text:
\section{Introduction}\label{sec:intro}
Delta-hedging, as the unique strategy to eliminate all hedging risk in the Black-Scholes model, has been one of the pillars of classical finance. By using the option-implied Delta, Delta-hedging is easily adapted to models beyond Black-Scholes. However, it has been well understood that other risk factors, such as changes in volatility (Vega risk), also have to be taken into account in effective hedging strategies. Within the setting of local or stochastic volatility models, the derivation of such adjustments to Delta-hedging has been repeatedly considered in the literature. Crep\'ey, for example, proposes in \cite{crepey2004delta} an adjustment of the form
\begin{equation}\label{eq:adj_Crepey}
\theta_t^{Crepey} = \mathsf{Delta}_t + \mathsf{Vega}_t \pd{\Sigma_t}{K},
\end{equation}
where $\pd{\Sigma_t}{K}$ is the derivative of implied volatility with respect to strike. Hull and White consider in \cite{hull2017optimal} an adjustment of the form
\begin{equation}\label{eq:HW}
\theta_t^{HW} = \mathsf{Delta}_t + \frac{\mathsf{Vega}_t}{S_t \sqrt{T-t}} \left(a + b\,\mathsf{Delta}_t + c \,\mathsf{Delta}_t^2 \right),
\end{equation}
where the coefficients $a$, $b$ and $c$ are determined by regression. Importantly, they emphasize that their regression approach corresponds to \emph{minimizing the variance} of the hedging error. Here, we also follow this variance-optimality approach, which has originally been introduced and thoroughly explored in a general setting by F\"ollmer and Schweizer \cite{schweizer1984varianten, follmer1988hedging}. We assume that a description of the dynamic evolution of the option-implied volatility $\Sigma$ is available, and we obtain a general expression for the variance-minimizing hedging strategy and its mean-square error in Section~\ref{sec:bg}. We then focus on the application of these theoretical results to optimal hedging in the SABR model \cite{hagan2002managing} and in its extension to `rough volatility' (see \cite{gatheral2018volatility}), the rough Bergomi model introduced in \cite{bayer2016pricing}.\\
Already in \cite{hagan2002managing}, an adjusted Delta hedge for the SABR model is proposed by Hagan-Kumar-Lesniewski-Woodward (HKLW), taking the form
\begin{equation}\label{eq:HKLW_hedge}
\theta_t^{HKLW} = \mathsf{Delta}_t + \mathsf{Vega}_t \frac{\partial \Sigma_t}{\partial S}.
\end{equation}
Aiming to improve the HLKW strategy, Bartlett proposes in \cite{bartlett2006hedging} the adjustment\footnote{All parameters refer to the SABR model as discussed in Section~\ref{sec:SABR} below.}
\begin{equation}\label{eq:Bartlett_hedge}
\theta_t^{Bartlett} = \mathsf{Delta}_t + \mathsf{Vega}_t \left(\frac{\rho}{2 \eta} \frac{\partial \Sigma_t}{\partial \alpha} + \frac{\partial \Sigma_t}{\partial S}\right),
\end{equation}
which is subsequently discussed as `Bartlett's Delta' in \cite{hagan2017bartlett}. In Section~\ref{sec:SABR}, we show mathematically that Bartlett's hedge corresponds exactly to the variance-optimal hedge in the SABR model. Moreover, we derive analytic expressions for the mean-square hedging errors of the classic Delta-hedge, the HLKW-hedge and the Bartlett/variance-optimal hedge, allowing us to compare the strategies on a theoretical basis, in addition to their numerical evaluation. Subsequently, we show in Section~\ref{sec:rough} that our `dynamic implied volatility' framework for variance-optimal hedging can also be applied to rough stochastic volatility models, which are neither Markovian nor semi-martingales. Combining our framework with the implied-volatility approximations of Fukasawa and Gatheral \cite{fukasawa2022rough} for the rough Bergomi model \cite{bayer2016pricing}, we derive analytic expressions for the variance-optimal hedging strategy and its mean-square error. In particular, we discover an interesting interplay between the leverage parameter $\rho$ and the roughness parameter $H$ in determining the efficiency of the variance-optimal strategy. In section~\ref{sec:numerics} we report numerical results for the SABR and the rough Bergomi model, which confirm our theoretical findings.

\section{Background}\label{sec:bg}
\subsection{Variance-optimal hedging}
Let $S = (S_t)_{t \geq 0}$ be the stock price and $C = (C_t)_{t \in [0,T]}$ be the price of a contingent claim with maturity $T$. For simplicity we set interest rates to zero and we assume that there exists a risk-neutral measure $\mathbb{Q}$ under which both $S$ and $C$ are square-integrable martingales. Let $\theta$ be a strategy to hedge the payoff $C_T$. Then we can write
\[C_t = w + \int_0^t \theta_u dS_u+ L_t,\]
where the terms on the right hand denote, respectively: the initial capital $w \in \mathbb{R}$, the accumulated value of the hedging portfolio and the \emph{pathwise hedging error} $L$. Note the similarities to linear regression: We are regressing (the stochastic process) $C$ onto the hedging portfolios in $S$, with regression coefficient $\theta$, intercept $w$, and residual $L$. In a complete market we can find a perfect hedging strategy $\theta^\text{perf}$ such that the terminal pathwise hedging error $L_T$ is identical to zero and the unique initial capital needed is $w = \mathbb{E}^\mathbb{Q}(C_T)$. However, in an incomplete market model, such as the SABR model or other rough and non-rough stochastic volatility models, there is no perfect hedging strategy. Thus, we have to decide on another criterion of optimality to select a suitable strategy $\theta$. Here, we focus on the principle of \emph{variance-optimality}, introduced by \cite{schweizer1984varianten, follmer1988hedging}, which aims to find the initial capital $w$ and the variance-optimal strategy $\theta^{VO}$, which minimizes the risk-neutral\footnote{One can also start from a non-risk-neutral measure $\mathbb{P}$ and minimize the MSHE under $\mathbb{P}$. This leads to the rich subject of mean-variance hedging, cf. \cite{schweizer1992mean}, which we do not consider here.} mean-square hedging error (MSHE)
\begin{equation}\label{eq:MSHE}
\mathrm{MSHE}(\theta) = \Ex{\mathbb{Q}}{L_T^2} = \Ex{\mathbb{Q}}{\left(C_T -  w + \int_0^T \theta_u dS_u \right)^2}.
\end{equation}
The proper mathematical framework is to assume that $S$ belongs to the Hilbert space (cf. \cite[I.4a]{jacod2013limit})
\[\mathcal{H}_2 = \set{M = (M_t)_{t \in [0,T]}: \text{$M$ is a martingale with } \E{M_T^2}< \infty}\]
of square integrable continuous martingales with norm $\norm{M}_2^2 = \E{M_T^2}$ and inner product $M \cdot N = \E{M_T N_T}$; and to restrict the hedging strategies to
\[L_S^2 = \set{\theta \text{ adapted}: \E{\int_0^T \theta_t^2 d\scal{S}{S}_t} < \infty}.\]
Finally, two martingales $M,N \in \mathcal{H}_2$ are called \emph{orthogonal in the strong sense}, if their quadratic covariation vanishes, i.e., if $\scal{N}{M} = 0$. If $M_0 N_0 = 0$, then this strong orthogonality implies orthogonality in the Hilbert-space $\mathcal{H}_2$.
 The key observations are that
\begin{itemize}
\item equation \eqref{eq:MSHE} is minimized by the orthogonal projection in $\mathcal{H}_2$ of the martingale $C = (C_t)_{t \in [0,T]}$ onto the closed subspace spanned by the integrals $\int_0^t \theta_u dS_u$ where $\theta$ ranges through $L_S^2$; and that
\item the residual  $L$ of this projection must be orthogonal to $S$; even in the strong sense of $\scal{L}{S} = 0$ (cf. \cite{kunita1967square}).
\end{itemize}
Strong orthogonality of $S$ and $L$ implies that
\[0 = \scal{L}{S}_t = \scal{C}{S}_t + \int_0^t \theta_s d\scal{S}{S}_s\]
for all $t \in [0,T]$. Rearranging yields the
variance-optimal strategy as
\[\theta_t^{VO} = \frac{d\scal{S}{C}_t}{d\scal{S}{S}_t},\]
where the right-hand side has to be read as the Radon-Nikodym derivative of the finite-variation processes $\scal{S}{C}_t$ and $\scal{S}{S}_t$. Note the similarity to linear regression, where the regression coefficient is obtained as the covariance between the dependent and the independent variable, divided by the variance of the independent variable.\\
Above, and in what follows, we make frequent use of the following calculation rules for quadratic covariations of continuous Ito processes $X,Y$ (see \cite[\S I.4d-e]{jacod2013limit})  of dimensions $m$ and $n$ respectively:
\begin{enumerate}[{Q}-I.]
\item If $f \in C^2(\mathbb{R}^m; \mathbb{R})$ and $g \in C^2(\mathbb{R}^n; \mathbb{R})$, then $d\scal{f(X)}{g(Y)} = \nabla f(X)^\top d\scal{X}{Y} \nabla g(X)$, \label{eq:Q1}
\item If $\theta \in L_Y^2$, then $d\scal{\int \theta\,dY}{X} = \theta\,d\scal{Y}{X}$ \label{eq:Q2}
\item $\scal{X}{Y} = 0$ if $X$ or $Y$ has finite variation. \label{eq:Q3}
\end{enumerate}

\subsection{Variance-optimal hedging and implied volatility}
We restrict our attention to the case where $C$ is the price process of a call option with maturity $T$ and strike price $K$. To the call-price $C_t$ we can associate its (time-$t$) implied volatility $\Sigma_t$ and rewrite $C_t$ as
\begin{equation}\label{eq:BS}
C_t = c_\text{BS}(T-t,S_t,\Sigma_t),
\end{equation}
where $c_\text{BS}$ is the Black-Scholes price in dependence on time-to-maturity, underlying and implied volatility. As a smooth function of the martingales $S$ and $C$, the process $\Sigma$ must be a semi-martingale. We introduce the following notation for the Black-Scholes Greeks and related quantities, given $\Sigma$:
\begin{align*}
\mathsf{d}_t^{\pm} &= \frac{\log(S_t/K)}{\Sigma_t \sqrt{T-t}} \pm \frac{\Sigma_t \sqrt{T-t}}{2}\\
\mathsf{Delta}_t &= \Phi(\mathsf{d}_t^+)\\
\mathsf{Vega}_t &= S_t \phi(\mathsf{d}_t^+) \sqrt{T-t}
\end{align*}
Combining \eqref{eq:BS} with the variance-optimal hedging framework from above, we get a representation of the variance-optimal strategy in terms of Delta and Vega:

\begin{thm}\label{thm:VO_generic}
The variance-optimal hedging strategy for a call option $C$ with implied volatility process $\Sigma$ is
\begin{equation}\label{eq:VO_generic}
\theta_t^{VO} = \mathsf{Delta}_t   + \mathsf{Vega}_t \frac{d\scal{\Sigma}{S}_t}{d\scal{S}{S}_t}
\end{equation}
with initial capital $w =\E{C_T}$. The mean-square hedging error of this strategy is
\begin{equation}\label{eq:VO_MSHE_generic}
\mathrm{MSHE}(\theta^{VO}) = \E{\int_0^T \mathsf{Vega}_t^2 dQ^\Sigma_t}
\end{equation}
where the `orthogonal vol-of-vol' process $Q^\Sigma$ is given by
\begin{equation}\label{eq:volovol}
dQ^\Sigma_t = d\scal{\Sigma}{\Sigma}_t - \frac{d\scal{\Sigma}{S}_t}{d\scal{S}{S}_t} d\scal{\Sigma}{S}_t.
\end{equation}
\end{thm}
\begin{rem}
We add some intuition to the nature of the orthogonal vol-of-vol process $Q^\Sigma$. First, define
\[\Sigma^\perp := \Sigma - \frac{d\scal{\Sigma}{S}}{d\scal{S}{S}}S,\]
which is the component of $\Sigma$ which is orthogonal to $S$, i.e., we have
$d\scal{\Sigma^\perp}{S} = 0$. The process $Q^\Sigma$ can now be written as
$dQ^\Sigma = d\scal{\Sigma^\perp}{\Sigma^\perp}$, hence `orthogonal vol-of-vol'.
\end{rem}
In addition, we can derive the following formula to evaluate the mean-square hedging error of \emph{any} (not necessarily variance-optimal) hedging strategy:
\begin{thm}\label{thm:MSHE}
Let $\theta$ be a hedging strategy for a call option $C$ with initial capital $w = \Ex{\mathbb{Q}}{C_T}$ and implied volatility process $\Sigma$. Then the mean-square-hedging error of the strategy is given by
\begin{equation}\label{eq:MSHE_general}
\mathrm{MSHE}(\theta) = \E{\int_0^T \Big((\theta_t - \mathsf{Delta}_t)^2 d\scal{S}{S}_t - 2(\theta_t - \mathsf{Delta}_t) \mathsf{Vega}_t d\scal{\Sigma}{S}_t + \mathsf{Vega}_t^2 d\scal{\Sigma}{\Sigma}_t\Big) dt}
\end{equation}.
\end{thm}
Applying this formula to the simple Delta-hedge we obtain the following corollary:
\begin{cor}\label{cor:MSHE_Delta}
The mean-square hedging error of the Delta-Hedging strategy $\Delta_t := \mathsf{Delta}_t$
is
\begin{equation}\label{eq:Delta_MSHE_generic}
\mathrm{MSHE}(\Delta) = \E{\int_0^T \mathsf{Vega}_t^2 d\scal{\Sigma}{\Sigma}},
\end{equation}
and the difference to the error of the variance-optimal strategy \eqref{eq:VO_MSHE_generic} is equal to
\begin{equation}\label{eq:Diff_MSHE_generic}
\mathrm{MSHE}(\Delta)  - \mathrm{MSHE}(\theta^{VO})= \E{\int_0^T \mathsf{Vega}_t^2 \left(\frac{d\scal{\Sigma}{S}_t}{d\scal{S}{S}_t} \right)^2 d\scal{S}{S}_t} \ge 0.
\end{equation}
\end{cor}
\begin{rem}
We note that the only difference between expressions \eqref{eq:VO_MSHE_generic} and \eqref{eq:Delta_MSHE_generic} for the hedging error of the variance-optimal and the Delta-hedging strategy is that the integral is taken with respect to \emph{orthogonal} vol-of-vol in the former, and with respect to ordinary vol-of-vol in the latter.
\end{rem}

\begin{proof}[Proof of Thms. \ref{thm:VO_generic}, \ref{thm:MSHE} and Cor. \ref{cor:MSHE_Delta}]
Using property \textbf{Q-I} from above, we have
\begin{align*}
\theta_t^{VO} &= \frac{d\scal{S}{C}_t}{\scal{S}{S}_t} = \pd{c}{S}(T-t,S_t,\Sigma_t) \frac{d\scal{S}{S}_t}{d\scal{S}{S}_t} + \pd{c}{\Sigma}(T-t,S_t,\Sigma_t) \frac{d\scal{\Sigma}{S}_t}{d\scal{S}{S}_t} = \\
&= \mathsf{Delta}_t   + \mathsf{Vega}_t \frac{d\scal{\Sigma}{S}_t}{d\scal{S}{S}_t}.
\end{align*}
To compute the MSHE of an arbitrary strategy $\theta \in L_S^2$, note that
\[\E{L_T^2} = \E{\scal{L}{L}_T} = \E{\scal{C - w - \int_0 \theta dS}{C - w - \int_0 \theta dS}_T}\]
and calculate, using property \textbf{Q-II},
\begin{align}
d\scal{L}{L}_t &= d\scal{C}{C}_t - 2 \scal{C}{\int \theta dS}_t + \scal{\int \theta dS}{\int \theta dS}_t = \notag \\
&= d\scal{C}{C}_t - 2 \theta_t \scal{C}{S}_t + \theta_t^2 \scal{S}{S}_t \label{eq:L_inter}.
\end{align}
Using property \textbf{Q-I} again, we obtain
\begin{align*}
d\scal{L}{L}_t &= \mathsf{Delta}_t^2 d\scal{S}{S}_t + 2\,\mathsf{Delta}_t \mathsf{Vega}_t d\scal{S}{\Sigma}_t + \mathsf{Vega}_t^2 d\scal{\Sigma}{\Sigma}_t - \\
&\phantom{=}-2 \theta_t \mathsf{Delta}_t d\scal{S}{S}_t - 2 \theta_t \mathsf{Vega}_t d\scal{S}{\Sigma}_t + \theta_t^2 d\scal{\Sigma}{\Sigma}_t  = \\
&= (\theta_t - \mathsf{Delta}_t)^2 d\scal{S}{S}_t - 2(\theta_t - \mathsf{Delta}_t) \mathsf{Vega}_t d\scal{\Sigma}{S}_t + \mathsf{Vega}_t^2 d\scal{\Sigma}{\Sigma}_t,
\end{align*}
which gives \eqref{eq:MSHE_general}. Inserting the variance-optimal strategy $\theta^{VO}$ yields
\begin{align*}
d\scal{L}{L}_t &= \mathsf{Vega}_t^2 \Big( \left(\frac{d\scal{\Sigma}{S}_t}{d\scal{S}{S}_t}\right)^2 d\scal{S}{S}_t - 2
\frac{d\scal{\Sigma}{S}_t}{d\scal{S}{S}_t} d\scal{\Sigma}{S}_t + d\scal{\Sigma}{\Sigma}_t \Big) = \\
&= \mathsf{Vega}_t^2 \Big(d\scal{\Sigma}{\Sigma}_t - \frac{d\scal{\Sigma}{S}_t}{d\scal{S}{S}_t} d\scal{\Sigma}{S}_t\Big)
\end{align*}
which gives \eqref{eq:VO_MSHE_generic}. Inserting the Delta-hedging strategy, on the other hand, yields \eqref{eq:Delta_MSHE_generic}.
\end{proof}

To compute the variance-optimal strategy from Theorem~\ref{thm:VO_generic}, we need a tractable description of the dynamic implied volatility process $\Sigma = (\Sigma_t)_{t \in [0,T]}$, which is rarely available in stochastic volatility models. However, in certain models, such as the SABR and the rough Bergomi model, accurate \emph{dynamic approximations} $\hat \Sigma_t$ of $\Sigma_t$ are available, due to \cite{hagan2002managing, balland2006forward, fukasawa2022rough}. This is our key to obtaining explicit approximate variance-optional strategies for these models in the following sections.

\section{The SABR model}\label{sec:SABR}
We consider the SABR model of \cite{hagan2002managing} in its conditionally lognormal form, i.e., with $\beta = 1$, which takes the form
\begin{align*}
dS_t &= S_t \alpha_t dB_t\\
d\alpha_t &= \frac{\eta}{2}\alpha_t dW_t
\end{align*}
where $d\scal{B}{W}_t = \rho dt$ with $\rho \in [-1,1]$. We parameterize vol-of-vol by $\tfrac{\eta}{2}$ for consistency with the rough Bergomi model, as discussed in Section~\ref{sec:rough}. An asymptotically arbitrage-free approximation $\hat \Sigma$ of $\Sigma$ in the lognormal SABR model is given by \cite{fukasawa2022rough} (see also \cite{balland2006forward}) as
\begin{equation}\label{eq:AAA_SABR}
\hat \Sigma_t = \alpha_t f(Y_t), \qquad Y_t = \frac{\eta}{\alpha_t} \log \left(K/S_t\right),
\end{equation}
where $f$ is given by the famous SABR formula of Hagan et al.\ \cite{hagan2002managing}:
\begin{equation}\label{eq:f}
f(y) = \frac{y}{g(y)}, \qquad g(y) = - 2 \log\left(\frac{\sqrt{1 + \rho y  + y^2/4} - \rho - y/2}{1 - \rho}\right).
\end{equation}
We denote by $\widehat \mathsf{Delta}_t$, $\widehat \mathsf{Vega}_t$, $\widehat \mathsf{d}^\pm$, etc. the Black-Scholes Greeks (and related quantities) evaluated at the approximation $\hat \Sigma$ rather than the exact implied volatility $\Sigma$.

\subsection{Bartlett's Delta is the variance-optimal strategy}
We consider and compare the following three hedging strategies for the SABR model:
\begin{itemize}
\item The (classic) Delta hedging strategy $\Delta_t = \Phi(\widehat \mathsf{d}^+_t)$,
which uses the Black-Scholes-Delta evaluated at the SABR-implied volatility;
\item The Hagan-Kumar-Lesniewski-Woodward-(HKLW) adjusted Delta strategy \eqref{eq:HKLW_hedge} proposed in \cite{hagan2002managing};
\item Bartlett's adjusted Delta strategy \eqref{eq:Bartlett_hedge} which was proposed in \cite{bartlett2006hedging} (see also \cite{hagan2017bartlett}) as an improvement of the HKLW strategy.
\end{itemize}

Our first result shows that Bartlett's strategy is variance-optimal (up to the approximation error induced by the approximation $\hat \Sigma \approx \Sigma$) , i.e., no other hedging strategy can obtain a smaller hedging error in the mean-square sense.
\begin{thm}\label{thm:SABR}The approximate variance-optimal strategy $\theta^{AVO}$ for the SABR model, obtained by substituting $\hat \Sigma$ for $\Sigma$ in \eqref{eq:VO_generic}, coincides with Bartlett's adjusted Delta strategy \eqref{eq:Bartlett_hedge} and is given by
\begin{align}
\theta_t^{AVO} &= \theta_t^{Bartlett} = \widehat \mathsf{Delta}_t + \frac{\eta}{2S_t} \widehat \mathsf{Vega}_t \Big(\rho F_1(Y_t)  + F_2(Y_t)\big) =\notag\\
&= \Phi(\widehat \mathsf{d}_t^+) + \frac{\eta}{2}\phi(\widehat \mathsf{d}_t^+) \sqrt{T-t} \big(\rho F_1(Y_t)  + F_2(Y_t)\big),\label{eq:AVO_SABR}
\end{align}
where
\begin{align*}
F_1(y) &= F_1(y;\rho) = f(y) - y f'(y)\\
F_2(y) &=F_2(y;\rho) = -2 f'(y)
\end{align*}
with $f$ given by \eqref{eq:f}.
\end{thm}
\begin{rem}
\begin{enumerate}[(a)]
\item Note that $F_1(Y_t) = \pd{\Sigma_t}{\alpha}$ and $F_2(Y_t) = \frac{2S_t}{\eta}\pd{\Sigma_t}{S}$, hence \eqref{eq:AVO_SABR} is the same as Bartlett's Delta \eqref{eq:Bartlett_hedge}.
\item For implementations, the following representations of $F_1$ and $F_2$ are useful:
\begin{align*}
F_1(y) &= f(y)^2 / \sqrt{1 + \rho y  + y^2/4}\\
F_2(y) &= \tfrac{2}{y} \left(F_1(y) - f(y)\right).
\end{align*}
\end{enumerate}
\end{rem}

\begin{thm}\label{thm:SABR_MSHE}
The approximate mean-squared hedging errors for the Bartlett/variance-optimal strategy, the Hagan-Kumar-Lesniewski-Woodward strategy and the simple Delta strategy are
\begin{align}\label{eq:AVO_MSHE_SABR}
\widehat \mathrm{MSHE}(\theta^{AVO}) &= \frac{\eta^2}{4} ( 1- \rho^2) \int_0^t  \widehat \mathsf{Vega}_s^2\;\alpha_s^2 \;F_1(Y_s)^2 ds,\\
\widehat \mathrm{MSHE}(\theta^{HKLW})  &= \frac{\eta^2}{4} \int_0^t  \widehat \mathsf{Vega}_s^2\;\alpha_s^2 F_1(Y_s)^2 ds,\\
\widehat \mathrm{MSHE}(\Delta)  &= \frac{\eta^2}{4} \int_0^t  \widehat \mathsf{Vega}_s^2\;\alpha_s^2 (F_1(Y_s)^2 + 2 \rho F_1(Y_s) F_2(Y_s) + F_2(Y_s)^2) ds.
\end{align}
\end{thm}
\begin{rem}
\begin{enumerate}[(a)]
\item It can be seen that in the `complete market limit' $|\rho| \to 1$, the MSHE vanishes only for the Bartlett/variance-optimal strategy. The other two strategies are not able to fully exploit the correlation of stock price and stochastic variance.
\item The formulas for the MSHE are calculated `within-approximation', i.e. the approximation $\Sigma \approx \widehat \Sigma$ is used to both calculate the strategy and to evaluate its error. A more honest evaluation would be to calculate the strategy using $\hat \Sigma$ and to evaluate the error using $\Sigma$. We expect the error approximations to be biased towards zero (that is, a little too optimistic) compared to this honest evaluation.
\end{enumerate}
\end{rem}

