too large, it somehow indicates an inaccuracy in our approximations.

Secondly, we observe that the errors using ImpVol(AppPrice(3,S0)) and ImpVol(AppPrice(3,K)) have roughly the same magnitude (but with opposite signs). Then, if we define the average Av.ImpVol(AppPrice(3, .))

= 1 2 ImpVol(AppPrice(3, S0)) + ImpVol(AppPrice(3, K)) , (1.21)
we expect to obtain a much better implied volatility estimate. The errors for Av.ImpVol(AppPrice(3, .)) for β = 0.8 and β = 0.2 are reported in Tables 1.6 and 1.7. Observe that for maturities smaller than 5Y, the accuracy is truly excellent (i.e. smaller than few bps) for a widened range of strikes.

INSERT TABLES 1.6 and 1.7 ABOUT HERE

We have compared our approximations with the known implied volatility approximation in the CEV model (with zero interest rates and zero dividend) (see [Hen08, formula (5.41) p.141]):

σ I (0, S 0 ; T, K) ≈ ν (1 -β) ln(K/S 0 ) K 1-β -S 1-β 0 1 + (β -1) 2 ν 2 T 24 ( S 0 + K 2 ) 2β-2 .
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This latter approximation yields a slightly better numerical accuracy compared to ours (and it is quicker to evaluate). However, our approximations are also able to deal naturally with general time-dependent local volatility (with piecewise continuity in time), as a difference with [Hen08, Chapter 5] for instance, or with stochastic interest rates [START_REF] Benhamou | Analytical formulas for local volatility model with stochastic rates[END_REF]. This may be a significant advantage compared to other approaches, while maintaining tight error estimates.

A Proof of Theorem 2

We apply Theorem 1, by taking h(x) = e - R T 0 rsds (C T e x -K) + and a(t, x) = σ(t, C t exp(x)). The required assumptions on h and a are satisfied owing to assumptions (E) and (R). By simple computations, we easily check that M Y,0 = M 0 and M Y,1 = M 1 . The proxy of X used in Theorem 1 now writes

X P t = log(S 0 ) -1 2 t 0 σ 2 s ds + t 0 σ s dW s .
Main term and correction terms. From this, we deduce that the main term E(h(X P T )) in the expansion is equal to

E(e - R T 0 rsds (C T e X P T -K) + ) = Call BS (0, S 0 ; T, K; (σ t ) 0≤t≤T , (r t ) 0≤t≤T , (q t ) 0≤t≤T ).
In the following, for the sake of brevity, we omit to indicate in the Black-Scholes formula the dependence w.r.t. (σ t , r t , q t ) 0≤t≤T . For computing the sensitivities

Greek h i (X P T ) = ∂ i x E(h(X P T + x))| x=0
, we proceed similarly to the main term. First, we have E(h(X P T +x)) = Call BS (0, S 0 e x ; T, K). By successive differentiations, we obtain (using matrix notation)

         Greek h 1 (X P T ) Greek h 2 (X P T ) Greek h 3 (X P T ) Greek h 4 (X P T ) Greek h 5 (X P T ) Greek h 6 (X P T )          =        
1 0 0 0 0 0 1 1 0 0 0 0 1 3 1 0 0 0 1 7 6 1 0 0 1 15 25 10 1 0 1 31 90 65 15 1

                 S 0 ∂ S Call BS (0, S 0 ; T, K) S 2 0 ∂ 2 S Call BS (0, S 0 ; T, K) S 3 0 ∂ 3 S Call BS (0, S 0 ; T, K) S 4 0 ∂ 4 S Call BS (0, S 0 ; T, K) S 5 0 ∂ 5 S Call BS (0, S 0 ; T, K) S 6 0 ∂ 6 S Call BS (0, S 0 ; T, K)          . (1.22)
Regarding the summation of the correction terms, it implies that

