Pricing of American Options under the Rough Heston model

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Pricing of American Options under the Rough Heston model

Elizabeth Zúñiga

Supervisors:

Etienne Chevalier Sergio Pulido

12th European Summer School in Financial Mathematics

LaMME and Fondation de Mathématiques Jacques Hadamard

September 5, 2019

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Pricing of American Options under the Rough Heston model

1 Motivation

2 Heston and Rough Heston Model

3 Affine Volterra Processes

4 Markovian structure of Affine Volterra Processes

5 Pricing American Options

6 Numerical Results

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1 Motivation

6 Numerical Results

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Motivation

Classical models in finance assume that the log-asset priceXtis given by:

dXt=µtdt +σtdWt

where µtis the drift, Wtis a BM and σtis the volatility of the asset.

I Stochastic volatility models(Heston model): These models reproduce correctly the term structure of ATM skew

ψ(τ ) =

∂

∂kσBS(k, τ ) _k=0

for large maturities, but they fail to explain theconvexity to very short maturities

IEmpirical studies indicatevolatility is rougher than BM [Gatheral et al, 2014].

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Motivation

dXt=µtdt +σtdWt

ψ(τ ) =

∂

∂kσBS(k, τ ) _k=0

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Motivation

dXt=µtdt +σtdWt

ψ(τ ) =

∂

∂kσBS(k, τ ) _k=0

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For small τ , in a model where the volatility is driven by afractional Brownian motionwith Hurst parameter H, ψ(τ ) = τ^H−1/2with H = 0.1.

(see Figure of S&P ATM volatility skews)

I New class of models: fractional stochastic volatility models(rough volatility models).

match roughness of time series data fit implied volatility smiles remarkably well

Drawback: Loss of tractability,neither Markov nor semi-martingales.

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For small τ , in a model where the volatility is driven by afractional Brownian motionwith Hurst parameter H, ψ(τ ) = τ^H−1/2with H = 0.1.

(see Figure of S&P ATM volatility skews)

I New class of models: fractional stochastic volatility models(rough volatility models).

match roughness of time series data fit implied volatility smiles remarkably well

Drawback: Loss of tractability,neither Markov nor semi-martingales.

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1 Motivation

6 Numerical Results

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Heston Model

TheHestonmodel is a stochastic volatility model where:

dSt= St

pVtdWt

Vt= V0+ Z t

0

λ(θ − Vs)ds + Z t

0

λν√ VsdBs

λ, θ, V0 and ν positive. W and B are two BM with correlation ρ.

Proposition (The characteristic function in the Heston model) The characteristic function of the log-price Xt= log(St/S₀) satisfies:

E[e^iaX^t] = exp(g(a, t) + V0h(a, t)), where h is the solution of the followingRiccati equation

∂th=1

2(−a²− ia) + λ(iaρν − 1)h(a, s) +(λν)²

2 h²(a, s), h(a, 0) = 0 and

g(a, t) = θλ Z t

0

h(a, s)ds

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Heston Model

TheHestonmodel is a stochastic volatility model where:

dSt= St

pVtdWt

Vt= V0+ Z t

0

λ(θ − Vs)ds + Z t

0

λν√ VsdBs

λ, θ, V0 and ν positive. W and B are two BM with correlation ρ.

Proposition (The characteristic function in the Heston model) The characteristic function of the log-price Xt= log(St/S₀) satisfies:

E[e^iaX^t] = exp(g(a, t) + V0h(a, t)), where h is the solution of the followingRiccati equation

∂th=1

2(−a²− ia) + λ(iaρν − 1)h(a, s) +(λν)²

2 h²(a, s), h(a, 0) = 0 and

g(a, t) = θλ Z t

0

h(a, s)ds

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Rough Heston Model

dSt= St

pVtdWt, Vt= V₀+

Z t 0

(t − s)^α−1

Γ(α) (λ(θ − Vs)ds + λν√ VsdBs) where α ∈ (1/2, 1) governs the smoothness of the volatility.

