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The microstructural foundations of leverage effect and rough volatility

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Abstract

We show that typical behaviors of market participants at the high frequency scale generate leverage effect and rough volatility. To do so, we build a simple microscopic model for the price of an asset based on Hawkes processes. We encode in this model some of the main features of market microstructure in the context of high frequency trading: high degree of endogeneity of market, no-arbitrage property, buying/selling asymmetry and presence of metaorders. We prove that when the first three of these stylized facts are considered within the framework of our microscopic model, it behaves in the long run as a Heston stochastic volatility model, where a leverage effect is generated. Adding the last property enables us to obtain a rough Heston model in the limit, exhibiting both leverage effect and rough volatility. Hence we show that at least part of the foundations of leverage effect and rough volatility can be found in the microstructure of the asset.

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Notes

  1. A large tick asset is an asset whose bid–ask spread is almost always equal to one tick and which therefore essentially moves by one tick jumps; see [23].

  2. Notice that the limit price is actually a “Bachelier” version of the Heston model. Furthermore, we remark that the definition of the rough Heston model is not a well-established one and other types of fractional Heston models can be defined; see for example [33].

  3. We use the fact that \([M^{T},M^{T}]_{t} = \mathrm{diag} (N_{t}^{T})\) and \(N^{T}_{t} = M ^{T}_{t} + \int_{0}^{t} \lambda^{T}_{s}\,ds\).

  4. Notice that for all \((x,y) \in(0, \infty)^{2}\), we have \(|\frac{x-\beta y}{\sqrt{x+y}\sqrt{x+ \beta^{2} y}}| \leq1\) by the Cauchy–Schwarz inequality.

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Acknowledgements

We thank Neil Shephard for inspiring discussions and Jim Gatheral and Kasper Larsen for very relevant comments. Omar El Euch and Mathieu Rosenbaum gratefully acknowledge the financial support of the ERC Grant 679836 Staqamof and the Chair Analytics and Models for Regulation.

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Appendix

Appendix

1.1 A.1 Proof of Lemma 4.1

This result has already been proved in [45] in dimension one. We need to generalize it to \(d \geq2\). Inspection of the proof of Proposition 4.1 in [45] shows that the tightness of

$$ H^{T}_{t}=\int_{0}^{t} f^{T}(t-s)\,d\overline{M}_{s}^{T}, \quad t \in[0,1], $$

with respect to the Skorokhod topology holds in the same way when the dimension is larger than one. So we just need to check the finite-dimensional convergence of \(H^{T}\) to zero. Using that \(\langle M^{T},M^{T} \rangle= \int_{0}^{.} \lambda^{T}_{s}\,ds\), we get

$$\begin{aligned} {\mathbb{E}} \big[\|H_{t}^{T}\|_{2}^{2}\big] &= \frac{1}{T^{2}} {\mathbb{E}} \bigg[ \int_{0}^{tT} f^{T}(t- s/T)^{2} \sum_{i=1}^{d} \lambda_{s,i}^{T}\,ds \bigg] \\ &= \frac{1}{T^{2}} \int_{0}^{tT} f^{T}(t- s/T)^{2} \sum_{i=1}^{d} {\mathbb{E}} [\lambda_{s,i}^{T}]\,ds. \end{aligned}$$

Using (2.2) and the fact that \(v_{i}^{\top}\varPsi^{T} = \psi_{i}^{T} v_{i}^{\top}\), we obtain for any \(i \in\{1, \dots,d\}\) and \(s\geq0\) that

$$ {\mathbb{E}} [v_{i}^{\top}\lambda_{s}^{T}] = \mu(v_{i}^{\top} \mathbf{1})\bigg(1+ \int_{0}^{s} \psi_{i}^{T}(u)\,du\bigg) . $$

Thus

$$ |{\mathbb{E}} [v_{i}^{\top}\lambda_{s}^{T}]|\leq\mu| v_{i}^{\top} \mathbf{1} | \bigg(1+ \sum_{k \geq1} \int_{0}^{\infty} a_{T}^{k} | \lambda_{i}|^{*k}(u)\,du\bigg) \leq\mu| v_{i}^{\top}\mathbf{1} | \frac{1}{1-a _{T} \|\lambda_{i}\|_{1}} \leq c T. $$

