Option pricing under fast-varying and rough stochastic volatility

Abstract

Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the origin. Another classic stylistic feature often assumed for the volatility is that it is mean reverting. In this paper it is shown that the price impact of a rapidly mean reverting rough volatility model coincides with that associated with fast mean reverting Markov stochastic volatility models. This reconciles the empirical observation of rough volatility paths with the good fit of the implied volatility surface to models of fast mean reverting Markov volatilities. Moreover, the result conforms with recent numerical results regarding rough stochastic volatility models. It extends the scope of models for which the asymptotic results of fast mean reverting Markov volatilities are valid. The paper concludes with a general discussion of fractional volatility asymptotics and their interrelation. The regimes discussed there include fast and slow volatility factors with strong or small volatility fluctuations and with the limits not commuting in general. The notion of a characteristic term structure exponent is introduced, this exponent governs the implied volatility term structure in the various asymptotic regimes.

Appendices

Technical lemmas

We denote

$$\begin{aligned} G(z) = \frac{1}{2} \big ( F(z)^2 - \overline{\sigma }^2\big ) . \end{aligned}$$

(54)

The martingale $\psi ^\varepsilon _t$ defined by (33) has the form

$$\begin{aligned} \psi _t^\varepsilon = \mathbb {E}\Big [ \int _0^T G(Z_s^\varepsilon ) ds \big | \mathcal{F}_t\Big ] . \end{aligned}$$

(55)

Lemma 2

$(\psi _t^\varepsilon )_{t\in [0,T]}$ is a square-integrable martingale and

$$\begin{aligned} d \left\langle \psi ^\varepsilon , W\right\rangle _t = \vartheta ^\varepsilon _{t} dt , \quad \quad \vartheta ^\varepsilon _{t} = \sigma _{\mathrm{ou}} \int _t^T \mathbb {E}\big [ G'(Z_s^\varepsilon )|\mathcal{F}_t \big ]\mathcal{K}^\varepsilon (s-t) ds . \end{aligned}$$

(56)

Proof

See Lemma B.1 in Garnier and Sølna (2016). $\square$

The important properties of the random process $\vartheta ^\varepsilon _{t}$ are stated in the following lemma.

Lemma 3

• 1. The exists a constant $K_T$ such that, for any $t \in [0,T]$, we have almost surely

  $$\begin{aligned} \big | \sigma _t^\varepsilon \vartheta ^\varepsilon _{t} \big | \le K_T \varepsilon ^{1/2} . \end{aligned}$$

  (57)

• 2. For any $t \in [0,T]$, we have

  $$\begin{aligned} \mathbb {E}[ \sigma _t^\varepsilon \vartheta ^\varepsilon _{t} ] = \varepsilon ^{1/2} \overline{D} + \widetilde{D}^\varepsilon _{t}, \end{aligned}$$

  (58)

  where $\overline{D}$ is the deterministic constant (27) and $\widetilde{D}^\varepsilon _{t}$ is smaller than $\varepsilon ^{1/2}$:

  $$\begin{aligned} \sup _{\varepsilon \in (0,1]} \sup _{t \in [0,T]} \varepsilon ^{-1/2} \big | \widetilde{D}^\varepsilon _{t}\big | < \infty , \end{aligned}$$

  (59)

  and

  $$\begin{aligned} \forall t \in [0,T),\quad \lim _{\varepsilon \rightarrow 0} \varepsilon ^{-1/2} \big | \widetilde{D}^\varepsilon _{t}\big | =0. \end{aligned}$$

  (60)

• 3. For any $0\le t< t' < T$, we have

  $$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \varepsilon ^{-1} \big | \mathrm{Cov} \big ( \sigma _t^\varepsilon \vartheta ^\varepsilon _{t},\sigma _{t'}^\varepsilon \vartheta ^\varepsilon _{t'}\big ) \big | =0. \end{aligned}$$

  (61)

Proof

Using the expression (56) of $\vartheta ^\varepsilon _t$:

$$\begin{aligned} \big | \vartheta ^\varepsilon _t \sigma ^\varepsilon _t \big | \le \sigma _\mathrm{ou} \Vert F\Vert _\infty \Vert G'\Vert _\infty \int _0^\infty |\mathcal{K}^\varepsilon (s)| ds \end{aligned}$$

The proof of the first item follows from the fact that $\mathcal{K}^\varepsilon (t)=\mathcal{K}(t/\varepsilon )/\sqrt{\varepsilon }$, $\mathcal{K} \in L^1(0,\infty )$.