Comparing the hedging errors of the strategies in Thm.~\ref{thm:SABR_MSHE}, we can compute the difference of the mean-squared hedging error of the simple Delta strategy and the Bartlett/variance-optimal strategy as
\begin{equation}\label{eq:difference_SABR}
\widehat \mathrm{MSHE}(\Delta) - \widehat\mathrm{MSHE}(\theta^{Bartlett}) = \frac{\eta^2}{4} \int_0^t  \widehat \mathsf{Vega}_s^2\;\alpha_s^2 (\rho F_1(Y_s) + F_2(Y_s)) ^2 ds \ge 0.
\end{equation}
Moreover, it is easy to see that
\[\widehat \mathrm{MSHE}(\theta^{HKLW}) = \frac{\widehat \mathrm{MSHE}(\theta^{Bartlett})}{1 - \rho^2},\]
i.e. the error of the HKLW-strategy is always larger than the error of the Bartlett/variance-optimal strategy by a factor of $1/(1 - \rho^2)$, independent of all other model parameters.
Finally, taking the difference of HKLW and the simple Delta strategy yields
\begin{equation}\label{eq:Delta_vs_HKLW}
\widehat\mathrm{MSHE}(\Delta) -  \widehat \mathrm{MSHE}(\theta^{HKLW})  = \frac{\eta^2}{4} \int_0^t  \widehat \mathsf{Vega}_s^2\;\alpha_s^2 \left(F_2^2(Y_t) + 2 \rho F_1(Y_s) F_2(Y_s) \right)  ds.
\end{equation}
For $\rho \approx 0$, the HKLW strategy improves upon the simple Delta hedge, which was the original intention of Hagan et al. For larger $|\rho|$, the conclusion from \eqref{eq:Delta_vs_HKLW} becomes unclear; the first-order analysis below shows that we should expect  the simple Delta hedge to have the smaller error.

\begin{proof}[Proof of Thms.~\ref{thm:SABR} and \ref{thm:SABR_MSHE}]
Applying Ito's formula to \eqref{eq:AAA_SABR} we obtain
\begin{align*}
d \hat \Sigma_t &= \pd{\widehat \Sigma_t}{\alpha} d\alpha_t + \pd{\widehat \Sigma_t}{S} dS_t + \textit{drift} = \frac{\eta}{2} \alpha_t \left(f(Y_t) - f'(Y_t) Y_t\right) dW_t - \eta \alpha_t f'(Y_t) dB_t + \textit{drift} = \\&= \frac{\eta}{2} \alpha_t \left(F_1(Y_t)dW_t + F_2(Y_t)dB_t\right) + \textit{drift}.
\end{align*}
Hence, we have
\[d \scal{\hat \Sigma}{S}_t = \frac{\eta}{2}\alpha_t^2 S_t \left(\rho F_1(Y_t)  + F_2(Y_t)\right) dt.\]
Together with
\[d \scal{S}{S}_t = S_t^2 \alpha_t^2 dt,\]
and inserting into \eqref{eq:VO_generic} we obtain the approximate variance-optimal strategy \eqref{eq:AVO_SABR}; coinciding with \eqref{eq:Bartlett_hedge}. For the mean-square hedging error, we calculate
\[d \scal{\hat \Sigma}{\hat \Sigma}_t = \frac{\eta^2}{4}\alpha_t^2  \left(F_1(Y_t)^2 + 2 \rho F_1(Y_t) F_2(Y_t)  + F_2(Y_t)^2\right) dt.\]
Inserting into \eqref{eq:volovol} we obtain the orthogonal vol-of-vol process
\[dQ^{\hat \Sigma} = \frac{\eta^2}{4} \alpha_t^2 \left\{(F_1(Y_t)^2 + 2\rho F_1(Y_t)F_2(Y_t) + F_2(Y_t)) - (\rho F_1(Y_t) + F_2(Y_t))^2\right\} = \frac{\eta^2}{4} \alpha_t^2 (1 - \rho^2) F_1(Y_t)^2;\]
from Thm.~\ref{thm:VO_generic} we obtain \eqref{eq:AVO_MSHE_SABR}. The expressions for the mean-square hedging errors of the HKLW- and the simple Delta strategy follow by applying Thm.~\ref{thm:MSHE}.
\end{proof}

\subsection{First-order analysis}
We simplify the comparison of the different hedging strategies, using the first-order approximation
\[f(y) \approx 1 + \frac{\rho}{4} y,\]
of the SABR implied volatility from \cite{fukasawa2022rough}. Under this approximation
\begin{equation}
F_1(y) \approx 1, \qquad F_2(y) \approx -\frac{\rho}{2},
\end{equation}
and the Bartlett/variance-optimal strategy is approximated by
\[\theta_t^{Bartlett} \approx \widehat \mathsf{Delta}_t + \frac{\rho \eta}{4S_t}  \widehat \mathsf{Vega}_t.\]
The mean-squared hedging errors of the Bartlett/variance-optimal, the HKLW- and the Delta strategy become
\begin{align}\label{eq:SABR_approx1}
\widehat \mathrm{MSHE}(\theta^{Bartlett}) &\approx  \frac{\eta^2}{4} ( 1- \rho^2) \int_0^t  \widehat \mathsf{Vega}_s^2\;\alpha_s^2 ds,\\
\widehat \mathrm{MSHE}(\theta^{HKLW}) &\approx  \frac{\eta^2}{4} \int_0^t  \widehat \mathsf{Vega}_s^2\;\alpha_s^2 ds,\label{eq:SABR_approx2}\\
\widehat \mathrm{MSHE}(\Delta) &\approx  \frac{\eta^2}{4} \left( 1- \frac{3\rho^2}{4}\right) \int_0^t  \widehat \mathsf{Vega}_s^2\;\alpha_s^2 ds,\label{eq:SABR_approx3}
\end{align}
such that our first-order analysis suggests
\begin{equation}\label{eq:ordering}
\widehat \mathrm{MSHE}(\theta^{Bartlett}) \le \widehat \mathrm{MSHE}(\Delta) \lesssim \widehat \mathrm{MSHE}(\theta^{HKLW}).
\end{equation}
Note, however, that the non-asymptotic result \eqref{eq:Delta_vs_HKLW} shows an advantage of the HKLW-strategy over the Delta hedge for $\rho \approx 0$, which gets lost in the first-order approximation.\\
An interesting quantity to analyze is the \emph{relative reduction} in root-mean-square error of the Bartlett/variance-optimal strategy in comparison to the Delta hedge. Using \eqref{eq:SABR_approx1} and \eqref{eq:SABR_approx3} we obtain
\begin{equation}\label{eq:relred_SABR}
\textsf{RelRed}_\rho = \frac{\sqrt{\mathrm{MSHE}(\Delta)} - \sqrt{\mathrm{MSHE}(\theta^{Bartlett}) }}{\sqrt{\mathrm{MSHE}(\Delta)}} = 1 - 2 \sqrt{\frac{1 - \rho^2}{4 - 3\rho^2}}.
\end{equation}
Note that this expression is independent of the option's strike and its time-to-maturity, but loses accuracy for ITM/OTM options. The quantity is shown as a function of $\rho$ in Figure~\ref{fig:relred_theory}. It is flat around zero and only starts to rise steeply in vicinity of the endpoints $\rho = \pm 1$. This suggest that the advantage of the Bartlett/variance-optimal hedge over the Delta hedge only starts to manifest in models with a leverage $\rho$ close to $\pm 1$. A half-way reduction of the root-mean-squared error, for example, is achieved only at $\rho = \pm \sqrt{\frac{12}{13}} \approx \pm 0.96$.

\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{relred_theory.png} \label{fig:relred_theory}
\caption{\textbf{Relative Reduction in hedging error - first order approximation.} The first-order approximation \eqref{eq:relred_rough} (see also \eqref{eq:relred_SABR}) of the relative reduction in root-mean-squared hedging error for the variance-optimal strategy compared to the simple Delta hedge is shown in dependence on correlation $\rho$ and roughness parameter $H$.}
\end{figure}

\section{The rough Bergomi model}\label{sec:rough}
We consider the rough Bergomi model, introduced in \cite{bayer2016pricing} to generalize Bergomi's model \cite{bergomi2008smile}, with price process $S$ and forward variance $\xi_t(s) = \Econd{\alpha^2_s}{\mathcal{F}_t}$ given by
\begin{align*}
dS_t &= S_t \alpha_t dB_t\\
d\xi_t(s) &= \xi_t(s) \kappa(s-t)dW_t˛\quad s > t.
\end{align*}
Here, $d\scal{B}{W}_t = \rho dt$ with $\rho \in [-1,1]$; $\kappa$ is the power-law kernel $\kappa(r) = \eta \sqrt{2H} r^{H - 1/2}$ with $H \in [0,1/2]$, and $\alpha_t = \sqrt{\xi_t(t)}$. Note that the Hurst parameter $H$ controls the roughness of the volatility process $\alpha$ and the kernel $\kappa$ reproduces the power-law behavior of the volatility skew, cf. \cite{gatheral2018volatility}.
An asymptotically arbitrage-free approximation $\hat \Sigma$ of $\Sigma$ is given by \cite{fukasawa2022rough} as
\begin{equation}\label{eq:AAA_Bergomi}
\hat \Sigma_t = U_t f(Y_t), \qquad Y_t = \frac{\kappa(T-t)}{U_t} \log \left(K/S_t\right),
\end{equation}
where
\[U_t = \sqrt{\frac{1}{T-t}\int_t^T \xi_t(s)ds}\]
and $f$ is the solution of the ODE
\begin{equation}\label{eq:Bergomi_ODE}
\left(1 - \frac{yf'(y)}{f(y)}\right) \left(1- 2\rho \frac{y}{2H+1}  + \left(\frac{y}{2H+1}\right)^2 \right) = f(y)^2 \left(1 - (1- 2H)\frac{yf'(y)}{f(y)}\right), \qquad f(0) = 1.
\end{equation}
We will also need the auxilliary process
\[R_t = \frac{\int_t^T \kappa (s - t) \xi_t(s) ds}{\kappa(T-t) \int_t^T \xi_t(s) ds}\]
and use the approximations
\begin{equation}\label{eq:approx}
\frac{U_t}{\alpha_t} \approx 1, \quad \text{and} \quad R_t \approx \frac{1}{H + 1/2},
\end{equation}
which become exact in the limit $t \to T$, see \cite{fukasawa2022rough}. Note that they are also exact in the $H \to \tfrac{1}{2}$ limit, i.e., in the SABR model, see Sec.~\ref{sec:SABR}.

