bla

###### Contents

  1. 1 Estimation of Historical volatility
    1. 1.1 Introduction
    2. 1.2 First assumptions
    3. 1.3 Volatility estimators
      1. 1.3.1 Moving Average
      2. 1.3.2 Autoregressive models
      3. 1.3.3 Modification of the GARCH model
      4. 1.3.4 Advanced volatility measures
    4. 1.4 A case study
      1. 1.4.1 Normality test
      2. 1.4.2 Autocorrelation test
      3. 1.4.3 Ljung - Box test
      4. 1.4.4 Volatility of the estimators
    5. 1.5 GARCH and EWMA on a daily estimation of the parameters
    6. 1.6 Spurious effects in Moving Average estimation
    7. 1.7 Conclusion
  2. 2 Allocation strategies using Variance Swaps
    1. 2.1 Why derivatives on volatility
    2. 2.2 Variance Swaps
    3. 2.3 Replicating Variance Swaps
      1. 2.3.1 Approximated replication and Corridor Variance Swap
    4. 2.4 Variance Swaps under stochastic volatility
      1. 2.4.1 The SABR model
      2. 2.4.2 The Heston model
      3. 2.4.3 The Stein Stein model
      4. 2.4.4 The $\lambda$-SABR model
    5. 2.5 A brief comparison of the pricing methods
    6. 2.6 Mark to Market price
    7. 2.7 Back test on a long strategy
    8. 2.8 Stationarity test on implied volatilities and Variance swaps
      1. 2.8.1 Correlation between implied volatilities and Variance Swap
      2. 2.8.2 Stationarity test on implied volatilities: the Dickey - Fuller test
    9. 2.9 Spread on Variance Swaps
      1. 2.9.1 A case study
    10. 2.10 Final analysis
      1. 2.10.1 Risk premium on Variance Swaps
      2. 2.10.2 Mean reversion on spreads of Variance Swaps

## Description of the company

During my internship, I was part of the Quantitative team of the Société
Générale Cross Asset Research division. I was based in London, while other
parts of the team were in Paris, Hong Kong and Bangalore.
The internship period was from 2016, May $2^{nd}$ to 2016, November, $4^{th}$.
My supervisors were Julien Turc, the head of the team, based in Paris, and
Lorenzo Ravagli, based in London.
The division focuses on equity, cross asset, commodities, economic, credit
analysis, fixed income and forex research. It employs a Cross-Asset Research
approach involving analysis of the impact of major events on different asset
classes, assessment of the links between asset classes, and strategic
synthesis of this key information to provide an integrated view. The firm has
global expertise on economics, quantitative research, strategy and asset
allocation, commodities, credit, currencies, rates, emerging markets, equity,
derivatives, technical analysis and Socially-Responsible Investment. The team
offers daily, weekly and monthly multi asset research publications.
It offers reports dedicated to the entire asset class range, as well as
regional and thematic focused reports. The firm caters to asset managers.
Société Générale Cross Asset Research is based in Paris, France with
additional offices in Bangalore, India; Hong Kong, Hong Kong; London, The
United Kingdom; Madrid, Spain; Milan, Italy; Moscow, Russia; New York, New
York; Prague, Czech Republic; Singapore; Tokyo, Japan and Warsaw, Poland.

bla

##  1 Estimation of Historical volatility

###  1.1 Introduction

A common comparison in the banking system is between the implied volatility of
an option and the historical / realized volatility of the underlying. This is
a useful tool that gives a hint on the fact that the market is pricing in a
good way the option, with respect to the behavior of the underlying in the
physic world.
Basically, if the implied volatility is higher than the historical volatility,
it means that the derivative is more expensive then what it should be, so
selling it would be a good trading choice, while if the implied volatility is
lower than the historical volatility, the derivative is underestimated by the
market, so buying it would be a good strategy.
This trading strategy, which works only on single assets, is not something
completely formal or based on technical conditions, but is able to give an
immediate and easy to understand trading opportunity, based only on what the
market price is and what the behavior of an asset in the real world is.
While there is only one implied volatility, which is given by the market,
there are many different techniques for the estimation of the historical
volatility. This fact raises a question on which is the best method for the
estimation of the volatility, and what properties should a good estimator
have.
Recently, the development of the IT infrastructure in the banking world has
given a lot of popularity and interest to the High Frequency environment.
Nowadays, big prices database can be used in real time, and more information
usually means a better estimation of the desired quantity.
Nevertheless, in this section we will not focus on the High Frequency
estimation of historical volatility, but we will study different techniques
which use intra day prices. We will compare them with more common techniques,
as the numerous modifications of the autoregressive GARCH model, and we will
focus on some of the properties of these estimators.
This comprehensive study could be also used in the view of forecasting of
volatility, even if the possibility of predict volatility is out of the scope
of this part, and so will not be developed.
The number of historical days for the historical volatility calculation
changes the calculation, in addition to the estimate of the drift (or average
amount stocks are assumed to rise). There should, however, be no difference
between the average daily or weekly historical volatility. We also examine
different methods of historical volatility calculation, including close-to-
close volatility and exponentially weighted volatility, in addition to
advanced volatility measures.
A comprehensive study of this topic can be found in [11] and [10]

###  1.2 First assumptions

Volatility is defined as the annualized standard deviation of log returns, and
the usual measure for historical volatility is given by the log returns from
the close-to-close prices.
The calculation for standard deviation calculates the deviation from the
average log return (or drift). This average log return has to be estimated
from the sample, which can cause problems if the return over the period
sampled is very high or negative. As over the long term very high or negative
returns are not realistic, the calculation of volatility can be corrupted by
using the sample log return as the expected future return.
In theory, the expected average value of an underlying at a future date should
be the value of the forward at that date, because of non arbitrage conditions.
As for all normal interest rates the forward return should be close to $100\%$
(for any reasonable sampling frequency i.e. daily/weekly/monthly). Hence for
simplicity reasons it is easier to assume a zero log return as $\ln(100\%)=0$.
Since, in addition, returns are normally close to $100\%$ (the ratio between
the value of today and the value of the previous trading day), the log returns
are similar to the return minus $1$, due to the fact that $\ln(1+x)\sim x$
when $x$ is small.
If the return over the period is assumed to be the same for all periods, and
if the mean return is assumed to be zero , that is confirmed by empirical
data, since it is normally very close to zero, the standard deviation of the
percentage change is simply the absolute value of the percentage return. Hence
an underlying which moves $1\%$ has a volatility of $1\%$ for that period. As
volatility is usually quoted on an annualized basis, this volatility has to be
multiplied by the square root of the number of samples in a year, that is
$252$ trading days in a year, the usual assumption in the industry.
While historical volatility can be measured monthly, quarterly or yearly it is
usually measured daily or weekly. Normally, daily volatility is preferable to
weekly volatility as $5$ times as many data points are available. However, if
volatility over a long period of time is being examined between two different
markets, weekly volatility could be the best measure to reduce the influence
of different public holidays (and trading hours). If stock price returns are
independent, then the daily and weekly historical volatility should on average
be the same. If stock price returns are not independent, there could be a
difference. Autocorrelation is the correlation between two different returns
so independent returns have an autocorrelation of $0\%$.

###  1.3 Volatility estimators

####  1.3.1 Moving Average

The first and most simple estimator for the volatility that we will analyze is
the moving average estimator (MA). This basic technique consists in using a
moving window for the estimation of the volatility: at day $i$, the volatility
is the standard deviation of the returns of the $N$ previous days. A common
choice is $N=21$ days or $N=63$ days. The drawback of this estimator is the
great instability in presence of shocks in the returns. Once a shock enters or
leaves the moving window, there will be correspondingly a great volatility,
even if in that precise day nothing has happened.
Because of this fact, and because of the lack of flexibility of this method,
moving average techniques are not used in practice, but we will study them to
see their behavior in different conditions.
When the window is as large as the data set, the moving average estimator
becomes the constant volatility estimator.

