Asymptotics of Implied Volatility in the Gatheral Model

School of Education, Culture and Communication

Division of Applied Mathematics

MASTER (1 YEAR) THESIS IN MATHEMATICS / APPLIED MATHEMATICS

Asymptotics of implied volatility in the Gatheral model

by

Finnan Tewolde and Jiahui Zhang

Magisterarbete i matematik / tillämpad matematik

MÄLARDALEN UNIVERSITY SE-721 23 VÄSTERÅS, SWEDEN

School of Education, Culture and Communication

Division of Applied Mathematics

Master (1 year) thesis in mathematics / applied mathematics

Date:

2019-05-27

Project name:

Asymptotics of implied volatility in the Gatheral model

Author(s):

Finnan Tewolde and Jiahui Zhang

Supervisor(s): Anatoliy Malyarenko Version: 4th June 2019 Reviewer: Olha Bodnar Examiner: Ying Ni Comprising: 15 ECTS credits

Abstract

The double-mean-reverting model by Gatheral is motivated by empirical dynamics of the vari-ance of the stock price. No closed-form solution for European option exists in the above model. We study the behaviour of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the-money. Using the method by Pagliarani and Pascucci, we calcu-late explicitly the first few terms of the asymptotic expansion of the implied volatility within a parabolic region.

Contents

1 Introduction 4 2 Asymptotic expansions 6 2.1 O-symbols . . . 6 2.2 Asymptotic sequences . . . 8 2.3 Asymptotic expansions . . . 9

3 The expansion for the European call price 11 3.1 The method of differential operatorA . . . 11

3.1.1 Assumptions . . . 13

3.1.2 The price expansion . . . 13

3.2 Operator expansion . . . 14

4 The expansion for the implied volatility 19

5 Conclusions 25

Acknowledgements

Thanks to our thesis instructor – Professor Anatoliy Malyarenko, who has showed enormous pa-tience and dedication for our theis work. We are really grateful of being students of Mälardalen University, and we cherish each course we have learnt in our bachelor and master program re-lated to mathematics and finance since they do enrich our knowledge in this field.

Chapter 1

Introduction

The history of implied volatility can be traced back at least to Latané and Rendleman Jr (1976), where it appeared under the name “implied standard deviation”, that is, the standard deviation of asset returns, which are implied in actual European call option prices when investors price options according to the Black–Scholes model. For a recent review of different approaches in determining the implied volatility, see Orlando and Taglialatela (2017).

In brief, determining the implied volatility uses several interesting mathematical theories. In particular, J.-P. Fouque and his collaborators used perturbation methods, see their books Fouque et al. (2000), Fouque et al. (2011), and recent papers Fouque et al. (2014), Fouque et al. (2016), and Fouque et al. (2017).

The theory of large deviations is used in the papers by A. Jacquier and his collaborators, see Forde and Jacquier (2009), Forde et al. (2010), Forde and Jacquier (2011), Jacquier et al. (2013), Jacquier and Roome (2015), Jacquier and Lorig (2015), Alòs et al. (2019).

Methods of partial differential equations are used by H. Berestycki and his collaborators, see Berestycki et al. (2002), Berestycki et al. (2004).

A good general reference is the book Gulisashvili (2012).

In order to briefly explain our contribution to the subject, introduce some notation. Let d ≥ 2 be a positive integer, let T0> 0 be a time horizon, let T ≤ T0be the maturity of a financial product,

and let { Zt: 0 ≤ t ≤ T } be a continuous Rd-valued adapted Markov stochastic process on a

probability space (Ω, F, P) with a filtration { Ft: 0 ≤ t ≤ T }. Assume that the first coordinate

St of the process Zt represents the risk-neutral price of a financial asset, and the d − 1 remaining

components Yt represent stochastic factors in the market with zero interest rate and no dividends.

On the one hand, the time t no-arbitrage price of a European call option with strike price K> 0 and maturity T ∈ (0, T0] is Ct,T,K= v(t, St, Yt, T, K), where

v(t, s, y, T, K) = E[max{0, ST− K} | Ft, St = s, Yt = y],

and where (t, s, y) ∈ [0, T ] × (0, ∞) × Rd−1. We change to logarithmic variables and define the option price by

u(t, x, y, T, k) = v(t, ex, y, T, ek),

where x is the log-price of the underlying asset, k is the log-strike of the option, and (t, x, y) ∈ [0, T ] × R × Rd−1.

On the other hand, by assuming that we have zero interest rate and zero dividend yield as pre-condition, the Black–Scholes price is

uBS(σ , τ, x, k) = exN (d+) − ekN (d−), where d±= 1 σ √ τ  x− k ±σ 2 τ 2  , τ := T − t and x, k ∈ R, s ∈ (0, ∞), τ ∈ (0, T ), and N (·) is the cumulative distribution function of a standard normal random variable.

Definition 1 (Pagliarani and Pascucci (2017)). The implied volatility σ = σ (t, x, y, T, k) is the unique positive solution of the equation

uBS(σ , T − t, x, k) = u(t, x, y, T, k).

Pagliarani and Pascucci (2012) derived a fully explicit approximation for the implied volat-ility at any given order N ≥ 0 for the scalar case. Lorig et al. (2017) extended this result to the multidimensional case. Denote the above approximation by σN(t, x, y, T, k).

Pagliarani and Pascucci (2017) proved that under some mild conditions, the following limits exist: ∂q ∂ Tq ∂m ∂ kmσN(t, x, y) :=(T,k)→(t,x)lim ∂q ∂ Tq ∂m ∂ kmσN(t, x, y, T, k), where the limit is taken as (T, k) approaches (t, x) within the parabolic region

Pλ = { (T, k) ∈ (0, T0] × R : |x − k| ≤ λ

√ T− t } for an arbitrary positive real number λ and nonnegative integers m and q.

Moreover, Pagliarani and Pascucci (2017) established an asymptotic expansion of the im-plied volatility in the following form:

σ (t, x, y, T, k) =

_∑

2q+m≤N 1 q!m! ∂q ∂ Tq ∂m ∂ kmσN(t, x, y)(T − t) q (k − x)m + o((T − t)N/2+ |k − x|N), as (T, k) approaches (t, x) withinPλ.

We apply the above described theory to the double-mean-reverting model by Gatheral (2008), see also Bayer et al. (2013), to fit in the setting of Pagliarani and Pascucci (2017) (later described in Sect. 3.3.1 ), and given by the following system of stochastic differential equations

dSt = √ vtStdWt1, dvt = κ1(vt0− vt) dt + ξ1vαt1dWt2, dvt0= κ2(θ − v0t) dt + ξ2v0αt2dWt3, (1.1)

where the Wiener processes Wti are correlated: E[WsiW j

t ] = ρi jmin{s,t}, and where S0, v0, and

v0₀are known positive real numbers.

The explanation behind the reason why the model in (1.1) fits in the setting of Pagliarani and Pascucci (2017) is described later in our section 3.1.1.

Chapter 2

Asymptotic expansions

This section gives a brief introduction and description of Asymptotic expansions as preliminaries for our later on calculations.