The expectation of $\sigma _t^\varepsilon \vartheta ^\varepsilon _{t}$ is equal to

$$\begin{aligned} \mathbb {E}\big [ \sigma _t^\varepsilon \vartheta ^\varepsilon _{t} \big ]= & {} \sigma _{\mathrm{ou}} \int _t^{T} \mathbb {E}\big [ F(Z^\varepsilon _t) G'(Z_s^\varepsilon ) \big ] \mathcal{K}^\varepsilon (s - t) ds \\= & {} \sigma _{\mathrm{ou}} \varepsilon ^{1/2} \int _0^{(T-t)/\varepsilon } \mathbb {E}\big [ F(Z_0^\varepsilon ) G'(Z_{\varepsilon s}^\varepsilon ) \big ] \mathcal{K}(s) ds \\= & {} \sigma _{\mathrm{ou}} \varepsilon ^{1/2} \int _0^{(T-t)/\varepsilon } \Big [\iint _{\mathbb {R}^2} F(\sigma _{\mathrm{ou}} z ) G'(\sigma _{\mathrm{ou}} z') p_{\mathcal{C}_Z(s)} (z,z') d z dz'\Big ] \mathcal{K}(s) ds, \end{aligned}$$

with $p_C$ defined in Proposition 1.

Therefore the difference

$$\begin{aligned} \mathbb {E}\big [ \sigma _t^\varepsilon \vartheta ^\varepsilon _{t} \big ] - \varepsilon ^{1/2} \overline{D} = \sigma _{\mathrm{ou}} \varepsilon ^{1/2} \int _{(T-t)/\varepsilon }^\infty \mathbb {E}\big [ F(Z_0^\varepsilon ) G'(Z_{\varepsilon s}^\varepsilon )\big ] \mathcal{K}(s) ds \end{aligned}$$

can be bounded by

$$\begin{aligned} \big | \mathbb {E}\big [ \sigma _t^\varepsilon \vartheta ^\varepsilon _{t} \big ] - \varepsilon ^{1/2} \overline{D} \big | \le \Vert F\Vert _\infty \Vert G'\Vert _\infty \sigma _{\mathrm{ou}} \varepsilon ^{1/2} \int _{(T-t)/\varepsilon }^\infty |\mathcal{K}(s)| ds, \end{aligned}$$

(62)

which gives the second item since $\mathcal{K} \in L^1(0,\infty )$.

Let us consider $0 \le t \le t'\le T$. We have

$$\begin{aligned} \mathbb {E}\big [ \sigma _t^\varepsilon \vartheta ^\varepsilon _{t}\sigma _{t'}^\varepsilon \vartheta ^\varepsilon _{t'} \big ]= & {} \sigma _{\mathrm{ou}}^2 \int _t^{T} ds \mathcal{K}^\varepsilon (s-t) \int _{t'}^T ds' \mathcal{K}^\varepsilon (s' -t') \\&\times \, \mathbb {E}\Big [ \mathbb {E}\big [ F(Z^\varepsilon _t) G'(Z_s^\varepsilon )|\mathcal{F}_t \big ] \mathbb {E}\big [ F(Z^\varepsilon _{t'}) G'(Z_{s'}^\varepsilon )|\mathcal{F}_{t'} \big ] \Big ] , \end{aligned}$$

so we can write

$$\begin{aligned} \mathrm{Cov} \big ( \sigma _t^\varepsilon \vartheta ^\varepsilon _{t} , \sigma _{t'}^\varepsilon \vartheta ^\varepsilon _{t'} \big )= & {} \sigma _{\mathrm{ou}}^2 \int _t^{T} ds \mathcal{K}^\varepsilon (s-t) \int _{t'}^T ds' \mathcal{K}^\varepsilon (s' - t') \\&\times \, \Big ( \mathbb {E}\Big [ \mathbb {E}\big [ F(Z^\varepsilon _t) G'(Z_s^\varepsilon )|\mathcal{F}_t \big ] \mathbb {E}\big [ F(Z^\varepsilon _{t'}) G'(Z_{s'}^\varepsilon )|\mathcal{F}_{t} \big ] \Big ] \\&-\, \mathbb {E}\Big [ \mathbb {E}\big [ F(Z^\varepsilon _t) G'(Z_s^\varepsilon )|\mathcal{F}_t \big ] \mathbb {E}\big [ F(Z^\varepsilon _{t'}) G'(Z_{s'}^\varepsilon ) \big ] \Big ] \Big ) , \end{aligned}$$