####  1.3.2 Autoregressive models

The second model we will look at is the ARCH model, which stands for Auto
Regressive Conditional Heteroscedasticity model. The AR comes from the fact
that these models are autoregressive models in squared returns, which we will
see later in this section. The conditional comes from the fact that, in these
models, the information of the following period depends conditionally on the
information present at this period. Heteroscedasticity means non constant
volatility, which is a fundamental point.
Let us assume that the return on an asset is
$r_{t}=\mu+\sigma_{t}\varepsilon_{t}$, where $\mu$ is the mean return,
$\sigma_{t}$ is the volatility at time $t$, and $\varepsilon_{t}$ is a
sequence of standard Gaussian random variables, independent of $\sigma_{t}$.
The residual return at time $t$ is
$a_{t}=r_{t}-\mu=\sigma_{t}\varepsilon_{t}$.
In the first ARCH model, developed in [1], the volatility is modeled in this
way:

$\sigma_{t}^{2}=\alpha_{0}+\alpha_{1}a_{t-1}^{2}.$ (1)

We can see right away that a time varying $\sigma_{t}^{2}$ will lead to fatter
tails, relative to a normal distribution, in the unconditional distribution of
$a_{t}$.
In fact, it is easy to prove that the kurtosis of the residual returns is
greater than $3$, that corresponds to the kurtosis of a Normal random
variable:

$kurt(a_{t})=\frac{\mathbb{E}[\sigma_{t}^{4}]\mathbb{E}[\varepsilon_{t}^{4}]}{(\mathbb{E}[\sigma_{t}^{2}])^{2}(\mathbb{E}[\varepsilon_{t}^{2}])^{2}}=3\frac{\mathbb{E}[\sigma_{t}^{4}]}{(\mathbb{E}[\sigma_{t}^{2}])^{2}}>3,$

where the last inequality is given by the Jensen inequality, and

$Var(a_{t})=\frac{\alpha_{0}}{1-\alpha_{1}}$

In an ARCH(1) model, the variance of the next period only depends on the
squared residual of the last period so a crisis that caused a large residual
would not have the sort of persistence that we observe after actual crises.
This has led to an extension of the ARCH model to a GARCH, or Generalized ARCH
model, first developed in [2], which is similar in spirit to an ARMA model. In
a GARCH(1,1) model,

$\sigma_{t}^{2}=\alpha_{0}+\alpha_{1}a_{t-1}^{2}+\beta_{1}\sigma_{t-1}^{2}.$
(2)

Since the unconditional variance of returns is

$\mathbb{E}(\sigma_{t}^{2})=\frac{\alpha_{0}}{1-\alpha_{1}-\beta_{1}},$

we can write the GARCH(1,1) equation yet another way:

$\sigma_{t}^{2}=(1-\alpha_{1}-\beta_{1})\mathbb{E}(\sigma^{2})+\alpha_{1}a_{t-1}^{2}+\beta_{1}\sigma_{t-1}^{2}.$

Written in this form, it is easy to see that next period’s conditional
variance is a weighted combination of the unconditional variance of returns,
$\mathbb{E}(\sigma^{2})$, squared residuals of the last period, $a_{t-1}^{2}$
, and the conditional variance of the last period, $\sigma_{t-1}^{2}$ , with
weights $(1-\alpha_{1}-\beta_{1}),\alpha_{1},\beta_{1}$ which sum to one.

An alternative measure could be to use an exponentially weighted moving
average model, which is shown below.

$\sigma_{t}^{2}=\lambda\sigma_{t-1}^{2}+(1-\lambda)a_{t-1}^{2}.$

The parameter $\lambda$ is between $0$ (effectively $1$ day volatility) and
$1$ (ignore current vol and keep vol constant). Normally, values of $0.94$ or
$0.97$ are used.
Exponentially weighted volatilities are rarely used, partly due to the fact
they do not handle regular volatility driving events such as earnings very
well. Previous earnings jumps will have least weight just before an earnings
date , when future volatility is most likely to be high, and most weight just
after earnings, when future volatility is most likely to be low. It could,
however, be of some use for indexes and equities.
Exponential weighted volatility has the advantage over standard historical
volatility in that the effect of a spike in volatility gradually fades, as
opposed to suddenly disappearing causing a collapse in historic volatility.
For example, if we are looking at the historical volatility over the past
month and a spike in realized volatility suddenly occurs the historical
volatility will be high for a month, then collapse. Exponentially weighted
volatility will rise at the same time as historical volatility, and then
gradually decline to lower levels, arguably in a similar way to how implied
volatility spikes, then mean reverts.

####  1.3.3 Modification of the GARCH model

One empirical observation is that, in many markets, the impact of negative
price moves on future volatility is different from that of positive price
moves. This is particularly true in equity markets.
It is clear that not only does the magnitude of $a_{t}^{2}$ affects future
volatility, but the sign of the residual return also affects future
volatility, at least for equities. It is not clear why volatility should
increase more when the level of stock prices drop compared to a stock price
rise. In fact, on one hand, as stocks drop, the debt/equity ratios increase
and stocks become more volatile with higher leverage ratios. But the changes
in volatility associated with stock market drops are much larger than that
which could be explained by leverage alone. One model which can take in
account this asymmetry is the Threshold GARCH (TGARCH) model, also known as
the GJR-GARCH model, based on the work in [3]:

$\sigma_{t}^{2}=\alpha_{0}+\alpha_{1}a_{t-1}^{2}+\gamma_{1}1_{a_{t-1}^{2}<0}a_{t-1}^{2}+\beta_{1}\sigma_{t-1}^{2}.$
(3)

This method is of course more flexible than the original GARCH model, but it
is also more difficult to estimate, since we add a parameter in the model.
Another variant on GARCH to account for the asymmetry between up and down
moves described above is the EGARCH model of [4]. In an EGARCH

$\ln(\sigma_{t}^{2})=\alpha_{0}+\frac{\alpha_{1}a_{t-1}^{2}+\gamma_{1}|a_{t-1}|}{\sigma_{t-1}}+\beta_{1}\ln(\sigma_{t-1}^{2}).$
(4)

There are different other variations of asymmetric GARCH models which can be
found in the literature, but we will focus only on these ones, analyzing their
properties.

####  1.3.4 Advanced volatility measures

The increasing availability of high-frequency price time series has permitted
the development of new integrated variance estimators that are more efficient
than the realized variance estimator.
Close-to-close volatility is usually used as it has the benefit of using the
closing auction prices only. Should other prices be used, then they could be
vulnerable to manipulation or a "fat fingered" trade. However, a large number
of samples need to be used to get a good estimate of historical volatility,
and using a large number of closing values can obscure short-term changes in
volatility. There are, however, different methods of calculating volatility
using some or all of the open (O), high (H), low (L) and close (C).
For most markets, intra day volatility is greatest just after the open (as
results are often announced around the open) and just before the close
(performance is often based upon closing prices). Intra day volatility tends
to sag in the middle of the day, due to the combination of a lack of
announcements and reduced volumes/liquidity due to lunch breaks. For this
reason using an estimate of volatility more frequent than daily tends to be
very noisy. Traders who wish to take into account intra day prices should
instead use an advanced volatility measure.
Before introducing new volatility measures, it is important also to have a
measure of the goodness of the estimation. The efficiency is the number used
to test this property, and it is defined as the variance of the most basic
estimator, the close to close, divided by the variance of the new estimator:

$eff_{\text{estimator}}=\frac{var\left(\sigma_{\text{close to
close}}\right)}{var\left(\sigma_{\text{estimator}}\right)}.$

Since the aim of the estimators is to have the biggest efficiency, we are
looking for low variance estimators.
A comprehensive study of this estimators can be found in [11], [12] and [9].
In the following, $c_{i},l_{i},h_{i},o_{i}$ are, respectively, the closing,
low, high and closing price at day $i$, $N$ is the number of trading days
considered.

  * •

Close to close (C): The most common type of calculation that benefits from
only using reliable prices from closing auctions. By definition its efficiency
is $1$ at all times.

$\sigma_{\text{close to
close}}=\sqrt{\frac{1}{N}}\sqrt{\sum_{i=1}^{N}\left(\ln\left(\frac{c_{i}}{c_{i-1}}\right)^{2}\right)}.$
(5)

  * •

Parkinson (HL), see [5]: The first advanced volatility estimator was created
by Parkinson in 1980, and instead of using closing prices it uses the high and
low price. One drawback of this estimator is that it assumes continuous
trading, hence it underestimates the volatility as potential movements when
the market is shut are ignored. As this estimate only uses the high and low
price for an underlying, it is less sensitive to differences in trading hours.
For example, as the time of the EU and US closes are approximately half a
trading day apart, they can give very different returns. Using the high and
low means the trading over the whole day is examined, and the days overlap. As
it does not handle jumps, on average it underestimates the volatility, as it
does not take into account highs and lows when trading does not occur
(weekends, between close and open). Although it does not handle drift, this is
usually small. The Parkinson estimate is up to $5.2$ times more efficient than
the close to close estimate. While other measures are more efficient based on
simulated data, some studies have shown it to be the best measure for actual
empirical data.

