Then we have, with an error of order $\epsilon$

$$ \mathbb{E}[\log(\sigma_{t+\Delta}^2) | \mathcal{F}_t] \approx \frac{\cos(\mathrm{H}\pi)}{\pi} \int_0^r \frac{\log(\sigma_{t-\Delta u}^2)}{(u+1)u^{\mathrm{H}+1/2}} du. $$

This prediction formula says that future volatility depends on the whole path of the volatility process. However, since the weights decrease with time, one does not need to go to $-\infty$. It suffice to consider a time to go down to in order to forecast the future. This is roughly defined by setting the error margin $\epsilon$. For example, in order to forecast $\Delta$ in the future, it is common practice by practitioners to take $\Delta$ in the past. This corresponds to $r = 1$, and $\epsilon = 0.35$ which is not so unreasonable.

### 5.5.2 Predicting the variance

Based on the same approximation of the fOU volatility process in the RFSV model, we rewrite $\sigma_t^2 = \exp(2\nu W_t^H + C)$ for some constants $\nu$ and $C$. The prediction of the variance knowing the information at time $t$ is:

$$ \begin{aligned} \widehat{\sigma_{t+\Delta}^2} &= \mathbb{E} \left[ \sigma_{t+\Delta}^2 \mid \mathcal{F}_t \right] \\ &= \mathbb{E} \left[ \exp(2\nu W_{t+\Delta}^H + C) \mid \mathcal{F}_t \right]. \end{aligned} $$

Since $W_{t+\Delta}^H$ is conditionally Gaussian (as shown by [135]) with conditional variance $\mathrm{Var}[W_{t+\Delta}^H \mid \mathcal{F}_t] = c\Delta^{2H}$ (where $c = \frac{\Gamma(3/2-H)}{\Gamma(H+1/2)\Gamma(2-2H)}$) and using the fact that $\log(\widehat{\sigma_t^2}) \approx \mathbb{E}[2\nu W_t^H + C \mid \mathcal{F}_t] = \mathbb{E}[\log(\sigma_{t+\Delta}^2) \mid \mathcal{F}_t]$, we have:

$$ \widehat{\sigma_{t+\Delta}^2} = \exp\left(\log(\widehat{\sigma_{t+\Delta}^2}) + 2c\nu^2\Delta^{2H}\right) $$

Note that this expression uses the estimation of $\log(\widehat{\sigma_{t+\Delta}^2}) = \mathbb{E}[\log(\sigma_{t+\Delta}^2) \mid \mathcal{F}_t]$ which we have seen in Section 5.5.1 and $\nu^2$ which is the exponential of the intercept in the linear regression of $\log(m(2, \Delta))$ on $\log(\Delta)$.

Once again, we compare the performance of the RFSV predictor to the AR, HAR and the GARCH predictors expressed as the following:

— AR(p):

$$ \widehat{\sigma_{t+\Delta}^2} = K_0^\Delta + \sum_{i=0}^p C_i^\Delta \sigma_{t-i}^2 $$

— HAR:

$$ \widehat{\sigma_{t+\Delta}^2} = K_0^\Delta + C_0^\Delta \sigma_t^2 + C_5^\Delta \frac{1}{5} \sum_{i=0}^p C_i^\Delta \sigma_{t-i}^2 + C_{20}^\Delta \frac{1}{20} \sum_{i=0}^{20} \sigma_{t-i}^2 $$

— GARCH(1,1):

$$ \widehat{\sigma_{t+\Delta}^2} = \alpha_0 \left( 1 + \sum_{i=1}^{\Delta-1} (\alpha_1 + \beta_1)^i \right) + (\alpha_1 + \beta_1)^\Delta \sigma_t^2 $$

Results on the variance prediction are given in Table 5.7. We can see that the RFSV model outperforms other predictors on all the considered time horizons. GARCH model performs poorly on the other hand.
<table><thead><tr><th>Ticker</th><th>AR(5)</th><th>AR(10)</th><th>HAR(3)</th><th>GARCH(1,1)</th><th>RFSV</th></tr></thead><tbody><tr><td>SP100 Δ=1</td><td>0.901</td><td>1.01</td><td>0.769</td><td>0.873</td><td>0.655</td></tr><tr><td>SP100 Δ=5</td><td>1</td><td>0.96</td><td>1.06</td><td>1.14</td><td>0.76</td></tr><tr><td>SP100 Δ=21</td><td>1.42</td><td>1.33</td><td>0.989</td><td>1.72</td><td>0.898</td></tr><tr><td>IBEX35 Δ=1</td><td>0.62</td><td>0.632</td><td>0.587</td><td>0.675</td><td>0.694</td></tr><tr><td>IBEX35 Δ=5</td><td>1</td><td>1.01</td><td>0.846</td><td>1.27</td><td>0.808</td></tr><tr><td>IBEX35 Δ=21</td><td>1.43</td><td>1.45</td><td>1.03</td><td>2.03</td><td>0.975</td></tr><tr><td>HSI Δ=1</td><td>0.993</td><td>1.16</td><td>0.847</td><td>1.37</td><td>0.794</td></tr><tr><td>HSI Δ=5</td><td>0.875</td><td>1.16</td><td>0.919</td><td>1.59</td><td>0.851</td></tr><tr><td>HSI Δ=21</td><td>1.1</td><td>1.56</td><td>0.977</td><td>1.81</td><td>0.932</td></tr><tr><td>MEXBOL Δ=1</td><td>0.58</td><td>0.591</td><td>0.566</td><td>0.686</td><td>0.659</td></tr><tr><td>MEXBOL Δ=5</td><td>0.938</td><td>0.866</td><td>0.808</td><td>1.26</td><td>0.767</td></tr><tr><td>MEXBOL Δ=21</td><td>1.32</td><td>1.31</td><td>0.965</td><td>1.77</td><td>0.928</td></tr><tr><td>FTSE100 Δ=1</td><td>0.65</td><td>0.67</td><td>0.618</td><td>0.776</td><td>0.646</td></tr><tr><td>FTSE100 Δ=5</td><td>0.833</td><td>0.908</td><td>0.808</td><td>1.25</td><td>0.721</td></tr><tr><td>FTSE100 Δ=21</td><td>1.15</td><td>1.21</td><td>0.926</td><td>1.69</td><td>0.872</td></tr><tr><td>ASX200 Δ=1</td><td>0.789</td><td>0.834</td><td>0.688</td><td>1.08</td><td>0.656</td></tr><tr><td>ASX200 Δ=5</td><td>0.826</td><td>0.845</td><td>0.721</td><td>1.25</td><td>0.725</td></tr><tr><td>ASX200 Δ=21</td><td>1</td><td>1.05</td><td>0.851</td><td>1.8</td><td>0.837</td></tr><tr><td>TOTAL Δ=1</td><td>0.519</td><td>0.554</td><td>0.497</td><td>0.587</td><td>0.568</td></tr><tr><td>TOTAL Δ=5</td><td>0.804</td><td>0.855</td><td>0.77</td><td>1.01</td><td>0.695</td></tr><tr><td>TOTAL Δ=21</td><td>1.29</td><td>1.39</td><td>0.997</td><td>1.59</td><td>0.885</td></tr><tr><td>XIN9I Δ=1</td><td>0.847</td><td>0.86</td><td>0.816</td><td>1.24</td><td>0.787</td></tr><tr><td>XIN9I Δ=5</td><td>0.943</td><td>0.945</td><td>0.86</td><td>1.54</td><td>0.862</td></tr><tr><td>XIN9I Δ=21</td><td>1.1</td><td>1.11</td><td>0.932</td><td>1.87</td><td>0.926</td></tr><tr><td>SHSZ300 Δ=1</td><td>0.841</td><td>0.853</td><td>0.806</td><td>1.17</td><td>0.767</td></tr><tr><td>SHSZ300 Δ=5</td><td>0.964</td><td>0.963</td><td>0.859</td><td>1.51</td><td>0.849</td></tr><tr><td>SHSZ300 Δ=21</td><td>1.06</td><td>1.07</td><td>0.918</td><td>1.96</td><td>0.903</td></tr><tr><td>BCOM Δ=1</td><td>0.823</td><td>0.82</td><td>0.776</td><td>1.28</td><td>0.824</td></tr><tr><td>BCOM Δ=5</td><td>0.884</td><td>0.851</td><td>0.818</td><td>1.52</td><td>0.812</td></tr><tr><td>BCOM Δ=21</td><td>0.967</td><td>0.961</td><td>0.86</td><td>1.95</td><td>0.834</td></tr><tr><td>INDU Δ=1</td><td>1.22</td><td>1.4</td><td>0.883</td><td>0.938</td><td>0.677</td></tr><tr><td>INDU Δ=5</td><td>1.04</td><td>1.05</td><td>1.35</td><td>1.22</td><td>0.779</td></tr><tr><td>INDU Δ=21</td><td>1.43</td><td>1.42</td><td>0.986</td><td>1.76</td><td>0.904</td></tr><tr><td>USDEUR Δ=1</td><td>0.711</td><td>0.73</td><td>0.663</td><td>0.945</td><td>0.673</td></tr><tr><td>USDEUR Δ=5</td><td>0.878</td><td>0.855</td><td>0.762</td><td>1.25</td><td>0.716</td></tr><tr><td>USDEUR Δ=21</td><td>1.05</td><td>1.03</td><td>0.816</td><td>1.46</td><td>0.823</td></tr><tr><td>IBOV Δ=1</td><td>0.899</td><td>1.04</td><td>0.687</td><td>0.894</td><td>0.686</td></tr><tr><td>IBOV Δ=5</td><td>1.05</td><td>0.896</td><td>0.908</td><td>1.11</td><td>0.782</td></tr><tr><td>IBOV Δ=21</td><td>1.74</td><td>1.33</td><td>1.06</td><td>1.75</td><td>0.932</td></tr><tr><td>MICROSOFT Δ=1</td><td>0.656</td><td>0.67</td><td>0.644</td><td>0.925</td><td>0.659</td></tr><tr><td>MICROSOFT Δ=5</td><td>0.926</td><td>0.894</td><td>0.91</td><td>1.29</td><td>0.746</td></tr><tr><td>MICROSOFT Δ=21</td><td>1.12</td><td>1.15</td><td>0.95</td><td>1.61</td><td>0.866</td></tr><tr><td>GOOGLE Δ=1</td><td>0.652</td><td>0.661</td><td>0.611</td><td>0.887</td><td>0.586</td></tr><tr><td>GOOGLE Δ=5</td><td>0.715</td><td>0.737</td><td>0.66</td><td>1.08</td><td>0.651</td></tr><tr><td>GOOGLE Δ=21</td><td>1.09</td><td>1.11</td><td>0.87</td><td>1.43</td><td>0.831</td></tr><tr><td>SP400 Δ=1</td><td>0.704</td><td>0.761</td><td>0.634</td><td>0.916</td><td>0.626</td></tr><tr><td>SP400 Δ=5</td><td>0.961</td><td>0.882</td><td>0.778</td><td>0.986</td><td>0.725</td></tr></tbody></table>
<table><caption>TABLEAU 5.7 – Ratio P for the AR, HAR and RFSV predictors for $\sigma_{t+\Delta}^2$</caption><tr><td>SP400 Δ = 21</td><td>1.27</td><td>1.19</td><td>0.882</td><td>1.56</td><td>0.866</td></tr></table>