2.1

O-symbols

The order symbols O and o forming the order relations as described below Erdélyi (1956). : Let R be a subset of a set R. In most applications, R is the set R of reals, but we are also interested in the case ofR = R2, the two-dimensional real linear space. In both cases, the definition of the limit

x0= lim

n→∞xn, xn∈ R, x0∈R (2.1)

is well-known from the course of calculus.

Definition 2. A point x0 ∈R is called a limit point of a set R ⊆ R if there is a sequence

{ xn: n ≥ 1 } of points in R satisfying (2.1).

Let φ (x) and ψ(x) be the real- or complex-valued functions of x in R, and let x0be a limit

point of R.

Definition 3. For all x in R we write

φ (x) = O(ψ (x)), x→ x0

if there are two positive constants A and δ such that for all x ∈ R located at a distance at most δ from x0we have

|φ (x)| ≤ A|ψ(x)|. Definition 4. For all x in R we write

φ (x) = o(ψ (x)), x→ x₀

if for any positive real number ε there is a positive constant δ such that for all x ∈ R located at a distance at most δ from x0we have

To formula the above cases more clearly when ψ 6= 0 in R: φ =

(

O(ψ), if_ψφ is bounded in R [as x → x0]

o(ψ), if φ

ψ → 0 [as x → x0]

Example 1. We prove that sin x = O(x), x → 0. Proof:

Let φ (x) = sin x, and ψ(x) = x, we then have, lim x→0 φ (x) ψ (x)= limx→0 sin x x = limx→0 cos x 1 = 1

which proves that

φ = O(ψ ) if φ

ψ is bounded in R [as x → x0] i.e. sin x = O(x), x → 0 is true.

Example 2. We prove that x2= O(x), x → 0. Proof:

Let φ (x) = x2, ψ(x) = x, x0= 0 and δ = 1, if |x| ≤ 1 we then have,

|φ (x)| = |x2| = x2= |x| · |x| ≤ 1 · |x| = 1 · |ψ(x)| = A · |ψ(x)| which proves the condition

|φ (x)| ≤ A · |ψ(x)| i.e. x2= O(x), x → 0 is true.

Example 3. We prove that x2= o(x), x → 0. Proof:

Let φ (x) = x2, and ψ(x) = x, we then have, lim x→0 φ (x) ψ (x)= limx→0 x2 x = limx→0 x 1 = 0 which proves that

o(ψ)ifφ

ψ → 0[as x → x0] i.e. x2= o(x), x → 0 is true.

Example 4. We prove that x − sin x = o(x), x → 0. Proof:

Let φ (x) = x − sin x, and ψ(x) = x, we then have, lim x→0 φ (x) ψ (x)= limx→0 x− sin x x = limx→0 1 − cos 0 1 = 1 − 1 = 0 which proves that

o(ψ), ifφ

ψ → 0[as x → x0] i.e. x − sin x = o(x), x → 0 is true.

2.2

Asymptotic sequences

Author Erdélyi introduces various methods for the asymptotic evaluation and solutions of ordin-ary linear differential equations by means of asymptotic expansions in Erdélyi (1956). Before we come to asymptotic expansions, we first introduce the definition of the asymptotic sequence Erdélyi (1956).

Definition 5. A sequence of functions { φn: n ≥ 1 } is called an asymptotic sequence for x → x0

in R if for each n, φnis defined in R and φn+1= o (φn) as x → x0in R.

Example 5. Put x0= 0, R = (0, ∞). The sequence

φn(x) = xn, n≥ 1

is an asymptotic sequence as x → 0. Indeed, | xn+1|≤ ε(xn_{). The proof is as following;}

xn+1= o(xn) Fix ε > 0 and δ := ε; then by 4

| xn+1| =| x | · | xn_|

| xn+1| ≤ δ | xn_{|= ε | x}n_{|, 0 < x < δ}

Definition 6. A sequence of functions { φq,m(x,t) : q ≥ 1 m ≥ 1 } is called an asymptotic

se-quence for x → x0and t → t0in R if for all q and m φq,mis defined in R and φq+1,m= o (φq,m) as

x→ x₀and φq,m+1= o (φq,m) as t → t0in R.

Example 6. A great example we are going to provide is as following, we use double asymptote sequences in: (T − t)q(k − x0)m T → t, k → x0 and as, (T − t)q+1(k − x0)m (T − t)q_{(k − x} 0)m → 0, (T − t) q_{(k − x} 0)m+1 (T − t)q_{(k − x} 0)m → 0, where both T and k are variables such that,

T → t, k → x₀ hence we write:

(T − t)q+1(k − x0)m+1= o ((T − t)q(k − x0)m)

2.3

Asymptotic expansions

The article Erdélyi (1956) has provided us with definition of the asymptotic expansions as fol-lowing;

Let { φn: n ≥ 1 } be an asymptotic sequence for x → x0in R.

Definition 7. The right hand side of the equation

f(x) =

_∑

k∈{k1,k2,...,kN}

akφk(x) + o(φkN(x))

is called an asymptotic expansion of the function f as x → x0 with respect to the asymptotic

sequence { φn: n ≥ 1 } if ak6= 0 and k1< k2< · · · < kN.

We now present an interesting example of calculating complicated limits using asymptotic expansions. Example 7. Calculate lim x→+∞x 3/2 _√ x+ 3 −√x− 3 2√x  .

To do that, consider the function f (x) =√x+ 3 defined on the set R = (−3, ∞). Define a16= 0

and α1∈ R such that

lim x→+∞ √ x+ 3 a1xα1 = 1. We have a1= 1, α1=1₂. It follows that

√

x+ 3 =√x+ o(√x), x→ +∞.

Put f (x) =√x+ 3 −√xon the set R = (0, ∞) and define a26= 0 and α2∈ R in such a way

that lim x→+∞ √ x+ 3 −√x a2xα2 = 1. We have lim x→+∞ √ x+ 3 −√x a2xα2 = lim x→+∞ (√x+ 3 −√x)(√x+ 3 +√x) a2xα2( √ x+ 3 +√x) = lim x→+∞ 3 a2xα2( √ x+ 3 +√x). Put a2=3₂, α2= −1₂. Then we have

lim x→+∞ √ x+ 3 −√x a₂xα2 = lim_x→+∞ 2 x−1/2(√x+ 3 +√x)= limx→+∞ 2 1 +p1 + 3/x = 1. It follows that √ x+ 3 =√x+ 3 2√x+ o  1 √ x  , x→ +∞.

Put f (x) =√x+ 3 −√x− 3

2√x on the set R = (0, ∞) and define a36= 0 and α3∈ R in such a

way that lim x→+∞ √ x+ 3 −√x− 3 2√x a3xα3 = 1. We have lim x→+∞ √ x+ 3 −√x− 3 2√x a₃xα3 = lim_x→+∞ (√x+ 3 −√x− 3 2√x)2 √ x 2a3xα3+1/2 = lim x→+∞ 2√x2_{+ 3x − (2x + 3)} 2a3xα3+1/2 = lim x→+∞ (2√x2_{+ 3x − (2x + 3))(2}√_x2_{+ 3x + (2x + 3))} 2a3xα3+1/2(2 √ x2_{+ 3x + (2x + 3))} = lim x→+∞ −9 2a3xα3+1/2(2 √ x2_{+ 3x + (2x + 3))}.