6 i=1 η i,T Greek h i (X P T ) = 6 i=1 ηi,T S i 0 ∂ i S Call BS (0, S 0 ; T, K) where η1,T =0, η2,T = 3 2 c 1,T + 1 2 c 2,T + 1 2 c 3,T + 9 4 c 4,T + 9 4 c 5,T + 13 2 c 6,T + 9c 7,T + 9 2 c 8,T , η3,T =c 1,T + 4c 4,T + 4c 5,T + 12c 6,T + 66c 7,T + 33c 8,T , η4,T =c 4,T + c 5,T + 3c 6,T + 153 2 c 7,T + 153 4 c 8,T ,
η5,T =24c 7,T + 12c 8,T , η6,T =2c 7,T + c 8,T .

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The expressions of the coefficients (c i,T ) 1≤i≤8 are given in Theorem 1, but in order to specify them in the current case a(t, x) = σ(t, C t exp(x)), we denote them by (α i,T ) 1≤i≤8 instead of (c i,T ) 1≤i≤8 . Easy computations show that these definitions coincide with those given in Theorem 2. Then, the second order expansion formula is obtained by keeping only the first coefficient α 1,T , while all the coefficients are taken for the third order expansion formula.

Error estimates. We have already observed that M Y,0 = M 0 and M Y,1 = M 1 . It remains to estimate the factor h (1) (vX T + (1v)X P T ) 2 arising in the error bounds of Theorem 1. For v ∈ [0, 1], define

σ v t := vσ(t, X t ) + (1 -v)σ t ∈ [σ inf , |σ| ∞ ] and σ 2,v t := vσ 2 (t, X t ) + (1 -v)σ 2 t ∈ [σ 2 inf , |σ| 2 ∞ ]: clearly we have d(vX t + (1 -v)X P t ) = σ v t dW t - 1 2 σ 2,v t dt.
We denote by P v the probability measure under which W v t = W t -2 t 0 σ v s ds is a Brownian motion. Then, putting d 0 = log(S 0 C T /K) and using h ′ (x) = e - R T 0 qsds e x 1 x-log(S0)>-d0 , we obtain

E([h ′ (vX T + (1 -v)X P T )] 2 ) = S 2 0 e -2 R T 0 qsds E(e 2 R T 0 σ v s dWs- R T 0 σ 2,v s ds 1 R T 0 σ v s dWs-1 2 R T 0 σ 2,v s ds>-d0 ) = S 2 0 e -2 R T 0 qsds E v (e 2 R T 0 [σ v s ] 2 ds- R T 0 σ 2,v s ds 1 R T 0 σ v s dW v s +2 R T 0 [σ v s ] 2 ds-1 2 R T 0 σ 2,v s ds>-d0 ) ≤ S 2 0 e -2 R T 0 qsds+2|σ| 2 ∞ T P v ( T 0 σ v s dW v s + 2|σ| 2 ∞ T > -d 0 ). (1.23)
If -d 0 > 2|σ| 2 ∞ T , one can apply the Bernstein exponential inequality to show that the above probability is bounded by exp(-

(-d0-2|σ| 2 ∞ T ) 2 2|σ| 2 ∞ T
). Using the inequality (ab) 2 ≥ 1 2 a 2b 2 , it follows that

sup v∈[0,1] E([h ′ (vX T + (1 -v)X P T )] 2 ) ≤ CS 2 0 exp(-
d 2 0 4|σ| 2 ∞ T ) (1.24)
where the constant C depends in an increasing way on the bounds on the coefficients and on the maturity. Note that the inequality (1.24) is also valid if