## 5.6 Conclusion

We aimed through this analysis to investigate the scaling behavior of the volatility range-based proxy. Gather and co-authors have already studied high-frequency based realized-volatility in this sense in [89]. They found that it exhibits a rough scaling behavior and that its logarithm behaves like a fractional Brownian motion with Hurst exponent of order 0.14. We applied their analysis using the range-based volatilities as proxies for spot volatility. The latter use only information about range prices (open, close, high, low) for their estimation. We also find that the volatility process is monofractal with a small Hurst exponent (lower than 0.1) and can indeed be as low as 0.014.

Further tests justify that log-volatility increments are approximately Gaussian. This allows us to model them using the RFSV model. To ensure that the RFSV is consistent with the results on data, we simulate intraday and range prices using this model, recover the realized-volatility, and compute again the range-based estimators. We perform again the statistical checking only to find that simulations give similar results to the data. This reinforces that the hypothesis of rough volatility can not be rejected at this point, while other models like FSV seem to be misleading and, to a certain extent, wrong.

Finally, to measure its prediction power, we compare RFSV with other models such as AR or HAR for the log-volatility prediction and with AR, HAR and GARCH for the variance prediction. The RFSV shows good performance compared to other models and its prediction power is at least comparable to that of the HAR. GARCH, on the other hand, shows very weak performance for these estimators.
# Chapitre 6

## Rough volatility : evidence from option prices

**Abstract**— It has been recently shown that spot volatilities can be very well modeled by rough stochastic volatility type dynamics. In such models, the log-volatility follows a fractional Brownian motion with Hurst parameter smaller than 1/2. This result has been established using high frequency volatility estimations from historical price data. We revisit this finding by studying implied volatility based approximations of the spot volatility. Using at-the-money options on the S&P500 index with short maturity, we are able to confirm that volatility is rough. The Hurst parameter found here, of order 0.3, is slightly larger than that usually obtained from historical data. This is easily explained from a smoothing effect due to the remaining time to maturity of the considered options.

**Keywords** : Rough volatility; fractional Brownian motion; implied volatility; Medvedev-Scaillet approximation.

### 6.1 Introduction

Since the seminal work of Black and Scholes [30], the most classical way to model the behavior of the price $S_t$ of a financial asset is to use continuous semi-martingale dynamics of the form

$$d \log S_t = \mu_t dt + \sigma_t dW_t,$$

with $\mu_t$ a drift process and $W_t$ a Brownian motion. The coefficient $\sigma_t$ is referred to as the volatility process. As is well-known, it is the key ingredient in the model when one is interested in derivatives pricing and hedging.

Historically, following the pioneering approach of [30], practitioners have first considered the case where the process $\sigma_t$ is constant or deterministic, that is the Black and Scholes model. However, in the late eighties, it became clear that such specification for the volatility is inadequate. In particular, the Black and Scholes model is inconsistent with observed prices for liquid European options. Indeed the implied volatility, that is the volatility parameter that should be plugged into the Black-Scholes formula to retrieve a market option price, depends in practice on the strike and maturity of the considered option, whereas it is constant in the Black-Scholes framework.

Hence more sophisticated models have been introduced. A first possible extension, proposed by Dupire [69] and Derman and Kani [63], is to take $\sigma_t$ as a deterministic function of time and asset price. Such models, called local volatility models, enable us to perfectly reproduce a given implied volatility
surface. However, its dynamic is usually quite unrealistic under local volatility. Another approach is to consider the volatility $\sigma_t$ itself as an Ito process driven by an additional Brownian motion, typically correlated to W. Doing so one obtains less accurate static fits for the implied volatility surface but more suitable dynamics. Among the most famous of these stochastic volatility models are the Hull and White model [101], the Heston model [97] and the SABR model [95]. More recent market practice is to use so-called local-stochastic volatility models, see for example [24], which both fit the market exactly and generate reasonable dynamics.

In all the Brownian volatility models mentioned above, the smoothness of the sample path of the volatility is the same as that of a Brownian motion, namely $1/2 - \epsilon$ Hölder continuous, for any $\epsilon > 0$. However, it is shown in [88] that in practice, spot volatility is much rougher than this. This result in [88] is based on a statistical analysis of historical data using sophisticated high frequency estimation methods. More precisely, it is established in [88] that the dynamic of the log-volatility process is very close to that of a fractional Brownian motion with Hurst parameter $H \in (0, 1)$ is a Gaussian process with stationary increments such that

$$ \mathrm{Cov}[W_t^H, W_s^H] = \frac{1}{2} (|t|^{2H} + |s|^{2H} - |t-s|^{2H}). $$

The Hölder regularity of $W^H$ is $H-\epsilon$ for any $\epsilon > 0$ and for $H = 1/2$ we retrieve the classical Brownian motion. Therefore models where the volatility is driven by a fractional Brownian motion with $H < 1/2$ are called rough volatility models. Beyond fitting almost perfectly historical volatility time series, rough volatility models enable us to reproduce important stylized facts of liquid option prices that local or stochastic (or local-stochastic) volatility models typically fail to generate. In particular, the exploding term structure when maturity goes to zero of the at-the-money skew (the derivative of the implied volatility with respect to strike) is readily obtained, see [20, 79]. Other developments about rough volatility models can be found in [22, 23, 70, 71, 74, 80, 94, 105, 134].

The goal of this paper is to revisit the finding in [88] using implied volatility data. Indeed in [88], the authors work with historical price data from underlyings to estimate spot volatility. Here we use a spot volatility proxy which is not based on historical data, but on implied volatility. More precisely, we approximate the spot volatility by the implied volatility of an at-the-money liquid option with short maturity (or a refined version of it). This idea can be justified by the fact that in most models, the at-the-money implied volatility tends to the spot volatility as maturity goes to zero, see for example [132]. Our main result is a confirmation of that in [88] : When using alternate spot volatility measurement methods based on option prices, we can still conclude that volatility is rough.

The paper is organized as follows. We investigate in Section 6.2 the roughness of time series of spot volatility approximations given by implied volatilities of at-the-money options on the S&P500 index, with maturity one month. In Section 6.3, instead of using raw implied volatilities, we compute spot volatilities from implied ones through a correction formula due to Medvedev and Scaillet, see [124]. We then carry the same analysis as in Section 6.2. The results in Sections 6.2 and 6.3 are very similar to those in [88]. However, the estimated values for the Hurst parameter, although smaller than 1/2, are actually larger than those obtained in [88]. We show numerically and analytically in Section 6.4 that this upward bias comes from a regularizing effect due to the remaining time to maturity of the considered options.
## 6.2 At-the-money implied volatility with short maturity as spot volatility proxy

As explained in the introduction, our goal is to study the behavior of the spot volatility and to show that it is well approximated by a rough process. Of course this is a difficult task since volatility is a latent, unobserved variable. In [88], the authors use recent estimation methods based on ultra high frequency price data to estimate spot volatility. In this work, instead of using historical data as in [88], we wish to use option price data. This idea is reasonable if we use at-the-money options for which the time to maturity is short. Indeed, it is well-known that in most models, the at-the-money implied volatility converges to the spot volatility as maturity goes to zero, see for example [132].

### 6.2.1 Data description

In this section, we use a data set from Bloomberg<fnref n="1" />, made of daily observations of the implied volatility of the option with maturity one month on the S&P500 index, from January 5, 2006 to May 5, 2011². Note that the data are in fact already interpolated internally by the data provider (using quoted options at 4 PM) and do not necessarily exactly correspond to transaction data, see [32]. In Section 6.3, we present a method enabling us to derive spot volatilities from observed option prices with various maturities. Here we rely on the data provider approach to get option prices with the same maturity. This is not an issue since our aim in this work is to show that a rough dynamic for the volatility is obtained from any reasonable spot volatility proxy.