and therefore

$$\begin{aligned} \big | \mathrm{Cov} \big ( \sigma _t^\varepsilon \vartheta ^\varepsilon _{t} , \sigma _{t'}^\varepsilon \vartheta ^\varepsilon _{t'} \big ) \big |\le & {} \sigma _{\mathrm{ou}}^2 \Vert F\Vert _\infty \Vert G'\Vert _\infty \int _t^{T} ds |\mathcal{K}^\varepsilon (s-t)| \int _{t'}^T ds' |\mathcal{K}^\varepsilon (s'-t')| \\&\times \, \mathbb {E}\Big [ \big ( \mathbb {E}\big [ F(Z^\varepsilon _{t'}) G'(Z_{s'}^\varepsilon )|\mathcal{F}_{t} \big ] - \mathbb {E}\big [ F(Z^\varepsilon _{t'}) G'(Z_{s'}^\varepsilon ) \big ] \big )^2 \Big ]^{1/2} . \end{aligned}$$

We can write for any $\tau >t$:

$$\begin{aligned}&Z^\varepsilon _{\tau } = A^\varepsilon _{t\tau } + B^\varepsilon _{t\tau } ,\quad A^\varepsilon _{t\tau } =\, \sigma _\mathrm{ou} \int _{-\infty }^t \mathcal{K}^\varepsilon (\tau -u)dW_u, \quad B^\varepsilon _{t\tau } = \sigma _\mathrm{ou}\int _t^{\tau } \mathcal{K}^\varepsilon (\tau -u)dW_u , \end{aligned}$$

where $A^\varepsilon _{t\tau }$ is $\mathcal{F}_t$ adapted while $B^\varepsilon _{t\tau }$ is independent from $\mathcal{F}_t$. Therefore ($s'\ge t'\ge t$)

$$\begin{aligned}&\mathbb {E}\Big [ \big ( \mathbb {E}\big [ F(Z^\varepsilon _{t'}) G'(Z_{s'}^\varepsilon )|\mathcal{F}_{t} \big ] - \mathbb {E}\big [ F(Z^\varepsilon _{t'}) G'(Z_{s'}^\varepsilon ) \big ]\big )^2 \Big ]\\&= \mathbb {E}\Big [ \mathbb {E}\big [ F(Z^\varepsilon _{t'}) G'(Z_{s'}^\varepsilon )|\mathcal{F}_{t} \big ]^2 \Big ] - \mathbb {E}\big [ F(Z^\varepsilon _{t'}) G'(Z_{s'}^\varepsilon ) \big ]^2 \\&= \mathbb {E}\Big [ F( A^\varepsilon _{tt'} + B^\varepsilon _{tt'}) G'( A^\varepsilon _{ts'} + B^\varepsilon _{ts'}) F( A^\varepsilon _{tt'} + \tilde{B}^\varepsilon _{tt'}) G'( A^\varepsilon _{ts'} + \tilde{B}^\varepsilon _{ts'}) \\&\quad - F( A^\varepsilon _{tt'} + B^\varepsilon _{tt'}) G'( A^\varepsilon _{ts'} + B^\varepsilon _{ts'}) F( \tilde{A}^\varepsilon _{tt'} + \tilde{B}^\varepsilon _{tt'}) G'( \tilde{A}^\varepsilon _{ts'} + \tilde{B}^\varepsilon _{ts'}) \Big ] , \end{aligned}$$

where $(\tilde{A}^\varepsilon _{tt'} , \tilde{B}^\varepsilon _{tt'}, \tilde{A}^\varepsilon _{ts'} , \tilde{B}^\varepsilon _{ts'})$ is an independent copy of $({A}^\varepsilon _{tt'} , {B}^\varepsilon _{tt'}, {A}^\varepsilon _{ts'} , {B}^\varepsilon _{ts'})$. We can then write

