Spot estimation for fractional Ornstein–Uhlenbeck stochastic volatility model: consistency and central limit theorem

Abstract

There has been an increasing interest for rough stochastic volatility models. However, little is known about the statistical inference for such models, especially for high frequency data. This paper investigates estimation of the fractional spot volatility from discrete observations of the price process on a grid with a time interval $\Delta _n\rightarrow 0$ as $n\rightarrow \infty$. Namely, the model with fractional Ornstein–Uhlenbeck log-volatility and Itô-semimartingale log-price processes is considered. In this setup both consistency and central limit theorem are proven for truncated and non-truncated spot volatility estimators. Then, asymptotic confidence intervals are derived for a finite number of spot volatility estimators at different estimation times. Consequently, the highest possible rate of convergence achieved in the central limit theorem $\Delta _n^{H/(2H+1)}$ is a function of the Hurst parameter H of the fractional Brownian motion driving the volatility. This rate coincides with the already known highest convergence rate for the Brownian case when $H=0.5$. Furthermore, simulations in this paper validate the consistency and central limit theorem numerically. Article class.

Appendix

In this chapter we prove some auxiliary lemmas and Step 4 of Theorem 1.

Proof of Lemma 1

This proof is a modified version of Lemma 13.3.10 of Jacod and Protter (2012). Remembering that $i_n(t)=i_n=i$ if $t\in \left( (i-1)\Delta _n , i\Delta _n \right]$:

$$\begin{aligned} {\mathbb {P}}(c_t^{n}(k_n,v_n,X)\ne & {} c_t^{n}(k_n,X))\le \sum _{i=1}^{k_n}{\mathbb {P}}\left( \left| \Delta ^n_{i_n+i}X\right|>v_n\right) \nonumber \\\le & {} \sum _{i=1}^{k_n}{\mathbb {P}}\left( \left| \Delta ^n_{i_n+i}X'\right|>v_n/2\right) +{\mathbb {P}}\left( \left| \Delta ^n_{i_n+i}X''\right| >v_n/2\right) . \end{aligned}$$

(47)

Setting $Q:=\Gamma *p$ if $r\le 1$ in (C2-r) and $Q:=X''$ otherwise, it is possible to apply (13.2.22) and (13.2.23) from Jacod and Protter (2012) for $i=1,\ldots ,k_n$ and $m>0$ arbitrary, we get:

$$\begin{aligned} {\mathbb {P}}\left( \left| \Delta ^n_{i_n+i}X'\right|>v_n/2\right)\le & {} K_m\Delta _n^{-m\varpi }{\mathbb {E}}\left[ \left| \Delta ^n_{i_n+i}X'\right| ^m \right] \le K_m\Delta _n^{m(1/2-\varpi )}\nonumber \\ {\mathbb {P}}\left( \left| \Delta ^n_{i_n+i}X''\right| >v_n/2\right)\le & {} K{\mathbb {E}}\left[ \min \left( 1,\frac{\Delta ^n_{i_n+i}Q}{\Delta _n^\varpi }\right) ^r \right] \le K\Delta _n^{1-r\varpi }\phi _n, \end{aligned}$$

(48)

where $\phi _n\rightarrow 0$ as $n\rightarrow \infty$. Setting m large enough, (13) of Lemma 1 is proved for $X=X'$, since $\varpi <\frac{1}{2}$ and $k_n\Delta _n^{m(1/2-\varpi )}$. Otherwise, if condition (7) of Theorem 1 and $\varpi \le \frac{1-\tau }{r}$ hold, $k_n\Delta _n^{1-r\varpi }\phi _n\le K \phi _n\rightarrow 0$.