### 6.2.2 Scaling property

#### Reminder about the statistical methodology

We first recall the strategy used in [88] to investigate the smoothness of volatility sample paths. There is no novelty in term of statistical device here since we take the very same approach as in [88], but based on implied volatilities. Our contribution in this section is on the empirical side.

Let $\sigma_{t_0}^{imp}, \dots, \sigma_{t_N}^{imp}$ be the time series of implied volatilities extracted from our data base. Here for $i \ge 0$, $t_{i+1} - t_i$ corresponds to one business day. In the spirit of [88], we wish to review the behavior of the so-called structure function $m(q, \Delta)$ given by

$$m(q, \Delta) = \frac{1}{N} \sum_{k=0}^{\lfloor(N-1)/\Delta\rfloor} \left| \log(\sigma_{t_{(k+1)}\Delta}^{imp}) - \log(\sigma_{t_k\Delta}^{imp}) \right|^q$$

for various $q > 0$ and lags $\Delta$ going from 1 to about 40 days<fnref n="3" />. Through the quantity $m(q, \Delta)$, our goal is to revisit the finding in [88] that the (spot) log-volatility is well approximated by a fractional Brownian motion with Hurst parameter H smaller than 1/2. In this case, assuming spot and implied volatilities coincide, we should observe the following relationship :

$$m(q, \Delta) \sim c_q \Delta^{qH}, \qquad (6.1)$$

with $c_q$ a constant depending on $q$. Indeed, we have for $t \ge 0$ and $\Delta > 0$

$$\mathbb{E}[|W_{t+\Delta}^{H} - W_{t}^{H}|^q] = \bar{c}_q \Delta^{qH},$$

<footnote n="1">1. Data obtained from AXA Group Risk Management.</footnote>

<footnote n="2">2. Data around the third Friday of each month (settlement date) are removed from the data base. We have 1166 points in total.</footnote>

<footnote n="3">3. Of course when computing $m(q, \Delta)$ we in fact also average over the possible starting points $t_0, \dots, t_{\Delta-1}$.</footnote>
with $\tilde{c}_q$ the absolute moment of order q of a standard Gaussian random variable.

## Results

To investigate the validity of (6.1), we plot in Figure 6.1 the logarithm of $m(q, \Delta)$ against the logarithm of $\Delta$, for several values of $q$.

<figure><img src="image_71.png" /><figcaption>FIGURE 6.1 – Scaling property of log-volatility increments.</figcaption></figure>

For every $q$, the points with coordinates $(\log(\Delta), \log(m(q, \Delta)))$ are almost perfectly on the same line, and this for a wide range of $\Delta$. Figure 6.1 is actually very similar to that obtained from historical volatility measurements in [88]. Thus we can deduce that indeed, for a given $q$,

$$m(q, \Delta) \sim c_q \Delta^{\zeta(q)},$$

for some $\zeta(q)$.

Now we want to check whether $\zeta(q)$ can be taken of the form $qH$ for some $H$, as suggested in [88]. This would lead to the same monofractal scaling as that of the fractional Brownian motion with Hurst parameter $H$. To answer this, we plot in Figure 6.2 the points with coordinates $(q, \zeta(q))$, where $\zeta(q)$ is taken as the slope of the line in Figure 6.1 corresponding to the power $q$, and the points with coordinates $(q, 0.32q)^4$.

<footnote n="4">4. The value 0.32 is simply obtained from a standard linear fit of the points given by the slopes of the lines in Figure 6.1</footnote>
<figure><img src="image_72.png" /><figcaption>FIGURE 6.2 – Monofractal scaling.</figcaption></figure>

We see that the two graphs on Figure 6.2 can hardly be distinguished. This means that (6.1) almost
perfectly holds, with H around 0.32. Note that such value for H corresponds to rough volatility since
it is smaller than 1/2. However, it is larger than those reported in [88]. This is actually due to the fact
that our options have a significant remaining time to maturity of one month. This induces a smoo-
thing phenomenon in the estimation of the Hurst parameter. This effect is of the same nature as that
described and explained in [88] caused by the discrepancy between spot and integrated volatility over
a short time interval. We quantify this measurement bias numerically and analytically in Section 6.4.

**6.2.3 Distribution of log-volatility increments**

Recall that it is suggested in [88] that the log-volatility process is well modeled by a fractional Brownian motion with Hurst parameter smaller than 1/2. This implies monofractal scaling as investigated above but also a Gaussian behavior of the log-volatility increments. This feature is indeed satisfied when using historical estimates as measurements for spot volatility, see [88]. Here we wish to study whether such property also holds when the volatility proxies are given by our short term at-the-money implied volatilities. To this end, we display in Figure 6.3 histograms of log-volatility increments over different time intervals, together with a Gaussian density fit and the Gaussian density associated to the increments of a fractional Brownian motion with Hurst parameter equal to 0.32.

<figure><img src="image_73.png" /><figcaption>FIGURE 6.3 – Distribution of the log-volatility increments when using implied volatility as spot volatility proxy. The Gaussian fit is in blue and the density associated to the increments of a fractional Brownian motion with Hurst parameter equal to 0.32 is in red.</figcaption></figure>

From these graphs, we obtain that empirical distributions of log-volatility increments are reasonably
approximated by Gaussian laws. However, we can remark that the empirical distributions are slightly more concentrated around their center. Finally, the Gaussian fits almost exactly coincide with those associated to the fractional Brownian motion with Hurst parameter equal to 0.32. Note that this would of course probably no longer be true if considering higher frequencies than the daily scale.

In conclusion, using at-the-money implied volatilities with maturity one month as spot volatility proxies, we obtain that log-volatility is well approximated by a rough fractional Brownian motion. This confirms the finding in [88].

## 6.3 A refined implied volatility based proxy for the spot volatility

In this section, we wish to study the robustness of the results obtained in Section 6.2. To do so, we work with another spot volatility proxy based on at-the-money options with short maturity. More precisely, we use the approximation formula from Medvedev and Scaillet, see [124]. This correction formula enables us to compute a spot volatility proxy from an at-the-money implied volatility with any (short) maturity. This is an advantage compared to what is done in Section 6.2 where only options with one month maturity are considered<fnref n="5" />. The drawback of Medvedev-Scaillet formula is that it is proved to be valid only within a restricted class of stochastic volatility models, which does not include rough volatility models. However our goal here is to see whether a proxy obtained from a Brownian volatility model still exhibits a rough behavior.

### 6.3.1 Data description and processing

Here our data set is provided by OptionMetrics and consists in daily close bid/ask prices of European puts and calls on the S&P500 index, from September 5, 2001 to January 31, 2012, for various strikes and maturities, together with the daily traded volumes. We discard options with price less than 2.5 cents of dollars or with zero trading volume. Besides, as in Section 6.2, prices corresponding to settlement dates are removed, so as obvious outliers.

We then want to compute implied volatilities from put and call prices. Thus we have to invert (evently) the Black-Scholes formula. Therefore we need to fix for any time to maturity $τ$ an underlying forward price $F(τ)$ and a zero coupon bond price $D(τ)$. To do so, we use the following classical approach based on put-call parity. The values of $F(τ)$ and $D(τ)$ are taken as solutions of the minimization problem

$$ \arg\min_{D,F} \left\{ \sum_i w_i \left( \frac{1}{2}(C_i^a - P_i^b) + \frac{1}{2}(C_i^b - P_i^a) - D(\tau)(F(\tau) - K_i) \right) \right\}, $$

where $C_i^{a,b}$ and $P_i^{a,b}$ are respectively the call and put market prices (a standing for ask, b for bid) quoted at strike level $K_i$. The weights $w_i$ are given by

$$ w_i = \frac{\sqrt{\min\{V_i^C, V_i^P\}}}{\frac{1}{2}(C_i^a - C_i^b) + \frac{1}{2}(P_i^a - P_i^b)}, $$

with $V_i^C$ and $V_i^P$ the trading volumes of call and put options at strike $K_i$. Finally, our implied volatility is taken as that of a call whose price would be the midprice between the bid and ask prices.

Recall that for our approximations to be valid, we focus on at-the-money implied volatilities with short maturity. Following [124], we only select implied volatilities of options with time to maturity ranging

<footnote n="5">5. Mixing various maturities without any correction would have been very arguable.</footnote>
from 15 to 60 days. Shorter term options are discarded because quotes can be noisy. Moreover, we restrict our data to log forward moneyness belonging to the interval [-0.03, 0.03]. Such procedure yields a total number of 34842 implied volatilities over 2569 days.

### 6.3.2 The Medvedev-Scaillet correction formula

In [124], the authors consider a general modeling framework encompassing most of the classical parametric price models. They use a two factors jump-diffusion stochastic volatility model of the form

$$ \begin{cases} dS_t = (r - \mu(\sigma_t))S_t dt + \sigma_t S_t dZ_t + S_t dJ_t \\ d\sigma_t = a(\sigma_t) dt + b(\sigma_t)(\rho dZ_t + \sqrt{1-\rho^2} dW_t), \end{cases} \qquad (6.2) $$

where $Z_t$ and $W_t$ are two independent Brownian motions and $J_t$ is a Poisson-type jump process, independent of $Z_t$ and $W_t$. Both $r$ and the correlation coefficient $\rho$ are assumed to be constant. The expected jump size $\mathbb{E}[\Delta J]$ is also constant, but the jump intensity $\lambda(\sigma_t)$ may depend on the volatility in a deterministic way. Here, as in the numerical experiments in [124], we consider the following parametric forms:

$$ b(\sigma_t) = \beta\sigma_t^{\phi}, \quad \lambda(\sigma_t) = \lambda_0\sigma_t^{\psi}, $$

for some non-negative constants $\beta, \phi, \lambda_0$ and $\psi$.