Proposition (Characteristic function of the rough Heston [El Euch and Rosenbaum (2016)])

The characteristic function of the log-price Xt= log(St/S0) satisfies: E[e^iaX^t] = exp(g1(a, t) + V0g2(a, t))

g1(a, t) = θλ Z t

0

h(a, s)ds, g2(a, t) = I^1−αh(a, t) where h is the solution of the followingfractional Riccati equation:

D^αh=1

2(−a²− ia) + λ(iaρν − 1)h(a, s) +(λν)²

2 h²(a, s), I^1−αh(a, 0) = 0 Riemann-Liouville fractional integral and differential operators:

I^rf (t) = 1 Γ(r)

Z t 0

(t − s)^r−1f (s)ds, D^rf (t) = 1 Γ(1 − r)

d dt

Z t 0

(t − s)^−rf (s)ds r ∈ (0, 1],

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Rough Heston Model

dSt= St

pVtdWt, Vt= V₀+

Z t 0

(t − s)^α−1

The characteristic function of the log-price Xt= log(St/S0) satisfies:

E[e^iaX^t] = exp(g1(a, t) + V0g2(a, t)) g1(a, t) = θλ

Z t 0

D^αh=1

2(−a²− ia) + λ(iaρν − 1)h(a, s) +(λν)²

2 h²(a, s), I^1−αh(a, 0) = 0

Riemann-Liouville fractional integral and differential operators: I^rf (t) = 1

Γ(r) Z t

0

(t − s)^r−1f (s)ds, D^rf (t) = 1 Γ(1 − r)

d dt

Z t 0

(t − s)^−rf (s)ds r ∈ (0, 1],

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Rough Heston Model

dSt= St

pVtdWt, Vt= V₀+

Z t 0

(t − s)^α−1

The characteristic function of the log-price Xt= log(St/S0) satisfies:

E[e^iaX^t] = exp(g1(a, t) + V0g2(a, t)) g1(a, t) = θλ

Z t 0

D^αh=1

2(−a²− ia) + λ(iaρν − 1)h(a, s) +(λν)²

2 h²(a, s), I^1−αh(a, 0) = 0 Riemann-Liouville fractional integral and differential operators:

I^rf (t) = 1 Γ(r)

Z t 0

(t − s)^r−1f (s)ds, D^rf (t) = 1 Γ(1 − r)

d dt

Z t 0

(t − s)^−rf (s)ds r ∈ (0, 1], Elizabeth Zúñiga Pricing Options under the Rough Heston model

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1 Motivation

6 Numerical Results

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Affine Volterra Processes

We callstochastic Volterra equationsto the stochastic convolution equations of the form:

Xt= X0+ Z t

0

K(t − s)b(Xs)ds + Z t

0

K(t − s)σ(Xs)dWs, (1)

W is a multi-dimensional BM. K ∈ L²_loc(R+, R^d×d) is called kernel. b and σ continuous and satisfy for some constant cLGthe linear growth conditions:

|b(x)|∨|σ(x)| ≤ c_LG(1+|x|), x ∈ R^d (2)

We refer to the solutions of (1) asaffine Volterra processesif a(x) = σ(x)σ(x)^T and b(x) areaffinemaps given by:

a(x) = A⁰+ x1A¹+ . . . x_dA^d b(x) = b⁰+ x1b1 + . . . x_db^d for some d−dimensional symmetric matrices Aⁱ and vectors bⁱ.

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Affine Volterra Processes

Xt= X0+ Z t

0

K(t − s)σ(Xs)dWs, (1)

W is a multi-dimensional BM. K ∈ L²_loc(R+, R^d×d) is called kernel.

b and σ continuous and satisfy for some constant cLGthe linear growth conditions:

|b(x)|∨|σ(x)| ≤ cLG(1+|x|), x ∈ R^d (2)

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Affine Volterra Processes

Xt= X0+ Z t

0

K(t − s)σ(Xs)dWs, (1)

W is a multi-dimensional BM. K ∈ L²_loc(R+, R^d×d) is called kernel.

b and σ continuous and satisfy for some constant cLGthe linear growth conditions:

|b(x)|∨|σ(x)| ≤ cLG(1+|x|), x ∈ R^d (2)

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1 Motivation

6 Numerical Results

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Laplace representation of the kernel

Theorem (Bernstein–Widder theorem)

If K iscompletely monotone, there exists µ positive such that

K(t) = Z +∞

0

e^−tγµ(dγ) (3)

In thefractional casewith: K(t) =^t_Γ(α)^α−1, µ(dγ) = _{Γ(α)Γ(1−α)}^γ^−α dγ, α ∈ (1/2, 1).

We approximateµ(dγ)byPn

i=1ciδγi(dγ) [Carmona et al, 2000], then K by Kⁿ

Kⁿ(t)=

n

X

i=1

c_ie^−γⁱ^t.