Now assume that conditions (7) and (10) of Theorem 1 hold. Applying Lemma 13.2.6 from Jacod and Protter (2012) to $c_t^{n}(k_n,v_n,X) - c_t^{n}(k_n,v_n,X')$ with $m=1$ yields:

$$\begin{aligned} {\mathbb {E}}\left[ \left| c_t^{n}(k_n,v_n,X) - c_t^{n}(k_n,v_n,X')\right| \right] \le \left( \Delta _n^{\frac{2-r}{2}\min \left( 1,\frac{1}{r}\right) -\theta }+\Delta _n^{(2-r)\varpi - \theta }\right) \phi _n, \end{aligned}$$

(49)

where $\theta >0$ is arbitrary small and $\phi _n\rightarrow 0$ as $n\rightarrow \infty$. Under condition (10) of Theorem 1, $(2-r)\min (1,\frac{1}{r})-2\theta \ge \tau$ and $2(2-r)\varpi -2\theta \ge \tau$. Then (14) of Lemma 1 holds true, as long as (7) of Theorem 1 is satisfied. $\square$

Proof of Lemma 2

This proof is a modified version of Lemma 13.3.11 of Jacod and Protter (2012). Strengthening (C1-r) by localization, it is possible to assume (C2-r) and, hence, Lemma 1 holds true.

Assume that case (a) of Theorem 1 holds for the non-truncated version $c_t^{n}(k_n)$. A first application of (13) (with $X=X'$) yields that (8) holds as well for the truncated version $c_t^{n}(k_n,v_n)$, when X is continuous.

Next assume that that X is discontinuous, and (7) and (10) of Theorem 1 hold. Then the validity of (8) for $X'$ and the truncated version yields, together with (14), that (8) holds for X and the truncated version.

Finally, assume that X is discontinuous and (7) and (9) of Theorem 1 hold. Then the first part of (10) is true. It is possible to find $\varpi \in (0,\frac{1}{2})$, such that the second part of (10) is true as well together with $\varpi \le \frac{1-\tau }{r}$. Then, setting $v_n=\Delta _n^\varpi$ for this particular value of $\varpi$, (8) is proved for X and the truncated version by (14), and for the non-truncated version by (13). $\square$

Finally, we prove Step 4 of Theorem 1 and show that $\sqrt{k_n}\sum _{j=1}^{5}{\zeta _{n}^{(j)}} {\mathop {\longrightarrow }\limits ^{{\mathbb {P}}}} 0$.

Estimation of ${\mathbb {E}}\left[ \left| \zeta _{n}^{(1)}\right| \right]$

We prove here ${\mathbb {E}}\left[ \left| \sigma _{s_n}-\sigma _{t}\right| ^2 \right] \le K\Delta _n^{2H}$ instead of of ${\mathbb {E}}\left[ \left| c_{s_n}-c_{t}\right| ^2 \right] \le K_1\Delta _n^{2H}$, since the former inequality will be used further throughout the proof, whereas the proof of the latter one is very similar and differs only by the constant $K_1$ (replacing $\sigma _{s_n}-\sigma _{t}$ by $\sigma _{s_n}^2-\sigma _{t}^2$).

With $x={\tilde{\sigma }}(B_{s_n}-B_{t}-\alpha (exp\{-\alpha s_n\})\int \nolimits _{t}^{s_n}{B_s exp\{\alpha s\}ds}+\alpha (exp\{-\alpha t\}-exp\{-\alpha s_n\})\int \nolimits _{-\infty }^{t}{B_s exp\{\alpha s\}ds})$, the first-order Taylor approximation of $exp\{x\}$ around 0 yields for some $\xi \in (-|x|,|x|)$:

$$\begin{aligned} {\mathbb {E}}\left[ \left| \sigma _{s_n}-\sigma _{t}\right| ^2 \right] = {\mathbb {E}}\left[ c_t\left| exp\{x\}-1\right| ^2 \right] \le K{\mathbb {E}}\left[ c_t x^2 \right] +K{\mathbb {E}}\left[ c_t exp\{2\xi \}x^4 \right] . \end{aligned}$$

(50)

It holds that ${\mathbb {E}}\left[ \left| B_{s_n}-B_{t}\right| ^p \right] \le K_p \Delta _n^{pH}$ for $p\ge 1$ due to the stationarity of increments and selfsimilarity of $B^H$.