0 ≤ -d 0 ≤ 2|σ| 2 ∞ T : indeed, from (1.23), we write E([h ′ (vX T +(1-v)X P T )] 2 ) ≤ S 2 0 e -2 R T 0 qsds+2|σ| 2 ∞ T ≤ S 2 0 e -2 R T 0 qsds+2|σ| 2 ∞ T exp( (2|σ| 2 ∞ T ) 2 4|σ| 2 ∞ T ) exp(- d 2 0 4|σ| 2 ∞ T ) ≤ CS 2 0 exp(- d 2 0 4|σ| 2 ∞ T ).
To sum up we obtain

sup v∈[0,1] [h ′ (vX T + (1 -v)X P T ) 2 ≤ CS 0 exp(- [log(S 0 C T /K)] 2 8|σ| 2 ∞ T ) (1.25)
for any d 0 ≤ 0, or equivalently for any K ≥ S 0 C T . Thus, the announced estimates on Error 2 and Error 3 are valid for any Out of The Money calls. Using a similar analysis, the same estimates hold for Out of The Money puts (K ≤ S 0 C T ). But, since the call/put parity is preserved within these expansions, error estimates are equal for call/put with the same characteristics. Thus, estimates for Out of The Money puts transfer to In The Money calls. This completes the proof. ⊓ ⊔ A careful inspection of the current proof and that of Theorem 1 reveals that the factor 8 in the exponential (1.25) can be improved and actually, it can be taken strictly larger than 2: this gives presumably better error estimates for K ≪ S 0 or K ≫ S 0 .

B Proof of Theorem 3

The derivation of the expansion is obtained following the same lines as those for Theorem 2. We detail only the main arguments. The proxy for the process (Y t ) 0≤t≤T is defined by Y P t = log(K) +

E(e - R T 0 q T -t dt (S 0 - C 0 C T e Y P T ) + ) = E(e - R T 0 rsds e Y P T K (S 0 C T K e Y P T -K) + ) = Ẽ(e - R T 0 rsds (S 0 e R T 0 (rs-qs)ds e - R T 0 σsd Ws- 1 2 R T 0 σ2 s ds -K) + ) = Call BS (0, S 0 ; T, K; (σ t ) 0≤t≤T , (r t ) 0≤t≤T , (q t ) 0≤t≤T ).
This gives the main term in the expansion. Regarding the computation of the sensitivities Greek h i (Y P T ), observe that E(h(Y P T + x)) = Call BS (0, S 0 ; T, Ke x ), omitting the last parameters (σ t , r t , q t ) 0≤t≤T . Thus, we easily relate the sensitivities Greek h i (Y P T ) to the Greeks of Call BS (0, S 0 ; T, K) w.r.t. K (instead of S 0 in the Theorem 2). The relation is affine and is similar to (1.22). The other steps of the proof are analogous to that of Theorem 2, replacing S 0 and σ by K and σ in most places. ⊓ ⊔ C Computations of derivatives of the Black-Scholes price function w.r.t. S and K

In the following proposition, we make explicit the formulas for the six first derivatives of Call BS (0, S; T, K; σ, r, q) (in short Call BS (0, S; T, K)) w.r.t. S and K, leaving the proofs to the reader. These formulas are necessary to implement the expansions of Theorems 2 and 3.

Proposition 1 (Black-Scholes Greeks). Using the notation from Definition 3, the sensitivities w.r.t. S are given by

∆ S (t, S; T, K) = ∂ ∂S Call BS (t, S; T, K) = e -q(T -t) N (d 1 ), Γ S (t, S; T, K) = ∂ 2 ∂S 2 Call BS (t, S; T, K) = e -q(T -t) N ′ (d 1 ) Sσ √ T -t , Speed S (t, S; T, K) = ∂ 3 ∂S 3 Call BS (t, S; T, K) = - Γ S S ( d 1 σ √ T -t + 1), ∂ 4 ∂S 4 Call BS (t, S; T, K) = Γ S S 2 (2 + 3d 1 σ √ T -t + d 2 1 -1 σ 2 (T -t) ), ∂ 5 ∂S 5 Call BS (t, S; T, K) = Γ S S 3 (-6 - 11d 1 σ √ T -t + 6(1 -d 2 1 ) σ 2 (T -t) + 3d 1 -d 3 1 σ 3 (T -t) 3 2
),