$$\begin{aligned}&\mathbb {E}\Big [ \big ( \mathbb {E}\big [ F(Z^\varepsilon _{t'}) G'(Z_{s'}^\varepsilon )|\mathcal{F}_{t} \big ] - \mathbb {E}\big [ F(Z^\varepsilon _{t'}) G'(Z_{s'}^\varepsilon ) \big ]\big )^2 \Big ]\\&\quad \le \Vert F\Vert _\infty \Vert G'\Vert _\infty \mathbb {E}\Big [ \big ( F( A^\varepsilon _{tt'} + \tilde{B}^\varepsilon _{tt'}) G'( A^\varepsilon _{ts'} + \tilde{B}^\varepsilon _{ts'}) - F( \tilde{A}^\varepsilon _{tt'} + \tilde{B}^\varepsilon _{tt'}) G'( \tilde{A}^\varepsilon _{ts'} + \tilde{B}^\varepsilon _{ts'}) \big )^2 \Big ]^{1/2} \\&\quad \le C \Big ( \mathbb {E}\big [ (A^\varepsilon _{tt'} - \tilde{A}^\varepsilon _{tt'})^2 \big ]^{1/2}+ \mathbb {E}\big [ (A^\varepsilon _{ts'} - \tilde{A}^\varepsilon _{ts'})^2 \big ]^{1/2}\Big ) \\&\quad \le 2 C \Big ( \mathbb {E}\big [ (A^\varepsilon _{tt'} )^2 \big ]^{1/2}+ \mathbb {E}\big [ (A^\varepsilon _{ts'} )^2 \big ]^{1/2}\Big ) \\&\quad \le 2 C \Big [ \Big ( \sigma _\mathrm{ou}^2 \int _{-\infty }^t \mathcal{K}^\varepsilon (t'-u)^2 du \Big )^{1/2} + \Big ( \sigma _\mathrm{ou}^2 \int _{-\infty }^t \mathcal{K}^\varepsilon (s'-u)^2 du \Big )^{1/2} \Big ] \\&\quad \le 4C \sigma ^\varepsilon _{t'-t,\infty } \le {C_1} \big ( 1 \wedge (\varepsilon /(t'-t))^{1-H}\big ) , \end{aligned}$$

where we used Lemma  7 in the last inequality. Then, using the fact that $\mathcal{K}\in L^1$, this gives

$$\begin{aligned} \big | \mathrm{Cov} \big ( \sigma _t^\varepsilon \vartheta ^\varepsilon _{t} , \sigma _{t'}^\varepsilon \vartheta ^\varepsilon _{t'} \big ) \big |\le & {} C_2 \int _t^{T} ds |\mathcal{K}^\varepsilon (s-t)| \int _{t'}^T ds' |\mathcal{K}^\varepsilon (s'-t')| \big ( 1 \wedge (\varepsilon /(t'-t)))^{(1-H)/2}\big )\\\le & {} C_3 \varepsilon \big ( 1 \wedge (\varepsilon /(t'-t)))^{(1-H)/2}\big ) , \end{aligned}$$

which proves the third item.

Lemma 4

For any smooth function f with bounded derivative, we have

$$\begin{aligned} \mathrm{Var} \big ( \mathbb {E}\big [ f(Z_{t}^\varepsilon ) |\mathcal{F}_0 \big ]\big ) \le \Vert f'\Vert _\infty ^2 (\sigma _{t,\infty }^\varepsilon )^2 . \end{aligned}$$

(63)

Proof

The conditional distribution of $Z_t^\varepsilon$ given $\mathcal{F}_0$ is Gaussian with mean

$$\begin{aligned} \mathbb {E}\big [ Z_t^\varepsilon |\mathcal{F}_0 \big ] = \sigma _{\mathrm{ou}} \int _{-\infty }^0 \mathcal{K}^\varepsilon (t-u) dW_u \end{aligned}$$

and variance

$$\begin{aligned} \mathrm{Var} \big ( Z_t^\varepsilon |\mathcal{F}_0\big ) = (\sigma _{0,t}^\varepsilon )^2 =\sigma _{\mathrm{ou}}^2 \int _0^{t} \mathcal{K}^\varepsilon (u)^2 du . \end{aligned}$$

