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.
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References
Aït-Sahalia Y, Jacod J (2014) High frequency financial econometrics. Princeton University Press, Princeton
Aït-Sahalia Y, Jacod J (2017) Semimartigale: Itô or Not? Stoch Proces Their Appl 128(1):233–254
Aldous DJ, Eagleson GK (1978) On mixing and stability of limit theorems. Ann Probab 6(2):325–331
Bayer C, Fritz PK, Gulisashvili A, Horvath B, Stemper B (2019) Short-time near-the-money skew in rough fractional volatility models. Quant Finance 19(5):779–798
Bennedsen M, Lunde A, Pakkanen MS (2016) Decoupling the short- and long-term behavior of stochastic volatility. ar**v:1610.00332
Cheridito P, Kawaguchi H, Maejima M (2003) Fractional Ornstein–Uhlenbeck processes. Electron J Probab 8(3):1–14
Comte F, Renault E (1998) Long memory in continuous-time stochastic volatility models. Math Finance 8(4):291–323
Delbaen F, Schachermayer W (1994) A general version of the fundamental theorem of asset pricing. Math Ann 300(1):463–520
El Euch O, Fukasawa M, Rosenbaum M (2018) The microstructural foundations of leverage effect and rough volatility. Financ Stoch 22(2):241–280
Figueroa-López J, Li C (2016) Optimal kernel estimation of spot volatility of stochastic differential equations. https://pages.wustl.edu/figueroa/publications. Accessed July 2018
Gatheral J, Jaisson T, Rosenbaum M (2018) Volatility is rough. Quant Finance 18(6):933–949
Jacod J, Protter P (2012) Discretization of processes. Springer, Berlin
Liu Y (2017) Semiparametric inference for integrated volatility functionals using high-frequency financial data. https://cdr.lib.unc.edu/concern/dissertations/h128nd92c. Accessed June 2019
Mandelbrot B, Van Ness JW (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev 10(4):422–437
Mishura YS (2008) Stochastic calculus for fractional Brownian motion and related processes. Springer, Berlin
Nourdin I (2012) Selected aspects of fractional brownian motion. Springer, Berlin
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Yaroslav Eumenius-Schulz: LPSM-UPMC. I am deeply grateful to my PhD supervisors Jean Jacod (UPMC) and Mathieu Rosenbaum (École Polytechnique) for their support and helpful discussions.
Appendix
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 \(t\in \left( (i-1)\Delta _n , i\Delta _n \right] \):
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:
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:
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 \(\xi \in (-|x|,|x|)\):
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 \(p\ge 1\):
For \(p\ge 1\), the normal distribution of \(\int \limits _{-\infty }^{t}{B_s exp\{\alpha s\}ds}\) allows the following estimation:
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
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:
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:
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
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 \(i=1,\ldots ,k_n\)
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 \(exp\{x_i\}\):
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 \(i=1,\ldots ,k_n\):
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:
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:
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:
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:
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:
Thus, \(\sqrt{k_n}\sum _{j=1}^{5}{\zeta _{n}^{(j)}} {\mathop {\longrightarrow }\limits ^{{\mathbb {P}}}} 0\).
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Eumenius-Schulz, Y. Spot estimation for fractional Ornstein–Uhlenbeck stochastic volatility model: consistency and central limit theorem. Stat Inference Stoch Process 23, 355–380 (2020). https://doi.org/10.1007/s11203-020-09209-1
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DOI: https://doi.org/10.1007/s11203-020-09209-1