Put a3= −9₈, α3= −3₂. Then we have

lim x→+∞ √ x+ 3 −√x− 3 2√x a₃xα3 = lim_x→+∞ 4 x−1(2√x2_{+ 3x + (2x + 3))} = lim x→+∞ 4 2p1 + 3/x + 2 + 3/x= 1. It follows that √ x+ 3 =√x+ 3 2√x− 9 8x√x+ o  1 x√x  , as x→ +∞.

This expansion is enough to calculate the limit. Indeed, x3/2 √ x+ 3 −√x− 3 2√x  = x3/2 _√ x+ 3 2√x− 9 8x√x+ o  1 x√x  −√x− 3 2√x  = x3/2  − 9 8x√x+ o  1 x√x  = −9 8+ o(x−3/2) x−3/2 , x→ +∞. And finally lim x→+∞x 3/2 _√ x+ 3 −√x− 3 2√x  = −9 8+ limx→+∞ o(x−3/2) x−3/2 = − 9 8.

The examples of double asymptotic expansions are described later in our Chapter 3 and Chapter 4.

Chapter 3

The expansion for the European call

price

3.1

The method of differential operator

A

Before we calculate our price expansions we first need to understand the differential operator ¯

A , while A is the differential operator when we make transition into log–variables.

The differential operator A defined in Pagliarani and Pascucci (2017) has the following¯ form: ¯ A = 1 2 d

∑

i, j=1 ¯ ai j(t, z) ∂2 ∂ zi∂ zj + d

∑

i=1 ¯ ai(t, z) ∂ ∂ zi (3.1) while the operatorA , after switching into logarithmic variables is defined as:

A =1 2 d

∑

i, j=1 ai j(t, z) ∂2 ∂ zi∂ zj + d

∑

i=1 ai(t, z) ∂ ∂ zi . (3.2)

In order to find the operatorA , we should first find operator ¯A . Both operators are described in Pagliarani and Pascucci (2017). We need the generator A , because after switching to log-¯ variables, it is transformed into another operator, which isA . To calculate ¯A , we must first find the coefficients , ¯aiand ¯ai j, for i, j = 1, 2, . . . , d such that;

Given a model, in general case;

dS= η1(t, S,Y )SdW1

dYi= µi(t, S,Y )dt + ηi(t, S,Y )dWi

¯

a₁= 0; a¯i= µi; a¯11= ρ11η12x2; a¯i1= ¯a1 j= ρ1iηiη1x; a¯i j= ¯aji= ρi jηiηj.

In our case, from (1.1), we get;

ηi=   √ νt ξ1νtα1 ξ2(νt0)α2  , µi=   0 κ1(v0t− vt) κ2(θ − v0t)  ,

therefore, we denote z1= x z2= y z3= z and for the generator, we obtain; ¯ a1= 0, a¯2= κ1(z − y), a¯3= κ2(θ − z) ¯ a₁₁= x2y, a¯₂₂= ξ₁2y2α1_{, a¯} 33= ξ22z2α2 ¯ a12= ρ12ξ1xy(α1+ 1 2)_{, a¯13= ρ13ξ2x}√_yzα2_{, a¯} 23= ρ23ξ1ξ2yα1zα2

and the operatorA is;¯

¯ A = 1 2 3

∑

i, j=1 ¯ ai j(t, z) ∂2 ∂ zi∂ zj + 3

∑

i=1 ¯ ai(t, z) ∂ ∂ zi = 1 2  x2y ∂ 2 ∂ x2+ ξ 2 1y2α1 ∂2 ∂ y2+ ξ 2 2z2α2 ∂2 ∂ z2  + ρ12ξ1xy(α1+ 1 2) ∂ 2 ∂ x∂ y+ ρ13ξ2x √ yzα2 ∂ 2 ∂ x∂ z + ρ23ξ1ξ2yα1zα2 ∂2 ∂ y∂ z+ κ1(z − y) ∂ ∂ y+ κ2(θ − z) ∂ ∂ z. Finally, we use (3.2) to calculate the operatorA ,

By the transition equations from ¯ato a, described in Pagliarani and Pascucci (2017), that is, for i, j = 2, . . . , d; a₁(t, x, y, z) = −e −2x 2 a¯11(t, e x_{, y, z), a} 11(t, x, y, z) = e−2xa¯11(t, ex, y, z) a1i(t, x, y, z) = e−xa¯1i(t, ex, y, z), ai(t, x, y, z) = ¯ai(t, ex, y, z) ai j(t, x, y, z) = ¯ai j(t, ex, y, z). so we get; a₁= −y 2, a2= κ1(z − y), a3= κ2(θ − z) a₁₁= y, a₂₂= ξ₁2y2α1_{, a} 33= ξ22z2α2 a12= ρ12ξ1y(α1+ 1 2)_{, a13= ρ13ξ2}√_yzα2_{, a} 23= ρ23ξ1ξ2yα1zα2. and A =1 2 3

∑

i, j=1 ai j(t, z) ∂2 ∂ zi∂ zj + 3

∑

i=1 ai(t, z) ∂ ∂ zi =1 2  y ∂ 2 ∂ x2+ ξ 2 1y2α1 ∂2 ∂ y2+ ξ 2 2z2α2 ∂2 ∂ z2  + ρ12ξ1y(α1+ 1 2) ∂ 2 ∂ x∂ y + ρ13ξ2 √ yzα2 ∂ 2 ∂ x∂ z+ ρ23ξ1ξ2y α1_zα2 ∂ 2 ∂ y∂ z− y 2 ∂ ∂ x+ κ1(z − y) ∂ ∂ y+ κ2(θ − z) ∂ ∂ z. (3.3)

3.1.1 Assumptions

The methods for calculating the asymptotic expansion described below require that some condi-tions (Assumpcondi-tions 2.1, 2.4, 2.5 in Lorig et al. (2017)) should be satisfied.

We do not reproduce here rather long Assumption 2.1. Instead, we use (Lorig et al., 2017, Lemma 2.3). The above lemma essentially tells us the following: if the coefficients of the system (1.1) are continuous and bounded, then Assumption 2.1 is satisfied. This is indeed the case.

Assumption 2.4 essentially tells that the matrix of coefficients of the second derivatives of the operator (3.3) must be positive-definite and the coefficients themselves must be smooth. The latter is obvious, to prove the former, write down the above matrix, say A:

A=    y ρ12ξ1y(α1+ 1 2) _ρ13ξ2√yzα2 ρ12ξ1y(α1+ 1 2) _ξ2 1y2α1 ρ23ξ1ξ2yα1zα2 ρ13ξ2 √ yzα2 _ρ 23ξ1ξ2yα1zα2 ξ22z2α2   .