Therefore

$$\begin{aligned} \mathrm{Var} \big ( \mathbb {E}\big [ f(Z_{t}^\varepsilon ) |\mathcal{F}_0 \big ]\big ) = \mathrm{Var} \Big ( \int _\mathbb {R}f\big (\mathbb {E}\big [ Z_{t}^\varepsilon |\mathcal{F}_0 \big ] +\sigma _{0,t}^\varepsilon z \big ) p(z) dz \Big ) , \end{aligned}$$

where p(z) is the pdf of the standard normal distribution. The random variable $\mathbb {E}\big [ Z_{t}^\varepsilon |\mathcal{F}_0 \big ]$ is Gaussian with mean zero and variance $(\sigma _{t,\infty }^\varepsilon )^2$ so that

$$\begin{aligned} \mathrm{Var} \big ( \mathbb {E}\big [ f(Z_{t}^\varepsilon ) |\mathcal{F}_0 \big ]\big )= & {} \frac{1}{2} \int _\mathbb {R}\int _\mathbb {R}dz dz' p(z) p(z') \int _\mathbb {R}\int _\mathbb {R}du du' p(u) p(u') \\&\times \Big [ f\big (\sigma _{t,\infty }^\varepsilon u +\sigma _{0,t}^\varepsilon z \big ) - f\big (\sigma _{t,\infty }^\varepsilon u' +\sigma _{0,t}^\varepsilon z \big )\Big ] \\&\times \Big [ f\big (\sigma _{t,\infty }^\varepsilon u +\sigma _{0,t}^\varepsilon z' \big ) - f\big (\sigma _{t,\infty }^\varepsilon u' +\sigma _{0,t}^\varepsilon z' \big )\Big ] \\\le & {} \Vert f'\Vert _\infty ^2 (\sigma _{t,\infty }^\varepsilon )^2 \frac{1}{2} \int _\mathbb {R}\int _\mathbb {R}du du' p(u) p(u')(u-u')^2 \\= & {} \Vert f'\Vert _\infty ^2 (\sigma _{t,\infty }^\varepsilon )^2 , \end{aligned}$$

which is the desired result.

The random term $\phi ^\varepsilon _{t}$ defined by (32) has the form

$$\begin{aligned} \phi _{t}^\varepsilon = \mathbb {E}\Big [ \int _t^T G (Z_s^\varepsilon ) ds \big | \mathcal{F}_t \Big ] , \end{aligned}$$

(64)

with G defined in (54).

Lemma 5

For any $t \le T$, $\phi _{t}^\varepsilon$ is a zero-mean random variable with standard deviation of order $\varepsilon ^{1-H}$:

$$\begin{aligned} \sup _{\varepsilon \in (0,1]} \sup _{t \in [0,T]}\varepsilon ^{2H-2} \mathbb {E}[ (\phi _{t}^\varepsilon )^2] < \infty . \end{aligned}$$

(65)

Proof

For $t\in [0,T]$ the second moment of $\phi _{t}^\varepsilon$ is:

$$\begin{aligned} \mathbb {E}\big [ (\phi _{t}^\varepsilon )^2\big ]= & {} \mathbb {E}\Big [ \mathbb {E}\Big [ \int _t^T G (Z_s^\varepsilon ) ds \big | \mathcal{F}_t \Big ]^2 \Big ]\\= & {} \int _0^{T-t} ds \int _{0}^{T-t} ds' \mathrm{Cov}\big ( \mathbb {E}\big [ G(Z_s^\varepsilon )|\mathcal{F}_0\big ] ,\mathbb {E}\big [ G(Z_{s'}^\varepsilon )|\mathcal{F}_0\big ] \big ) . \end{aligned}$$

We have by Lemma 4

$$\begin{aligned} \mathbb {E}\big [ (\phi _{t}^\varepsilon )^2\big ]\le & {} \Big ( \int _0^{T-t} ds \mathrm{Var}\big ( \mathbb {E}\big [ G(Z_s^\varepsilon )|\mathcal{F}_0\big ]\big )^{1/2} \Big )^2\le \Vert G'\Vert _\infty ^2\Big ( \int _0^{T-t} ds \sigma _{s,\infty }^\varepsilon \Big )^2 . \end{aligned}$$

In view of Lemma 7 we then have

$$\begin{aligned} \mathbb {E}\big [ (\phi _{t}^\varepsilon )^2\big ]\le C_{T} \big ( \varepsilon +\varepsilon ^{1-H}\big )^2 \le 4 C_{T} \varepsilon ^{2-2H} , \end{aligned}$$

uniformly in $t \le T$ and $\varepsilon \in (0,1]$ for some constant $C_{T}$.