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Laplace representation of the kernel

Theorem (Bernstein–Widder theorem)

If K iscompletely monotone, there exists µ positive such that

K(t) = Z +∞

0

e^−tγµ(dγ) (3)

In thefractional casewith: K(t) =^t_Γ(α)^α−1, µ(dγ) = _{Γ(α)Γ(1−α)}^γ^−α dγ, α ∈ (1/2, 1).

We approximateµ(dγ)byPn

i=1c_iδγi(dγ) [Carmona et al, 2000], then K by Kⁿ

Kⁿ(t)=

n

X

i=1

c_ie^−γⁱ^t.

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Multi-factor model

For n ∈ N, we define the following process:

Xtⁿ= X0+ Z t

0

Kⁿ(t − s)b(Xsⁿ)ds + Z t

0

Kⁿ(t − s)σ(Xsⁿ)dWs, t ∈ [0, T ] (4)

For n ∈ N and t ≥ 0, we have

X_tⁿ= X0+

n

X

i=1

c_iY_t^γⁱ

where for any i ∈ {1, ..., n}, Y^γⁱ is the affine process given by:

dY_t^γⁱ=



−γiY_t^γⁱ+ b0+ b1X₀+ b1 n

X

j=1

c_jY_t^γ^j



dt+σ



X₀+

n

X

j=1

c_jY_t^γ^j



dWt

(5) Y = (Y^(γⁱ⁾)ⁿ_i=1 isaffine, Markovian and can be simulated.

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Multi-factor model

Xtⁿ= X0+ Z t

0

Kⁿ(t − s)σ(Xsⁿ)dWs, t ∈ [0, T ] (4) For n ∈ N and t ≥ 0, we have

X_tⁿ= X0+

n

X

i=1

c_iY_t^γⁱ

dY_t^γⁱ=



−γiY_t^γⁱ+ b0+ b1X₀+ b1 n

X

j=1

c_jY_t^γ^j



dt+σ



X₀+

n

X

j=1

c_jY_t^γ^j



dWt

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Y = (Y^(γⁱ⁾)ⁿ_i=1 isaffine, Markovian and can be simulated.

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Multi-factor model

Xtⁿ= X0+ Z t

0

Kⁿ(t − s)σ(Xsⁿ)dWs, t ∈ [0, T ] (4) For n ∈ N and t ≥ 0, we have

X_tⁿ= X0+

n

X

i=1

c_iY_t^γⁱ

dY_t^γⁱ=



−γiY_t^γⁱ+ b0+ b1X₀+ b1 n

X

j=1

c_jY_t^γ^j



dt+σ



X₀+

n

X

j=1

c_jY_t^γ^j



dWt

(5) Y = (Y^(γⁱ⁾)ⁿ_i=1 isaffine, Markovian and can be simulated.

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Multi-factor model

Proposition (Convergence in law of Xⁿ to X: Abi Jaber and El Euch (2018) )

Let Xⁿ= (X_tⁿ)t leT continuous weak solutions to (4), with the kernel Kⁿ such that there exist γ > 0 and C > 0:

sup

n≥1

Z T −h 0

|Kⁿ(u + h) − Kⁿ(u)|²du + Z h

0

|Kⁿ(u)|²du

!

≤ Ch^2γ, (6)

and

Z t 0

|K(u) − Kⁿ(u)|²du → 0 (7)

for any t ∈ [0, T ] as n goes to infinity. Then(Xⁿ)_n≥1is tightfor the uniform topology and any point limit X is a solution of (1).

Proposition states theconvergence in lawof the familyXⁿ to X whenever (1) admits a unique weak solution.

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1 Motivation

6 Numerical Results

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Adjusted forward process

Theadjusted forward process(ut)t≥0and its factor-representation (uⁿ_t)t≥0

are defined as follows:

uⁿ_t(x) = E

X_t+xⁿ −

Z x 0

Kⁿ(x − s)b(X_t+sⁿ )ds

Ft

Lemma

Assume that Kⁿ satisfies (6). Let T > 0 and p > 2/γ be such that sup_n≥1,t≤TE[|Xtⁿ|^p] is finite. Then for every s, t ∈ [0, T ] and x, y ∈ [0, M ]

E[|uⁿt(y) − uⁿs(x)|^p] ≤ C(max (|t − s|, |y − x|))^pγ

where C = C(p, K, M, T ). As a consequence uⁿ admits a Hölder continuous modification of order α ∈ [0, γ − p/2[. Denoting this version by uⁿ, we have

E

max

(t,y)6=(s,x)

|uⁿt(y) − uⁿ_s(x)|

|(t − s, y − x)|^α

p

< ∞

Proposition (Convergence of (uⁿ))

uⁿconverges in law to u under the uniform topology when n goes to infinity.