Recall that x > 0, y > 0, and z > 0. That is, (x, y, z)>∈ D, where D= { (x, y, z)>∈ R3: x > 0, y > 0, z > 0 }. For the first determinant we have: y > 0. For the second determinant:

det y ρ12ξ1y (α1+12) ρ12ξ1y(α1+ 1 2) _ξ2 1y2α1 ! = (1 − ρ₁₂2 )ξ₁2y2α1+1_{> 0.}

For the third one:

det A = ξ₁2ξ₂2y2α1+1z2α2det   1 ρ12 ρ13 ρ12 1 ρ23 ρ13 ρ23 1  > 0.

Assumption 2.5 tells the following: the process (St, vt, vt0) is a Feller process, that is, for any

T > 0 and for any continuous function ϕ of 3 variables equal to 0 outside a closed bounded subset of the set D, the function f (t, z) defined on the set [0, T ) × D by

f(t, z) = E[ϕ(ST, vT, v0T) | Ft, (ST, vT, v0T) >

= z]

is continuous. To prove this, we directly apply Theorem 2.1 in Ethier and Kurtz (1986).

3.1.2 The price expansion

As Equation 3.6 in Pagliarani and Pascucci (2017), we assume that the pricing function u has the following expansion:

u= ∞

∑

n=0 u(¯z)n .

Theorem 1. The price expansion of order 0 has the form u(t, x, y, z, T, k) = exN (d+) − ekN (d−) + O(T − t), where d±=p 1 ν0(T − t)  x− k ±ν0(T − t) 2  .

Proof. Definition 3.3 in Pagliarani and Pascucci (2017) defines the Nth order approximations of price u(., .; T, k), with maturity T and log– strike k, as

¯ uN(t, z; T, k) = N

∑

n=0 u(z)n (t, z; T, k), t∈ [0, T ], z∈ R × Rd−1 (3.4)

so, for N = 0, we obtain;

¯ u0(t, z; T, k) = 0

∑

n=0 u(z)n (t, z; T, k) = u(z)₀ (t, z; T, k) which is the approximation of u(·, ·; T, k) of order 0.

For explicit definition of functions of u(z)n , see Pagliarani and Pascucci (2017).

3.2

Operator expansion

We fix ¯z = ( ¯x, ¯y, ¯z), with values (s0, ν0, ν00) and we expand the operator A by replacing the

functions ai j(t, ·) and ai(t, ·) with their Taylor series around ¯z. That is we have;

A =

_∑

∞

n=0

A¯z n.

We implementA₁(¯z)to our model in (1.1), that is; A(¯z) 1 = 1 2 d

∑

i, j=1|βββ |=1

∑

Dβββ_a i j(t,¯z) β β β ! (z − ¯z) β β β ∂zizj+ d

∑

i=1|βββ |=1

∑

Dβββ_a i(t,¯z) β ββ ! , where βββ = (β1, β2, β3) and Dβββ = ∂β1+β2+β3

∂ xβ1∂ yβ2∂ zβ3. Note the typing mistake in (Pagliarani and

Pas-cucci, 2017, Equation 3.4), the coefficient1₂ is missing! For βββ = (1, 0, 0); ∂ ∂ x(ai j) = ∂ ∂ x(ai) = 0 For βββ = (0, 1, 0), we obtain;

∂ ∂ y(a1) = − 1 2, ∂ ∂ y(a2) = −κ1, ∂ ∂ y(a3) = 0 ∂ ∂ y(a11) = 1, ∂ ∂ y(a22) = 2α1ξ 2 1y2α1−1, ∂ ∂ y(a33) = 0 ∂ ∂ y(a12) = (α1+ 1 2)ρ12ξ1y (α1−12)_, ∂ ∂ y(a13) = 1 2√yρ13ξ2z α2_, ∂ ∂ y(a23) = α1ρ23ξ1ξ2y α1−1_zα2_.

By implementing the same method, for βββ = (0, 0, 1), we get; ∂ ∂ z(a1) = 0, ∂ ∂ z(a2) = κ1, ∂ ∂ z(a3) = −κ2 ∂ ∂ z(a11) = 0, ∂ ∂ z(a22) = 0, ∂ ∂ z(a33) = 2α2ξ 2 2z2α2−1 ∂ ∂ z(a12) = 0, ∂ ∂ z(a13) = α2ρ13ξ2 √ yzα2−1_, ∂ ∂ z(a23) = α2ρ23ξ1ξ2y α1_zα2−1_.

Then, the corresponding term of the Taylor expansion becomes A(¯z) 1 (z) = 1 2 d

∑

i, j=1|βββ |=1

∑

Dβββ_a i j(t,¯z) β ββ ! (z − ¯z) β ββ ∂zizj+ d

∑

i=1|βββ |=1

∑

Dβββ_a i(t,¯z) βββ ! =1 2(y − ν0) ∂2 ∂ x2+ α1ξ 2 1ν 2α1−1 0 (y − ν0) ∂2 ∂ y2+ α2ξ 2 2ν 0 0 2α2−1 (z − ν₀0)∂ 2 ∂ z2 + (α1+ 1 2)ρ12ξ1ν (α1−12) 0 (y − ν0) ∂2 ∂ x∂ y +  1 2√ν0 ρ13ξ2ν00 α2 (y − ν0) + α2ρ13ξ2 √ ν0ν00 α2−1 (z − ν₀0)  ∂2 ∂ x∂ z + h α1ρ23ξ1ξ2ν₀α1−1ν₀0α2(y − ν0) + α2ρ23ξ1ξ2ν₀α1ν₀0α2−1(z − ν₀0) i _∂2 ∂ y∂ z −1 2(y − ν0) ∂ ∂ x+−κ1(y − ν0) + κ1(z − ν 0 0)  ∂ ∂ y− κ2(z − ν 0 0) ∂ ∂ z. (3.5)

Following Lorig et al. (2015), write the operatorA₁(¯z)in the form A(¯z) 1 (z) = 2

∑

|βββ |=1 a_βββ(z)∂ |βββ | ∂ zβββ .

The next task is to calculate the operatorA₁(¯z)(s, z − ¯z + m(¯z)(t, s) +C(¯z)(t, s)∇z).

To do that, we perform the following substitution to (3.5):   x− s0 y− ν0 z− ν0 0  →     x− s₀− (s − t)ν0/2 + (s − t)(ν0_{∂ x}∂ + ρ12ξ1ν (α1+12) 0 ∂ y∂ + ρ13ξ2ν 1/2 0 (ν 0 0)α2∂ z∂ ) y− ν0+ (s − t)κ1(ν00− ν0) + (s − t)(ρ12ξ1ν(α1+ 1 2) 0 ∂ ∂ x+ ξ 2 1ν 2α1 0 ∂ ∂ y+ ρ23ξ1ξ2ν α1 0 (ν 0 0)α2∂ z∂ ) z− ν0 0+ (s − t)κ2(θ − ν00) + (s − t)(ρ13ξ2ν₀1/2(ν00)α2∂ x∂ + ρ23ξ1ξ2ν α1 0 (ν 0 0)α2∂ y∂ + ξ 2 2(ν00)2α2∂ z∂ )     .