The mean-value theorem and the finiteness of ${\mathbb {E}}\left[ (\sup _{s \in [0,1]}\left| B_s\right| )^p \right]$ yield for $p\ge 1$:

$$\begin{aligned}&{\mathbb {E}}\left[ \left| \alpha (exp\{-\alpha s_n\})\int \limits _{t}^{s_n}{B_s exp\{\alpha s\}ds}\right| ^p \right] \nonumber \\&\quad \le \alpha ^p \left( \int \limits _{t}^{s_n}{exp\{\alpha s\}ds}\right) ^p{\mathbb {E}}\left[ \left| \sup _{s \in [s_n,t]}(B_s-B_{s_n}) + B_{s_n}\right| ^p \right] \nonumber \\&\quad \le K_p\Delta _n^p. \end{aligned}$$

(51)

For $p\ge 1$, the normal distribution of $\int \limits _{-\infty }^{t}{B_s exp\{\alpha s\}ds}$ allows the following estimation:

$$\begin{aligned} {\mathbb {E}}\left[ \left| \alpha (exp\{-\alpha t\}-exp\{-\alpha s_n\}) \int \limits _{-\infty }^{t}{B_s exp\{\alpha s\}ds}\right| ^p \right] \le K_p \Delta _n^p. \end{aligned}$$

(52)

Thus, Hölder inequality, finiteness of ${\mathbb {E}}\left[ c_t^p \right]$ and the inequality ${\mathbb {E}}\left[ exp\{p \xi \} \right] \le {\mathbb {E}}\left[ exp\{px\}+1 \right]$ for $p\ge 1$ yield that ${\mathbb {E}}\left[ \left| \sigma _{s_n}-\sigma _{t}\right| ^2 \right] \le K \Delta _n^{2H}$ and ${\mathbb {E}}\left[ \left| c_{s_n}-c_{t}\right| ^2 \right] \le K \Delta _n^{2H}$. Consequently, ${\mathbb {E}}\left[ \left| \zeta _{n}^{(1)}\right| \right] \le K \Delta _n^{H}$.

Estimation of ${\mathbb {E}}\left[ \left| \zeta _{n}^{(2)}\right| \right]$

Assume that

$$\begin{aligned} p^{i}_{n}= & {} \Delta _{i_n+i}^{n}X-\sigma _{s_n+(i-1)\Delta _n}\Delta _{i_n+i}^{n}W \nonumber \\= & {} \int \limits _{s_n+(i-1)\Delta _n}^{s_n+i\Delta _n}{b'_s ds}+\int \limits _{s_n+(i-1)\Delta _n}^{s_n+i\Delta _n}{}(\sigma _s- \sigma _{s_n+(i-1)\Delta _n)} dW_s \end{aligned}$$

(53)

and $d^{i}_{n}=\sigma _{s_n+(i-1)\Delta _n}\Delta _{i_n+i}^{n}W$.

Applying the Cauchy–Schwarz inequality in the first, Fubini’s theorem in the second and the finiteness of ${\mathbb {E}}\left[ (b'_s)^2 \right]$ in the third step yields:

$$\begin{aligned} {\mathbb {E}}\left[ (p^{i}_{n})^2 \right]\le & {} K_1\Delta _n\int \limits _{s_n+(i-1)\Delta _n}^{s_n+i\Delta _n}{{\mathbb {E}}\left[ ({b'}_s)^2 \right] ds}+ K_1\int \limits _{s_n+(i-1)\Delta _n}^{s_n+i\Delta _n}{{\mathbb {E}}\left[ (\sigma _s- \sigma _{s_n+(i-1)\Delta _n})^2 \right] ds}\nonumber \\\le & {} K \max (\Delta _n^{2H+1}, \Delta _n^{2}). \end{aligned}$$

(54)

The independence of $\Delta _{i_n+i}W$ from ${{\mathcal {F}}}_{s_n+(i-1)\Delta _n}$ and measurability of $\sigma _{s_n+(i-1)\Delta _n}$ with respect to ${{\mathcal {F}}}_{s_n+(i-1)\Delta _n}$ yield ${\mathbb {E}}\left[ (d^{i}_{n})^2 \right] \le K \Delta _n$.