\subsection{Variance-Optimal Hedging}
\begin{thm}\label{thm:rough}The approximate variance-optimal strategy $\theta^{AVO}$ for the rough Bergomi model, obtained by substituting $\hat \Sigma$ for $\Sigma$ in \eqref{eq:VO_generic} and using the approximations \eqref{eq:approx}, is given by
\begin{align}
\theta_t^{AVO} &= \widehat \mathsf{Delta}_t + \frac{\kappa(T-t)}{2} \frac{\widehat \mathsf{Vega}_t}{S_t} \Big(\frac{\rho}{H + 1/2} F_1(Y_t)  + F_2(Y_t)\Big) =\notag\\
&= \Phi(\widehat \mathsf{d}_t^+) + \frac{\eta}{2}\phi(\widehat \mathsf{d}_t^+) \sqrt{2H} (T-t)^H \Big(\frac{\rho}{H + 1/2} F_1(Y_t)  + F_2(Y_t)\Big),\label{eq:AVO_rough}
\end{align}
where
\begin{align*}
F_1(y) &= F_1(y;\rho,H) = f(y) - y f'(y)\\
F_2(y) &= F_2(y;\rho,H) = -2 f'(y),
\end{align*}
and $f$ the solution of \eqref{eq:Bergomi_ODE}. The approximate mean-squared hedging error, obtained by substituting $\hat \Sigma$ for $\Sigma$ in \eqref{eq:VO_MSHE_generic} and using the approximations \eqref{eq:approx} is given by
\begin{equation}\label{eq:AVO_MSHE_rough}
\widehat \mathrm{MSHE}(\theta^{AVO}) = \frac{(1- \rho^2)}{(H + 1/2)^2} \int_0^T  2H (T-s)^{2H - 1} \widehat \mathsf{Vega}_s^2  \alpha_s^2 F_1(Y_s)^2 ds.
\end{equation}
\end{thm}
\begin{rem}Explicit expressions for the solution $f$ of \eqref{eq:Bergomi_ODE} (and hence of $F_1(y)$ and $F_2(y)$) are only available for $H = 0$ and $H = \tfrac{1}{2}$, but not for intermediate values. For $H \in (0,\tfrac{1}{2})$, we use the approximation of \cite{fukasawa2022rough}; see Appendix~\ref{app}.
\end{rem}

\begin{cor}\label{cor:Delta_rough}Let $\Delta_t = \widehat \mathsf{Delta}_t$ denote the Delta-hedging strategy (using the approximate implied volatility $\hat \Sigma$).
The approximate mean-square hedging error of this strategy is
\begin{equation}
\widehat \mathrm{MSHE}(\Delta)  = \frac{\eta^2}{4} \int_0^t  \mathsf{Vega}_s^2\;\alpha_s^2 \left(\left(\frac{F_1(Y_s)}{H+\tfrac{1}{2}}\right)^2 + 2 \rho \frac{F_1(Y_s)}{H + \tfrac{1}{2}} F_2(Y_s) + F_2(Y_s)^2\right) ds
\end{equation}
and its difference to the error of the approximate variance-optimal strategy is given by
\begin{equation}
\widehat \mathrm{MSHE}(\Delta) - \widehat \mathrm{MSHE}(\theta^{AVO}) = \frac{\eta^2}{4} \int_0^t  \mathsf{Vega}_s^2\;\alpha_s^2 \left(\rho \frac{F_1(Y_s)}{H + \tfrac{1}{2}} + F_2(Y_s)\right) ^2 ds \ge 0.
\end{equation}
\end{cor}

\begin{proof}[Proof of Thm.~\ref{thm:rough} and Cor.~\ref{cor:Delta_rough}]
From \cite{fukasawa2022rough} we know that
\[d \hat \Sigma_t = \frac{\kappa(T-t)}{2} \left(R_t U_t F_1(Y_t)dW_t + \alpha_t F_2(Y_t)dB_t\right) + \textit{drift}.\]
Hence, we have
\[d \scal{\hat \Sigma}{S}_t = \frac{\kappa(T-t)}{2}\alpha_t S_t \left(\rho R_t U_t F_1(Y_t)  + \alpha_t F_2(Y_t)\right) dt.\]
Together with
\[d \scal{S}{S}_t = S_t^2 \alpha_t^2 dt,\]
and inserting into \eqref{eq:VO_generic} we obtain the approximate variance-optimal strategy \eqref{eq:AVO_rough}.
For the mean-square hedging error, we calculate
\[d \scal{\hat \Sigma}{\hat \Sigma}_t = \frac{\eta^2}{4}\left(R_t^2 U_t^2 F_1(Y_t)^2 + 2 \rho R_t U_t \alpha_t F_1(Y_t) F_2(Y_t)  + \alpha_t^2 F_2(Y_t)^2\right) dt.\]
Inserting into \eqref{eq:volovol} we obtain the orthogonal vol-of-vol process
\[dQ^{\hat \Sigma} = \frac{\kappa(T-t)^2}{4} (1 - \rho^2) R_t^2 U_t^2 F_1(Y_t)^2.\]
Together with \eqref{eq:VO_MSHE_generic} and using the approximations \eqref{eq:approx} we obtain \eqref{eq:AVO_MSHE_rough}. Cor.~\ref{cor:Delta_rough} now follows by an application of Cor.~\ref{cor:MSHE_Delta}.
\end{proof}

\subsection{First-order analysis}
We use the first-order approximation
\[f(y) \approx 1 + \frac{a_H}{2} \rho y, \qquad a_H = \frac{1}{(H+\frac{1}{2})(H + \frac{3}{2})}\]
of the rough Bergomi implied volatility, as given in \cite{fukasawa2022rough}. Under this approximation
\begin{equation}
F_1(y) \approx 1, \qquad F_2(y) \approx -a_H \rho.
\end{equation}
Using these approximations, the error of the approximate variance-optimal and the Delta strategy become
\begin{align}\label{eq:Bergomi_approx}
\widehat \mathrm{MSHE}(\theta^{AVO}) &\approx  \frac{( 1- \rho^2)}{(H + \tfrac{1}{2})^2}  \int_0^t  \kappa(t-s)^2 \widehat \mathsf{Vega}_s^2\;\alpha_s^2 ds, \\
\widehat \mathrm{MSHE}(\Delta) &\approx  \frac{1}{(H + \tfrac{1}{2})^2} \left( 1- 2\rho^2\frac{H+1}{(H + \tfrac{3}{2})^2}\right) \int_0^t  \kappa(t-s)^2 \widehat \mathsf{Vega}_s^2\;\alpha_s^2 ds.
\end{align}
The ratio of these quantities evaluates to
\[r = \frac{\widehat \mathrm{MSHE}(\theta^{AVO})}{\widehat \mathrm{MSHE}(\Delta)} = \Big(1 - \rho^2\Big)/\left(1 - 2\rho^2\frac{H+1}{(H + \tfrac{3}{2})^2}\right).\]
Taking derivatives, it is easy to see that this ratio is decreasing in $H$ on $(0,\tfrac{1}{2})$. Consequently, the relative reduction of the hedging error is given by
\begin{equation}\label{eq:relred_rough}
\textsf{RelRed}_{\rho, H} = \frac{\sqrt{\mathrm{MSHE}(\Delta)} - \sqrt{\mathrm{MSHE}(\theta^{AVO})}}{\sqrt{\mathrm{MSHE}(\Delta)}} = 1 -  (H + \tfrac{3}{2}) \sqrt{\frac{1 - \rho^2}{(H + \tfrac{3}{2})^2 - 2(H + 1)\rho^2}}
\end{equation}
and is increasing in $H$. We conclude that the effectiveness of variance-optimal hedging, relative to Delta hedging, is largest in the SABR case and smallest in very rough models with $H$ close to zero. A plot of $\textsf{RelRed}_{\rho, H}$ in dependency of $\rho$ and for different $H$ is shown in Figure~\ref{fig:relred_theory}. While the effect of $H$ is visible, the influence of $\rho$ dominates, and is qualitatively similar for all $H \in [0,\tfrac{1}{2}]$. The boundary case $H = 0$ can also be simplified; for $H =0$ the relative error reduction evaluates to
\[\textsf{RelRed}_{\rho,0} = 1 -  3 \sqrt{\frac{1 - \rho^2}{9 - 8\rho^2}};\]
compare with \eqref{eq:relred_SABR}. A half-way reduction of the root-mean-squared error in the case $H = 0$, is achieved only at $\rho = \pm \sqrt{\frac{27}{28}} \approx \pm 0.98$.

\section{Numerical Results}\label{sec:numerics}
\subsection{Method}
To verify our results by simulation, we have implemented the SABR model and the calculation of the variance-optimal strategies in Python. For the simulation of the rough Bergomi model, we use the turbocharged Monte-Carlo scheme of \cite{mccrickerd2018turbocharging}, publicly available at \url{http://https://github.com/ryanmccrickerd/rough_bergomi}. As parameters for the SABR/rough Bergomi model we use
\[\eta = 0.5 \quad \text{and} \quad \alpha_0 = 0.4\]
and a flat term structure of initial forward variance is assumed in the rough Bergomi case. The parameters $\rho$ and $H$ are varied over a grid of
\[\rho \in (-0.95,-0.9,-0.8,-0.6,0.0) \quad \text{and} \quad H \in (0.5, 0.35, 0.2, 0.1).\]
We have also explored positive values of $\rho$, with results that are similar (up to symmetry) to the results for negative $\rho$ and therefore not reported here. For the call options we use a time-to-maturity of one year and set today's stock price to $S_0 = 1$. The strike price is varied over a grid of
\[K \in (0.6,0.8,1.0,1.25,1.66).\]
For each parameter combination we simulate $10.000$ paths on a time grid of $1000$ steps. We evaluate the pathwise hedging error for both the variance-optimal and the Delta strategy and then calculate all relevant summary statistics, such as the mean-square hedging error or the relative error reduction.

\begin{figure}
\centering
\includegraphics[width=0.95\textwidth]{relred.png}
\caption{\textbf{Relative Reduction in hedging error - simulation results.} The relative reduction in root-mean-squared hedging error for the variance-optimal strategy compared to the simple Delta hedging strategy is show in dependence on option strike price (colored lines), correlation $\rho$ and roughness parameter $H$. The black dotted line indicates the first-order approximation \eqref{eq:relred_rough}. \label{fig:relred}}
\end{figure}

\subsection{Observations}
We first focus on the empirical reduction in hedging error, which is shown in Figure~\ref{fig:relred}, together with the first-order approximation \eqref{eq:relred_rough}. For the SABR model ($H = 0.5$) and for the rough Bergomi model with large $H$, the first-order approximation matches well with the empirically observed error reduction. For smaller $H$, the approximation and the empirical error start to diverge, in particular for
ITM options. Overall, the approximate variance-optimal hedge consistently leads to a reduction of the hedging error (in comparison to the Delta hedge), in particular for models with large $|\rho|$ and $H$ not too close to $0$. This is in line with our theoretical results from Sections~\ref{sec:SABR} and \ref{sec:rough}. Only for $H \le 0.2$ and for far ITM options a slightly negative reduction (i.e. an increase) of the empirical hedging error can be observed.\\

An even more detailed picture is painted by Figures~\ref{fig:densities_H05} and \ref{fig:densities_H02}, where we show estimated densities of the empirical pathwise hedging error for different combinations of $\rho$ and $K$. In particular for large $|\rho|$ it can be seen that the density of the variance-optimal strategies' error is narrower, more symmetric, and lighter-tailed then the density of the Delta strategies' error. The density of the Delta strategy error exhibits a skew that varies with strike $K$ and has heavier and asymmetric tails. However, already for the moderate values of $\rho = -0.6$ the effect is much diminished, in line with the theoretical results of Sections~\ref{sec:SABR} and \ref{sec:rough}, see also Figure~\ref{fig:relred_theory}. Comparing Figures~\ref{fig:densities_H05} and \ref{fig:densities_H02} it
can also be seen that the described differences are more pronounces in the (non-rough) SABR case $H = 0.5$ than in the rough Bergomi case with $H = 0.2$. Again, this observation confirms our theoretical results.