By collecting coefficients of the same partial derivatives, we obtain the following coefficients; ∂3 ∂ x3: 1 2(s − t)ρ12ξ1ν (α1+1₂) 0 ∂3 ∂ x2∂ y: (s − t)ξ 2 1ν 2α1 0 ( 1 2+ (α1+ 1 2)ρ 2 12) ∂3 ∂ x2∂ z: (s − t)  1 2ρ23ξ1ξ2ν α1 0 (ν 0 0)α2+ 1 2ρ12ρ13ξ1ξ2ν α1 0 (ν 0 0)α2+ α2ρ132ξ22ν0(ν00)(2α2−1)  ∂3 ∂ x∂ y2: (s − t)(2α1+ 1 2)ξ 3 1ρ12ν₀3α1−1/2 ∂3 ∂ y3: (s − t)α1ξ 4 1ν04α1−1 ∂3 ∂ y2∂ z: (s − t)ρ23ξ 2 1ξ2ν₀2α1(ν₀0)α2  2α1ξ1ν₀(α1−1)+ α2ρ23ξ2(ν00)(α2−1)  ∂3 ∂ x∂ z2: (s − t)ξ 2 2ν 1/2 0 (ν 0 0)2α2ρ13  2α2ξ2(ν00)(α2−1)+ 1 2ρ23ξ1ν (α1−1) 0  ∂3 ∂ y∂ z2: (s − t)ξ1ξ 2 2ρ23ν₀α1(ν₀0)(2α2)  2α2ξ2(ν00)(α2−1)+ α1ρ23ξ1ν₀(α1−1)  ∂3 ∂ z3: (s − t)α2ξ 4 2(ν 0 0)4α2−1 ∂3 ∂ x∂ y∂ z: (s − t)  (2α1+ 1 2)ρ12ρ23ξ 2 1ξ2ν (2α1−12) 0 (ν 0 0)α2 + (2α2)ρ13ρ23ξ1ξ22ν (2α1+1₂) 0 (ν 0 0)(2α2−1)+ 1 2ρ13ξ 2 1ξ2ν 2α1−1₂ 0 (ν 0 0)α2  ∂2 ∂ x2: 1 2 y− ν0+ (s − t)κ1(ν 0 0− ν0)
− 1 2(s − t)ρ12ξ1ν (α1+12) 0 ∂2 ∂ x∂ y: (α1+ 1 2)ρ12ξ1ν α1−12 0 y− ν0+ (s − t)κ1(ν00− ν0)
−1 2(s − t)ξ 2 1ν 2α1 0 − κ1(s − t)ρ12ξ1ν α1+12 0 + κ1(s − t)ρ13ξ2ν 1 2 0(ν 0 0)α2 ∂2 ∂ x∂ z: 1 2ρ13ξ2ν −1 2 0 (ν 0 0)α2 y− ν0+ (s − t)κ1(ν00− ν0)
+ α2ρ13ξ2ν 1 2 0(ν 0 0)α2−1 z− ν00+ (s − t)κ2(θ − ν00)
− (s − t) 1 2ξ1(ν 0 0)2α2+ κ2ρ13ξ2ν 1 2 0(ν 0 0)α2  ∂2 ∂ y2: α1ξ 2 1ν 2α1−1 0 y− ν0+ (s − t)κ1(ν00− ν0)
− κ1(s − t)ξ12ν 2α1 0 + κ1(s − t)ρ23ξ1ξ2ν α1 0 (ν 0 0)α2 ∂2 ∂ y∂ z: α1ρ23ξ1ξ2ν α1−1 0 (ν 0 0)α2 y− ν0+ (s − t)κ1(ν00− ν0)
+ α2ρ23ξ1ξ2ν₀α1(ν₀0)α2−1 z− ν₀0+ (s − t)κ2(θ − ν00)
− (s − t)ρ23ξ1ξ2ν₀α1(ν₀0)α2(κ1+ κ2) + (s − t)κ1ξ₂2(ν₀0)2α2 ∂2 ∂ z2: α2ξ 2 2(ν 0 0)2α2−1 z− ν 0 0+ (s − t)κ2(θ − ν00)
− κ2(s − t)ξ22(ν 0 0)2α2 ∂ ∂ x: − 1 2 y− ν0+ (s − t)κ1(ν 0 0− ν0)
∂ : − κ1 y− ν0+ (s − t)κ1(ν00− ν0)
+ κ1 z− ν00+ (s − t)κ2(θ − ν00)
16

By Definition 3.2 in Pagliarani and Pascucci (2017); uBS(σ ; T − t, x, k) = u(t, z; T, k) and by Definition 3.3 in Pagliarani and Pascucci (2017); for N = 1 ¯ uN(t, z; T, k) = u (z) 0 (t, z; T, k) + u (z) 1 (t, z; T, k).

Now, by Theorem D.1 in Pagliarani and Pascucci (2017); u(z)₁¯ =L₁(z)¯ u(z)₀¯ (t, z) and L(z)¯ 1 = Z T t G ¯ (z) 1 (t, s1, z)ds1,

whereL₁(z)¯ is a differential operator acting on variable z. OurG1consists of all the coefficients

of the partial derivatives and since;

u(z)₀¯ (t, z; T, k) = uBS(σ₀(z)¯ ; T − t, x, k)

all the coefficients of the partial derivatives disappear, except those dependent only with respect to x, therefore; L(z)¯ 1 = Z T t G ¯ (z) 1 (t, s1, z)ds1 = Z T t  1 2(s1− t)ρ12ξ1ν (α1+1₂) 0  ds₁ ∂ 3 ∂ x3 + Z T t  1 2 ν0− ν0+ (s1− t)κ1(ν 0 0− ν0)
− 1 2(s1− t)ρ12ξ1ν (α1+12) 0  ds1 ∂2 ∂ x2 − Z T t 1 2 ν0− ν0+ (s1− t)κ1(ν 0 0− ν0)
ds1 ∂ ∂ x = 1 4(T − t) 2 ρ12ξ1ν(α1+ 1 2) 0 ∂3 ∂ x3+ 1 4(T − t) 2  κ1(ν00− ν0) − ρ12ξ1ν(α1+ 1 2) 0  ∂2 ∂ x2 −1 4(T − t) 2 κ1(ν00− ν0) ∂ ∂ x and u(z)₁¯ =L₁(z)¯ u(z)₀¯ (t, z) =1 4(T − t) 2 ρ12ξ1ν(α1+ 1 2) 0 ∂3 ∂ x3u BS (σ₀(z)¯ ; T − t, x, k) +1 4(T − t) 2  κ1(ν00− ν0) − ρ12ξ1ν (α1+12) 0  ∂2 ∂ x2u BS_(σ(z)¯ 0 ; T − t, x, k) −1 4(T − t) 2 κ1(ν00− ν0) ∂ ∂ xu BS_(σ(z)¯ 0 ; T − t, x, k).