Let $\sigma$ be the spot volatility and $\hat{\sigma} = \hat{\sigma}(\tau)$ be the at-the-money implied volatility of an option with time to maturity $\tau$. Following [124], we build up our option-based spot volatility proxy in two steps. First, the chosen model is calibrated from the approximation formula in Proposition 7 in [124] using all our option prices over the entire time period. To retrieve the proxy for the spot volatility, we then consider the following expansion as $\tau$ goes to zero shown in [124]:

$$ \sigma = \hat{\sigma} - I_1(0, \hat{\sigma})\sqrt{\tau} + \left( I_1(0, \hat{\sigma}) \frac{\partial I_1(0, \hat{\sigma})}{\partial \sigma} - I_2(0, \hat{\sigma}) + \frac{1}{2} \rho b(\hat{\sigma}) \mathbb{E}[\Delta J] \frac{\partial \lambda(\hat{\sigma})}{\partial \sigma} \right) \tau + O(\tau \sqrt{\tau}). \quad (6.3) $$

The functions $I_1$ and $I_2$ are explicitly defined in [124] and depend only on $\beta, \rho, \phi, \lambda_0, \psi$ and $\mathbb{E}[\Delta J]$.

### 6.3.3 The scaling property revisited

We now wish to study the scaling property of spot volatility proxies based on the approximation formula (6.3). We consider two cases: The Heston case, where $\phi = 0$ and $\lambda_0 = 0$, and the general case, where all the parameters are calibrated. The calibration results are given in Table 6.1.

<table><caption>TABLEAU 6.1 – Parameters calibrated on quoted S&P500 option prices, from September 5, 2001 to January 31, 2012.</caption><thead><tr><th>PARAMETER</th><th>HESTON</th><th>GENERAL CASE</th></tr></thead><tbody><tr><td>βρ</td><td>-0.18 (0.00)</td><td>-3.27 (0.08)</td></tr><tr><td>ρ</td><td>-0.48 (0.00)</td><td>-0.39 (0.00)</td></tr><tr><td>φ</td><td>0</td><td>1.79 (0.02)</td></tr><tr><td>λ<sub>0</sub>E(ΔJ)</td><td>0</td><td>-0.6924 (0.03)</td></tr><tr><td>E(ΔJ)</td><td>--</td><td>-0.17 (0.00)</td></tr><tr><td>ψ</td><td>--</td><td>1.11 (0.01)</td></tr></tbody></table>
Once the parameters are obtained, we can implement Equation (6.3) to compute everyday a spot volatility proxy. Note that in Equation (6.3), we take for $\hat{\sigma}$ the implied volatility with shortest time to maturity. Then we conduct the same analysis as in Section 6.2.2. The results are given in Figure 6.4 for the Heston model and Figure 6.5 for the general case (notations are the same as in Section 6.2.2).

FIGURE 6.4 – Scaling property of log-volatility increments when based on Heston proxy. In the second graph H is taken equal to 0.33.

<figure><img src="image_74.png" /></figure>

FIGURE 6.5 – Scaling property of log-volatility increments when based on the general case proxy. In the second graph H is taken equal to 0.34.

<figure><img src="image_75.png" /></figure>

The results are very similar to those in Section 6.2.2. Here again we can confirm the fact that volatility is rough. This is even obtained although in the models in which the proxies are computed, volatility is of Brownian type and therefore not rough.

### 6.3.4 A control experiment

We have shown in the previous sections that when using market implied volatilities (directly or through Medvedev-Scaillet's formula in a Heston framework), we deduce that volatility is rough. However, one may wonder whether this effect is not just mechanical when considering implied volatilities. In other words, do implied volatility based estimators of the smoothness of volatility sample paths always lead to the conclusion that volatility is rough. This would of course question the validity of this finding.

To investigate this point, we consider the following Heston model for the stock price $S_t$ and instantaneous variance $v_t$:

$$dS_t = rS_t dt + \sqrt{v_t} S_t dZ_t \\ dv_t = \kappa(\theta - v_t) dt + \sigma_v \sqrt{v_t} dW_t,$$
where Z and W are two Brownian motions with constant correlation ρ. We use the arbitrary set of pa-
rameters κ = 6.4, r = 0.04, v₀ = θ = 0.25², σV = 0.5 and ρ = -0.53, and simulate the path of the stock price
and variance over the time interval [0, T], where T corresponds to 2520 days.

We first consider the scaling property of the (true) spot log-volatility increments in Figure 6.6.

<figure><img src="image_76.png" /><figcaption>FIGURE 6.6 – The scaling property of log-volatility increments for a simulated Heston model. In the second graph H is taken equal to 0.48, estimation based on the five smaller lags for each $q$.</figcaption></figure>

Contrary to what is observed on data, we observe two different behaviors for the empirical moments
in the Heston case. For small lags, a fractional Brownian motion type scaling is observed whereas for
large lags, stationarity kicks in. Still, estimation of the smoothness of volatility sample paths based on
the first lags leads to a value of H around 0.48 which is close to 0.5 the theoretical one in the Brownian
volatility case.

We now consider implied volatilities. Our goal is to see whether our methodology provides good esti-
mates of the smoothness of volatility sample paths or it spuriously leads to the conclusion that volati-
lity is rough in this Brownian volatility case. To do so, we build implied volatility time series for times
to maturity from 1 to 20 days (using the Fourier pricing approach to Heston model, see [47]), for each
day in the interval [0, T]. We then compute the scaling parameter for each time to maturity in the same
way as previously. The results are given in Figure 6.7.

<figure><img src="image_77.png" /><figcaption>FIGURE 6.7 – Estimated values of the Hurst parameter using implied volatilities as a function of time to maturity (in days).</figcaption></figure>

We see that the estimated smoothness is always around 0.5 (slightly below, notably for smaller times
to maturity). Thus, in the context of a Brownian volatility model, based on our methodology, one concludes that the smoothness of volatility sample paths is around 0.5. This means that the result obtained in this work that volatility is rough is not just a mechanical effect due to our statistical device or the nature of implied volatilities.

## 6.4 On the upward bias when estimating the Hurst parameter

We explain in this section why using implied volatility measures as spot volatility proxies induces an upward bias in the estimation of the Hurst parameter. We start with a numerical investigation of this phenomenon.

### 6.4.1 Monte Carlo study

To understand the extend of the bias when estimating the Hurst parameter, we simulate option prices in a rough volatility model. Then we compute the Hurst parameter based on these simulated data. Let $T > 0$. We consider the following model without leverage effect over the time interval $[0, T]$:

$$d \log S_t = \sigma_t dZ_t, \quad d \log \sigma_t = \eta dW_t^H.$$

Here $Z_t$ is a Brownian motion, $W_t^H$ a fractional Brownian motion independent of $Z_t$ and $\eta > 0$.

#### Simulation of fractional Brownian motion

We consider a time interval $[0, T]$ and fix an equidistant partition $0 = t_0 < t_1 < \dots < t_n = T$. We first wish to simulate $(W_{t_1}^H, \dots, W_{t_n}^H)$. For $i, j \in \{1, \dots, n\}$, we have

$$\mathbb{E}[W_{t_i}^H W_{t_j}^H] = \frac{1}{2} (t_i^{2H} + t_j^{2H} - |t_i - t_j|^{2H}).$$

Then we can use the Cholesky decomposition of the covariance matrix $\Sigma$ of $(W_{t_1}^H, \dots, W_{t_n}^H) : \Sigma = LL^T$, where $L = (l_{ij})_{i,j \in \{1, \dots, n\}}$ is lower-triangular. Thus simulating a sample path of the fractional Brownian motion at times $(t_i)$ can be done generating a vector $X = (X_1, \dots, X_n)$ of independent standard Gaussian random variables and setting $(W_{t_1}^H, \dots, W_{t_n}^H) = LX$, see for example [65] for details.

#### Simulating option prices under rough volatility

We place ourselves at time $t_i > 0$ and assume past spot volatilities and prices have been observed at times $t_1, \dots, t_i$. We want to compute the price at time $t_i$ of an option with expiration date $t_k = t_i + \tau$ for some $\tau > 0$. The procedure goes as follows:

- We generate M paths of the volatility process on the interval $[t_{i+1}, t_k]$. This is done simulating $(W_{t_j}^H)_{t_{i+1} \le t_j \le t_k}$ conditional on past information, that is the filtration generated by $(X_{t_1}, \dots, X_{t_i})$. Using the lower triangular form of L, these new values for the fractional Brownian motion at times $t_{i+1} \le t_j \le t_k$ can be obtained writing

$$W_{t_j}^H = \sum_{p=1}^{i} l_{jp} X_p + \sum_{p=i+1}^{j} l_{jp} X_p.$$

The $i$ first variables $X_p$ are those used to simulate the fractional Brownian motion up to time $t_i$, whereas $(X_{i+1}, \dots, X_j)$ is a sample of independent standard Gaussian random variables, independent from past values. Taking the exponential, we get our spot volatility sample path. We write $\sigma^m$ for the $m$-th volatility trajectory.
— The price at time $t_i$ of an at-the-money option with time to maturity $\tau$ is obtained computing

$$ \frac{1}{M} \sum_{m=1}^{M} C_{BS} \left( S_{t_i}, \tau, \sqrt{\frac{1}{\tau} \sum_{p=i+1}^{k} (\sigma_{t_p}^m)^2} \right), $$

where $C_{BS}(S_{t_i}, \tau, \sigma)$ is the price of an at-the-money option with time to maturity $\tau$ in a Black-Scholes model with volatility $\sigma$, zero interest rate, and underlying value $S_{t_i}$.