$\sigma_{\text{Parkinson}}=\sqrt{\frac{1}{N}}\sqrt{\frac{1}{4\ln
2}\sum_{i=1}^{N}\left(\ln\left(\frac{h_{i}}{l_{i}}\right)\right)^{2}}.$ (6)

  * •

Garman-Klass (OHLC), see [6]: Later in 1980 the Garman-Klass volatility
estimator was created. It is an extension of Parkinson which includes opening
and closing prices. If opening prices are not available the close from the
previous day can be used instead. As overnight jumps are ignored the measure
underestimates the volatility. This estimate is the most powerful for stocks
with Brownian motion, zero drift and no opening jumps, i.e. opening price is
equal to closing price of previous period. While it is up to $7.4$ times as
efficient as the close to close estimate, it also underestimates the
volatility, as like Parkinson it assumes no jumps.

$\sigma_{\text{Garman-
Klass}}=\sqrt{\frac{1}{N}}\sqrt{\frac{1}{2}\sum_{i=1}^{N}\left(\ln\left(\frac{h_{i}}{l_{i}}\right)\right)^{2}-(2\ln
2)\left(\ln\left(\frac{c_{i}}{o_{i}}\right)\right)^{2}}.$ (7)

  * •

Rogers-Satchell (OHLC), see [7]: Securities which have a drift, or non-zero
mean, require a more sophisticated measure of volatility. The Rogers-Satchell
volatility created in the early 1990s is able to properly measure the
volatility for securities with non-zero mean. It does not, however, handle
jumps, hence it underestimates the volatility. The efficiency of the Rogers-
Satchell estimate is similar to that for Garman-Klass, however it benefits
from being able to handle non-zero drift. Opening jumps are, however, not
handled well.

$\sigma_{\text{Rogers-
Satchell}}=\sqrt{\frac{1}{N}}\sqrt{\sum_{i=1}^{N}\ln\left(\frac{h_{i}}{c_{i}}\right)\ln\left(\frac{h_{i}}{o_{i}}\right)+\ln\left(\frac{l_{i}}{c_{i}}\right)\ln\left(\frac{l_{i}}{o_{i}}\right)}.$
(8)

  * •

Garman-Klass Yang-Zhang extension (OHLC), see [8]: Yang-Zhang modified the
Garman-Klass volatility measure in order to let it handle jumps. The measure
does assume a zero drift, hence it will overestimate the volatility if a
security has a non-zero mean return. As the effect of drift is small, the fact
continuous prices are not available usually means it underestimates the
volatility, but by a smaller amount than the previous alternative measures. It
has an efficiency of $8$ times the close to close estimate.

$\sigma_{\text{GKYZ}}=\sqrt{\frac{1}{N}}\sqrt{\sum_{i=1}^{N}\left(\ln\left(\frac{o_{i}}{c_{i-1}}\right)\right)^{2}+\frac{1}{2}\left(\ln\left(\frac{h_{i}}{l_{i}}\right)\right)^{2}-(2\ln
2)\left(\ln\left(\frac{c_{i}}{o_{i}}\right)\right)^{2}}.$ (9)

  * •

Yang-Zhang (OHLC), see [8]: The most powerful volatility estimator which has
minimum estimation error. In 2000 Yang-Zhang created a volatility measure that
handles both opening jumps and drift. It is the sum of the overnight
volatility (close-to-open volatility) and a weighted average of the Rogers-
Satchell volatility and the open-to-close volatility. The assumption of
continuous prices does mean the measure tends to slightly underestimate the
volatility. It is up to a maximum of 14 times as efficient as the close to
close estimate.

$\sigma_{\text{Yang-
Zhang}}=\sqrt{\sigma^{2}_{\text{overnight}}+k\sigma^{2}_{\text{open to
close}}+(1-k)\sigma^{2}_{\text{Rogers - Satchell}}},$ (10)

where

$\sigma^{2}_{\text{overnight}}=\frac{1}{N-1}\sum_{i=1}^{N}\left(\ln\left(\frac{o_{i}}{c_{i-1}}\right)-\overline{\ln\left(\frac{o_{i}}{c_{i-1}}\right)}\right)^{2},$
$\sigma^{2}_{\text{open to
close}}=\frac{1}{N-1}\sum_{i=1}^{N}\left(\ln\left(\frac{c_{i}}{o_{i}}\right)-\overline{\ln\left(\frac{c_{i}}{o_{i}}\right)}\right)^{2},$
$k=\frac{10.34}{1.34+\frac{N+1}{N-1}}.$

While efficiency (how volatile the measure is) is important, so too is the
bias of the estimator. Bias depends on the sample size, and the type of
distribution the underlying security has.
Generally, the close-to-close volatility estimator is too big, since it does
not model overnight jumps, while alternative estimators are too small, as they
assume continuous trading, and discrete trading will have a smaller difference
between the maximum and minimum.
The key variables which determine the bias are:

  * •

Sample size: As the standard close-to-close volatility measure suffers with
small sample sizes, this is where alternative measures perform best.

  * •

Volatility of volatility: While the close-to-close volatility estimate is
relatively insensitive to a changing volatility (vol of vol), the alternative
estimates are far more sensitive. This bias increases the more vol of vol
increases , i.e. more vol of vol means a greater underestimate of volatility.

  * •

Overnight jumps between close and open: Approximately one-sixth of equity
volatility occurs outside the trading day. Overnight jumps cause the standard
close-to-close estimate to overestimate the volatility, as jumps are not
modeled. Alternative estimates which do not model jumps (Parkinson, Garman
Klass and Rogers-Satchell) underestimate the volatility. Yang-Zhang estimates,
both YangZhang extension of Garman Klass and the Yang-Zhang measure itself,
will converge with standard close-to-close volatility if the jumps are large
compared to the overnight volatility.

  * •

Drift of underlying: If the drift of the underlying is ignored as it is for
Parkinson and Garman Klass, and the Yang Zhang extension of Garman Glass, then
the measure will overestimate the volatility. This effect is small for any
reasonable drifts, i.e. if we are looking at daily, weekly or monthly data.

  * •

Correlation daily volatility and overnight volatility: While Yang-Zhang
measures deal with overnight volatility, there is the assumption that
overnight volatility and daily volatility are uncorrelated. Yang-Zhang
measures will underestimate volatility when there is a correlation between
daily return and overnight return, and vice versa, but this effect is small.

###  1.4 A case study

While all the works in literature focus on the statistical, and so
theoretical, efficiency of the various estimators, we study the
autocorrelation of the residuals in order to test which method gives a better
non correlated volatility series. All the figures and the tables here are
related on the SPX. The Standard & Poor’s 500, often abbreviated as the S&P
500, or just "the S & P", is an American stock market index based on the
market capitalizations of 500 large companies having common stock listed on
the NYSE or NASDAQ. Data are from January 1999 to June 2016, and are
downloaded from Bloomberg.
We introduce different types of test, which are based on the estimated
residuals

$res_{t,i}=\frac{ret_{t}}{\sigma_{t,i}},$

where $i$ is the method chosen, $ret_{t}$ is the series of the log returns of
the spot price at time $t$, and $\sigma_{t,i}$ is the value of the volatility
estimator at time $t$:

  1. 1.

Constant volatility

  2. 2.

Parkinson

  3. 3.

Garman Klass

  4. 4.

Rogers Satchell

  5. 5.

German Klass Yang Zhang

  6. 6.

Yang Zhang

  7. 7.

Moving Average 21 days

  8. 8.

Moving Average 63 days

  9. 9.

Garch (1,1)

  10. 10.

Asymmetric Garch (1,1)

  11. 11.

Exponential Garch (1,1)

  12. 12.

Exponential Weighted Moving Average (1,1)

####  1.4.1 Normality test

We first check the normality of the series of the residuals, studying the QQ-
plot and the skewness and kurtosis parameters. Remember that, in the case of a
Gaussian series, the skewness would be $0$ and the kurtosis equal to $3$.

Figure 1: QQ plot, Skewness and Kurtosis parameters in the SPX asset for all
the estimators.

We can immediately see that all the Garch estimators and the Moving Average
estimators perform better in term of skewness and kurtosis. There are
estimators that perform better in term of skewness (see the Rogers Satchell
estimator), but really badly from a kurtosis point of view.
It is important to understand that the data frame is of 17 years. It could be
an interval too important, but the interest of the study is to give a more
general overview on the possible estimation techniques, rather than building
or choosing the best estimator for a given asset in a given time period.