Finally, ¯ u1(t, z; T, k) = u(z)0 (t, z; T, k) + u (z) 1 (t, z; T, k) = uBS(σ₀(z)¯ ; T − t, x, k) +1 4(T − t) 2 ρ12ξ1ν (α1+12) 0 ∂3 ∂ x3u BS_(σ(z)¯ 0 ; T − t, x, k) +1 4(T − t) 2  κ1(ν00− ν0) − ρ12ξ1ν (α1+12) 0  ∂2 ∂ x2u BS (σ₀(z)¯ ; T − t, x, k) −1 4(T − t) 2 κ1(ν00− ν0) ∂ ∂ xu BS (σ₀(z)¯ ; T − t, x, k).

The assumptions of calculating the operator in this chapter are actively used in our Chapter 4, however, the price expansion and implied volatility expansion can be calculated separately from each other.

Chapter 4

The expansion for the implied volatility

We begin by rewriting Definition 3.4 and Equation 3.15 in Pagliarani and Pascucci (2017). Definition 8. For a call option with log–strike k and maturity T , we define the Nth order of approximation of the implied volatility σ (t, x, k; T, k) as

¯ σN(t, x, y; T, k) := N

∑

n=0 σ(x,y)(t, x, y; T, k) where σn(¯z)= u(¯z)n ∂σuBS(σ (¯z) 0 ) − 1 n! n

∑

h=2 Bn,h  1!σ₁(¯z)), 2!σ₂(¯z), . . . , (n − h + 1)!σ_(n−h+1)(¯z) _∂h σu BS_(σ(¯z) 0 ) ∂σuBS(σ (¯z) 0 )

as Bn,h denote the so–called Bell polynomials and a σn(x,y) term is a polynomial in the log–

moneyness k − x [defined in Equation 3.15, Pagliarani and Pascucci (2017)] with ¯x, ¯y= x0, y0=

x, y, see Lorig et al. (2017), p 957.

Then we rewrite equation (5.3) by Pagliarani and Pascucci (2017);

σ (t, x0, y0; T, k) =

∑

2q+m≤N ∂q ∂ Tq ∂m ∂ kmσ¯N(t, x0, y0; T, k)× (T − t)q_{(k − x} 0)m q!m! +o(| T −t | N 2 + | k −x₀|N) where ∂q ∂ Tq ∂m ∂ kmσ¯N(t, x0, y0;t, x0) =(T,k)→(t,xlim 0) ∂q ∂ Tq ∂m ∂ kmσ¯N(t, x0, y0; T, k) (4.1) and | x0− k |≤ λ √ T− t.

et al. (2017), that is; σ₀x, ¯¯y=p2a11( ¯x, ¯y) ¯ σ0(t, x0, y0; T, k) = p 2a11(x, y) σ (t, x0, y0; T, k) = p 2a11(x0, y0) + o(1) =√ν0+ o(1) hence, σ0= √ ν0.

The second step is to calculate σ₁(x,y)(t, x, y; T, k). By Lorig et al. (2017, Equation (3.9)), we obtain: σ₁(x,y)(t, x, y; T, k) = u1 ∂ ∂ σu BS_(σ 0) . Lorig et al. (2017, Equation (3.13)) gives

u₁ ∂ ∂ σu BS_(σ 0) = ˜ L1(t, T )(∂ 2 ∂ x2− ∂ ∂ x)u BS_(σ 0) τ σ0(∂ 2 ∂ x2− ∂ ∂ x)u BS_(σ 0) . (4.2)

By (3.14) and example 2.3 in Lorig et al. (2017); ˜ L1(t, T ) = Z T t a11,1(Mx(t,t1),My(t,t1),Mz(t,t1))dt1 where a_11,1=1 2(y − ¯y) and Mx(t,t1) = x − 1 2y(t1− t) + y(t1− t) ∂ ∂ x+ ρ12ξ1y α1+1/2_(t 1− t) ∂ ∂ y, My(t,t1) = y + κ1(z − y)(t1− t) + ξ12y2α1(t1− t) ∂ ∂ y+ ρ12ξ1y α1+1/2_(t 1− t) ∂ ∂ x, Mz(t,t1) = z + κ2(θ − z)(t1− t) + ξ22z2α2(t1− t) ∂ ∂ z+ ρ13ξ2y 1/2_zα2_(t 1− t) ∂ ∂ x by Lorig et al. (2017, Equation (2.25)). Then we obtain

˜ L1(t, T ) = 1 2 Z T t  (y − ¯y) + (t1− t)κ1(¯z − ¯y) + (t1− t)(ρ12ξ1y¯(α1+ 1 2)∂ ∂ x+ ξ 2 1y¯2α1 ∂ ∂ y+ ρ23ξ1ξ2y¯ α1_¯zα2 ∂ ∂ z)  dt1 =1 2τ (y − ¯y) + 1 4τ 2 κ1(¯z − ¯y) + 1 4τ 2 ρ12ξ1y¯(α1+ 1 2) ∂ ∂ x+ . . .

The remaining terms are not relevant for our case, since they become 0 when they react on Black– Scholes, meaning they do not depend on x.

Now, by (4.2), we get; σ₁x, ¯¯y=(y − ¯y) 2σ0 +τ κ1(¯z − ¯y) 4σ0 +τ ρ12ξ1y¯ (α1+12) 4σ0 ∂ ∂ x( ∂2 ∂ x2− ∂ ∂ x)u BS_(σ 0) (∂2 ∂ x2 − ∂ ∂ x)u BS_(σ 0) (4.3)

To calculate the last term, we use (3.10) in Lorig et al. (2017) but first need to define Hermite Polynomial. The definition is written below;

Hm(x) := (−1)mex 2 dm dxm  e−x2  . (4.4)

In our case, for m = 0 and m = 1,H0(ζ ) andH1(ζ ) are respectively 1 and 2ζ , where

ζ :=

x− k −1₂σ₀2τ σ0

√

2τ .

Back to our calculation;

∂ ∂ x( ∂2 ∂ x2− ∂ ∂ x)u BS_(σ 0) (∂2 ∂ x2 − ∂ ∂ x)u BS_(σ 0) = − 1 σ0 √ 2τ·H1(ζ ) = − 1 σ0 √ 2τ· 2ζ = − 1 σ0 √ 2τ· 2 · x− k −1₂σ₀2τ σ0 √ 2τ = −2 ·x− k − 1 2σ 2 0τ σ₀22τ = −x− k − 1 2σ02τ σ₀2τ

By substituting the RHS of the equation above in 4.3, we get;

σ₁x, ¯¯y=(y − ¯y) 2σ0 +τ κ1(¯z − ¯y) 4σ0 +τ ρ12ξ1y¯ (α1+12) 4σ0 ∂ ∂ x( ∂2 ∂ x2 − ∂ ∂ x)u BS_(σ 0) (∂2 ∂ x2− ∂ ∂ x)u BS_(σ 0) =(y − ¯y) 2σ0 +τ κ1(¯z − ¯y) 4σ0 +τ ρ12ξ1y¯ (α1+12) 4σ0 (−x− k − 1 2σ 2 0τ σ₀2τ ) =(y − ¯y) 2σ0 +τ κ1(¯z − ¯y) 4σ0 −ρ12ξ1y¯ (α1+12)(x − k −1 2σ02τ ) 4σ₀3 hence, σ₁x,y(t, x, y; T, k) = (T − t)κ1(z − y) 4σ0 −ρ12ξ1y (α1+1₂)_{(x − k −}1 2σ 2 0(T − t)) 4σ3 0 .