— Eventually we invert Black-Scholes formula to obtain the implied volatility.

## Results

We consider the following set of parameters: $H = 0.04$, $\eta = 1.0$ and $T = 1000$ days. Such parameters are consistent with [20, 88]. We take $\tau \in \{1, \dots, 20\}$ days and run $M = 10^4$ simulations. Figure 6.8 displays the sample path of the spot volatility together with those of the implied volatilities associated to 5 and 20 days.

<figure><img src="image_78.png" /><figcaption>FIGURE 6.8 – Sample paths of spot volatility and implied volatilities for $\tau = 5$ and $\tau = 20$.</figcaption></figure>

At the visual level, it is already clear that implied volatility trajectories are not as rough as that of the spot volatility. Furthermore, the longer the time to maturity, the larger the smoothing effect.

As in Sections 6.2 and 6.3, we now consider Equation (6.1). Based on our simulation, for several values of $q$, we plot in Figure 6.9 the logarithm of $m(q, \Delta)$ against the logarithm of $\Delta$. This is done in two cases: when $m$ is obtained from spot volatility values and when $m$ is derived from implied volatility values, with $\tau = 5$ days.
<figure><img src="image_79.png" /><figcaption>FIGURE 6.9 – Scaling property of log-volatility increments : spot volatility and implied volatility with $τ = 5$.</figcaption></figure>

We see that for a given $q$, when $m(q, \Delta)$ is computed from implied volatilities, the points with coordinates $(\log(\Delta), \log(m(q, \Delta)))$ remain on the same line. However, the slope of this line is larger than that obtained when $m(q, \Delta)$ is computed from spot volatilities (which provides the true underlying $H$ up to small statistical error). Hence there is indeed a smoothing effect due to the remaining time to maturity of the considered options.

Finally, we give in Figure 6.10 the estimated values of $H$ when using implied volatilities from the simulation, for different times to maturity.

<figure><img src="image_80.png" /><figcaption>FIGURE 6.10 – Estimated values of the Hurst parameter using implied volatilities as a function of time to maturity (in days).</figcaption></figure>

Under our simulation framework, we see that using options with maturity 1 day, we obtain a quite accurate value for $H$ of 0.06, while the true parameter is equal to 0.04. Taking longer maturities leads to an increasing bias. With 20 days maturity, one gets an estimated Hurst parameter of about 0.27. These results are in line with those in Sections 6.2 and 6.3.
6.4.2 Analytical illustration of the upward bias

In the spirit of Appendix C in [88], we finally want to provide a more quantitative understanding of
the observed upward bias when estimating the Hurst parameter from implied volatilities. To do so, we
consider a very crude approximation. Indeed we suppose that the at-the-money implied variance at
time t of an option with time to maturity τ > 0, denoted by $\hat{v}^T(t)$, is given by

$$
\hat{v}^{\tau}(t) = \frac{1}{\tau} \int_{t}^{t+\tau} \mathbb{E}_{t}[v_{u}] du,
$$

where $v_u$ is the spot variance at time $u$ and $\mathbb{E}_t[\cdot]$ the conditional expectation operator with respect to information up to time $t$. Furthermore, we take a simplified rough volatility model assuming that for $u > 0$,

$$
v_u = v_0 + \nu W_u^H,
$$

for some $v_0 > 0$ and $\nu > 0$. These approximations are actually probably enough to shed light on the
bias phenomenon. Indeed it is due to the effects of the conditional expectation and integral operators
appearing in the implied volatility.

In this simplified setting, our goal is to illustrate the smoothing effect leading to the upward bias. To
do so, we compute a quantity very related to $m(2, \Delta)$, namely

$$
\hat{m}^{\tau}(2, \Delta) = \mathbb{E}[(\hat{v}^{\tau}(\Delta) - \hat{v}^{\tau}(0))^2].
$$

Indeed, under our assumptions, if the implied volatility were equal to the spot one, this quantity would be proportional to $\Delta^{2H}$. However, we now show that because of the use of implied volatility in $\hat{m}(2, \Delta)$, this relationship no longer holds, particularly for large $\tau/\Delta$.

We recall the Mandelbrot and Van Ness representation of fractional Brownian motion :

$$
W_t^H = c_H \left( \int_0^t (t-s)^{H-1/2} dW_s + \int_{-\infty}^0 ((t-s)^{H-1/2} - (-s)^{H-1/2}) dW_s \right),
$$

where $W_t$ is a two-sided Brownian motion and $c_H$ is so that the variance of $W_1^H$ is equal to 1. We easily have

$$
\hat{v}^{\tau}(\Delta) = v_0 + \frac{\nu}{\tau} c_H \int_0^{\tau} \int_{-\infty}^{0} ((\Delta + u - s)^{H-1/2} - (-s)^{H-1/2}) dW_s du \\
+ \frac{\nu}{\tau} c_H \int_0^{\tau} \int_{0}^{\Delta} (\Delta + u - s)^{H-1/2} dW_s du.
$$

Using stochastic Fubini theorem, this gives

$$
\hat{v}^{\tau}(\Delta) - \hat{v}^{\tau}(0) = \frac{\nu}{\tau} c_H \int_{-\infty}^{0} \int_{0}^{\tau} ((\Delta + u - s)^{H-1/2} - (u-s)^{H-1/2}) du \, dW_s \\
+ \frac{\nu}{\tau} c_H \int_{0}^{\Delta} \int_{0}^{\tau} (\Delta + u - s)^{H-1/2} du \, dW_s.
$$

Hence we easily deduce from Ito isometry that

$$
\hat{m}^{\tau}(2, \Delta) = A(h_1(\Delta, \tau) + h_2(\Delta, \tau)),
$$
with

$$A = \frac{c_H^2 v^2}{(H + 1/2)^2},$$

$$h_1(\Delta, \tau) = \frac{1}{\tau^2} \int_{-\infty}^{0} ((\Delta + \tau - s)^{H+1/2} - (\Delta - s)^{H+1/2} - (\tau - s)^{H+1/2} + (-s)^{H+1/2})^2 ds,$$

$$h_2(\Delta, \tau) = \frac{1}{\tau^2} \int_{0}^{\Delta} ((\Delta + \tau - s)^{H+1/2} - (\Delta - s)^{H+1/2})^2 ds.$$

We write $h_1(\Delta, \tau)$ under the form

$$\frac{1}{\tau^2} \Delta^{2H+2} \int_{-\infty}^{0} \left( (1 + \frac{\tau}{\Delta} - s)^{H+1/2} - (1 - s)^{H+1/2} - \left(\frac{\tau}{\Delta} - s\right)^{H+1/2} + (-s)^{H+1/2} \right)^2 ds.$$

Setting $\theta = \tau/\Delta$, we obtain

$$h_1(\Delta, \tau) = \Delta^{2H} f_1(\theta),$$

where

$$f_1(\theta) = \frac{1}{\theta^2} \int_{-\infty}^{0} \left( (1 + \theta - s)^{H+1/2} - (1 - s)^{H+1/2} - (\theta - s)^{H+1/2} + (-s)^{H+1/2} \right)^2 ds.$$

Similarly, we have

$$h_2(\Delta, \tau) = \Delta^{2H} f_2(\theta),$$

where

$$f_2(\theta) = \frac{1}{\theta^2} \int_{0}^{1} \left( (1 + \theta - s)^{H+1/2} - (1 - s)^{H+1/2} \right)^2 ds.$$

So

$$\hat{m}^{\tau}(2, \Delta) = A \Delta^{2H} (f_1(\theta) + f_2(\theta)).$$

Now remark that

$$\lim_{\theta \to 0} f_1(\theta) = (H + 1/2)^2 \int_{-\infty}^{0} \left( (1 - s)^{H-1/2} - (-s)^{H-1/2} \right)^2 ds$$

and

$$\lim_{\theta \to 0} f_2(\theta) = (H + 1/2)^2 \int_{0}^{1} (1 - s)^{2H-1} ds.$$

Consequently,

$$\lim_{\theta \to 0} (f_1(\theta) + f_2(\theta)) = (H + 1/2)^2 \frac{1}{c_H^2}.$$

Thus, when $\theta$ is small,

$$\hat{m}^{\tau}(2, \Delta) \sim v^2 \Delta^{2H}.$$

This means that the same scaling relationship as that associated to the spot volatility is approximately satisfied when considering implied volatilities with small enough times to maturity. Otherwise, one should add the multiplicative factor

$$f(\theta) = \frac{c_H^2}{(H + 1/2)^2} (f_1(\theta) + f_2(\theta))$$

on the right hand side of the above relationship. This disrupts the scaling property and implies biased estimations for the Hurst parameter. We draw in Figure 6.11 the graph of the function $f$ for $H = 0.04$.
<figure><img src="image_81.png" /><figcaption>FIGURE 6.11 – The function $f$ for $H = 0.04$.</figcaption></figure>

For fixed $\tau$ (as in Section 6.2), the function $f$ is increasing with $\Delta$. Therefore, when doing a regression analysis of the cloud of points with coordinates $(\log(\Delta), \log(\hat{m}^T(2, \Delta)))$, this implies an upward bias in the estimation of $H$ due to a higher slope.