####  1.4.2 Autocorrelation test

We study then the behavior of the series of the residuals, testing the
autocorrelation function. The autocorrelation of a random process is the
correlation between values of the process at different times, as a function of
the time lag. Since all the computations are done in R, the default time lag
is $10\log_{10}N$, where $N$ is the number of observations. Assuming the
hypothesis of the residuals to be close to white noise, we study the
autocorrelation of the series of the residuals and of the squared of the
residuals.

Figure 2: Autocorrelation test for the series of $res_{t,i}$ in the SPX asset
for all the estimators.

From the study of the autocorrelation of the returns in Figure 2 we cannot
have much information. The results are almost the same for every estimator, so
this test does not give much information about the goodness of the methods.

Figure 3: Autocorrelation test for the series of $(res_{t,i})^{2}$ in the SPX
asset for all the estimators.

If we focus on the autocorrelation of the squared returns in Figure 3, we see
immediately how the constant volatility is, as highly expectable, not a good
choice. Also the (static) moving average choice is really poor, since it
appears that the autocorrelation is influenced by the natural construction of
the estimator. Both advanced measures and GARCH based estimators seem
promising from this aspect.

####  1.4.3 Ljung - Box test

The Ljung - Box test (named for Greta M. Ljung and George E. P. Box) is a type
of statistical test of whether any of a group of autocorrelations of a time
series are different from zero. Instead of testing randomness at each distinct
lag, it tests the "overall" randomness based on a number of lags.
The Ljung - Box test may be defined as:
$H_{0}$: The data are independently distributed, i.e., the correlations in the
population from which the sample is taken are $0$, so that any observed
correlations in the data result from randomness of the sampling process.
$H_{a}$: The data are not independently distributed; they exhibit serial
correlation.
The test statistic is:

$Q=N(N+2)\sum_{i=1}^{h}\frac{\hat{\rho}^{2}_{i}}{N-i},$

where $N$ is the sample size, $\hat{\rho}^{2}_{i}$ is the sample
autocorrelation at lag $i$, and $h$ is the number of lags being tested. Under
$H_{0}$ the statistic $Q$ follows a $\chi_{(h)}^{2}$ distribution. For
significance level $\alpha$, the critical region for rejection of the
hypothesis of randomness is

$Q>\chi^{2}_{1-\alpha,h},$

where $\chi^{2}_{1-\alpha,h}$ is the $\alpha$-quantile of the chi-squared
distribution with $h$ degrees of freedom.
If the p value is under the threshold of $5\%$, the test accepts the
hypothesis of autocorrelation of the time series. The test is performed with
three different lags.

| lag = 10 | p value | lag = 15 | p value | lag = 20 | p value
---|---|---|---|---|---|---
Constvol | 3505.63 | 0.00 | 4913.57 | 0.00 | 6086.93 | 0.00
P | 103.92 | 0.00 | 113.66 | 0.00 | 115.56 | 0.00
GK | 31.32 | 0.00 | 35.96 | 0.00 | 37.12 | 0.01
RS | 2.05 | 1.00 | 2.42 | 1.00 | 3.53 | 1.00
GKYZ | 32.26 | 0.00 | 38.20 | 0.00 | 39.21 | 0.01
YZ | 21.71 | 0.02 | 22.38 | 0.10 | 23.27 | 0.28
MA21days | 44.66 | 0.00 | 60.92 | 0.00 | 97.31 | 0.00
MA63days | 415.37 | 0.00 | 438.82 | 0.00 | 446.16 | 0.00
Garch | 21.50 | 0.02 | 24.04 | 0.06 | 24.64 | 0.22
GJRGarch | 22.43 | 0.01 | 25.78 | 0.04 | 26.72 | 0.14
eGarch | 25.97 | 0.00 | 30.74 | 0.01 | 33.39 | 0.03
EWMA | 26.54 | 0.00 | 30.04 | 0.01 | 31.29 | 0.05
Table 1: Ljung - Box test with three different lags in the SPX asset for all
the estimators.

Analyzing Table 1, a high $p$ value seems to indicate absence of correlation
in the series of the residuals, but the test does not prove this fact.
We can see how the Moving Average estimation is really poor when it comes to
analyzing the autocorrelation of the residuals. All the Advanced volatility
measures, using intra day prices, perform quite well, and their ability to get
uncorrelated residuals is very interesting.
Rogers Satchell seems to perform incredibly well, and also Yang Zhang is a
good estimator, following the results of this test.

####  1.4.4 Volatility of the estimators

Even if the advanced estimators are constructed in order to reduce their own
volatility with respect to the close to close variance, the fact that they not
depend on the information of the past, but only on the prices of the day, make
them more volatile than the autoregressive estimators, which are constructed
in order to depend from the past in different ways.

Figure 4: Time series of $\sigma_{t,i}$ in the SPX asset for all the
estimators.

We can immediately see from Figure 4 that the advanced estimators are
incredibly volatile, if compared with the Moving Average estimators and the
GARCH estimators, since the last ones present a smoothing effect given by the
regressive part. Inspired by the figure, we introduce a measure of volatility
of the time series. Assuming Gaussian properties of the volatility series, we
compute the standard deviation of the "log returns of the volatilities" for
all estimators and all assets.

ConstVol | Parkinson | Garman Klass | Rogers Satchell | GKYZ | Yang Zhang
---|---|---|---|---|---
0.00 | 0.53 | 0.50 | 0.77 | 0.51 | 0.32
MA 21 days | MA 63 days | GARCH | GJR GARCH | eGARCH | EWMA
0.06 | 0.02 | 0.06 | 0.08 | 0.09 | 0.05
Table 2: Standard deviation of the time series of $\sigma_{t,i}$ in the SPX
asset for all the estimators.

One drawback of Intra day estimators, that can be immediately seen from Table
2, is that, even if the correlation of the residuals is very low, the
volatility of the time series is very high, providing some issues in the
eventual forecasting of the volatility series, even if this is not a matter in
this study.

###  1.5 GARCH and EWMA on a daily estimation of the parameters

Going back to Autoregressive models, we analyze how the parameters of these
autoregressive models are changing in time. Every calibration is done in an
increasing time window, and the last fit contains all the data set.
Basically, the volatility at time $t$ is given by the last value of the
calibrated series from day $1$ to day $t$. With this technique, every element
of the time series of volatility is obtained from a different calibration.

Figure 5: Time series of the parameters in a Garch and EWMA estimation of the
volatility.

Figure 5 gives an idea of the importance to recalibrate the model to the data,
and not to do it just once on the whole set. But it is also important to see
that a daily recalibration is almost useless, since the parameters are not
changing on a daily basis, but more on a monthly basis.
In Figure 5, $\omega,\alpha$ and $\beta$ are the parameters of the GARCH
model, while $\lambda$ is for the EWMA model.
If we test this new volatility series, where we use at every time step $t$ the
final value of the fitting on the data from the beginning to $t$, we can see
that the results on the squared residuals are not very encouraging

Figure 6: Autocorrelation test for the series of $(res_{t,i})^{2}$ in the SPX
asset for GARCH and EWMA estimators with an increasing time window.

In fact the ACF test in Figure 6 gives results basically similar to the ones
of the original GARCH, so using a floating window on the estimation of the
GARCH models is not a real gain, since it takes more computational time.

###  1.6 Spurious effects in Moving Average estimation

The first tests on autocorrelation of the Moving Averaged based estimation
show that the residuals are far from being white noise. Another analysis can
be made on the different sizes of the floating window used for the computation
of the standard deviation.
We here used different sizes for the windows, and analyzed an interesting
behavior of this estimator.
As we can see in the Figure 7, there is always negative correlation when the
lag is exactly the size of the window. This is quite intuitive, since this lag
corresponds to the fact that one return is going out from the time window, so
it is not considered anymore by the estimator.

Figure 7: Autocorrelation test for the series of $(res_{t,i})^{2}$ in the SPX
asset for Moving Average estimators with different window size.

The red line, present in every graph, corresponds to the autocorrelation with
lag exactly the size of the window of the Moving Average estimators.
As we can see, the red line presents always negative correlation, so this
gives us an hint that the fact that it always corresponds to the day when we
have new information coming, and old information exiting the window, this
effect is just a spurious correlation element given by the method and not by
the data.