\section{Conclusions}
In this article, we have derived analytic expressions for the variance-optimal hedging strategy and its mean-square error in the lognormal SABR and in the rough Bergomi model. For the SABR model, we find that the variance-optimal strategy coincides with Bartlett's Delta strategy from \cite{bartlett2006hedging}. Both theoretical results and simulations show that the variance-optimal strategy has lower hedging error than the implied Delta strategy, but this advantage only becomes substantial for strongly correlated models with large $|\rho|$. The relative efficiency of variance-optimal hedging is also affected by the roughness parameter $H$, with smaller $H$ (that is, increasing roughness) diminishing the advantage over the simple Delta hedge.\\
The general results of Section~\ref{sec:bg} on variance-optimal hedging in the `dynamic implied-volatilty' setting can likely be applied to other models beyond the lognormal SABR/rough Bergomi model, whenever a tractable expression for the evolution of implied volatility is available. In particular, in future work, we aim to generalize results to the SABR/rough Bergomi model with variable $\beta$, shedding further light on the stability of Bartlett's Delta with respect to variations in $\beta$, as discussed in \cite{hagan2017bartlett}.

\begin{figure}
\centering
\includegraphics[width=0.95\textwidth]{densities_H05.png}
\caption{\textbf{Densities of pathwise hedging error in the SABR model.}
This plot shows the simulated densities of the pathwise hedging error in the SABR model (Hurst parameter $H = 0.5$), comparing the Bartlett/variance-optimal strategy (red) and the Delta hedge (blue). Different combinations of correlation $\rho$ and option strike price $K$ are considered. \label{fig:densities_H05}}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=0.95\textwidth]{densities_H02.png}
\caption{\textbf{Densities of pathwise hedging error in the rough Bergomi model ($H = 0.2$).}
This plot shows the simulated densities of the pathwise hedging error in the rough Bergomi model (Hurst parameter $H = 0.2$), comparing the Bartlett/variance-optimal strategy (red) and the Delta hedge (blue). Different combinations of correlation $\rho$ and option strike price $K$ are considered.\label{fig:densities_H02} }
\end{figure}

\clearpage

\bibliographystyle{alpha}

redpajama_set_name: RedPajamaArXiv


Brij Mohan Amar Rahe is a Hindi film starring Arjun Mathur in the lead role. Arjun Mathur by now has appeared in more than 20 films like Luck By Chance, My Name is Khan and many others. As an actor he is been doing supporting roles for over a decade now and finally got a chance to do his solo lead. This is a lost opportunity and a failed film which is fortunate enough to see the light of projection in the first place. Brij Mohan is a man who is convicted by the law for his own murder and how he gets out of this confusion is the story.
The film has failed on layered as the director tried to make an unintentional comedy and comedy did not happen only. Nikhil Nagesh Bhat debuts with this film as a director and has proven absolutely no potential with his storytelling. Arjun Mathur is a good actor but a performance can only be as good as the role written. Yoodlee films have started a new policy to release one film a month for next four months.
All the films they are releasing are either very low budget films or the films with no credible name behind. One would like to commend their efforts to promote under rated people but their first film does not have a pinch of talent or hardwork. The goal should have been to promote credible content driven cinema and Yoodlee films did not pay enough attention, their plan is going down.


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Yes. From the buyer's perspective, a short sale is a lot like a regular sale. The difference is that it can be 90 days or more until closing. On top of that, there is no guarantee of a final approval, so buyers can wait around for months only to be told that the offer was rejected. That said, we are an excellent fit for short sale buyers because of our commission rebate and legal counsel throughout the transaction.


According to our records, Timbaland is possibly single.
Timbaland was previously married to Monique Idlett (2008).
Timbaland has had encounters with Karrine Steffans (2003) and Missy Elliott.
Timbaland is a 47 year old American Composer. Born Timothy Zachery Mosley on 10th March, 1972 in Virginia Beach, Virginia, U.S, he is famous for Timbaland & Magoo in a career that spans 1992–present. His zodiac sign is Pisces.
Timbaland is a member of the following lists: Grammy Award winners for dance and electronic music, Musicians from Virginia Beach, Virginia and Musicians from Norfolk, Virginia.
Help us build our profile of Timbaland! Login to add information, pictures and relationships, join in discussions and get credit for your contributions.
Timothy Zachary Mosley (born March 10, 1972), known professionally as Timbaland, is an American record producer, rapper, singer, songwriter and DJ.
Monique Idlett and Timbaland were married from 10th June, 2008 to 2004 (4 years).
Timbaland had an encounter with Missy Elliott.
Timbaland is sooooo hot...why is he dating Missy alone??
Timbaland has got to be one of the best songwriters of all time and the best producer. He is innovative in a way no one has been since Quincy Jones. Timbaland you are so fit! I love watching your videos!
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With Congress preparing for renewed debate over President Bush's Iraq war policies, a majority of Americans now support setting a deadline for withdrawing U.S. forces from the war-torn nation and also support putting new conditions on the military that could limit the number of personnel available for duty there, according to a new Washington Post-ABC News Poll.
Opposition to Bush's plan to send an additional 21,500 troops to Iraq remained strong, with two in three Americans registering their disapproval – 56 percent said they strongly object. The House recently passed a nonbinding resolution opposing the new deployments, but Republicans have successfully blocked consideration of such a measure in the Senate.
Of course, the handful of politicians and pundits who still support the war (including Australia's delusional Prime Minister) can only speak in meaningless rhetoric and sound-bites. The war is lost, children.
For at least one Iraqi blogger, the Bush administration's support for the corrupt Iraqi leadership is unforgivable.


text:
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The Australia Institute surveyed a nationally representative sample of Australians about which major party policies they thought would be better for reducing emissions, lowering electricity prices and energy reliability.
"Interestingly, while Labor is very clearly ahead with voters when looking at which major party has the best policies on addressing emissions, Labor also leads on reliability and prices as well," said Richie Merzian, Climate & Energy Program Director at the Australia Institute.
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Apology from AEC Required After Giving Wrong Information on New Senate Voting Rules
The Australia Institute is calling for an apology and explanation from AEC Commissioner Tom Rogers after the Commissioner gave wrong senate voting instructions to listeners of RN Breakfast on the morning pre-polls open for the 2019 Federal Election.
With early voting for the federal election starting today, Monday 29 April, mass public confusion still exists around Senate voting changes introduced in 2016 at the last election. Australia Institute research shows that almost half of voters are mistaken on how the new Senate voting rules work, which could have a big impact in a tight election contest.
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All five Tasmanian electorates are among the biggest losers of the Government's income tax cut plan, inner-city electorates in Sydney and Melbourne are the biggest winners, and South Australia and Central & North Queensland get at least 30% per capita less than electorates in Sydney and Melbourne.
The analysis by the Canberra-based think-tank's senior economist Matt Grudnoff reveals the entire population of Tasmania receives less from the tax cut than the top-ranked electorate: the electorate of Sydney.
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Solar surges as large-scale rooftop takes off
New research shows that medium to large rooftop solar systems are taking off, contributing with other solar and wind energy to the jump from 10% to 15% of the energy supplied to the grid over the last year.
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Unless national action is taken to meet the Paris Target to limit global warming to less than 2 degrees, Australia's GDP faces a hit of an average of $130 billion per year according to a new briefing note by The Australia Institute.
The analysis released today by the Canberra-based think tank shows that the current climate debate is largely missing three key points:
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New Australia Institute modelling shows that at least $77 billion of the lost revenue from the Morrison Government's top-end tax cut plan will benefit those earning more than $180,000, and $64 billion of that figure will go to those who earn over $200,000.
The Government has claimed that the difference between its tax cut plan and Opposition's is $230 billion over ten years. The analysis also reveals that after the Government's tax plan is fully implemented in 2024-25, high income earners will pay a smaller proportion of tax (4% less) and low and middle income earners will pay a larger proportion of tax.
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The much-hyped 'North/South divide' on climate action is a political misconception, according to new research from The Australia Institute.
The research shows that the majority of Australian voters across states and poltical allegiance are concerned by climate change, and want the Government to mobilise on the issue, "like they mobilised everyone during the world wars".
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January 26, 2016, 11:00 a.m.
The recent decision by the Obama administration to delay issuing new regulations under the landmark Americans with Disabilities Act (ADA) is outrageous. The regulations would have provided guidance on how businesses can meet their legal obligation to make their websites accessible to people with disabilities. These regulations have been in the works for over five years, but the Obama administration now proposes further delaying them until 2018, effectively washing its hands of the matter. This move is particularly shocking in light of the president's correct observation, made when he first announced his intention to issue the regulations in 2010, that such rules are "the most important updates to the ADA since its original enactment." The urgent need for these regulations has only increased, so why has the administration's position inexplicably changed?
because improperly designed websites can block our ability to effectively access all of the information. For example, if a website uses images to convey important information without also providing "alt tags" that a screen reader can read, then the screen reader will spit out gibberish because it can't "read" a picture in the way it can read text. And the inability to access websites is not merely an inconvenience; it is a barrier to education and employment. For example, the college graduation rate for people with disabilities is just thirty-four percent; inaccessible online technology used by today's colleges and universities undoubtedly contributes to this dismal statistic.
All of this is not due to hostility towards Americans with disabilities. While a few businesses simply refuse to provide equal access to their websites until a legal settlement or court order forces them to do so, many others simply don't know where to turn for guidance on how to make their websites accessible. Organizations like the National Federation of the Blind are doing all we can to educate business leaders and programmers, but by issuing clear and legally binding guidelines, the Obama administration could quickly bring reluctant businesses to the table and show other well-intentioned but uninformed players a clear path to providing equal service to their clients and customers with disabilities. The administration's continued refusal to do this is irresponsible. Its failure to act not only leaves disabled computer users on the wrong side of a real digital divide, but ensures that litigation, which is costly both for disability advocates and businesses, will continue for the foreseeable future.
Recently, the National Federation of the Blind and several other organizations representing Americans with all types of disabilities urged the immediate issuance of the proposed regulations in a letter sent directly to President Obama. From the business perspective, Microsoft and other business leaders have also written to the president calling for the release of the regulations. If the president ignores these requests, the inescapable conclusion will be that he is indifferent to the inequality that is part of everyday life for me and millions of other Americans. This indifference has an intolerably high cost: we are denied equal access to services that are readily available to everyone else, denied educational and employment opportunities, and denied first-class citizenship in twenty-first-century America. If the president is serious about the civil rights of all Americans, a recurring theme in his rhetoric, then he must not renege on the commitment to equal Internet access for Americans with disabilities that he made in 2010. Fortunately, he still has time to honor that commitment. I, along with millions of other people with disabilities, fervently hope that the president will do so immediately.
Mark A. Riccobono is president of the National Federation of the Blind. He lives in Baltimore, Maryland with his wife and three children.
Posted on February 5, 2016 by MamaRupp	• This entry was posted in Uncategorized. Bookmark the permalink.