Lemma 6

Let us define for any $t \in [0,T]$:

$$\begin{aligned} {\kappa }^\varepsilon _t = \frac{\varepsilon ^{1/2}}{2} \int _0^t \big ( (\sigma _s^\varepsilon )^2 -\overline{\sigma }^2\big ) ds = \varepsilon ^{1/2} \int _0^t G(Z^\varepsilon _s) ds , \end{aligned}$$

(66)

as in (41). We have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \sup _{t\in [0,T]} \varepsilon ^{-1/2} \mathbb {E}\big [ ( {\kappa }^\varepsilon _t )^2\big ]^{1/2} = 0. \end{aligned}$$

(67)

Proof

Since the expectation $\mathbb {E}[ G(Z^\varepsilon _0)]=0$, we have

$$\begin{aligned} \mathbb {E}\big [ ( {\kappa }^\varepsilon _t )^2\big ] = \varepsilon \mathbb {E}\Big [ \Big ( \int _0^t G(Z^\varepsilon _s) ds \Big )^2\Big ] = 2{\varepsilon } \int _0^t ds (t-s) \mathrm{Cov} \big ( G(Z^\varepsilon _s) , G(Z^\varepsilon _0) \big ) ds . \end{aligned}$$

We have moreover

$$\begin{aligned} \big | \mathrm{Cov} \big ( G(Z^\varepsilon _s) , G(Z^\varepsilon _0) \big )\big |= & {} \big | \mathbb {E}\big [ \big ( \mathbb {E}[G(Z^\varepsilon _s)|\mathcal{F}_0] -\mathbb {E}[G(Z^\varepsilon _s)]\big ) G(Z^\varepsilon _0) \big ]\big |\\\le & {} \Vert G\Vert _\infty \mathrm{Var} \big ( \mathbb {E}[G(Z^\varepsilon _s)|\mathcal{F}_0] \big )^{1/2} . \end{aligned}$$

By Lemma 4 we obtain

$$\begin{aligned} \big | \mathrm{Cov} \big ( G(Z^\varepsilon _s) , G(Z^\varepsilon _0) \big )\big |\le & {} \Vert G\Vert _\infty \Vert G'\Vert _\infty \sigma ^\varepsilon _{s,\infty } . \end{aligned}$$

In view of Lemma 7 we then have

$$\begin{aligned} \mathbb {E}\big [ ( {\kappa }^\varepsilon _t )^2\big ] \le C_T \varepsilon \big (\varepsilon + \varepsilon ^{1-H} \big ) \le 2 C_T \varepsilon ^{2-H} , \end{aligned}$$

uniformly in $t \in [0,T]$ and $\varepsilon \in (0,1]$, which gives the desired result.

Lemma 7

Define

$$\begin{aligned} \sigma ^\varepsilon _{t,\infty } = \sigma _\mathrm{ou} \left( \int _t^\infty \mathcal{K}^\varepsilon (s)^2 ds \right) ^{1/2} , \end{aligned}$$

(68)

Then there exists $C>0$ such that

$$\begin{aligned} \sigma ^\varepsilon _{t,\infty } \le C \big ( 1 \wedge (\varepsilon /t)^{1-H}\big ) . \end{aligned}$$

(69)

Proof

This follows from $|\mathcal{K}(s)| \le K s^{H-\frac{3}{2}}$ for $s \ge 1$ and $\mathcal{K}\in L^2$.