###  1.7 Conclusion

After these tests we can recollect some interesting information:

  * •

Constant volatility: The poorest choice for the estimation of volatility. The
Gaussianity of the residuals is given by the Gaussianity of the log returns,
but the acf test and the Ljung - Box test prove that the squared residuals are
far from a x squared. The volatility of the time series of the volatilities is
of course 0, by construction.

  * •

Intraday price volatility: The good results on the ACF graphs and Ljung - Box
test are balanced by very bad results in the QQplot. The volatility of the
time series of volatilities is high, only the case of Yang Zhuang is better,
because the statistical volatility is low by construction

  * •

Moving Average: Not a good choice, since it lacks both from the normality
point of view (especially in the squared of the residuals) and on the
additional correlation studied in the acf comparison for different windows.

  * •

Autoregressive models: Good results in all the tests, even if in the ACF and
Ljung - Box tests the results for intra day volatilities are better. The best
results are for GARCH and EWMA, and these may be the best choice. GARCH
because of its flexibility, EWMA because it depends only on one parameter.

bla

##  2 Allocation strategies using Variance Swaps

###  2.1 Why derivatives on volatility

Volatility swaps are forward contracts on future realized stock volatility.
Variance swaps are similar contracts on variance, the square of future
volatility. Both of these instruments provide an easy way for investors to
gain exposure to the future level of volatility.
Unlike a stock option, whose volatility exposure is contaminated by its stock-
price dependence, these swaps provide pure exposure to volatility alone. These
instruments can be used to speculate on future volatility levels, to trade the
spread between realized and implied volatility, or to hedge the volatility
exposure of other positions or businesses.
Investors are increasingly looking at the derivatives space as a way for
boosting returns of cross-asset portfolios, as depressed yields impact
negatively the possibility of generating attractive returns in the current
market.
Selling options is a popular strategy for extracting a positive carry, i.e.,
the return obtained from holding an asset, out of the derivatives space, given
the premium between implied and realized volatilities (and variances) on
different markets.
The strategy displays attractive features over long-term periods, can be
applied to several markets, and can deliver a positive return even in a low-
rates environment, provided that the volatility premium remains wide enough.
The drawback of the strategy is that systematic options sellers can suffer
large and sudden drawdowns during episodes characterized by spikes in
volatility levels, which requires the ability of actively risk-managing the
positions, ideally over short time-frames.
We will investigate a possibility for mitigating the losses associated with
short volatility trades, which considers implementing long/short trades in the
vol space. This implementation can allow removing the directional bias
naturally associated with short-vol trades and the aforementioned large
potential losses. A comprehensive study of volatility and variance derivatives
can be found in [13] and [14].

###  2.2 Variance Swaps

Although options market participants talk of volatility, it is variance, or
volatility squared, that has more fundamental theoretical significance. This
is so because the correct way to value a swap is to value the portfolio that
replicates it, and the swap that can be replicated most reliably (by
portfolios of options of varying strikes, as we show later) is a variance
swap.
A variance swap is a forward contract on annualized variance, the square of
the realized volatility. Its payoff at expiration is equal to

$(\sigma_{R}^{2}-K_{var})\times Notional,$ (11)

where $\sigma_{R}^{2}$ is the realized stock variance, computed as

$\sigma_{R}^{2}=100\times\frac{252}{N}\times\displaystyle\sum_{t=1}^{N}\left(\ln\left(\frac{P_{t}}{P_{t-1}}\right)\right)^{2},$

with $P_{t}$ the closing price at time $t$, $K_{var}$ is the delivery price
for variance.
The holder of a variance swaps receive at expiration an amount of dollars
equal to the $Notional$ multiplied by the points by which the realized
variance of the stock $\sigma^{2}_{R}$ has exceeded the variance delivery
price $K_{var}$.

###  2.3 Replicating Variance Swaps

Let us assume that the stock price evolution is given by

$\frac{dS_{t}}{S_{t}}=\mu(t,\dots)dt+\sigma(t,\dots,)dZ_{t},$ (12)

where $Z_{t}$ is a standard Brownian motion and the drift $\mu$ and the
continuously sampled volatility $\sigma$ are arbitrary functions of time and
other parameters.
The theoretical definition of realized variance is the continuous integral

$V=\frac{1}{T}\int_{0}^{T}\sigma^{2}(t,\dots)dt.$ (13)

This is a good approximation to the variance of daily returns used in the
contract terms of most variance swaps. The value of a forward contract $F$ on
future realized variance with strike $K_{var}$ and notional equal to $1$ is
the expected present value of the future payoff in the risk-neutral world:

$F=\mathbb{E}\left[e^{-rT}(V-K_{var})\right].$

We will consider the Notional equal to $1$ in the whole analysis.
Here $r$ is the risk-free discount rate corresponding to the expiration date
$T$, and $\mathbb{E}[\cdot]$ denotes the expectation under the risk neutral
measure.
The fair delivery value of future realized variance is the strike $K_{var}$
for which the contract has zero present value:

$K_{var}=\mathbb{E}[V]=\frac{1}{T}\mathbb{E}\left[\int_{0}^{T}\sigma^{2}(t,\dots)dt\right].$
(14)

We focus now on how to replicate the contract. We will focus after on how to
value the contract in the case of stochastic volatility models.
By applying Ito’s lemma to $\log(S_{t})$ we get that (here we omit the
dependence of $\mu$ and $\sigma$ from time and parameters)

$d\log(S_{t})=\left(\mu-\frac{\sigma^{2}}{2}\right)dt+\sigma dZ_{t}.$ (15)

If we subtract (15) from (12) we get

$\frac{dS_{t}}{S_{t}}-d\log(S_{t})=\frac{\sigma^{2}}{2}dt.$ (16)

Integrating (16) over time we can get the variance swap payoff:

$\frac{2}{T}\int_{0}^{T}\frac{dS_{t}}{S_{t}}-\frac{2}{T}\log\left(\frac{S_{t}}{S_{0}}\right)=\frac{1}{T}\int_{0}^{T}\sigma^{2}dt.$
(17)

Taking the risk neutral expectation in (17) we obtain

$K_{var}=\frac{2}{T}\mathbb{E}\left[\int_{0}^{T}\frac{dS_{t}}{S_{t}}\right]-\frac{2}{T}\mathbb{E}\left[\log\left(\frac{S_{T}}{S_{0}}\right)\right].$
(18)

In a risk neutral world with a constant risk free rate $r$, the stock price
evolves according to

$\frac{dS_{t}}{S_{t}}=rdt+\sigma(t,\dots,)dZ_{t},$

so we get that

$\mathbb{E}\left[\int_{0}^{T}\frac{dS_{t}}{S_{t}}\right]=rT+\mathbb{E}\left[\int_{0}^{T}\sigma(t,\dots,)dZ_{t}\right]=rT.$
(19)

We analyze the second part of the right hand side of (18). We can split the
logarithm in two, in order to eliminate the part present in (19):

$\log\left(\frac{S_{T}}{S_{0}}\right)=\log\left(\frac{S_{T}}{S_{0}e^{rT}}\right)+\log\left(\frac{S_{0}e^{rT}}{S_{0}}\right)=\log\left(\frac{S_{T}}{S_{0}e^{rT}}\right)+rT,$

so that

$K_{var}=\frac{2}{T}\mathbb{E}\left[-\log\left(\frac{S_{T}}{S_{0}e^{rT}}\right)\right].$
(20)

For any twice differentiable payoff we have that (see e.g. [15]):

$f(x)-f(y)=f^{\prime}(y)(x-y)+\int_{0}^{y}f^{\prime\prime}(K)(K-x)^{+}dK+\int_{y}^{+\infty}f^{\prime\prime}(K)(x-K)^{+}dK.$
(21)

Adapting the result in (21) to our case, with $f(x)=-\log(x)$, $x=S_{T}$ and
$y=S_{0}e^{rT}$ we have that

$\displaystyle-\log(S_{T})+\log(S_{0}e^{rT})=$
$\displaystyle-\frac{S_{T}-S_{0}e^{rT}}{S_{0}e^{rT}}$
$\displaystyle+\int_{0}^{S_{0}e^{rT}}\frac{1}{K^{2}}(K-S_{T})^{+}dK+\int_{S_{0}e^{rT}}^{+\infty}\frac{1}{K^{2}}(S_{T}-K)^{+}dK,$

that can be rewritten as

$\displaystyle-\log\left(\frac{S_{T}}{S_{0}e^{rT}}\right)=$
$\displaystyle-\frac{S_{T}}{S_{0}e^{rT}}+1$
$\displaystyle+e^{rT}\int_{0}^{S_{0}e^{rT}}\frac{1}{K^{2}}e^{-rT}(K-S_{T})^{+}dK$
(22)
$\displaystyle+e^{rT}\int_{S_{0}e^{rT}}^{+\infty}\frac{1}{K^{2}}e^{-rT}(S_{T}-K)^{+}dK.$