Since $(a^2-b^2)=(a-b)^2+2b(a-b)$, we get for every $i=1,\ldots ,k_n$ using Cauchy–Schwarz inequality:

$$\begin{aligned} {\mathbb {E}}\left[ \left| (\Delta _{i_n+i}^{n}X)^2-(\sigma _{s_n+(i-1)\Delta _n} \Delta _{i_n+i}^{n}W^1)^2\right| \right]\le & {} K\left( {\mathbb {E}}\left[ (p^{i}_{n})^2 \right] +\left( {\mathbb {E}}\left[ (d^{i}_{n})^2 \right] \right) ^{\frac{1}{2}}\right) \left( {\mathbb {E}}\left[ (p^{i}_{n})^2 \right] \right) ^{\frac{1}{2}}\nonumber \\\le & {} K \max (\Delta _n^{2H+1}, \Delta _n^{2})+K \max (\Delta _n^{H+1}, \Delta _n^{1.5}).\nonumber \\ \end{aligned}$$

(55)

Thus, by the triangle inequality, ${\mathbb {E}}\left[ \left| \zeta _{n}^{(2)}\right| \right] \le K (\Delta _n^{H}+\Delta _n^{0.5}$).

Estimation of ${\mathbb {E}}\left[ \left| \zeta _{n}^{(3)}\right| \right]$

The independence of $\Delta _{i_n+i}^{n}W$ from ${{\mathcal {F}}}_{s_n+(i-1)\Delta _n}$ and the measurability of $\sigma _{s_n+(i-1)\Delta _n}-\sigma _{s_n}$ with respect to ${{\mathcal {F}}}_{s_n+(i-1)\Delta _n}$ yield

$$\begin{aligned} \frac{1}{k_n\Delta _n}\sum _{i=1}^{k_n}{{\mathbb {E}}\left[ {\mathbb {E}}\left[ \left. \left| (\Delta _{i_n+i}^{n}W)^2 (\sigma _{s_n+(i-1)\Delta _n}-\sigma _{s_n})^2\right| \right| {{\mathcal {F}}}_{s_n+(i-1)\Delta _n} \right] \right] }\le K(k_n\Delta _n)^{2H}.\nonumber \\ \end{aligned}$$

(56)

Thus, ${\mathbb {E}}\left[ \left| \zeta _{n}^{(3)}\right| \right] \le K(k_n\Delta _n)^{2H}$.

Estimation of ${\mathbb {E}}\left[ \left| \zeta _{n}^{(4)}\right| \right]$

Define for every $i=1,\ldots ,k_n$

$$\begin{aligned} x_i= & {} {\tilde{\sigma }}\left( B_{s_n+(i-1)\Delta _n}-B_{s_n}-\alpha \left( \int \limits _{s_n}^{s_n+(i-1)\Delta _n}{B_s exp\left\{ \alpha (s-s_n-(i-1)\Delta _n)\right\} ds}\right. \right. \nonumber \\&\left. \left. +\,\left( exp\left\{ -(i-1)\Delta _n\right\} -1\right) \int \limits _{-\infty }^{s_n}{B_s exp\left\{ \alpha (s-s_n)\right\} ds}\right) \right) . \end{aligned}$$

(57)

Then, combining the independence of $\Delta _{i_n+i}^{n}W$ from ${{\mathcal {F}}}_{s_n+(i-1)\Delta _n}$ together with the first order Taylor approximation around 0 for some $\xi _i \in (-|x_i|,|x_i|)$ yields for $exp\{x_i\}$:

$$\begin{aligned}&{\mathbb {E}}\left[ {\mathbb {E}}\left[ \left. \left| (\Delta _{i_n+i}^{n}W)^2(\sigma _{s_n}\sigma _{s_n+(i-1)\Delta _n}-\sigma _{s_n}^2-{\tilde{\sigma }} \sigma _{s_n}^2(B_{s_n+(i-1)\Delta _n}-B_{s_n}))\right| \right| {{\mathcal {F}}}_{s_n+(i-1)\Delta _n} \right] \right] \nonumber \\&\quad =\Delta _n{\mathbb {E}}\left[ c_{s_n}\left| exp\{x_i\} - 1-{\tilde{\sigma }}(B_{s_n+(i-1)\Delta _n}-B_{s_n})\right| \right] \nonumber \\&\quad =\Delta _n{\mathbb {E}}\left[ c_{s_n}\left| -\alpha {\tilde{\sigma }}\left( \int \limits _{s_n}^{s_n+(i-1)\Delta _n}{B_s exp\left\{ \alpha (s-s_n-(i-1)\Delta _n)\right\} ds} \right. \right. \right. \nonumber \\&\qquad \left. \left. \left. +\,\left( exp\left\{ -(i-1)\Delta _n\right\} -1\right) \int \limits _{-\infty }^{s_n}{B_s exp\{\alpha (s-s_n)\}ds}\right) +x_i^2\frac{exp\{\xi _i\}}{2}\right| \right] . \end{aligned}$$

(58)

Similar to the estimation of ${\mathbb {E}}\left[ \left| \zeta _{n}^{(1)}\right| \right]$ with $p\ge 1$, the mean value theorem and the finiteness of ${\mathbb {E}}\left[ (\sup _{s \in [0,1]}\left| B_s\right| )^p \right]$ in the first, the normality of $\int \limits _{-\infty }^{t}{B_s exp\{\alpha s\}ds}$ in the second and the stationarity of the increments and the self-similarity of B in the third inequality yield for every $i=1,\ldots ,k_n$:

$$\begin{aligned}&{\mathbb {E}}\left[ \left| \int \limits _{s_n}^{s_n+(i-1)\Delta _n}{B_s exp\left\{ \alpha (s-s_n-(i-1)\Delta _n)\right\} ds}\right| ^{p} \right] \nonumber \\&\quad \le K\left( \int \limits _{s_n}^{s_n+(i-1)\Delta _n}{exp\{\alpha s \}ds}\right) ^p{\mathbb {E}}\left[ \left| \sup _{s\in [s_n,s_n+(i-i)\Delta _n]}(B_s-B_{s_n}+B_{s_n})\right| ^{p} \right] \nonumber \\&\quad \le K_p (k_n\Delta _n)^p, \end{aligned}$$

(59)

$$\begin{aligned}&{\mathbb {E}}\left[ (1-exp\{-(i-1)\Delta _n\})^p\left| \int \limits _{-\infty }^{s_n}{B_s exp\{\alpha (s-s_n)\}ds}\right| ^p \right] \le K_p (k_n\Delta _n)^p, \end{aligned}$$

(60)

$$\begin{aligned}&{\mathbb {E}}\left[ \left| (B_{s_n+(i-1)\Delta _n}-B_{s_n})\right| ^p \right] \le K(k_n\Delta _n)^{pH}. \end{aligned}$$

(61)

Hence, the utilization of triangle and Cauchy–Schwarz inequalities for $p\ge 1$, combined with the uniform estimate of ${\mathbb {E}}\left[ exp\{p \xi _i\} \right]<{\mathbb {E}}\left[ exp\{px_i\}+1 \right] <K$ and ${\mathbb {E}}\left[ c_{s_n}^p \right] <K$, yield:

$$\begin{aligned} {\mathbb {E}}\left[ \left| \zeta _{n}^{(4)}\right| \right] \le K((k_n\Delta _n)^{2H}+k_n\Delta _n). \end{aligned}$$

(62)