∂ 6 ∂S 6 Call BS (t, S; T, K) = Γ S S 4 (24 + 50d 1 σ √ T -t + 35(d 2 1 -1) σ 2 (T -t) + 10d 1 (d 2 1 -3) σ 3 (T -t) 3 2 + 3(1 -2d 2 1 ) + d 4 1 σ 4 (T -t) 2 ).
The sensitivities w.r.t. K are given by 89$:;<-=>$$+"?@&-=5/A)BB >$$89$:;<-=5/AB 89$:;<-=>$$+"?@&-=5/CBB >$$89$:;<-=5/CB 89$:;<-=>$$+"?@&-=4/A)BB 89$:;<-=>$$+"?@&-=4/CBB >DE-89$:;<=>$$+"?@&-=4/EBB

∆ K (t, S; T, K) = ∂ ∂K Call BS (t, S; T, K) = -e -r(T -t) N (d 2 ), Γ K (t, S; T, K) = ∂ 2 ∂K 2 Call BS (t, S; T, K) = e -r(T -t) N ′ (d 2 ) Kσ √ T -t , Speed K (t, S; T, K) = ∂ 3 ∂K 3 Call BS (t, S; T, K) = - Γ K K (1 - d 2 σ √ T -t ), ∂ 4 ∂K 4 Call BS (t, S; T, K) = Γ K K 2 (2 - 3d 2 σ √ T -t + d 2 2 -1 σ 2 (T -t) ), ∂ 5 ∂K 5 Call BS (t, S; T, K) = Γ K K 3 (-6 + 11d 2 σ √ T -t + 6(1 -d 2 2 ) σ 2 (T -t) + d 3 2 -3d 2 σ 3 (T -t) 3 2 ), ∂ 6 ∂K 6 Call BS (t, S; T, K) = Γ K K 4 (24 - 50d 2 σ √ T -t + 35(d 2 2 -1) σ 2 (T -t) - 10d 2 (d 2 2 -3) σ 3 (T -t) 3 2 + 3(1 -2d 2 2 ) + d 4 2 σ 4 (T -t) 2 ). 3M