Taking the expected value on both the sides of (2.3) we get that

$\displaystyle\mathbb{E}\left[-\log\left(\frac{S_{T}}{S_{0}e^{rT}}\right)\right]=$
$\displaystyle-\frac{\mathbb{E}\left[S_{T}\right]}{S_{0}e^{rT}}+1$
$\displaystyle+e^{rT}\mathbb{E}\left[\int_{0}^{S_{0}e^{rT}}\frac{1}{K^{2}}e^{-rT}(K-S_{T})^{+}dK\right]$
$\displaystyle+e^{rT}\mathbb{E}\left[\int_{S_{0}e^{rT}}^{+\infty}\frac{1}{K^{2}}e^{-rT}(S_{T}-K)^{+}dK\right].$

Using the fact that, under the risk neutral measure,
$\mathbb{E}\left[S_{T}\right]=S_{0}e^{r}T$, and that we can use Fubini -
Tonelli theorem to interchange integrals and expectations, since the integrand
functions are positive, we obtain

$\mathbb{E}\left[-\log\left(\frac{S_{T}}{S_{0}e^{rT}}\right)\right]=e^{rT}\left(\int_{0}^{S_{0}e^{rT}}\frac{1}{K^{2}}P_{T}(K)dK+\int_{S_{0}e^{rT}}^{+\infty}\frac{1}{K^{2}}C_{T}(K)dK\right),$
(23)

where $P(K)$ and $C(K)$ are, respectively, the price of a put and a call
option with strike $K$ and maturity $T$.
We can finally write a replication formula for the variance swap:

$K_{var}=\frac{2}{T}e^{rT}\left(\int_{0}^{S_{0}e^{rT}}\frac{1}{K^{2}}P_{T}(K)dK+\int_{S_{0}e^{rT}}^{+\infty}\frac{1}{K^{2}}C_{T}(K)dK\right).$
(24)

An important observation is that this replication formula assumes only that
the spot price can be written in the form of (12). This approach is not
compatible with jump diffusion processes, but it does not assume any
particular model in a diffusion environment. A study of the impact of jumps in
the replication formula can be found in [14], but it is out of the scope of
this study.

####  2.3.1 Approximated replication and Corridor Variance Swap

Due to the fact that it is impossible to have a continuous set of call and
option prices, because of lack of out of the money liquidity and market
standards, we can approximate the formula in (24) in order to adapt it to a
discrete set of strikes for put and call prices. As we can see in (24), the
replicating formula includes only out of the money options. The approximation
formula, which approximate the integral with a rectangle rule, is the
following:

$\displaystyle K_{var}\approx$
$\displaystyle\frac{2}{T}\frac{1}{S_{0}}\bigg{(}\sum_{i=1}^{N_{puts}}\frac{1}{(k_{i}^{put})^{2}}P_{T}(k_{i}^{put})\left(k_{i}^{put}-k_{i-1}^{put}\right)$
$\displaystyle+\sum_{i=1}^{N_{calls}}\frac{1}{(k_{i}^{call})^{2}}C_{T}(k_{i}^{call})\left(k_{i}^{call}-k_{i-1}^{call}\right)\bigg{)},$
(25)

where $P_{T}(k_{i}^{put})$ and $C_{T}(k_{i}^{call})$ are, respectively, the
market put and call prices with maturity $T$ and the strikes $k_{i}^{put}$ and
$k_{i}^{call}$ are in percentage of the forward value. In the equity space,
$k_{i}^{put}-k_{i-1}^{put}$ and $k_{i}^{call}-k_{i-1}^{call}$ are usually
constant and equal to $5\%$.
This approximation assumes that $k_{0}^{put}=0$ and
$k_{N_{calls}}^{call}=+\infty$, but since for such strikes the options are
deeply out of the money, and so their value really close to $0$, we can assume
that $k_{0}^{put}=50\%$ and $k_{N_{calls}}^{call}=150\%$ with a small (second)
approximation error.
Motivated by the replication technique, it is possible to introduce a new type
of derivative, that we present for completeness, but that we will not study in
this work. The Corridor Variance Swap is a variance derivative similar to the
variance swap, but it takes in account only a corridor of puts and calls in
the replication formula. For example, if we are interested in a Corridor
Variance Swap with corridor $\left[90\%,110\%\right]$, (the most liquid
derivatives in the market), the valuation formula would be:

$\displaystyle K^{corr}_{var}=$
$\displaystyle\frac{2}{T}\frac{1}{S_{0}}\bigg{(}\sum_{\begin{subarray}{c}i=1\\\
k_{i}^{put}\geq
90\%\end{subarray}}^{N_{puts}}\frac{1}{(k_{i}^{put})^{2}}P_{T}(k_{i}^{put})\left(k_{i}^{put}-k_{i-1}^{put}\right)$
$\displaystyle+\sum_{\begin{subarray}{c}i=1\\\ k_{i}^{call}\leq
110\%\end{subarray}}^{N_{calls}}\frac{1}{(k_{i}^{call})^{2}}C_{T}(k_{i}^{call})\left(k_{i}^{call}-k_{i-1}^{call}\right)\bigg{)},$
(26)

###  2.4 Variance Swaps under stochastic volatility

The general strategy discussed in the previous section can be used to
determine the fair variance and the hedging portfolio from the set of
available options and their implied volatilities. Here we discuss how it is
possible to find a pricing formula in different stochastic volatility models.
This is an alternative way to the hedging approach developed in the previous
section, since it gives only a pricing formula, without a flavor of the
hedging strategy. Nevertheless, this approach is really important since it
permits to price Variance Swaps from the calibrated parameters of a model, and
suggests the dependence of the price from the implied parameters. All the
formulas obtained are important also for the comparison between the market
price, obtained using implied parameters, and the statistical price, obtained
using parameters from the real world distribution. The formulas will allow a
simple estimation of the statistical premium embedded in variance swap prices.

####  2.4.1 The SABR model

The SABR (stochastic alpha, beta, rho) model, introduced in [16], is an
extension of Black’s model and of the CEV model, and assumes that the
volatility follows the dynamic of a Geometric Brownian motion. The equations
describing the model are

$\displaystyle dS_{t}$ $\displaystyle=\sigma_{t}S_{t}^{\beta}dW^{1}_{t}$
$\displaystyle d\sigma_{t}$ $\displaystyle=\nu\sigma_{t}dW^{2}_{t}$ (27)
$\displaystyle\mathbb{E}\left[dW_{t}^{1}dW^{2}_{t}\right]$ $\displaystyle=\rho
dt,$

where $S_{t}$ is the asset price, $\sigma_{t}$ is the volatility, $\nu$ is the
volvol parameter, $\rho$ the correlation between the two Brownian motions
$W_{t}^{1}$ and $W_{t}^{2}$, and initial values $S_{0}=S_{0}$ and
$\sigma_{0}=\alpha$.
As described in (14), the fair value of a Variance Swap can be computed as

$K_{var}=\frac{1}{T}\mathbb{E}\left[\int_{0}^{T}\sigma^{2}_{t}dt\right].$

Since the volatility follows the dynamic of a GBM, it is straightforward the
computation of the solution:

$\sigma_{t}=\alpha e^{\nu W_{t}-\frac{\nu^{2}}{2}t}.$

The computation of the Variance Swap is then

$\displaystyle K_{var}^{SABR}$
$\displaystyle=\frac{1}{T}\mathbb{E}\left[\int_{0}^{T}\alpha^{2}e^{2\nu
W_{t}-\nu^{2}t}dt\right]$
$\displaystyle=\frac{\alpha^{2}}{T}\mathbb{E}\left[\int_{0}^{T}e^{2\nu
W_{t}-\frac{(2\nu)^{2}}{2}t}e^{\nu^{2}t}dt\right]$
$\displaystyle=\frac{\alpha^{2}}{T}\int_{0}^{T}e^{\nu^{2}t}\mathbb{E}\left[e^{2\nu
W_{t}-\frac{(2\nu)^{2}}{2}t}\right]dt$
$\displaystyle=\frac{\alpha^{2}}{T}\int_{0}^{T}e^{\nu^{2}t}dt,$

where we used Fubini Tonelli theorem since the integrand is positive and the
fact that $e^{2\nu W_{t}-\frac{(2\nu)^{2}}{2}t}$ is a martingale. We obtain
that