An alternative model

In Comte and Renault (1998); Funahashi and Kijima (2017) the authors consider a stochastic volatility model that is a kind of fractional Orstein-Uhlenbeck process, but they consider the following representation of the fractional Brownian motion:

$$\begin{aligned} W^{H,0}_t= \frac{1}{\varGamma (H+\frac{1}{2})} \int _0^t (t-s)^{H-\frac{1}{2}}dW^0_s , \end{aligned}$$

(70)

where $(W^0_t)_{t \in \mathbb {R}^+}$ is a standard Brownian motion over $\mathbb {R}^+$. $(W^{H,0}_t)_{t \in \mathbb {R}^+}$ is a zero-mean self-similar Gaussian process, in the sense that $(\alpha ^H W^{H,0}_{t/\alpha })_{t \in \mathbb {R}^+}$ and $(W_t^{H,0})_{t \in \mathbb {R}^+}$ have the same distribution, but it is not stationary, nor does it have stationary increments. Its variance is

$$\begin{aligned} \mathbb {E}\big [ (W_{t}^{H,0})^2 \big ]= \frac{1}{2H \varGamma (H+\frac{1}{2})^2} t^{2H}, \end{aligned}$$

while the variance of its increment is (for $s>0$):

$$\begin{aligned} \mathbb {E}\big [ (W_{t+s}^{H,0}-W_t^{H,0})^2 \big ] = \frac{1}{\varGamma (H+\frac{1}{2})^2} \Big [ \int _0^{t/s} \big ( (1+u)^{H-\frac{1}{2}}-u^{H-\frac{1}{2}}\big )^2 du +\frac{1}{2H}\Big ] {s}^{2H} , \end{aligned}$$

which has the following behavior

$$\begin{aligned} \mathbb {E}\big [ (W_{t+s}^{H,0}-W_t^{H,0})^2 \big ] {\mathop {\longrightarrow }\limits ^{t \rightarrow +\infty }} \frac{1}{\varGamma (2H+1)\sin (\pi H)} {s}^{2H} . \end{aligned}$$

This model is special because time zero plays a special role, and we think it is desirable to deal with the stationary situation addressed in this paper. However, it turns out that the two models give the same result in the fast-varying case. Indeed, the modified fOU process corresponding to (70) is [to be compared with (4)]:

$$\begin{aligned} Z^{\varepsilon ,0}_t= & {} Z_0 e^{-t/\varepsilon } +\varepsilon ^{-H} \int _0^t e^{-\frac{t-s}{\varepsilon }} dW^{H,0}_s \nonumber \\= & {} Z_0 e^{-t/\varepsilon } +\varepsilon ^{-H} W^{H,0}_t - \varepsilon ^{-H-1} \int _0^t e^{-\frac{t-s}{\varepsilon }} W^{H,0}_s ds , \end{aligned}$$

(71)

where $Z_0$ is considered as a constant as in Comte and Renault (1998), Funahashi and Kijima (2017). In terms of the Brownian motion $W^0_t$ this reads:

$$\begin{aligned} Z^{\varepsilon ,0}_t =Z_0 e^{-t/\varepsilon }+ \sigma _{\mathrm{ou}} \int _0^t \mathcal{K}^\varepsilon (t-s) dW^0_s, \end{aligned}$$

(72)

where $\mathcal{K}^\varepsilon$ is defined in (10). It is a Gaussian process with the following covariance ($t,s\ge 0$):

$$\begin{aligned} \mathrm{Cov} \big ( Z^{\varepsilon ,0}_t , Z^{\varepsilon ,0}_{t+s} \big ) = \sigma ^2_{\mathrm{ou}} \mathcal{C}_{t/\varepsilon }^0 \Big (\frac{s}{\varepsilon }\Big ) , \end{aligned}$$

that is a function of $t/\varepsilon$ and $s/\varepsilon$ with

$$\begin{aligned} \mathcal{C}_t^0(s) = \int _0^t \mathcal{K}(u) \mathcal{K}(u+s)du. \end{aligned}$$

Note that

$$\begin{aligned} \mathcal{C}_t^0(s) {\mathop {\longrightarrow }\limits ^{t \rightarrow +\infty }} \int _0^\infty \mathcal{K}(u) \mathcal{K}(u+s)du = \mathcal{C}_Z(s) , \end{aligned}$$

with $\mathcal{C}_Z$ defined by (6). In other words, except for a small period of time just after time 0 which is of duration of the order of $\varepsilon$, the modified process has the same behavior as the one introduced in this paper. One can then check the detailed calculations carried out in this paper and find that Proposition 1 still holds true with the modified model $Z^{\varepsilon ,0}_t$.