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Adjusted forward process

Theadjusted forward process(ut)t≥0and its factor-representation (uⁿ_t)t≥0

are defined as follows:

uⁿ_t(x) = E

X_t+xⁿ −

Z x 0

Kⁿ(x − s)b(X_t+sⁿ )ds

Ft

Lemma

Assume that Kⁿ satisfies (6). Let T > 0 and p > 2/γ be such that sup_n≥1,t≤TE[|Xtⁿ|^p] is finite. Then for every s, t ∈ [0, T ] and x, y ∈ [0, M ]

E[|uⁿt(y) − uⁿs(x)|^p] ≤ C(max (|t − s|, |y − x|))^pγ

where C = C(p, K, M, T ). As a consequence uⁿ admits a Hölder continuous modification of order α ∈ [0, γ − p/2[. Denoting this version by uⁿ, we have

E

max

(t,y)6=(s,x)

|uⁿt(y) − uⁿ_s(x)|

|(t − s, y − x)|^α

p

< ∞

Proposition (Convergence of (uⁿ))

uⁿ converges in law to u under the uniform topology when n goes to infinity.

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Conditional Fourier Laplace transform of the process

Xt= ln(St) =√

VtdW t −¹₂Vtdt, ut(x) = V₀+Rt

0K(t − s + x) λ(θ − V s)ds + η√ VsdW s Theconditional Fourier Laplace transformof the process ({uⁿ_t(.)}, (X_tⁿ)_t≥0) is given by:

E

exp

wXn T+

Z ∞ 0

hn(x)un T(x)dx

Ft

= exp

wXn

t + φn(T − t, w) +

Z ∞ 0

Ψn(T − t, x, hn, w)un t (x)dx

fractionalRiccati equations:

∂tφⁿ(t, w) = R_φn ψⁿ(t, w)

, φⁿ(0) = 0

Ψⁿ(t, x, h, w) = hⁿ(x−t)1_{x≥t}+=_Ψn w, ψⁿ(t − x, w)

1_{x<t}, Ψⁿ(0, x) = hⁿ(x) where

R_φn ψⁿ(t, w)

= λθψⁿ(t, w), =_Ψn(w, ψⁿ(t)) =1

2(w²− w) +

ρηw − λ +η² 2 ψⁿ(t, w)

ψⁿ(t, w)

ψⁿ(t) = Z ∞

0

hⁿ(x)Kⁿ(t + x)dx + K ∗ =_Ψn ψⁿ(., w) (t)

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Pricing American Options

We denote the value of the American option at time i by Ui, i = 0, . . . .M .

Theorem

We will suppose that hⁿ→ h uniformly in n, Kⁿ→ K in L²_loc(R+), (uⁿ, Xⁿ) is tight and converges weakly to (u, X). Then,

U₀ⁿ→ U0

uniformly in n.

IIdea of the proof:

Approximate the American option with Bermuda (uniformly in n) Show the convergence of the Bermuda: We prove the result by induction on the number of dates M of the Bermudan option. The convergence in law of the forward process is used alongside an argument of density (which comes from the Stone-Weierstrass theorem) that allows us to reduce the problem to payoffs of the form exp(wXT+R

h(x)uT(x)dx).

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1 Motivation

6 Numerical Results

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Numerical Simulation in the Rough Heston model

Time grid: t_k= k∆t, k = 1, . . . , N , ∆t = T /N .

Euler schemefor the log priceXt= log(St) Xtk+1= Xtk+

r −Vtk

2

∆t +p

(Vtk)⁺∆t(ρN (0, 1) +p

1 − ρ²N (0, 1))˜ Explicit-implicit Euler schemefor the variance processVⁿ

Vtk= uⁿ₀(tk) +

n

X

i=1

cⁿ_iYtⁱ_k, Y₀ⁱ= 0 for all i = 1, . . . , n

Ytⁱ_k+1= 1 1 + γi∆t

Ytⁱ_k− λVt_k∆t + ηp

(Vt_k)⁺∆t

N (0, 1), i = 1, . . . , n with

uⁿ₀(t_k) = V0+ λθ

n

X

i=1

ci

1 − e^−γⁿⁱ^t^k γ_iⁿ

and parameters

cⁿ_i =(r^1−α_n − 1)r(α−1)(1+n/2) n

Γ(α)Γ(2 − α) r_n^(α−1)i, γ_iⁿ=1 − α 2 − α

r^2−α_n − 1

r^1−αn − 1r^i−1−n/2_n

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Numerical Simulation in the Rough Heston model