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Videos: Ray Lewis's real last dance will be in Ravens' loss to Broncos on Saturday
Michael Roberts January 7, 2013 7:18AM
Now we know who the Broncos will face in their 2013 playoffs debut: the Baltimore Ravens, by dint of the squad's 24-9 win over the Indianapolis Colts on Sunday. So...will the emotion from Ray Lewis's final home game (depicted in videos included in a big collection of clips below) help the Ravens defeat a squad that destroyed them in a December road game? Short answer: No.
The Ravens' margin of victory over the Colts suggests a blowout, but that's not how the game played out. Indianapolis rookie Andrew Luck looked like one for much of the contest, repeatedly getting away with passes that should have been intercepted -- including one heaved directly to Lewis, who should have been able to haul it in even with an enormous brace on his arm necessitated by the triceps injury that's kept him out on the shelf for the past several weeks. Yet Indy stayed close through the first half and well into the third quarter thanks to Baltimore's inability to cash in on loads of opportunities.
Granted, a Baltimore turnover that put the kibosh on a drive into the red-zone was based on a blown call that couldn't be reviewed under the NFL's increasingly arcane rules. But Ravens quarterback Joe Flacco hardly looked like the elite hurler he claims to be, and running back Ray Rice matched an ordinary performance with an unexpected case of fumble-itis.
Fortunately for Baltimore, the Colts' secondary is the very definition of mediocrity, and the mighty Anquan Boldin took advantage, turning several unimpressive Flacco heaves into big gainers and applying the dagger in the fourth quarter with a touchdown catch in which he snared both the ball and defender Darius Butler's entire arm.
Unsurprisingly, these achievements were overshadowed by Lewis, a surefire Hall of Famer whose announced retirement at season's end transformed the contest into his final home performance. He took advantage via an extended version of his trademark out-of-the-tunnel dance that he reprised in the waning seconds -- and the fact that he took part in more than a dozen tackles in between these displays more than justified them.
Don't expect to see a similar performance on Saturday in Denver, though.
Continue for more about the Broncos-Ravens match-up, including eight videos. As Denver demonstrated during a 34-17 mid-December victory in Baltimore, the Ravens' once vaunted defense is not what it was in Lewis's prime. The squad was able to rattle Luck via blitz overloads that Denver QB Peyton Manning is more than capable of turning into big gains because of the open spaces they create, and which his quick release and legendary field vision is ideal for exploiting.
As such, the Ravens may well unleash the hounds on fewer occasions and opt for a base defense instead. But doing so will allow Manning the time he needs to check down and find open receivers and/or hand off to Knowshon Moreno, whose late-season surge has created the sort of balanced offensive attack Denver established prior to Willis McGahee's injury, to he unexpected delight of doubting fans like yours truly.
Defensively, meanwhile, the Broncos have a lot more weapons -- not just pass rushers Von Miller and Elvis Dumervil (plus well-kept-secret Wesley Woodyard), but also a secondary anchored by Champ Bailey, who'll join Lewis in Canton five years after his retirement (several years from now, we hope), and bolstered by up-and-comers such as Chris Harris, a star in the making in our estimation.
That doesn't mean the Baltimore game will be a breeze. The team brings oodles of playoff experience and a lot of post-season wins, albeit usually in round one. But as long as the Broncos don't overlook the Ravens in light of a likely pairing against New England in the AFC championship game, they should find a way to send Lewis into his next life as an ESPN commentator less than a week from now. And a lack of focus isn't something Mr. Manning is going to allow.
Look below to see a clip of Lewis's first dance, a post-game interview and an ESPN analysis -- the first helping of a video buffet.
Continue for five more videos -- an interview with Manning after the defeat of the Ravens last month, Peyton's post-game press conference, a weird animated video by a disappointed Baltimore fan, veteran Raven Ed Reed admitting to frustration and an ESPN analysis.
More from our Sports archive: "Videos: Peyton Manning, Broncos prove Joe Flacco, Ravens aren't elite."
Michael Roberts has written for Westword since October 1990, serving stints as music editor and media columnist. He currently covers everything from breaking news and politics to sports and stories that defy categorization.
Contact: Michael Roberts
Facebook: Michael Roberts
Twitter: @mikerobertscolo
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EDITORIAL: No Half Measures in Responding to Clerical Crisis
Argentinians Living as Enemies Since Vote for Life
Catholics Tell Cdl. Cupich: Clerical Sex Abuse 'Has Everything to Do With Homosexuality'
By The Editors, National Catholic Register, 8/10/18
The revelations about Archbishop Theodore McCarrick's shocking sexual misconduct have rightfully shattered Catholic Americans' confidence in the collective leadership that has been provided by their own bishops.
And, in the face of a crisis of this gravity and magnitude, there is simply no room for half measures in how Church leaders, both in Rome and here in the United States, address this renewed crisis.
Any sound analysis of the McCarrick scandal — and other cases now coming to light that similarly involve clerical abusers accused of preying upon seminarians and other clergy — must acknowledge two fundamental dimensions: an almost incomprehensible absence of effective structures of episcopal accountability; and, even more fundamentally, an appalling lack of fidelity to what the Church teaches with respect to sexual morality.
It is therefore heartening that Cardinal Daniel DiNardo of Galveston-Houston, president of the U.S. Conference of Catholic Bishops, squarely addressed the concerns of outraged Catholics on both fronts in his Aug. 1 statement that sketched an initial outline of the conference leadership's response to the crisis. The U.S. bishops' Executive Committee will develop plans to be discussed at the conference's annual meeting in November, and Cardinal DiNardo signaled his intent to press for an investigation that would expose Archbishop McCarrick's enablers.
An insistence on a full examination of McCarrick's abusive behavior and the episcopal cover-up that protected him from censure is an essential first step and will help guide subsequent reforms. What's also promising is that the USCCB president didn't attempt to gloss over the second central problem brought to light by Archbishop McCarrick's misconduct.
"Our Church is suffering from a crisis of sexual morality," the cardinal stated bluntly. "The way forward must involve learning from past sins."
Nevertheless, in the weeks and months ahead, it will not be easy to reconcile the very real tensions between the cautious mindset of many American prelates and the fury of lay Catholics demanding change. Yet a patient, frank and respectful dialogue is needed to map out the best way to deal with the substance and scope of the problems before us.
Likewise, we need time to develop a plan of action that hews to the ecclesial nature of the Church, even as it harnesses the expertise of lay Catholics.
Thus far, there appears to be general agreement among U.S. bishops and the faithful on the need for a mechanism that will allow seminarians and priests to report allegations against their superiors without fear of reprisals. Some prelates have already proposed that the U.S. Conference of Catholic Bishops convene a panel to receive and vet allegations and even rumors involving bishops.
At the same time, Marie Collins and other lay authorities on clergy-abuse issues are lobbying for a more radical national response, such as mandated annual reporting of settlements involving bishops and priests, or even an external financial forensic review designed to expose previous cover-ups.
Philip Lawler, the author of The Faithful Departed: The Collapse of Boston's Catholic Culture, has pressed for a comprehensive investigation of the Church leaders who knew about McCarrick's behavior. Any such effort, said Lawler, should be led by Archbishop Charles Scicluna of Malta, the former prosecutor of clergy-abuse crimes for the Congregation for the Doctrine of the Faith, whom Pope Francis dispatched to address Chile's widening abuse scandal.
In the meantime, Lawler has called for bishops and others to forcefully challenge moral corruption within the Church. And some lay leaders are prepared to take action on their own, if Rome and the USCCB fail to secure a comprehensive investigation.
Lay leaders told the Register that several strategies for exposing immoral behavior by priests and bishops are under review and would likely begin with forwarding testimony about specific individuals to relevant Church authorities and new reporting offices. And if these initial measures did not bear fruit, the media would be contacted, and information would be posted on a designated website.
Again, such proposals underscore the need for prompt action. But it would also be a mistake for Church authorities to endorse limited solutions in the rush to tamp down this crisis.
Our leaders must tackle the major failures of the past, including the acceptance of widespread homosexual activity among the U.S. priesthood.
That aspect, unfortunately, was notably absent from the public responses of a pair of prominent U.S. cardinals in the week immediately following the release of Cardinal DiNardo's response as USCCB president.
In public statements, media appearances and in an interview with the Register, Cardinal Donald Wuerl of Washington was silent about the problem of active homosexuals in chanceries, rectories and seminaries, despite the revelations involving Archbishop McCarrick.
Meanwhile, in an interview with America magazine, Cardinal Blase Cupich went even further, asserting that it is "a diversion" to call attention to the homosexual dimension of the scandal, even though homosexual misconduct is known to have been preponderantly in play throughout the clergy-abuse crisis, and, indeed, it is exclusively involved when it comes to Church leaders who prey on seminarians and priests.
The McCarrick case surely confirms that powerful forces have tolerated and even fomented sexual immorality within the priesthood, and will resist exposure. Those who downplay this fact should consider a troubling irony. Even as the Church is grappling with multiple scandals involving homosexual predation and the abuse of seminarians and priests, well-placed Catholic leaders have signaled their sympathy and support for homosexual relationships as a legitimate alternative to the marriage of one man and one woman instituted by God.
"While the Church upholds the ideal of marriage as a permanent commitment between a man and a woman, other unions exist which provide mutual support to [couples]," reads a preparatory booklet for this month's 2018 World Meeting of Families in Dublin. The booklet, which was published last year, had featured a section on same-sex couples. It was subsequently revised after pushback.
This month, Jennifer Roback Morse, the founder of the Ruth Institute, a nonprofit educational organization dedicated to promoting lifelong married love, noted an equally disturbing development. On the "very day the news about Cardinal McCarrick broke, the Vatican released the working document for the upcoming Synod on Youth," she noted. "It used the 'LGBT' acronym, the first such use in a Vatican document."
While Pope Francis has rightly emphasized the need to respect the dignity of persons with same-sex attraction, he also has repeatedly warned that men with deep-seated homosexual inclinations should not enter seminaries or supervise the education of future priests. But a parallel campaign in Rome and elsewhere appears to be fueling moral confusion and complacency at the very moment that the new clergy-abuse scandals in the Church point to a need for moral clarity and vigilance.
This is why a partial response to the unfinished business of the 2002 abuse crisis won't pass muster with active Catholics. The Body of Christ rightly demands a thorough housecleaning. Nothing less will do.
Copyright © 2018 EWTN News, Inc. All rights reserved
http://www.ncregister.com/daily-news/no-half-measures-in-responding-to-clerical-crisis
Patriot Post: Honoring the Sanctity of Life
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I am a candidate for a Ph.D. in Architecture and a teaching assistant at The University of Manchester (UoM). My research focuses on the complexities of the Blanka tunnel project in Prague, Czech Republic. It asks how we can reimagine infrastructural landscapes of mobility as integral social and material constituents of an urban fabric; and why is it important to reread such infrastructures as multi-dimensional spaces of physical movement and social processes rather than three-dimensional structures built for the sole purpose of travel.
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Sonali Bendre Returns To Reality Television With Dance India Dance?
Sonali was last seen as the judge of Zee TV show India's Best Dramebaaz. However, she quit the show after she was diagnosed with cancer in 2018.
Sonali Bendre has won millions of hearts and was a charming sensation in the 90s. The Bollywood celebrity continues to rule the block with her enticing fashion statement and that million dollar smile! However, in a shocking turn of events, Sonali was diagnosed with metastatic cancer and spent quite a chunk of 2018 in the USA undergoing treatment. Taking #OneDayAtATime and #SwitchingOnTheSunshine, Sonali along with husband Goldie Behl put up a brave front.
In December 2018, Sonali returned to India and has been getting stronger than ever. Our sources tell us that the 44-year-old actress is set to return to reality television with dance show Dance India Dance. We hear Sonali will be seen as one of the judges on the show. Although we did reach out to Sonali for a comment, she did not respond to our messages. Guess we will have to wait for an official confirmation to figure out.
Notably, before the diagnosis of the big C, Sonali was one of the judges on India's Best Dramebaaz. It is a talent-search reality television show on Zee TV which has children between the ages of 4–15 years as the participants. After Sonali quit the show for her treatment, she was replaced by Huma Qureshi as the judge alongside Vivek Oberoi and Anurag Basu.
Meanwhile, a leading news portal reported that choreographer Bosco Martis of Bosco-Caesar duo has been approached for the show too. He even confirmed the development in a quote given to the portal. "I was offered the first season, but at that time, I wasn't sure of entering the world of television. I was focussed on choreographing film songs. The makers have changed the format and I felt the time was right to make a comeback on TV," shared Bosco.
It yet remains to be seen if Sonali and Bosco shall really feature on Dance India Dance as judges or all of this is a fragment of somebody's imagination.
While we keep our fingers crossed, tell us in the comments box below if you'd like to see Sonali on Dance India Dance.
In the meantime, don't miss any episodes of your favourite TV show and watch them for FREE on ZEE5.
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they all are delicious. Despite their diversity, they all have a very characteristic basic taste - unmistakably their own, quite different from traditional mustards, archaic. Something magical lies in it; like from a witch's cauldron, as if one could taste the strong winds of the Hebrides in it and hear the whispering of old spirits. Where did you get that from?
You are so kind and I think it comes through in your mustard which is a delight :-) I wouldn't buy any other now. Looking forward to tasting the new flavours.
We've nearly finished one of the wholegrain already! Had it with venison sausages- beautiful match! Thanks very much and hope your business is very successful!
Great idea, more importantly, great product - I had some of you jalapeño mustard and it was fantastic.
Loving my Slàinte Liquorice mustard, bought at the craft fair last Saturday!
... the friend who got the chilli salt is a fan, as she put it on her chips, and just said 'YUM'... I made a quiche last night and sprinkled liberally with the chilli, and it turned a rather ordinary meal into something worth eating!!
Enjoyed sweet delight on a local oatcake with Orkney cheddar whilst camping at Horgabost. yummy!
I just came home from my holiday on Harris and my first meal was a delicious salad made with a handful of greens, some cherry tomatos from my garden, half an avocado, two mushrooms, some feta cheese crumbles AND - most importantly- the dressing made from the recipe you gave me. I added the double amount of your Slainte Liquorice mustard and it turned out absolutely scrumptious! This jar will definitely not last long so I'd better start planning my next trip to Harris.
I was given a pot of your absolutely delicious Mustard Sea salt - as a present from a friend who visited Harris in June and got it by the road side. I am totally addicted to it and love it on Fish and meat and salads - in fact on everything.!!
So great to see your mustardly presence on the One Show! I met you at Tarbert craft fair where I bought and have so enjoyed since your amazing mustard!
Super tasty proper mustard and lovely people, highly recommended!
Marinated some trout fillets with your Liquorice and chilli mustards before smoking them. absolutely amazing!
Yesterday I took what I thought was a joint of beef out of the freezer but it turned out to be venison.
I cooked it in a pressure cooker with a generous slug of red wine, a tot of whisky, an onion, a carrot and your 'Christmas Edition' mustard. OH. MY. WORD. What a fantastic gravy it made!!
There was a hint of whisky and a hint of mustard and a hint of spice. I'll do that again.
Ur liquorice mustard is amazing!
I treated myself to some Mustheb Slainte Licorice and it is the most fantastic stuff! It's very unusual and you don't need to use a lot of it to make an impact, but the flavour is phenomenal. I use it to brighten up cheese and cracker lunches, as well as in cooking - it goes really well in a beef sausage, tomato and bean stew! It's also fantastic with beans on toast! The owner is incredibly knowledgeable and passionate about her product, I will definitely be picking up another jar or two next time I'm on the island!
your mustard is wonderful. I always put mustard in a white/cheese sauce but your xmas edition takes the flavour to a whole new level. We get fresh tuna steaks from the fish van and last night I made a white sauce with garlic butter and xmas mustard, it was wonderful, never had a sauce with tuna steaks before but this just complimented it so well.
just used my pesto in my batch of pasta sauce, fair lifts it!
My daughters love your mustard, as do I, so we could be ordering some.
I ❤❤❤❤❤ the Christmas edition mustard... I'll be back for more.
The pesto been in my steak pie and lasagne and I've used it on bruschetta Delish !
Tried for the first time today - absolutely delicious, and a really nice lady too!
Every time when I visit Harris, I make sure I will go to Craft Market in Tarbert on Saturday.
Tried the sweet mustard tonight with pork roast. Absolutely delicious.
I used Mustheb Mustard tonight to flavour a mash and it was delicious! Lovely flavours. It's equally good on its own or under grilled cheese on a toasted sandwich.
Heike´s mustards are without exception a culinary delight. I have tried them all and even use them as "spread on bread". They seem to be produced by someone who cares very much about good food and all that that entails, such as using healthy ingredients.
Heike have created a truly unique and delicious range of artisan mustards. They are a range of handmade artisan mustard with a quirkiness and originality that uses organic ingredients to create something truly special. I use them in sandwiches, salads, in dressings and one takes cauliflower cheese to a whole new level! Heike is passionate about her products and this shines through her whole range. You can't go wrong with this mustard range, the tastes are bold and flavourful, I loved them! Would definitely be ordering more.
Fantastic mustards which are sharp, and overwhelming full of flavour. I have tried three different mustards now and have to say I have never tasted anything quite like it, amazing condiment for any table. Great to know that they are organic too. They certainly beat anything I have tried and it's great to support someone local. Can't wait to get my hands other flavours.
The mustard that we had was pretty much the best mustard I've eaten so far. It was not too spicy to be overpowering, but spicy enough to give a good kick. You could taste the whisky without it being overpowering - it was a nice aftertaste.
This was our first taste of home-made organic mustard and we really enjoyed it. The mustard has quite a "kick" to it, but the addition of honey and balsamic vinegar give it a lovely sweet and sour taste. We tried the mustard in three different ways: on a roast ham sandwich; in some home-made vegetable broth and mixed with mayonnaise in a home-made coleslaw salad. It was lovely on the sandwich, and was a really tasty addition to the mayonnaise, but I felt that it didn't work as well in the soup as "ordinary" mustard (for us, this means mass-produced Dijon mustard!) as the sweet and sour flavours did not go with the soup so well. My favourite use, however, was in the mayonnaise - this was truly delicious! I've never reviewed anything before, so I'm not quite sure what you want. Please let me know if there is something else you would like us to comment on. I was trying very hard to be critical, so that it didn't just look like a glowing review from a friend!!!! I struggled to say anything even remotely negative about it - the lovely flavours were wasted on the soup, really. We both loved it, so hope that you DO manage to set up your business. It is always much better to enjoy quality home-made produce than mass-produced stuff.