## 6.5 Conclusion

Using implied volatility data to approach spot volatility, we were able to confirm that volatility is rough. First using the one month at-the-money implied volatility on the S&P500 index as a volatility proxy. Then through computing spot volatilities from implied ones using a correction formula given by Medvedev and Scaillet. Following [88] we uncovered the monofractal scaling with a Hurst exponent $H$ of order 0.32. This value corresponds to rough paths, but is larger than the one obtained in [88]. We also found that the distribution of the increments of log-volatility is close to Gaussian.

Given the value of $H$, we conducted a numerical analysis which consisted in estimating implied volatilities with different time to maturities, for a simulated rough volatility path with fixed $H$. We then estimated and compared the Hurst exponent for each time series. We found that the longer the time to maturity, the larger the smoothing effect. This upwards bias comes from a regularizing effect due to the remaining time to maturity. Using a crude but reasonable approximation, we were able to confirm this bias analytically.
Bibliographie

[1] F. Abergel and G. Loeper. Pricing and hedging contingent claims with liquidity costs and market impact. Available at SSRN: http://ssrn.com/abstract=2239498 or http://dx.doi.org/10.2139/ssrn.2239498, April 2013. 98

[2] P. Abry and F. Sellan. The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Mayer: Remarks and fast implementation. *Applied Computational Harmonic Analysis*, 3:377-383, 1996. 141

[3] M. Albizzati and H. Geman. Interest rate risk management and valuation of the surrender option in life insurance policies. *The Journal of Risk and Insurance*, 61(4):616-637, 1994. 31, 32

[4] S. Alizadeh, M. W. Brandt, and F. X. Diebold. Range-based estimation of stochastic volatility models. *Journal of Finance*, 57(3):1047-1091, 2002. 131

[5] R. Almgren. Optimal execution with nonlinear impact functions and trading-enhanced risk. *Applied Mathematical Finance*, 10:1-18, 2003. 99

[6] R. Almgren. Optimal trading with stochastic liquidity and volatility. *SIAM Journal on Financial Mathematics*, 3:163-181, 2012. 100, 110

[7] R. Almgren and N. Chriss. Optimal execution of portfolio transactions. *Journal of Risk*, 3:5-39, 2000. 15, 97, 99, 100, 110

[8] R. Almgren, C. Thum, E. Hauptmann, and H. Li. Direct estimation of equity market impact. *Risk*, 18(7):58-62, May 2005. Working paper. 99

[9] L. Andersen and V. Piterbarg. Interest rate modeling. *Atlantic Financial Press*, 2010. 82

[10] T. G. Andersen, T. Bollerslev, F. X. Diebold, and H. Ebens. The distribution of realized stock return volatility. *Journal of Financial Economics*, 61:43-76, 2001. 139

[11] A. R. Bacinello, P. Millossovich, A. Olivieri, and E. Pitacco. Variable annuities: A unifying valuation approach. *Insurance: Mathematics and Economics*, 49(3):285-297, 2011. 50

[12] E. Bacry and J. F. Muzy. Log-infinitely divisible multifractal processes. *Communications in Mathematical Physics*, 236(3):449-475, 2003. 149

[13] T. G. Bali and D. Weinbaum. A comparative study of alternative extreme-value volatility estimators. *Journal of Futures Markets*, 25(9):873-892, 2005. 132

[14] G. Barles, C. Daher, and M. Romano. Convergence of numerical schemes for parabolic equations arising in finance theory. *Mathematical Models and Methods in Applied Sciences*, 5(1):125-143, 1995. 39
[15] G. Barone-Adesi and R. Whaley. Efficient analytic approximation of American option value. *Journal of Finance*, XLII(2), 1987. 35, 36, 38

[16] Barra. Market impact model handbook. Technical report, Barra Inc., 1997. 99

[17] D. S. Bates. Empirical option pricing : a retrospection. *Journal of Econometrics*, 116 :387-404, 2003. 98

[18] D. Bauer, A. Kling, and J. Russ. A universal pricing framework for Guaranteed Minimum Benefits in Variable Annuities. Working paper, 2006. 28

[19] D. Bauer, A. Kling, and J. Russ. A universal pricing framework for guaranteed minimum benefits in variable annuities. *Astin Bulletin*, 38(2):621-651, 2008. 50

[20] C. Bayer, P. Friz, and J. Gatheral. Pricing under rough volatility. *Quantitative Finance*, 16(6):887-904, 2016. 19, 154, 163

[21] S. Beckers. Variance of security price returns based on high, low and closing prices. *Journal of Business*, 1983. 131

[22] M. Bennedsen, A. Lunde, and M. S. Pakkanen. Hybrid scheme for Brownian semistationary processes. *arXiv preprint arXiv:1507.03004*, 2015. 19, 154

[23] M. Bennedsen, A. Lunde, and M. S. Pakkanen. Decoupling the short-and long-term behavior of stochastic volatility. *Available at SSRN* 2846756, 2016. 19, 154

[24] L. Bergomi. Local, stochastic, local/stochastic, volatility models-and non-models. 2016. 154

[25] C. Bernard and M. Kwak. Semi-static hedging of variable annuities. *Insurance: Mathematics and Economics*, 67:173-186, September 2014. 14

[26] P. Bertrand and J. Prigent. Portfolio Insurance: The extreme value approach to the CPPI method. *Finance*, 23:68-86, 2002. 77

[27] D. Bertsimas and A. W. Lo. Optimal control of execution costs. *Journal of Financial Markets*, 1:1-50, 1998. 16, 97, 109

[28] P. Bjerksund and G. Stensland. Closed form valuation of american options. Working paper, 2002. 35

[29] F. Black and R. Jones. Simplifying portfolio insurance. *Journal of Portfolio Management*, 14(1):48-51, 1987. 76, 77

[30] F. Black and M. Scholes. The pricing of options and corporate liabilities. *The Journal of Political Economy*, 81:637-654, 1973. 18, 153

[31] F. Black and M. Scholes. The pricing of options and corporate liabilities. *The Journal of Political Economy*, 81(3):637-654, 1973. 98

[32] Bloomberg. Introduction into the new Bloomberg implied volatility calculations. 2008. 155

[33] T. Bollerslev, D. Osterrieder, N. Sizova, and G. Tauchen. Risk and return : long-run relations, fractional cointegration, and return predictability. *Journal of Financial Economics*, 108:409-424, 2013. 130
[34] O. Bondarenko. Why are put options so expensive? *Quarterly Journal of Finance*, 4(3):1-50, April 2014. 98

[35] B. Bouchard, A. Kalife, X. Tan, and L. Wang. Understanding guaranteed minimum withdrawal benefit : a study on financial risks and rational lapse strategy. Working paper, 2010. 26, 33, 34

[36] B. Bouchard and N. Touzi. Weak dynamic programming principle for viscosity solutions. *SIAM Journal on Control and Optimization*, 49(3):948-962, 2011. 7

[37] B. Bouchard and X. Warin. Monte-Carlo valorisation of American options : facts and new algorithms to improve existing methods. preprint submitted to Elsevier, 2010. 27, 34, 39, 40

[38] P. Boyle and W. Tian. The design of equity-indexed annuities. *Insurance : Mathematics and Economics*, 43(3):303-315, 2008. 50

[39] P. E. Boyle and M. R. Hardy. Reserving for maturity guarantees. *Insurance : Mathematics and Economics*, 21:113-127, 1997. 3

[40] P. P. Boyle and M. R. Hardy. Reserving for maturity guarantees, institute for insurance and pensions research. University of Waterloo, 1996. 25

[41] M. Brennan and E. Schwartz. The pricing of equity-linked life insurance policies with an asset value guarantee. *Journal of Financial Economics*, 3:195-213, 1976. 76

[42] M. J. Brennan and E. S. Schwartz. The valuation of American put options. *The Journal of Finance*, 1977. 29

[43] D. Brigo and F. Mercurio. Interest rate models - theory and practice : with smile, inflation and credit. Springer Finance, 2006. 82

[44] X. Brokmann, E. Sérié, J. Kockelkoren, and J.-P. Bouchaud. Slow decay of impact in equity markets. Technical report, Capital Fund Management, 2014. 99

[45] J. Brun and L. Prigneaux. VaR approach for credit CPPI and counterparty risk. Quant Congress, 2007. 77

[46] D. Bunch and H. Johnson. The American put option and its critical stock price. Journal Finance WFA Meetings, 1999. 35

[47] P. Carr and D. Madan. Option valuation using the fast Fourier transform. *Journal of Computational Finance*, 2(4):61-73, 1999. 161

[48] L. Chan and J. Lakonishok. The behavior of stock prices around institutional trades. *Journal of Finance*, 50:1147-1174, 1995. 107

[49] Z. Chen and P. Forsyth. A semi-Lagrangian approach for natural gas storage valuation and optimal operation. *SIAM Journal on Scientific Computing*, 30(1):339-368, 2007. 100, 124