Now, to calculate σ (t, x0, y0; T, k), we need to find ¯σ1(t, x, y; T, k) and resubstitute τ by T −t,

so we obtain; ¯ σ1(t, x, y; T, k) = σ₀x,y+ σ₁x,y =√ν0+ (T − t)κ1(z − y) 4σ0 −ρ12ξ1y (α1+12)(x − k −1 2σ02(T − t)) 4σ₀3 and therefore;

Theorem 2. The asymptotic expansion of order 1 has the form, σ (t, x0, y0; T, k) = √ ν0− ρ12ξ1ν (α1+1₂) 0 (x0− k −1₂σ02(T − t)) 4σ3 0 =√ν0+ 1 4· ρ12ξ1ν α1+12− 3 2 0 (k − x0) =√ν0+ 1 4· ρ12ξ1ν α1−1 0 (k − x0) + o( √ T− t+ | k − x0|) as T → t, k → x₀ |x₀− k| ≤ λ√T− t

Theorem 3. The asymptotic expansion of order 2 of the implied volatility has the form σ (t, x0, ν0, ν00; T, k) = √ ν0+ 1 8ρ12ξ1ν α1−1 0 (k − x0) + 1 128[32κ1ν −1/2 0 (ν 0 0− ν0) + 8ρ12ξ1ν0α1 + 3ρ₁₂2ξ₁2ν₀2α1−3/2](T − t) − 3 64ρ 2 12ξ12ν02α1−2(k − x0)2 + o(T − t + (k − x0)2). (4.5) as T → t, k → x0 |x0− k| ≤ λ √ T− t Proof. Equation (4.1) takes the form

σ₂(z)(t, x, y, z; T, k) = u(z)₂  ∂ ∂ σu BS σ₀(z) −1 −1 2(σ (z) 1 ) 2 ∂2 ∂ σ2u BS σ₀(z)  _∂ ∂ σu BS σ₀(z) −1 . (4.6)

Following Pagliarani and Pascucci (2017), define the vector m(z)(t, s) by m(z)_i (t, s) = (s − t)ai(z), 1 ≤ i ≤ 3,

the matrix C(z)(t, s) by

C_{i j}(z)(t, s) = (s − t)ai j(z), 1 ≤ i, j ≤ 3,

and the operatorGn(z)(t, s, z) by

G(z)

Define the set In,hby

In,h= { i = (i1, . . . , ih) : i1+ · · · + ih= n },

and the operatorLn(z)(t, T, z) as the differential operator acting on the z-variable and defined by

(Pagliarani and Pascucci, 2017, Equation D.2) as L(z) n (t, T, z) = n

∑

h=1 Z T t Z T t1 · · · Z T th−1_i∈I

∑

_n,h G(z) i1 (t,t1, z) · · ·G (z) ih (t,th, z) dth· · · dt1. (4.8)

The sets I2,h are I2,1= {(2)}, I2,2 = {(1, 1)}. We have a11,2(x, y, z) = 0. It follows that

Equa-tion (4.8) with n = 2 includes only summaEqua-tion over the set I2,2and takes the form

˜ L(z) 2 (t, T, z) = Z T t Z T t1 G(z) 1 (t,t1, z) × a11,1(z − z + m(z)(t,t2) +C(z)(t,t2)∇z) dt2dt1.

While calculating the operatorG₁(z)(t,t1, z) using Equation (4.7), we need to calculate only

the coefficients of the three partial derivatives with respect to the variable x. We obtain G(z) 1 (t,t1, z) = 1 4(t1− t)ρ12ξ1y α1+1/2 ∂ 3 ∂ x3 + 1 2(y − y) + 1 2(t1− t)κ1(z − y) − 1 4(t1− t)ρ12ξ1y α1+1/2  ∂2 ∂ x2 −1 2[(y − y) + (t1− t)κ1(z − y)] ∂ ∂ x+ · · · .

Calculation of the first term in the right hand side of Equation (4.6) using equation ∂m ∂ xm  ∂2 ∂ x2− ∂ ∂ x  uBS(σ0)  ∂2 ∂ x2− ∂ ∂ x  uBS(σ0) −1 =  − 1 σ0 √ 2τ m Hm(ζ ),

see (Lorig et al., 2017, Lemma 3.4), may be left to the reader. Next, we calculate the left hand side of Equation

∂h ∂ σhu BS_(σ 0)  ∂ ∂ σu BS_(σ 0) −1 = bh/2c

∑

q=0 c_h,h−2qσ₀h−2q−1τh−q−1 h−q−1

∑

p=0 h − q − 1 p  ×  1 σ0 √ 2τ p+h−q−1 Hp+h−q−1(ζ ),

see (Lorig et al., 2017, Proposition 3.5), for h = 2. Using the Hermite polynomials H0(ζ ) = 1,

H1(ζ ) = 2ζ , and H2(ζ ) = 4ζ2− 2, we obtain ∂2 ∂ σ2u BS_(σ 0)  ∂ ∂ σu BS_(σ 0) −1 =p2(T − t)ζ + 2σ₀−1ζ2.

Combining everything together, we obtain the formula for σ2(t, x, y, z; T, k): σ2(t, x, y, z; T, k) = √ ν0+ κ1(ν00− ν0) 4√ν0 (T − t) −1 8ρ12ξ1ν α1−1 0 (x0− k) + 1 16ρ12ξ1ν α1 0 (T − t) − 3 64ρ 2 12ξ12ν02α1−2(x0− k)2 + 3 128ρ 2 12ξ12ν 2α1−3/2 0 (T − t) + · · · , (4.9) as T → t, k → x₀ |x₀− k| ≤ λ√T− t

where the dots denote the terms satisfying the following condition: the limits of the term, its first partial derivative with respect to T , and its first two partial derivatives with respect to k as (T, k) approaches (t, x) withinPλ, are all equal to 0.

In the right hand side of Equation (4.9), the first term, the partial derivatives with respect to T of the second, fourth, and sixth terms, the first partial derivative with respect to k of the third term, and the second partial derivative with respect to k of the fifth term give non-zero contributions to the right hand side of the asymptotic expansion (4.5).

Chapter 5

Conclusions

Our thesis focuses on the asymptotics of the implied volatility applied in the Gatheral model, in the specified parabilic region. We started off by introducing the change in variables, meaning the price of the underlying asset as well as strike prices, to logarithmic–price and logarithmic– strike, respectively. In the second chapter of our thesis, we introduce basic the definitions of the Osymbols, which then lead us to the asymptotic sequences and expansions.

In the latter chapters, we then used an example of the European call and after stating three assumptions that validate the conditions to utilize definitions and theorems of the referenced art-icles, we computed the operator expansion and price expansion. Finally, we computed asymp-totic expansion of implied-volatility. The obtained results are relevant such that, by approximat-ing implied volatility, it is possible to find the price of European option.