3M

-12.3 -5.8 -2.4 -0.9 -0.3 -0.1 -0.1 -0.1 -0.2 -1.3 -2.6 -4.9 -8.5 -1.7 -0.9 -0.5 -0.3 -0.2 -0.1 -0.1 -0.1 -0.1 -0.3 -0.5 -0.6 -0.8 -17.1 -6.8 -2.6 -0.9 -0.3 -0.1 -0.1 -0.1 -0.2 -1.2 -2.4 -4.2 -6.8 -1.7 -0.9 -0.5 -0.3 -0.2 -0.1 -0.1 -0.1 -0.1 -0.3 -0.5 -0.6 -0.8 -1.4 -0.4 -0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.4 0.6 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.1 -0.4 6M -13.3 -3.4 -1.5 -0.6 -0.3 -0.2 -0.1 -0.2 -0.2 -0.4 -1.6 -4.4 -14.8 -1.9 -0.9 -0.6 -0.4 -0.2 -0.2 -0.1 -0.2 -0.2 -0.3 -0.5 -0.9 -1.5 -17.7 -3.7 -1.6 -0.7 -0.3 -0.2 -0.1 -0.2 -0.2 -0.4 -1.5 -4.0 -11.2 -2.1 -0.9 -0.6 -0.4 -0.2 -0.2 -0.1 -0.2 -0.2 -0.3 -0.5 -0.8 -1.4 -1.1 -0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.8 0.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.1 -0.7 1Y -23.5 -8.0 -2.3 -1.2 -0.4 -0.3 -0.3 -0.3 -0.5 -1.2 -3.9 -7.7 -36.8 -3.5 -1.9 -1.0 -0.7 -0.4 -0.3 -0.3 -0.3 -0.4 -0.7 -1.2 -1.6 -3.1 -34.1 -9.2 -2.4 -1.2 -0.4 -0.3 -0.3 -0.3 -0.5 -1.1 -3.6 -6.7 -23.2 -3.9 -2.0 -1.0 -0.7 -0.4 -0.3 -0.3 -0.3 -0.4 -0.6 -1.1 -1.5 -2.8 -2.1 -0.3 -0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.2 3.0 2.0 0.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.1 -0.2 -2.3 1.5Y -28.4 -10.6 -3.5 -2.0 -0.7 -0.4 -0.4 -0.5 -0.6 -1.6 -5.6 -12.1 -50.0 -4.7 -2.7 -1.5 -1.1 -0.6 -0.4 -0.4 -0.5 -0.5 -0.9 -1.7 -2.4 -4.3 -41.3 -12.2 -3.7 -2.1 -0.7 -0.4 -0.4 -0.5 -0.6 -1.5 -5.1 -10.2 -29.7 -5.3 -2.9 -1.6 -1.2 -0.6 -0.4 -0.4 -0.5 -0.5 -0.9 -1.6 -2.2 -3.8 -2.5 -0.4 -0.1 -0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.4 4.4 2.5 0.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.1 -0.4 -3.3 2Y -36.5 -14.5 -5.3 -1.9 -0.8 -0.6 -0.5 -0.6 -0.9 -2.0 -6.0 -17.6 -91.0 -6.2 -3.7 -2.2 -1.3 -0.8 -0.6 -0.5 -0.6 -0.8 -1.2 -2.1 -3.3 -6.2 -55.7 -17.2 -5.6 -1.9 -0.8 -0.6 -0.5 -0.6 -0.9 -2.0 -5.5 -14.2 -43.6 -7.1 -4.0 -2.3 -1.3 -0.8 -0.6 -0.5 -0.6 -0.8 -1.2 -1.9 -3.0 -5.3 -3.5 -0.6 -0.2 -0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.7 10.0 3.6 0.5 0.1 0.0 0.0 0.0 0.0 0.0 0.0 -0.1 -0.1 -0.6 -6.6 3Y -64.7 -17.8 -11.3 -2.9 -1.4 -0.9 -0.8 -0.8 -1.3 -3.1 -10.5 -27.0 -140.9 -10.5 -5.0 -3.9 -1.9 -1.2 -0.9 -0.8 -0.8 -1.1 -1.8 -3.2 -4.9 -8.8 -122.7 -21.1 -12.6 -3.0 -1.4 -0.9 -0.8 -0.8 -1.2 -3.0 -9.3 -20.7 -57.5 -12.6 -5.6 -4.3 -2.0 -1.3 -0.9 -0.8 -0.8 -1.1 -1.8 -3.0 -4.3 -7.3 -8.9 -0.8 -0.4 -0.1 -0.1 0.0 0.0 0.0 0.0 0.1 0.3 1.2 16.0 10.7 0.6 0.3 0.1 0.0 0.0 0.0 0.0 -0.1 -0.1 -0.3 -1.2 -10.1 5Y -106.7 -30.6 -13.2 -5.9 -2.2 -1.5 -1.2 -1.3 -2.1 -4.8 -17.3 -45.5 -471.9 -18.1 -8.6 -5.5 -3.6 -2.0 -1.5 -1.2 -1.3 -1.9 -2.9 -5.3 -7.9 -14.5 -256.0 -38.2 -14.5 -6.1 -2.2 -1.5 -1.2 -1.3 -2.1 -4.6 -14.9 -32.2 -88.5 -23.2 -10.0 -6.1 -3.8 -2.1 -1.5 -1.2 -1.3 -1.8 -2.7 -4.7 -6.7 -11.5 -18.8 -1.6 -0.5 -0.3 -0.1 -0.1 0.0 0.0 0.1 0.1 0.5 2.4 38.9 23.1 1.3 0.3 0.2 0.1 0.0 0.0 -0.1 -0.1 -0.2 -0.5 -2.4 -20.9 10Y -172.3 -69.5 -30.2 -10.0 -4.4 -2.7 -2.2 -2.4 -3.6 -9.1 -34.6 -103.0 ND -33.7 -19.2 -12.1 -6.7 -4.1 -2.8 -2.2 -2.4 -3.4 -5.6 -10.2 -15.5 -29.6 -472.8 -94.7 -34.0 -10.3 -4.4 -2.8 -2.2 -2.4 -3.6 -8.7 -28.2 -60.5 -159.3 -47.5 -24.3 -14.3 -7.3 -4.2 -2.8 -2.2 -2.4 -3.3 -5.1 -8.7 -12.3 -20.9 -33.9 -5.0 -1.1 -0.5 -0.3 -0.2 0.0 0.1 0.2 0.3 1.1 6.7 146.5 27.4 2.5 0.6 0.4 0.2 0.1 0.0 -0.1 -0.2 -0.3 -1.3 -7.0 -58.7