[50] Z. Chen, K. Vetzal, and P. A. Forsyth. The effect of modelling parameters on the value of gmwb guarantees. *Insurance : Mathematics and Economics*, 42(1):165-173, 2008. 50

[51] P. Cheridito, H. Kawaguchi, and M. Maejima. Fractional Ornstein-Uhlenbeck processes. *Electronic Journal of Probability*, 8(3):1-14, 2003. 130, 140

[52] P. Christoffersen, R. Goyenko, K. Jacobs, and M. Karoui. Illiquidity premia in the equity options market. Available at SSRN 1784868, 2014. 98
[53] T. F. Coleman, Y. Kim, Y. Li, and M. Patron. Robustly hedging variable annuities with guarantees under jump and volatility risks. *The Journal of Risk and Insurance*, 74(2):347-376, 2007. 14

[54] T. F. Coleman, Y. Li, and M.-C. Patron. Hedging guarantees in variable annuities under both equity and interest rate risks. *Insurance: Mathematics and Economics*, 38:215-228, 2006. 14

[55] F. Comte, L. Coutin, and E. Renault. Affine fractional stochastic volatility models. *Annals of Finance*, 8:337-378, 2012. 130

[56] F. Comte and E. Renault. Long memory in continuous-time stochastic volatility models. *Mathematican Finance*, 8(4):291-323, 1998. 20, 130, 141

[57] R. Cont and P. Tankov. Constant proportion portfolio insurance in the presence of jumps in asset prices. *Mathematical Finance*, 19(3):379-401, July 2009. 12, 77

[58] F. Corsi. A simple approximate long-memory model of realized volatility. *Journal of Financial Econometrics*, 7(2):174-196, 2009. 147, 148

[59] J. Crank and P. Nicolson. A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type. *Advances in Computational Mathematics*, 6(1):207-226, 1996. 39

[60] M. Dai, Y. Kuen Kwok, and J. Zong. Guaranteed minimum withdrawal benefit in variable annuities. *Mathematical Finance*, 18(4):595-611, 2008. 60, 62

[61] M. Dai, Y. K. Kwok, and J. Zong. Guaranteed minimum withdrawal benefit in variable annuities. *Mathematical Finance*, 18(4):595-611, 2007. 8, 50

[62] G. Deelstra and G. Rayée. Pricing variable annuity guarantees in a local volatility framework. *Insurance: Mathematics and Economics*, 53(3):650-663, 2013. 51

[63] E. Derman and I. Kani. Riding on a smile. *Risk*, 7(2):139-145, 1994. 18, 129, 153

[64] Y. D'Halluin, P. A. Forsyth, and G. Labahn. A semi-Lagrangian approach for American Asian options under jump diffusion. *SIAM Journal on Scientific Computing*, 27:315-345, 2005. 100

[65] T. Dieker. Simulation of fractional brownian motion. *MSc theses*, University of Twente, Amsterdam, The Netherlands, 2004. 162

[66] G. Dionne, editor. *Handbook of insurance*. Springer, 2000. 1

[67] J. Duchon, R. Robert, and V. Vargas. Forecasting volatility with the multifractal random walk model. *Mathematical Finance*, 1:83-108, 2012. 149

[68] D. J. Duffy. A critique of the Crank-Nicolson scheme, strengths and weakness for financial instrument pricing. *Wilmott*, 4:68-76, 2004. 62

[69] B. Dupire. Pricing with a smile. *Risk*, 7(1):18-20, 1994. 18, 129, 153

[70] O. El Euch, M. Fukasawa, and M. Rosenbaum. The microstructural foundations of leverage effect and rough volatility. *arXiv preprint arXiv:1609.05177*, 2016. 19, 154

[71] O. El Euch and M. Rosenbaum. The characteristic function of rough Heston models. *Mathematical Finance*, to appear, 2016. 19, 154
[72] R. F. Engle and A. J. Patton. What good is a volatility? *Quantitative Finance*, **1**:237-245, 2001.
130

[73] W. Feller. The asymptotic distribution of the range of sums of random variables. *Annals of Mathematical Statistics*, **3**(22):427-432, 1951. 131

[74] M. Forde and H. Zhang. Asymptotics for rough stochastic volatility and Lévy models. *Preprint*, 2015. 19, 154

[75] P. A. Forsyth. A Hamilton Jacobi Bellman approach to optimal trade execution. *Applied Nume-rical Mathematics*, **61**(2):241-265, 2010. Working paper. 100, 110, 124

[76] P. A. Forsyth, J. S. Kennedy, S. T. Tse, and H. Windcliff. Optimal trade execution : A mean qua-dratic variation approach. *Journal of Economic Dynamics and Control*, **36**(12):1971-1991, De-cember 2012. 110

[77] R. Frey. Perfect option hedging for a large trader. *Finance and Stochastics*, **2**:115-141, 1998. 98

[78] R. Frey and A. Stremme. Market volatility and feedback effects from dynamic hedging. *Mathematical Finance*, **7**(4):351-374, October 1997. 15, 98

[79] M. Fukasawa. Short-time at-the-money skew and rough fractional volatility. *Quantitative Finance, to appear*, 2016. 19, 154

[80] H. Funahashi and M. Kijima. Does the Hurst index matter for option prices under fractional volatility? *Annals of Finance*, pages 1-20, 2015. 19, 154

[81] M. C. Fung, K. Ignatieva, and M. Sherris. Systematic mortality risk: An analysis of guaranteed lifetime withdrawal benefits in variable annuities. *Insurance: Mathematics and Economics*, 58:103-115, 2014. 51

[82] J. Gao and E. R. Ulm. Optimal consumption and allocation in variable annuities with guaranteed minimum death benefits. *Insurance: Mathematics and Economics*, 51:586-598, 2012. 50

[83] J. Garcia and S. Goosens. The art of credit derivatives: Demystifying the Black Swan. *The Wiley Finance Series*, 2009. 87

[84] J. Garcia, S. Goosens, and W. Schoutens. Let's jump together: Pricing credit derivatives. *Risk*, pages 130-133, September 2008. 77, 87

[85] N. Gârleanu, L. H. Pedersen, and A. M. Poteshman. Demand-based option pricing. *The Review of Financial Studies*, **22**(10):4259-4299, 2009. 98

[86] M. B. Garman and M. J. Klass. On the estimation of security price volatility from historical data. *Journal of Business*, 53:67-78, 1980. 19, 132

[87] J. Gatheral. *The volatility surface: A practitioner's guide*. John Wiley & Sons, 2006. 129

[88] J. Gatheral, T. Jaisson, and M. Rosenbaum. Volatility is rough. Available at SSRN 2509457, 2014.
19, 20, 21, 154, 155, 156, 157, 158, 163, 165, 167

[89] J. Gatheral, T. Jaisson, and M. Rosenbaum. Volatility is rough. Available at SSRN 2509457, 2014.
129, 130, 131, 134, 135, 139, 140, 141, 142, 152
[90] J. Gatheral and A. Schied. Optimal trade execution under geometric Brownian motion in the Almgren and Chriss framework. *The International Journal of Theoretical and Applied Finance*, 14(3):353–368, 2011. 110

[91] H. U. Gerber and E. S. Shiu. Pricing lookback options and dynamic guarantees. *North American Actuarial Journal*, 7(1):48–66, 2003. 50

[92] S. Graf, A. Kling, and J. Ruß. Risk analysis and valuation of life insurance contracts : Combining actuarial and financial approaches. *Insurance : Mathematics and Economics*, 49(1):115–125, 2011. 51

[93] S. J. Grossman and J.-L. Vila. Optimal dynamic trading with leverage constraints. *The Journal of Financial and Quantitative Analysis*, 27(2):151–168, 1992. 11

[94] H. Guennoun, A. Jacquier, and P. Roome. Asymptotic behaviour of the fractional Heston model. *Available at SSRN 2531468*, 2014. 19, 154

[95] P. S. Hagan, D. Kumar, A. S. Lesniewski, and D. E. Woodward. Managing smile risk. *Wilmott Magazine*, pages 84–108, 2002. 18, 129, 154

[96] M. Hardy. *Investment Guarantees : Modeling and Risk Management for Equity-Linked Life Insurance*. Wiley Finance, 2003. 14

[97] S. L. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. *Review of Financial Studies*, 6(2):327–343, 1993. 18, 129, 154

[98] D. Holz, A. Kling, and J. Ruß. Gmwb for life: an analysis of lifelong withdrawal guarantees. *Zeitschrift für die gesamte Versicherungswissenschaft*, 101(3):305–325, 2012. 50

[99] Y. Huang, P. Forsyth, and G. Labahn. Iterative methods for the solution of a singular control formulation of a GMWB pricing problem. *Numerische Mathematik*, 122:133–167, 2012. 60

[100] G. Huberman and W. Stanzl. Price manipulation and quasi-arbitrage. *Econometrica*, 72(4):1247–1275, 2004. 99

[101] J. Hull and A. White. The pricing of options on assets with stochastic volatilities. *The Journal of Finance*, 42(2):281–300, 1987. 18, 129, 154

[102] J. Hull and A. White. Pricing interest-rate-derivative securities. *The Review of Financial Studies*, 3(4):573–592, 1990. 6

[103] G. Iacurci. How insurers are losing when it comes to variable annuities. http://www.investmentnews.com/article/20160830/FREE/160839998/how-insurers-are-losing-when-it-comes-to-variable-annuities, August 30 2016. 4