Appendix A

Criteria for a One-year Master Thesis

The requirements of Swedish National Agency of Higher Education to Master thesis in math-ematics, mathematical statistics, financial mathematics and actuarial science have provided the authors the standard objectives for the writers to fullfill. The authors of this thesis have full-filled the following criteria: the criteria 1, which is the capability of demonstration and deeper understanding of the main field of the thesis topic; criteria 2–to demonstrate deeper knowledge of the topic; criteria 3–to show that the integration of knowledge applied to analysis and to solve complex problems;criteria 4–to formulate questions and solve the advanced problems easily; criteria 5– to be able to orally as well as with respect to writing to present the problems and to have discussions with different groups; last but not the least, the authors have reached the criteria 6–to be able to have the subjective analyses and judgements towards the major field of study with relative scientific paper and materials.

On May 27th 2018, the authors will defend this thesis. The demonstration of understanding the study will be conducted on this day, and questions regarding the thesis can be raised by the exterminator or other listeners.

Bibliography

Elisa Alòs, Antoine Jacquier, and Jorge A. León. The implied volatility of forward-start options: ATM short-time level, skew and curvature. Stochastics, 91(1): 37–51, 2019. ISSN 1744-2508. doi: 10.1080/17442508.2018.1499105. URL https://doi.org/10.1080/17442508.2018.1499105.

Christian Bayer, Jim Gatheral, and Morten Karlsmark. Fast Ninomiya–Victoir calibration of the double-mean-reverting model. Quantitative Finance, 13(11):1813–1829, 2013.

H. Berestycki, J. Busca, and I. Florent. Asymptotics and calibration of local volatility mod-els. Quant. Finance, 2(1):61–69, 2002. ISSN 1469-7688. doi: 10.1088/1469-7688/2/1/305. URL https://doi.org/10.1088/1469-7688/2/1/305. Special issue on volatil-ity modelling.

Henri Berestycki, Jérôme Busca, and Igor Florent. Computing the implied volatility in stochastic volatility models. Comm. Pure Appl. Math., 57(10):1352–1373, 2004. ISSN 0010-3640. doi: 10.1002/cpa.20039. URL https://doi.org/10.1002/cpa.20039.

A. Erdélyi. Asymptotic expansions. Dover Publications, Inc., New York, 1956.

Stewart N. Ethier and Thomas G. Kurtz. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986.

Martin Forde and Antoine Jacquier. Small-time asymptotics for implied volat-ility under the Heston model. Int. J. Theor. Appl. Finance, 12(6):861–

876, 2009. ISSN 0219-0249. doi: 10.1142/S021902490900549X. URL

https://doi.org/10.1142/S021902490900549X.

Martin Forde and Antoine Jacquier. Small-time asymptotics for an uncor-related local-stochastic volatility model. Appl. Math. Finance, 18(6):517–

535, 2011. ISSN 1350-486X. doi: 10.1080/1350486X.2011.591159. URL

https://doi.org/10.1080/1350486X.2011.591159.

Martin Forde, Antoine Jacquier, and Aleksandar Mijatovi´c. Asymptotic formulae for im-plied volatility in the Heston model. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466(2124):3593–3620, 2010. ISSN 1364-5021. doi: 10.1098/rspa.2009.0610. URL

Jean-Pierre Fouque, George Papanicolaou, and K. Ronnie Sircar. Derivatives in financial mar-kets with stochastic volatility. Cambridge University Press, Cambridge, 2000. ISBN 0-521-79163-4.

Jean-Pierre Fouque, George Papanicolaou, Ronnie Sircar, and Knut Sø lna. Multiscale stochastic volatility for equity, interest rate, and credit derivatives. Cambridge University Press, Cambridge, 2011. ISBN 978-0-521-84358-4. doi: 10.1017/CBO9781139020534. URL https://doi.org/10.1017/CBO9781139020534.

Jean-Pierre Fouque, Yuri F. Saporito, and Jorge P. Zubelli. Multiscale stochastic volatility model for derivatives on futures. Int. J. Theor. Appl. Finance, 17(7): 1450043, 31, 2014. ISSN 0219-0249. doi: 10.1142/S0219024914500435. URL https://doi.org/10.1142/S0219024914500435.

Jean-Pierre Fouque, Matthew Lorig, and Ronnie Sircar. Second order multiscale stochastic volatility asymptotics: stochastic terminal layer analysis and calibration. Finance Stoch., 20(3):543–588, 2016. ISSN 0949-2984. doi: 10.1007/s00780-016-0298-y. URL https://doi.org/10.1007/s00780-016-0298-y.

Jean-Pierre Fouque, Ronnie Sircar, and Thaleia Zariphopoulou. Portfolio optimization and stochastic volatility asymptotics. Math. Finance, 27(3):704–745, 2017. ISSN 0960-1627. doi: 10.1111/mafi.12109. URL https://doi.org/10.1111/mafi.12109.

Jim Gatheral. Consistent modelling of SPX and VIX options, 2008. Bachelier Congress, Lon-don.

Archil Gulisashvili. Analytically tractable stochastic stock price

mod-els. Springer Finance. Springer, Heidelberg, 2012. ISBN

978-3-642-31213-7; 978-3-642-31214-4. doi: 10.1007/978-3-642-31214-4. URL

https://doi.org/10.1007/978-3-642-31214-4.

Antoine Jacquier and Matthew Lorig. From characteristic functions to implied volatil-ity expansions. Adv. in Appl. Probab., 47(3):837–857, 2015. ISSN 0001-8678. doi: 10.1239/aap/1444308884. URL https://doi.org/10.1239/aap/1444308884. Antoine Jacquier and Patrick Roome. Asymptotics of forward implied volatility. SIAM J.

Financial Math., 6(1):307–351, 2015. ISSN 1945-497X. doi: 10.1137/140960712. URL https://doi.org/10.1137/140960712.

Antoine Jacquier, Martin Keller-Ressel, and Aleksandar Mijatovi´c. Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models. Stochastics, 85(2):321–345, 2013. ISSN 1744-2508. doi: 10.1080/17442508.2012.720687. URL https://doi.org/10.1080/17442508.2012.720687.

Henry A. Latané and Richard J. Rendleman Jr. Standard Deviations of Stock Price Ratios Im-plied in Option Prices. Journal of Finance, 31(2):369–381, May 1976.

Matthew Lorig, Stefano Pagliarani, and Andrea Pascucci. Analytical expansions for parabolic equations. SIAM J. Appl. Math., 75(2):468–491, 2015.

Matthew Lorig, Stefano Pagliarani, and Andrea Pascucci. Explicit implied volatilities for multi-factor local-stochastic volatility models. Math. Finance, 27(3):926–960, 2017.

Giuseppe Orlando and Giovanni Taglialatela. A review on implied volatility calculation. J. Comput. Appl. Math., 320:202–220, 2017.

Stefano Pagliarani and Andrea Pascucci. Analytical approximation of the transition density in a local volatility model. Cent. Eur. J. Math., 10(1):250–270, 2012.

Stefano Pagliarani and Andrea Pascucci. The exact Taylor formula of the implied volatility. Finance Stoch., 21(3):661–718, 2017.