Estimation of${\mathbb {E}}\left[ \left| \zeta _{n}^{(5)}\right| \right]$

The Cauchy–Schwarz inequality together with the finiteness of ${\mathbb {E}}\left[ c_{s_n}^2 \right]$ yield:

$$\begin{aligned}&{\mathbb {E}}\left[ \left| \frac{2\nu c_{s_n}}{k_n\Delta _n}\sum _{i=1}^{k_n}{(B_{s_n+(i-1)\Delta _n}-B_{s_n}) ((\Delta _{i_n+i}^{n}W)^2-\Delta _n)}\right| \right] \nonumber \\&\quad \le \frac{K}{k_n\Delta _n}{\mathbb {E}}\left[ \left( \sum _{i=1}^{k_n}{(B_{s_n+(i-1) \Delta _n}-B_{s_n})((\Delta _{i_n+i}^{n}W)^2-\Delta _n)}\right) ^2 \right] ^\frac{1}{2}\nonumber \\&\quad =\frac{K}{k_n\Delta _n}\Bigg (\sum _{i=1}^{k_n}{{\mathbb {E}}\left[ (B_{s_n+(i-1)\Delta _n}-B_{s_n})^2 ((\Delta _{i_n+i}^{n}W)^2-\Delta _n)^2 \right] }\nonumber \\&\qquad +\,2\sum _{i=1}^{k_n}{}\sum _{r=1}^{k_n-i}{}{\mathbb {E}}\left[ (B_{s_n+(i-1)\Delta _n}-B_{s_n}) (B_{s_n+(i+r-1)\Delta _n}-B_{s_n})\right. \nonumber \\&\qquad \left. ((\Delta _{i_n+i}^{n}W)^2-\Delta _n)((\Delta _{i_n+i+r}^{n}W)^2-\Delta _n) \right] \Bigg )^\frac{1}{2}. \end{aligned}$$

(63)

For every $i=1,\ldots ,k_n$, the independence of $\Delta _{i_n+i}^{n}W$ from ${{\mathcal {F}}}_{s_n+(i-1)\Delta _n}$ yields:

$$\begin{aligned} {\mathbb {E}}\left[ {\mathbb {E}}\left[ \left. (B_{s_n+(i-1)\Delta _n}-B_{s_n})^2((\Delta _{i_n+i}^{n}W)^2-\Delta _n)^2 \right| {{\mathcal {F}}}_{s_n+(i-1)\Delta _n} \right] \right] \le K(k_n\Delta _n)^{2H}\Delta _n^2.\nonumber \\ \end{aligned}$$

(64)

For every $i=1,\ldots ,k_n$ and $r=1,\ldots ,k_n-i$, the independence of $\Delta _{i_n+i+r}^{n}W$ from ${{\mathcal {F}}}_{s_n+(i+r-1)\Delta _n}$ yields:

$$\begin{aligned}&{\mathbb {E}}\left[ (B_{s_n+(i-1)\Delta _n}-B_{s_n})(B_{s_n+(i+r-1)\Delta _n}-B_{s_n})\right. \nonumber \\&\quad \left. ((\Delta _{i_n+i}^{n}W)^2-\Delta _n)((\Delta _{i_n+i+r}^{n}W)^2-\Delta _n) \right] =0. \end{aligned}$$

(65)

Thus, by the triangle inequality, ${\mathbb {E}}\left[ \left| \zeta _{n}^{(5)}\right| \right] \le K\frac{(k_n\Delta _n)^{H}}{\sqrt{k_n}}$.

Finally, by the triangle inequality the estimation of ${\mathbb {E}}\left[ \left| \zeta _{n}^{(j)}\right| \right]$ for $j=1,\ldots ,5$ yields:

$$\begin{aligned} \sqrt{k_n}\sum _{j=1}^{5}{{\mathbb {E}}\left[ \left| \zeta _{n}^{(j)}\right| \right] }= O \left( \sqrt{k_n}\Delta _n^H+\sqrt{k_n \Delta _n}+k_n^{2H+0.5}\Delta _n^{2H}+k_n^{1.5}\Delta _n+(k_n\Delta _n)^{H}\right) .\nonumber \\ \end{aligned}$$

(66)

Thus, $\sqrt{k_n}\sum _{j=1}^{5}{\zeta _{n}^{(j)}} {\mathop {\longrightarrow }\limits ^{{\mathbb {P}}}} 0$.