My Journey to Japanese Fluency!
Five years of studying Japanese, with tons of procrastination in between, had lead me to believe that language acquisition is not a walk in the park. At first I thought that since I'm quite interested in anime and J-pop, it would be as easy as learning English in elementary school (I do not even remember how I learned English). One day I don't know it. Next day I'm learning it in class. And then after sometime I can speak it! But unfortunately Japanese is nothing like that. I had to consciously drag myself out of the bed and away from the TV and beat myself up to go and focus on studying. I don't remember doing that in my English classes at all.
But then there's more bad news for me. Time went by and at some point I realized that I'm no longer interested in just Japanese. I wanted to learn Spanish, French, Dutch and Finnish – all while taking up Japanese. And so I did! "Some Dutch here and Spanish there will do the trick!" I keep on telling myself, and initially I thought I was saving time and effort juggling all of them up. But then in the end it dawned on me that I was making everything worse for myself. I wasn't having enough improvement in my Japanese and I was mixing up words of different languages.
My dear reader, don't let the same thing happen to you. In this article I will give you my ultimate list of do's and don'ts when learning a language or two, so you won't end up doing what I did wrong.
1. DO HAVE 1 TARGET LANGUAGE. It doesn't matter if you'd like to learn two, three or four languages. What is important is that you only have one main language to learn in a time frame you set for yourself. For example, if you wish to be fluent in Japanese in 4 years, make that your goal. Do not worry if after 4 years you're still not fluent in French and Korean. What matters is that you already have a mastery of one language under your belt. After 4 years, you can then start focusing on French, if that is the second language you want to master. Giving the same amount of time and attention to all languages you are interested in will not just confuse you and slow your pace, it will leave you unable to converse in any of the languages with fluency.
2. DO SET UP REALISTIC GOALS. It is common for language learners to imagine being fluent in a language in the shortest period of time possible, and when they don't meet their goal they get lazy and demotivated. Remember readers, there is nothing worse in language learning than lack of motivation and laziness! If you want to keep your drive up, it is better to set a realistic goal for yourself. Consider the time you have to study, all your commitments, and all your resources. One trick could be instead of keeping the overall goal "Be fluent in 4 years!" on top of your head (and get overwhelmed), you can divide the goal into smaller friendlier goals. For example you can learn 3 to 5 new words a day and use them in different sentences for 30 minutes on your way to school or during your lunch time. Then at night, make sure to spend at least 15 minutes reviewing those words in your head and thinking of songs and lines that use them so you can make sure that they will be stored in your long-term memory. That approach to language learning is better than chanting a mantra in your head that you have a 4 year deadline to follow.
3. DO CONSIDER TAKING REAL CLASSES. I know some of you would like to learn languages all by yourself to save money and time going to classes, but I highly suggest that you take a real language class at least in the beginning. I say this because the first few weeks of learning is the most crucial part. It is the time when you will learn all the basic rules and points that will guide you to compose more complex sentences. If you don't have a teacher or a professional to guide you on this stage, you might end up stumbling and falling along the way, thinking you got all the grammar right but actually saying the wrong things. Granted that people can still learn a language without a teacher around, I cannot stress enough the value of a real teacher. If you do opt to enroll in a class and find it beneficial, you can also consider taking classes till the end. Real time lessons surrounded by peers who are learning the same language can help you practice conversing with others using your target language – a chance you won't get if you are studying alone in the comfort of your room.
4. DO HAVE A NOTEBOOK WITH YOU AND A PEN. Most online sites would tell you that they can make you fluent in a language after 3 months just by subscribing to their program and listening to their podcasts. Though podcasts are great (and I use them a lot), do not underestimate the power of good old pen and paper. There are tons of researches out there that prove that those who write the information they gather on paper tend to remember more than those who just read and scroll articles and lessons over the internet. The combination of motion when writing and the image of a text or a drawing people themselves put on paper, tie information to the idea or image in their heads. The idea behind this is that we cannot ever say that we know something completely if we have never touched it or encountered it in real time and space.
Though a notebook and a pen can be a bit boring, trust me and use them when learning. If you find the plain old black pen dull, then use different colors when writing! You can even use crayons and water colors. Be unique. Play around with the words you're learning. Do some lettering or make your own jigsaw puzzle. Be creative!
5. DO HAVE FUN. Language learning is not supposed to be boring. Spice it up! Every Monday you can schedule listening to foreign songs and mimicking them. Pick up a word or two from the lyrics, look them up in the dictionary, see how they're used in the lyrics and then sing with the context in mind. Every Tuesday you can play with self-made flashcards of words you already learned in the past. And then on Wednesday you can try and translate a nursery rhyme to your target language. There are tons of ways to play around. All you have to do is think like a child and avoid thinking like language is a nemesis you have to defeat. Think of it as a game or as a puzzle. You can also choose to reward yourself every time you exceed your goals!
6. DO MAKE USE OF YOUR PHONE. I know for sure that you are married to your phone. You cannot go anywhere without it or you'll die. So since you're practically connected hips to hips with it, why not make the most out of it? Instead of watching a girl twerk and dance around half naked, why not look for fun language stuff in Youtube? Download songs and podcasts! You can listen to the songs while in the bus or while chilling at home. Instead of doing Candy crush (not that I'm against it), why not download word applications and games that focus on your target language? There are tons of apps out there! You just have to look for the right ones.
7. DO GRAB EVERY CHANCE YOU GET TO PRACTICE TALKING. The internet is littered with people who speak your target language as their native language. Without looking like a stalker or a creeper online, you can figure out a way to befriend some of those people and make them help you learn. There are many reliable language exchange (not dating site) websites out there that you can try and check. You can also join Facebook groups of other people like you who are learning the same language. The key is to only use your target language when communicating with people. Do not use the language you are comfortable using or the effort will be for naught. The best trick I have to share here is to look for someone who do not speak any of the languages you know, aside from your target. That will force you! You can also record yourself talking (or singing) and you might want to upload it on Youtube and let the netizens fry you with praise (or criticism). It all depends on you what road you want to take and how far you want to push yourself. The only thing I want you to remember is to never stop, and never slow down. Keep the engine running!
1. DON'T PROCRASTINATE. Procrastination is a language learner's number one enemy. Well, it's everyone's enemy, be it at work, at school, or at home. It never brought anyone any good and for sure you've seen it by experience. The more you procrastinate and leave a work undone, the more the work piles up, and the more overwhelmed you get when you realize that you haven't done anything. It's a giant snowball that hits no one else but you, and once you're buried, you're paralyzed and won't be able to do a thing but surrender. And we don't want that! So be strict with yourself. If you've made your language learning journey fun enough for your taste, you have no other excuse but to give it all your best to meet your daily and weekly goals. Use a calendar, a planner, set your phone on alarm, make your goal your computer or phone's wallpaper. It is your call! Just be sure that you do not put yourself in a situation where you will be the victim of your own laziness.
2. DON'T STAGNATE. After some time of studying, you'll find yourself quite comfortable with constructing sentences and conversing using basic language tools. Around that time you'd feel like you reached a learning plateau. From going up and up suddenly there's a halt and everything starts to move in slow-motion. It is like walking on the moon. Keep in mind that such thing happens in all situations. Learning curve drops around the middle and the only way to keep the pace up and running is by stepping up the challenge and adding some more unique twists and turns to your learning style. You can increase the number of words you target to learn every week, or you can increase your study time from 30 minutes a day to 1 hour day. The strategy is up to you. The key here is to challenge yourself. Your brain is already used to the way you're learning and it is bored. You just have to make it more interesting. But then read up on #3.
3. DON'T OVERDO IT. Sometimes when we feel like there's still enough space and memory in our heads to store information, we use it and stuff things to fill that space. Though there is nothing wrong with that, especially when we are on a learning curve plateau, sometimes we fail to notice that there really is no space left. It is just us feeling like there is still room for more (either because we're in a rush or we're having too much fun). No matter what the case is, do not overexert yourself. Think of your head as a glass of water. It can only hold so much information. If you pour more than it can handle, it will overflow and you're just going to waste all the time you've invested in learning for the day. Make sure that you're only taking what you can and leave some space for yourself to breathe and process all the information. It is better than feeling like you got more vocabulary under your belt, only to wake up the next say not remembering half of them.
4. DON'T BEAT YOURSELF UP. While I told you not to procrastinate and not to stagnate, I do not mean that you should go sulk and pull your hair out if you ever miss your goal for the day or the week. That will not help you a bit. Be kind and considerate to yourself, but promise that you will make up for the lessons you've missed. I would still stress that you should not procrastinate though. If you start thinking that you could always be considerate and loving to yourself, you will definitely procrastinate. There is a reason why I put procrastination in #1. Keep that in mind.
5. DON'T BE AFRAID TO MAKE MISTAKES. Mistake is the mother of success. No language learner ever in history has never made a mistake. All language students, teachers, and researchers make them even when they are already fluent. If you do not go out of your comfort zone and daringly use the language, you will never get the feedback from people around you. Remember, it is better for your mistakes to be noticed and corrected, than not be heard at all (because you will never know if they were wrong or right in the first place).
6. DON'T GIVE UP. Don't even think about it! The moment you do, it will definitely be the end. No language is easy to learn and definitely there will be bumps on the road, but do not let those discourage you. Remember that there are tons of people who are on the same journey with you. It is not just you who is struggling to learn. There are millions of others too. Trust me. There are also tons of other people who have been in the same situation as you but are now successful and fluent in the language they chose to learn! There are thousands of polyglots out there! They wouldn't be successful if they gave up.
If you ever find yourself in a slump, all you have to do is to try and remember what made you study the language in the first place. Was it because it is your dream to move to a country that speaks it? Was it because of a loved one who's fluent in it? Whatever it is that gave you the drive to even start learning, keep that in mind! Look back to the past and see how far you've come from being a beginner with zero knowledge to the present version of yourself who can speak the language fairly. Would you have it any other way?
I can tell you by experience that I've been there too, and I wanted to give up. But as silly as it sounds what made me go on was Dory (as in Dory in Finding Nemo). She has very poor memory, but because she is playful and with a positive outlook, she made it a long way! And according to her, when you're lost and don't know where to go, then all you have to do is just keep swimming. As silly as it sounds, that seriously is what made me continue. And even up to this day, I am still trying to learn languages with that Dory lesson in mind.
Alright? Always remember it can be done, so don't give up!
Asians and their Oh So Beloved: Rice?