Time grid: t_k= k∆t, k = 1, . . . , N , ∆t = T /N . Euler schemefor the log priceXt= log(St)

Xtk+1= Xtk+

r −Vt_k

2

∆t +p

(Vtk)⁺∆t(ρN (0, 1) +p

1 − ρ²N (0, 1))˜

Explicit-implicit Euler schemefor the variance processVⁿ

Vtk= uⁿ₀(tk) +

n

X

i=1

Ytⁱ_k+1= 1 1 + γi∆t

(Vt_k)⁺∆t

N (0, 1), i = 1, . . . , n with

n

X

i=1

ci

and parameters

cⁿ_i =(r^1−α_n − 1)r(α−1)(1+n/2) n

Γ(α)Γ(2 − α) r_n^(α−1)i, γ_iⁿ=1 − α 2 − α

r^2−α_n − 1

r^1−αn − 1r^i−1−n/2_n

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Numerical Simulation in the Rough Heston model

Xtk+1= Xtk+

r −Vt_k

2

∆t +p

(Vtk)⁺∆t(ρN (0, 1) +p

Vtk= uⁿ₀(t_k) +

n

X

i=1

Ytⁱ_k+1= 1 1 + γi∆t

(Vt_k)⁺∆t

N (0, 1), i = 1, . . . , n with

n

X

i=1

ci

and parameters

cⁿ_i =(r^1−α_n − 1)r(α−1)(1+n/2) n

Γ(α)Γ(2 − α) r_n^(α−1)i, γ_iⁿ=1 − α 2 − α

r^2−α_n − 1

r^1−αn − 1r^i−1−n/2_n

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Numerical Simulation in the Rough Heston model

Xtk+1= Xtk+

r −Vt_k

2

∆t +p

(Vtk)⁺∆t(ρN (0, 1) +p

Vtk= uⁿ₀(t_k) +

n

X

i=1

Ytⁱ_k+1= 1 1 + γi∆t

(Vt_k)⁺∆t

N (0, 1), i = 1, . . . , n with

n

X

i=1

ci

and parameters

cⁿ_i =(r_n^1−α− 1)r(α−1)(1+n/2) n

Γ(α)Γ(2 − α) r_n^(α−1)i, γ_iⁿ=1 − α 2 − α

r^2−αn − 1

r^1−αn − 1r^i−1−n/2_n

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Pricing American Put options in the Rough Heston Model

Longstaff-Schwartz methodology.

T = 1, strike = 100, ρ = −0.7, θ = 0.02, λ = 0.3, ν = 0.3, V0= 0.02 α = 0.6

Figure:Put options with 10⁵simulations, 50 time step, 20 factors and r20= 2.5

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Optimal exercise surface

(a) α = 0.6 (b) α = 0.8

(c) α = 1

Figure:Optimal exercise surface of American Put options in the Rough Heston

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References

E.Abi Jaber and O. El Euch.

Markovian structure of the Volterra Heston model.

arXiv:1803.00477, 2018.

E. Abi Jaber, M. Larsson and S. Pulido Affine Volterra processes.

The Annals of Applied Probability, 2019.

P. Carmona and L. Coutin and G. Montseny Approximation of Some Gaussian Processes.

Statistical Inference for Stochastic Processes, volume: 3, number: 1, pages: 161-171, 2000.

P. Carr and Dilip B. Madan

Option Valuation Using the Fast Fourier Transform

Journal of Computational Finance, 1999. volume: 2, pages: 61-73.

G.Drimus

Options on realized variance by transform methods: A non-affine stochastic volatility model

September, 2009.

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References

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The characteristic function of rough Heston models.

Mathematical Finance, 2019.

D. Filipovié and E. Mayerhofer E.

Affine Diffusion Processes: Theory and Applications.

arXiv:0901.4003v3. 2009.

Fukasawa and Masaaki

Asymptotic analysis for stochastic volatility: martingale expansion.

Finance and Stochastics, 2011. volume: 15, number: 4, pages: 635-654.

J. Gatheral, T. Jaisson and M. Rosenbaum Volatility is rough.

Available at SSRN 2509457, 2014.

C. Li and C. Tao

On the fractional Adams method.

Computers and Mathematics with Applications. July, 2009.

D. Revuz and M.Yor.

Continuous martingales and Brownian motion Springer-Verlag, 1991.