[104] S. Jaimungal and D. Kinzebulatov. Optimal execution with a price limiter. *Risk*, pages 49–54, July 2014. 112, 114

[105] T. Jaisson and M. Rosenbaum. Rough fractional diffusions as scaling limits of nearly unstable heavy-tailed Hawkes processes. *The Annals of Applied Probability*, 26(5):2860–2882, 2016. 19, 154

[106] C. Jiang, Y. Ma, and Y. An. The effectiveness of the VaR-based portfolio insurance strategy : An empirical analysis. *International Review of Financial Analysis*, 18:185–197, 2009. 11, 84
[107] N. Ju and R. Zhong. An approximate formula for pricing American options. *Journal of Derivatives*, 72(3):31–40, 1999. 35, 36, 37, 38

[108] D. B. Keim and A. Madhavan. Execution costs and investment performance: An empirical analysis of institutional equity trades. Working paper, School of Business Administration, University of Southern California, 1995. 107

[109] A. Kling, F. Ruez, and J. Russ. The impact of stochastic volatility on pricing, hedging, and hedge efficiency of withdrawal benefit guarantees in variable annuities. *Astin Bulletin*, 41(2):511–545, 2011. 14

[110] A. Kling, F. Ruez, and J. Russ. The impact of policyholder behavior on pricing, hedging, and hedge efficiency of withdrawal benefit guarantees in variable annuities. *European Actuarial Journal*, 4(2):281–314, 2014. 8, 14

[111] A. Kolkiewicz and Y. Liu. Semi-static hedging for gmwb in variable annuities. *North American Actuarial Journal*, 16(1):112–140, 2012. 14

[112] S. G. Kou. A jump-diffusion model for option pricing. *Management Science*, 48(8):1086–1101, August 2002. 12, 80

[113] N. Kunitomo. Improving the parkinson method of estimating security price volatilities. *Journal of Business*, 65:295–302, 1992. 132

[114] R. D. Lee and L. R. Carter. Modeling and forecasting u.s. mortality. *Journal of the American Statistical Association*, 87(419):659–671, September 1992. 56

[115] H. E. Leland. Option pricing and replication with transaction costs. *The Journal of Finance*, 40(5):1283–1301, 1985. 15, 98, 99, 100

[116] H. E. Leland and M. Rubinstein. The evolution of portfolio insurance. Published in *Dynamic Hedging: A Guide to Portfolio Insurance*, edited by Don Luskin (John Wiley and Sons, 1988), 1976. 76

[117] E. Lépinette and T. T. Quoc. Approximate hedging in a local volatility model with proportional transaction costs. *Applied Mathematical Finance*, 21(4):313–341, 2014. 16, 98, 102, 123

[118] T. M. Li and R. Almgren. A fully-dynamic closed-form solution for δ-hedging with market impact. *To appear Operations Research*, 2013. 98

[119] T. M. Li and R. Almgren. Option hedging with smooth market impact. *Working paper*, October 2014. 98

[120] F. Longstaff and E. Schwartz. Valuing American options by simulation: A simple least-squares approach. *The Review of Financial Studies*, 14(1):113–147, February 2001. 40

[121] J. Lorenz and R. Almgren. Mean-variance optimal adaptive execution. *Applied Mathematical Finance*, 18:311–323, 2011. 100, 111

[122] K. Lott. *Ein Verfahren zur Replikation von Optionen unter Transaktionkosten in stetiger Zeit*. PhD thesis, Universität der Bundeswehr München. Institut für Mathematik und Datenverarbeitung, 1993. 102
[123] C. Marshall, M. Hardy, and D. Saunders. Measuring the effectiveness of static hedging strategies for a guaranteed minimum income benefit. *North American Actuarial Journal*, 16(2):143–182, 2012. 14

[124] A. Medvedev and O. Scaillet. Approximation and calibration of short-term implied volatilities under jump-diffusion stochastic volatility. *Review of Financial Studies*, 20(2):427–459, 2007. 21, 154, 158, 159

[125] R. C. Merton. Optimum consumption and portfolio rules in a continuous-time model. *Journal of Economic Theory*, 3:373–413, 1971. 11, 76

[126] R. C. Merton. On the pricing of corporate debt: The risk structure of interest rates. *The Journal of Finance*, 29(2):449–470, 1973. 98

[127] M. A. Milevsky and S. E. Posner. The titanic option: Valuation of the guaranteed minimum death benefit in variable annuities and mutual funds. *Journal of Risk and Insurance*, 68(1):93–128, 2001. 50

[128] M. A. Milevsky and T. S. Salisbury. Financial valuation of guaranteed minimum withdrawal benefits. *Insurance: Mathematics and Economics*, 2005. 50

[129] M. A. Milevsky and T. S. Salisbury. Financial valuation of Guaranteed Minimum Withdrawal Benefits. *Insurance: Mathematics and Economics*, 38(1):21–38, February 2006. 28, 31

[130] M. A. Milevsky and T. S. Salisbury. Financial valuation of guaranteed minimum withdrawal benefits. *Insurance: Mathematics and Economics*, 38(1):21–38, 2006. 73

[131] T. Moenig and D. Bauer. Policyholder exercise behavior for variable annuities including guaranteed minimum withdrawal benefits. Working paper, 2011. 8

[132] J. Mühle-Karbe and M. Nutz. Small-time asymptotics of option prices and first absolute moments. *Journal of Applied Probability*, 48(4):1003–1020, 2011. 19, 154, 155

[133] U. Nations. World population ageing report. Technical report, United Nations, 2015. 1

[134] A. Neuenkirch and T. Shalaiko. The order barrier for strong approximation of rough volatility models. *arXiv preprint arXiv:1606.03854*, 2016. 19, 154

[135] C. J. Nuzman and V. H. Poor. Linear estimation of self-similar processes via Lamperti's transformation. *Journal of Applied Probability*, 37(2):429–452, 2000. 20, 147, 150

[136] L. Ornstein and G. E. Uhlenbeck. On the theory of the Brownian motion. *Physical Review*, 36, 1930. 140

[137] J. F. B. P. d'Andria and L. Elie. Modèle analytique d'évaluation des options de remboursement anticipé. *Finance*, 12(2), 1991. 31

[138] D. Pain and J. Rand. Recent developments in portfolio insurance. *Bank of England Quarterly Bulletin*, Q1, 2008. 77

[139] M. Parkinson. The extreme value method for estimating the variance of the rate of return. *Journal of Business*, 53:61–65, 1980. 19, 130, 131, 132

[140] A. F. Perold. Constant portfolio insurance. *Harvard Business School*, 1986. 76, 77
[141] A. F. Perold and R. Salomon. The right amount of assets under management. *Financial Analysts Journal*, 47:31-39, 1991. 99

[142] J. Pézier. Rationalization of investment preference criteria. *ICMA Centre Discussion Papers in Finance DP2011-12*, 2011. 84

[143] G. Piscopo and S. Haberman. The valuation of guaranteed lifelong withdrawal benefit options in variable annuity contracts and the impact of mortality risk. *North American Actuarial Journal*, 15(1):59-76, 2011. 8, 51

[144] J.-L. Prigent and F. Tahar. CPPI with cushion insurance. Available at SSRN eLibrary http://ssrn.com/paper=675824, 2005. 77

[145] L. Rogers and S. Satchell. Estimating variance from high, low and closing prices. *Annals of Applied Probability*, 1:504-512, 1991. 132

[146] J. M. Romero and J. Bautista. Exact solutions for optimal execution of portfolios transactions and the Riccati equation. eprint arXiv:1601.07961, January 2016. 116

[147] M. Rubinstein. Implied binomial trees. *Journal of Finance*, 49:771-818, 1994. 117

[148] M. Rubinstein and H. E. Leland. Replicating options with positions in stock and cash. *Financial Analysts Journal*, 37(4):63-72, 1981. 5

[149] P. Shah and D. Bertsimas. An analysis of the guaranteed withdrawal benefits for life option. Available at SSRN 1312727, 2008. 8, 50

[150] P. V. Shevchenko and X. Luo. A unified pricing of variable annuity guarantees under the optimal stochastic control framework. Available at SSRN 2773384, 2016. 51, 56

[151] J. Shu and J. E. Zhang. Testing range estimators of historical volatility. *Journal of Futures Markets*, 26(3), March 2006. 131

[152] P. Tankov. Pricing and hedging gap risk. *Journal of Computational Finance*, 13(3):33-59, Spring 2010. 81, 86

[153] S. T. Tse, P. A. Forsyth, J. S. Kennedy, and H. Windcliff. Comparison between the mean-variance optimal and the mean-quadratic-variation optimal trading strategies. *Applied Mathematical Finance*, 20(5):415-449, 2013. 111

[154] J. Wang and P. Forsyth. Maximal use of central differencing for Hamilton-Jacobi-Bellman PDEs in finance. *Journal on Numerical Analysis*, 46:1580-1601, 2008. 125

[155] D. Yang and Q. Zhang. Drift-independent volatility estimation based on high, low, open, and close prices. *Journal of Business*, 73:477-491, 2000. 132

[156] S. S. Yang and T.-S. Dai. A flexible tree for evaluating guaranteed minimum withdrawal benefits under deferred life annuity contracts with various provisions. *Insurance: Mathematics and Economics*, 52(2):231-242, 2013. 51

[157] J. Yong and X. Y. Zhou. *Stochastic controls: Hamiltonian Systems and HJB Equations*. Springer, 1999. 112