Hence \({\mathbb{E}}[\lambda_{s,i}^{T}] \leq cT\) for any \(i \in\{1, \dots,d\}\). Therefore

$$ {\mathbb{E}}\big[\|H_{t}^{T}\|_{2}^{2}\big]\leq c \int_{0}^{\infty} f ^{T}(s)^{2}\,ds \mathop{\longrightarrow}^{T \rightarrow\infty} 0 $$

and so \(H_{t}^{T}\) tends to zero in probability, giving the finite-dimensional convergence of the process.  □

1.2 A.2 Wiener–Hopf equations

The following result is used extensively in this work to solve Wiener–Hopf type equations; see for example [7, Lemma 3].

Lemma A.1

Let \(g :\mathbb{R} \to\mathbb{R}^{d}\) be a measurable and locally bounded function and \(\varPhi: \mathbb{R}_{+} \rightarrow\mathcal {M}^{\textbf{d}}(\mathbb{R}) \) have integrable components such that \(\mathcal{S}(\int_{0}^{\infty}\varPhi(s)\,ds) < 1\). Then there exists a unique locally bounded function \(f: \mathbb{R}_{+} \to\mathbb{R}^{d}\) which solves

$$ f(t) = g(t) + \int_{0}^{t} \varPhi(t-s) f(s)\,ds,\quad t \geq0. $$

It is given by

$$ f(t) = g(t) + \int_{0}^{t} \varPsi(t-s) g(s)\,ds,\quad t \geq0, $$

where \(\displaystyle \varPsi= \sum_{k \geq1} \varPhi^{*k}\).

1.3 A.3 Fractional integrals and derivatives

The fractional integral of order \(r \in(0,1]\) of a function \(f\) is defined by

$$ I^{r} f(t) = \frac{1}{\varGamma(r)} \int_{0}^{t} (t-s)^{r-1} f(s)\,ds, $$

whenever the integral exists. The fractional derivative of order \(r \in[0,1)\) of \(f\) is given by

$$ D^{r} f(t) = \frac{1}{\varGamma(1-r)} \frac{d}{dt} \bigg(\int_{0}^{t} (t-s)^{-r} f(s)\,ds\bigg), $$

whenever it exists.

1.4 A.4 Mittag-Leffler functions

Let \((\alpha,\beta) \in(0, \infty)^{2}\). The Mittag-Leffler function \(E_{\alpha,\beta}\) is defined for \(z \in\mathbb{C}\) by

$$ E_{\alpha,\beta}(z) = \sum_{n \geq0} \frac{z^{n}}{\varGamma(\alpha n + \beta)}. $$

For \((\alpha,\lambda) \in(0,1)\times\mathbb{R}_{+}\), we also define

$$\begin{aligned} f^{\alpha,\lambda}(t) &= \lambda t^{\alpha- 1} E_{\alpha,\alpha}(- \lambda t^{\alpha}),\quad t>0, \end{aligned}$$
(A.1)
$$\begin{aligned} F^{\alpha,\lambda}(t) &= \int_{0}^{t} f^{\alpha,\lambda}(s)\,ds, \quad t \geq0. \end{aligned}$$
(A.2)

The function \(f^{\alpha,\lambda}\) is a density function on \(\mathbb{R}_{+}\), called Mittag-Leffler density function.

For \(\alpha\in(1/2,1)\), \(f^{\alpha,\lambda}\) is square-integrable, and its Laplace transform is given for \(z \geq0\) by

$$ \hat{f}^{\alpha,\lambda}(z)= \int_{0}^{\infty}f_{\alpha,\lambda}(s) e^{-zs}\,ds = \frac{\lambda}{\lambda+ z^{\alpha}}. $$

Finally, one can show that

$$ I^{1-\alpha}f^{\alpha,\lambda}(t) = \lambda\big(1-F^{\alpha, \lambda}(t)\big). $$

Further properties of \(f^{\alpha,\lambda}\) and \(F^{\alpha,\lambda }\) can be found in [36, 50] and [51, Chap. 2].

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El Euch, O., Fukasawa, M. & Rosenbaum, M. The microstructural foundations of leverage effect and rough volatility. Finance Stoch 22, 241–280 (2018). https://doi.org/10.1007/s00780-018-0360-z

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