A SELECTION OF (CURRENTLY – 19/04/18) 8 NEW-BUILD, DETACHED VILLAS WITH PRIVATE SWIMMING POOLS FROM 303.500€ TO 330.800€ WITH BOTH SINGLE STOREY AND TWO STOREY ON OFFER. PLEASE ASK ABOUT CURRENT AVAILABILITY! COMPLETION DUE JULY AND SEPTEMBER 2018.
All properties will come equipped with high-quality finishes throughout, including a fully fitted kitchen with appliances, bathrooms include sanitary devices Duravit brand, an extra flat shower plate and the brand for the taps is Hansgrohe, all top quality, preinstallation for air conditioning, triple glazed windows, low-energy LED lighting and more, so you can be assured that the property is going to be a solid build and practically ready to move in with the only requirement being to furnish the property.
Inside, the properties boast a very modern and open-plan feel, with the main space, consisting of living room, dining room, and kitchen, being dominated by large patio doors leading outside to the garden allowing for copious amounts of light to flood in. All properties have a total of three spacious bedrooms and two bathrooms.
To reserve, a 6.000€ deposit is required, then this must be increased to 40% before notary. The 40% can be paid in stages at different times or in one lump sum and the rest to be paid upon signing at the notary.


372552 is the high school code for Community Christian School. Use this code in registering for the ACT or SAT test. Scroll down for ACT and SAT testing dates.
ACT has implemented two new opportunities for students to gain information about colleges and the ACT test. First is the ACT College Search app for iPhone and iPod users (free download) to search for colleges and universities through five filters: locations, type, size, admission, and cost. Once users have entered their preferences on those criteria, the application returns results based on their choices. Users can add colleges to their favorites list as they move through the results. Each college or university page provides a map of the school's location and links to the school's website and its social media networks. Other information provided includes degrees offered, tuition, student demographics, admission requirements, average first-year student ACT scores, and housing options.
Another free application - the ACT Student app - offers clusters of information about the ACT test. Users can access practice questions and helpful tips. They can also access limited information specific to them about their registration and score status. Short videos providing more information, such as which items to bring to the test center on test day, are available on this mobile application.
OK City Community College is offering several low cost ACT workshops. $20.00 fee. Students can register on line for the 2018-2019 Workshops. Are you ready to take the ACT®? Gain confidence in all four components of the test by taking this course designed to review your skills. Community Outreach and Education offers ACT® preparation workshops that will review terminology, concepts and test-taking tips. Students will also become familiar with the test structure through practice tests. The attached PDF has all the information for the student to register.
PRE-ACT will be given to all 9th and10th graders at a cost of $15.00. The test will be given in the spring, Monday, April 8, 2019.
ACT ASSESSMENT AND ACT ASSESSMENT PLUS WRITING - Choose either of two test options. The ACT Assessment Plus Writing includes a 30 minute writing test. Not all colleges and universities require a Writing Test. Check with the colleges you are considering or go to www.actstudent.org for a list of institutions that require or recommend the Writing Test.
ACT ACADEMY - ACT Academy™ is a free online learning tool and test practice program designed to help you get the best score possible on the ACT test, and well on your way to college and career success.
ACT ONLINE PREP CLASSES - Available at www.actstudent.org for $39.95 for a six month subscription.
ACT PRACTICE BOOKS - Practice books may be picked up outside Mrs. Stephens' office. Several Peterson ACT Practice Books are available for check out from Mrs. Stephens for a two week period.
This workshop includes instruction on the English, Math, Critical Reading, and Science Reasoning sections of the exam, the ACT Handbook with the "secrets" to scoring high on the exam, and guided practice over all sections.
ACT Practice Exam – Students will take a practice ACT from beginning to end with actual time and environment restrictions. Utilize the results provided from the workshop to immediately see how the new practices Ms. Harris provides help you. You will also develop an understanding of areas to focus on for needed improvement and learning.
ACT PREPARATION COURSES - Precollegiate programs at OU help ensure each student's performance on the ACT test is an accurate measure of his or her acquired knowledge and skill in the content areas. The course will: Explain how to use time effectively, understand directions, read the questions, and review answer Suggest strategies that are particular to the types of questions Feature lectures with practice tests to review content, topics and types of questions found on the ACT test. Course dates:: Registration forms may be found at youth.ou.edu.
SAT Skills Insight helps students familiarize themselves with and practice for the SAT. By visiting www.collegeboard.com/satskillsinsight, students can access this online tool that categorizes the types of skills demonstrated within a particular score band and also provides suggestions for improvement.
My SAT Online Score Report shows students the meaning behind their numbers. It includes information about percentiles, a copy of the student's essay and details about her or his performance on different types of questions.
SAT PREP PLAN SITE - Free SAT study content including streaming video, a custom study plan creator, and many practice problems are located at www.satpreplan.com.
SAT PRACTICE BOOKS - Practice books may be picked up outside Mrs. Stephens' office. A Collegeboard SAT Practice Book is available for check out from Mrs. Stephens for a two week period.


Bird Flu. Seasonal Flu. Avian Flu. Pandemic. Preparedness or paranoia?
About every 50 years, the world is exposed to a pandemic form of the flu virus - a virus which has evolved into something much more serious than seasonal influenza. Past pandemics were lethal to healthy, young people as well as the old and infirm. Explore how one of our tiniest enemies can evolve from benign germ to killer.
Learn the difference between seasonal, bird and pandemic flu. Understand what you can do to protect your family from all types of flu viruses. The statistics are real. The questions are many.
HealthSmart Flu: Nothing to Sneeze At will educate you on what you can do to prepare for this year's cold and flu season.
Check out the program at the bottom of this post!
Perspective. Insight. Reality TV for the greater good. That's HealthSmart. WITF's Mid-Atlantic Emmy nominated, in-studio, locally produced 30-minute health show features the medical topics that are on everyone's mind.
Local experts give practical advice on how to beat the odds and live healthier and happier, everyday. And local survivors share their touching stories to help our audience understand various diseases and conditions. Intended to appeal to health-conscious viewers, HealthSmart targets anyone who is looking for practical "real life" ways to live healthier everyday. The focus of the series is on useful information that can be incorporated into daily life. Watch the program at the end of this post.
HealthSmart is made possible by Capital BlueCross.


Mabuhay Holdings Corporation has full confidence in the continuing rapid growth of the Philippine economy and envisions a bright future for investing in the real property and equity markets in the country.
only to the benefit of the company but to the investing public as a whole.
With the vision and leadership of the Management, dedication of the employees and the continuing support of the Board of Directors, every challenge shall be met and overcome, and success attained as the transformation towards a stronger Mabuhay continues.