Table 1.4. CEV model (β = 0.8): errors in bps on the implied volatility using the 6 approximations ImpVol(AppPrice(2,S0)), AppImpVol(2,S0), ImpVol(AppPrice(2,K)), AppImpVol(2,K), ImpVol(AppPrice(3,S0)) and ImpVol(AppPrice(3,K)).

Fig. 1 . 1 .

 11 Fig. 1.1. CEV model (β = 0.8): errors in bps on the implied volatility using the 7 approximations ImpVol(AppPrice(2,S0)), AppImpVol(2,S0), ImpVol(AppPrice(2,K)), AppImpVol(2,K), ImpVol(AppPrice(3,S0)), ImpVol(AppPrice(3,K)) and Av. ImpVol(AppPrice(3,.))

  Error 3 ,

	where
	π2,T =	3 2	α1,T +	1 2	α2,T +	1 2	α3,T +	9 4	α4,T +	9 4	α5,T +	13 2	α6,T + 9α 7,T +	9 2	α8,T ,
	π3,T = α1,T + 4α 4,T + 4α 5,T + 12 α6,T + 66 α7,T + 33 α8,T ,
	π4,T = α4,T + α5,T + 3α 6,T +	153 2	α7,T +	153 4	α8,T ,
	π5,T = 24α 7,T + 12 α8,T ,
	π6,T = 2α 7,T + α8,T ,

  We interpret e Y P T /K as the Radon-Nikodym derivative of a new measure P w.r.t. P on F T , under which Wt = W t -

	t 0 σs dW s -1 2 s ds. t t 0 σ2 0 σs ds is a standard BM; then we obtain

  28.755 28.003 27.312 26.673 26.080 25.528 25.010 24.535 24.074 23.232 22.845 22.477 22.128 6M 29.590 28.017 27.325 26.686 26.092 25.539 25.021 24.535 24.078 23.646 22.851 22.133 21.177 1Y 31.537 29.624 28.046 27.352 26.116 25.561 25.042 24.555 23.664 22.867 21.814 21.189 19.602 1.5Y 32.706 30.568 28.831 28.075 26.736 25.583 25.062 24.115 23.681 22.513 21.202 20.359 18.733 2Y 34.034 31.618 29.692 28.103 26.761 26.163 25.083 24.133 23.288 22.177 20.921 19.621 17.

													619
	3Y 37.339 32.840 31.698 28.924 27.459 26.209 25.124 24.170 22.930 21.547 19.882 18.555 16.406
	5Y 42.069 35.797 33.000 30.816 28.271 26.908 25.205 23.802 22.262 20.709 18.589 17.011 14.382
	10Y 47.850 41.604 37.460 33.144 30.082 27.758 25.378 23.535 21.407 19.089 16.346 14.325 10.993
		Table 1.3. CEV model (β = 0.2): implied volatilities in %.
	7
	5
	)
	)/0)	)/1)	)/2)	)/20	)/*0	)/30	./))	./.)	./.0	./4)	./0)	./10	5/))
	65
	67
	61

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