$K_{var}^{SABR}=\frac{\alpha^{2}}{\nu^{2}T}\left(e^{\nu^{2}T}-1\right).$ (28)

####  2.4.2 The Heston model

The Heston model, introduced in [17], is one of the most famous stochastic
volatility models, and assumes that the variance follows the dynamic of a Cox
- Ingersoll - Ross process:

$\displaystyle dS_{t}$ $\displaystyle=rS_{t}dt+\sigma_{t}S_{t}dW^{1}_{t}$
$\displaystyle d\sigma^{2}_{t}$
$\displaystyle=\kappa(\theta-\sigma_{t}^{2})dt+\nu\sigma_{t}dW^{2}_{t}$ (29)
$\displaystyle\mathbb{E}\left[dW_{t}^{1}dW^{2}_{t}\right]$ $\displaystyle=\rho
dt,$

where $S_{t}$ is the asset price, $\sigma^{2}_{t}$ is the variance, $\kappa$
is the variance mean reversion parameter, $\theta$ is the long term variance,
$\nu$ is the volvol parameter, $\rho$ the correlation between the two Brownian
motions $W_{1}^{t}$ and $W_{2}^{t}$, and initial values $S_{0}=S_{0}$ and
$\sigma^{2}_{0}=\sigma^{2}_{0}$.
The solution of the CIR process is more tricky to obtain, but can still be
computed. Applying Ito Lemma to $e^{\kappa t}\sigma^{2}_{t}$ we get

$\displaystyle de^{\kappa t}\sigma^{2}_{t}$ $\displaystyle=e^{\kappa
t}\kappa\sigma^{2}_{t}dt+e^{\kappa t}\kappa\theta dt-e^{\kappa
t}\kappa\sigma^{2}_{t}dt+e^{\kappa t}\nu\sigma_{t}^{2}dW^{2}_{t}$
$\displaystyle=e^{\kappa t}\kappa\theta dt+e^{\kappa
t}\nu\sigma_{t}dW^{2}_{t},$

so we have that

$\displaystyle e^{\kappa t}\sigma^{2}_{t}$
$\displaystyle=\sigma^{2}_{0}+\int_{0}^{t}e^{\kappa s}\kappa\theta
ds+\int_{0}^{t}e^{\kappa s}\nu\sigma_{s}dW^{2}_{s}$
$\displaystyle=\sigma^{2}_{0}+\theta\left(e^{\kappa
t}-1\right)+\int_{0}^{t}e^{\kappa s}\nu\sigma_{s}dW^{2}_{s},$

and the solution is

$\sigma^{2}_{t}=\theta+e^{-\kappa
t}\left(\sigma^{2}_{0}-\theta\right)+\int_{0}^{t}e^{\kappa(s-t)}\nu\sigma_{s}dW^{2}_{s}.$
(30)

We can then compute the value of the Variance Swap:

$\displaystyle K_{var}^{Heston}$
$\displaystyle=\frac{1}{T}\mathbb{E}\left[\int_{0}^{T}\left(\theta+e^{-\kappa
t}\left(\sigma^{2}_{0}-\theta\right)+\int_{0}^{t}e^{\kappa(s-t)}\nu\sigma_{s}dW^{2}_{s}\right)dt\right]$
$\displaystyle=\frac{1}{T}\int_{0}^{T}\mathbb{E}\left[\theta+e^{-\kappa
t}\left(\sigma^{2}_{0}-\theta\right)\right]dt+\int_{0}^{T}\mathbb{E}\left[\int_{0}^{t}e^{\kappa(s-t)}\nu\sigma_{s}dW^{2}_{s}\right]dt$
$\displaystyle=\theta+\frac{\sigma^{2}_{0}-\theta}{T}\int_{0}^{T}e^{-\kappa
t}dt,$

where we used Fubini Tonelli since the integrand are positive, and we assumed
a regularity in the process $\sigma_{t}$ in order to have the second
expectation equal to $0$. We obtain that

$K_{var}^{Heston}=\theta+\frac{\sigma^{2}_{0}-\theta}{\kappa
T}\left(1-e^{-\kappa T}\right).$ (31)

####  2.4.3 The Stein Stein model

The Stein Stein model, introduced in [18], is historically the first
stochastic volatility model that presents correlation between the price and
the volatility. The volatility follows the dynamic of an Ornstein - Uhlenbeck
process:

$\displaystyle dS_{t}$ $\displaystyle=rS_{t}dt+\sigma_{t}S_{t}dW^{1}_{t}$
$\displaystyle d\sigma_{t}$ $\displaystyle=\kappa(\theta-\sigma_{t})dt+\nu
dW^{2}_{t}$ (32) $\displaystyle\mathbb{E}\left[dW_{t}^{1}dW^{2}_{t}\right]$
$\displaystyle=\rho dt,$

where $S_{t}$ is the asset price, $\sigma_{t}$ is the volatility, $\kappa$ is
the volatility mean reversion parameter, $\theta$ is the long term volatility,
$\nu$ is the volvol parameter, $\rho$ the correlation between the two Brownian
motions $W_{1}^{t}$ and $W_{2}^{t}$, and initial values $S_{0}=S_{0}$ and
$\sigma_{0}=\sigma_{0}$.
The solution of the Ornstein - Uhlenbeck process can be obtained similarly to
the one of the CIR process in the previous section. We still need to apply the
Ito Lemma to $e^{\kappa t}\sigma_{t}$ :

$\displaystyle de^{\kappa t}\sigma_{t}$ $\displaystyle=e^{\kappa
t}\kappa\sigma_{t}dt+e^{\kappa t}\kappa\theta dt-e^{\kappa
t}\kappa\sigma_{t}dt+e^{\kappa t}\nu dW^{2}_{t}$ $\displaystyle=e^{\kappa
t}\kappa\theta dt+e^{\kappa t}\nu dW^{2}_{t},$

so we have that

$\displaystyle e^{\kappa t}\sigma_{t}$
$\displaystyle=\sigma_{0}+\int_{0}^{t}e^{\kappa s}\kappa\theta
ds+\int_{0}^{t}e^{\kappa s}\nu dW^{2}_{s}$
$\displaystyle=\sigma_{0}+\theta\left(e^{\kappa
t}-1\right)+\nu\int_{0}^{t}e^{\kappa s}\sigma_{s}dW^{2}_{s},$

so the solution is

$\sigma_{t}=\theta+e^{-\kappa
t}\left(\sigma_{0}-\theta\right)+\nu\int_{0}^{t}e^{\kappa(s-t)}dW^{2}_{s}.$
(33)

In the computation of the Variance Swap, we will need the expected value of
the variance, so we make the computations in a separated part:

$\displaystyle\mathbb{E}\left[\sigma_{t}^{2}\right]=$
$\displaystyle\left(\theta+e^{-\kappa
t}\left(\sigma_{0}-\theta\right)\right)^{2}+2\nu\left(\theta+e^{-\kappa
t}\left(\sigma_{0}-\theta\right)\right)\mathbb{E}\left[\int_{0}^{t}e^{\kappa(s-t)}dW^{2}_{s}\right]$
$\displaystyle+\nu^{2}\mathbb{E}\left[\left(\int_{0}^{t}e^{\kappa(s-t)}dW^{2}_{s}\right)^{2}\right]$
$\displaystyle=$ $\displaystyle\theta^{2}+e^{-2\kappa
t}\left(\sigma_{0}-\theta\right)^{2}+2\theta e^{-\kappa
t}\left(\sigma_{0}-\theta\right)+\nu^{2}\int_{0}^{t}e^{2\kappa(s-t)}ds,$

where we used Fubini Tonelli as usual, and the Ito isometry. We have that

$\mathbb{E}\left[\sigma_{t}^{2}\right]=\theta^{2}+e^{-2\kappa
t}\left(\sigma_{0}-\theta\right)^{2}+2\theta e^{-\kappa
t}\left(\sigma_{0}-\theta\right)+\nu^{2}\frac{1-e^{-2\kappa t}}{2\kappa}.$
(34)

We can then compute the value of the Variance Swap:

$\displaystyle K_{var}^{Stein}$
$\displaystyle=\frac{1}{T}\int_{0}^{T}\left(\theta^{2}+e^{-2\kappa
t}\left(\sigma_{0}-\theta\right)^{2}+2\theta e^{-\kappa
t}\left(\sigma_{0}-\theta\right)+\nu^{2}\frac{1-e^{-2\kappa
t}}{2\kappa}\right)dt$
$\displaystyle=\theta^{2}+\frac{\nu^{2}}{2\kappa}+2\theta\left(\sigma_{0}-\theta\right)\frac{1}{T}\int_{0}^{T}e^{-\kappa
t}dt+\left(\left(\sigma_{0}-\theta\right)^{2}-\frac{1}{2\kappa}\right)\frac{1}{T}\int_{0}^{T}e^{-2\kappa
t}dt,$

and we get that

$K_{var}^{Stein}=\theta^{2}+\frac{\nu^{2}}{2\kappa}+2\theta\left(\sigma_{0}-\theta\right)\frac{1-e^{-\kappa
T}}{\kappa
T}+\left(\left(\sigma_{0}-\theta\right)^{2}-\frac{\nu^{2}}{2\kappa}\right)\frac{1-e^{-2\kappa
T}}{2\kappa T}.$ (35)

####  2.4.4 The $\lambda$-SABR model

The $\lambda$-SABR model, introduced in [19], extends the SABR model of [16],
introducing mean reversion in the volatility, which follows the dynamic of a
modified Ornstein - Uhlenbeck process:

$\displaystyle dS_{t}$ $\displaystyle=\sigma_{t}S_{t}^{\beta}dW^{1}_{t}$
$\displaystyle d\sigma_{t}$
$\displaystyle=\kappa(\theta-\sigma_{t})dt+\nu\sigma_{t}dW^{2}_{t}$ (36)
$\displaystyle\mathbb{E}\left[dW_{t}^{1}dW^{2}_{t}\right]$ $\displaystyle=\rho
dt,$

where $S_{t}$ is the asset price, $\sigma_{t}$ is the volatility, $\kappa$ is
the volatility mean reversion parameter, $\theta$ is the long term volatility,
$\nu$ is the volvol parameter, $\rho$ the correlation between the two Brownian
motions $W_{1}^{t}$ and $W_{2}^{t}$, and initial values $S_{0}=S_{0}$ and
$\sigma_{0}=\alpha$.
The computations of the expected value of the variance and of the Variance
Swap are long and tedious, so we will present only the final result:

$\displaystyle K_{var}^{\lambda-SABR}=$
$\displaystyle\left(\frac{\kappa\theta}{\kappa+\frac{\nu}{2}}\right)^{2}+\frac{\alpha^{2}}{(2\kappa-\nu^{2})T}\left(1-e^{-(2\kappa-\nu^{2})T}\right)$
$\displaystyle+\frac{\kappa^{2}\theta^{2}}{2\left(\kappa+\frac{\nu^{2}}{2}\right)^{3}T}\left(1-e^{-2\left(\kappa+\frac{nu^{2}}{2}\right)T}\right)+\frac{2\alpha\theta}{\kappa\left(\kappa+\frac{\nu^{2}}{2}\right)T}\left(1-e^{-\kappa
T}\right)$
$\displaystyle+\frac{2\alpha\kappa\theta}{\left(\kappa+\frac{\nu^{2}}{2}\right)\left(2\kappa+\frac{\nu^{2}}{2}\right)T}\left(1-e^{-\left(2\kappa+\frac{\nu^{2}}{2}\right)T}\right).$
(37)

###  2.5 A brief comparison of the pricing methods

A first test on the goodness of the pricing techniques is made on the SPX 1
month Variance Swaps. Since the SABR model is the easiest and most
parsimonious model to calibrate, we will use it as a proxy for the pricing via
stochastic volatility models, and we will also analyze the pricing via
replication.
In the SABR pricing formula we calibrate a SABR model with $\beta=1$ and we
use the parameters that we need, i.e., $\alpha$ and $\nu$. The calibration
routine computes different parameters for different maturities, from $1$ week
to $1$ year. In the example we will focus on maturity $1$ month. The period
goes from March 2006 to March 2016, and we compare one variance swap per
month, so $12$ per year.

Figure 8: Pricing of Variance Swap via replication and via the SABR formula,
comparison with market data.

As we can see, the SABR formula gives an almost perfect fit, where the
differences are probably given by the calibration routine. In fact, the SABR
model is mostly used for FX and rates derivatives, while other models are used
for the Equity space. The objective of this numerical study was not to find
the best calibration of the market smile, but just a first proxy for the
pricing of Variance Swap. The difference is pricing is mostly due to the
incapacity of the SABR smile to fit perfectly the market smile, mostly for
deep out of the money options. Nevertheless, we are quite happy of the result,
as the SABR formula can be used as a good pricing estimation for Variance
Swap.
On the other side, the results given by the replication formula for the
corridor swap are quite poor but the motivation is easy to understand. The
first error is given by the fact that we approximate an integral with a finite
sum, and that is an error that will always be done, unless we try to bootstrap
the missing strikes of the replication formula. Since we wanted to use only
market data, we did not fit the market smile with artificial values. In
addition, we only used strikes from $90\%$ to $110\%$ of the moneyness. This
is due to the fact that Equity puts and calls are not as liquid as the FX
vanilla options, so for deep out of the money options the bid-ask spread was
too big to be considered. As we can see the biggest difference is for the peak
in volatility of August 2008, while for the others values the pricing is still
a good proxy.
The replication price is obtained using the replication formula, and keeping
constant volatility for deep out of the money options. Practically, we keep
the volatility constant to the value of $90\%$ of the moneyness for the
strikes from $50\%$ to $90\%$, and the value of $110\%$ for the strikes from
$110\%$ to $150\%$. Even if this choice of constant volatility outside the
range $[90\%,110\%]$ is quite poor and does not respect non arbitrage
conditions, the pricing formula seems fitting the values of the market data.

###  2.6 Mark to Market price

Since Variance Swaps can be exercised only at maturity, with the final payoff
given by the total realized variance minus the strike fixed at the beginning
of the contract, it is important, in terms of P&L of strategies using Variance
Swaps, to consider also the Mark to Market value of this derivative. We know
that the Mark to Market value of a Variance Swap at the beginning of the
contract is of course $0$, while at maturity $T$, counting $N$ days, is (in
the case we are long)

$100\times\frac{252}{N}\times\sum_{i=1}^{N}\ln\left(\frac{P_{i}}{P_{i-1}}\right)^{2}-K_{var},$

and we remember that, as usual,

$K_{var}=100\times\frac{252}{N}\times\sum_{i=1}^{N}\mathbb{E}\left[\ln\left(\frac{P_{i}}{P_{i-1}}\right)^{2}\right].$

Let us call $imp_{t}^{T}$ the value of a (synthetic) Variance Swap at time $t$
with maturity $T$, so such that

$imp_{t}^{T}=100\times\frac{1}{(T-t)}\times\mathbb{E}\left[\sum_{i=1+252\times
t}^{252\times T}\ln\left(\frac{P_{i}}{P_{i-1}}\right)^{2}\right].$ (38)

It can be easily seen that $K_{var}=imp_{0}^{T}$.
The Mark to Market value of a Variance Swap contract is then , assuming the
interest rates equal to $0$, a convex combination of the realized variance up
to time $t$ and the predicted variance from time $t$, minus the Variance Swap
strike :

$MtM_{t}=\frac{t}{T}\sum_{i=1}^{252\times
t}\ln\left(\frac{P_{i}}{P_{i-1}}\right)^{2}+\frac{T-t}{T}imp_{t}^{T}-K_{var},$
(39)

where we assume that $\sum_{i=1}^{0}\cdot=0$. We can easily see that this
definition is what we are looking for, in the case of $t=0$ and $t=T$, since

$\displaystyle MtM_{0}$ $\displaystyle=\frac{0}{T}\sum_{i=1}^{252\times
0}\ln\left(\frac{P_{i}}{P_{i-1}}\right)^{2}+\frac{T}{T}imp_{0}^{T}-K_{var}$
$\displaystyle=imp_{0}^{T}-K_{var}=0,$

and

$\displaystyle MtM_{T}$ $\displaystyle=\frac{T}{T}\sum_{i=1}^{252\times
T}\ln\left(\frac{P_{i}}{P_{i-1}}\right)^{2}+\frac{T-T}{T}imp_{t}^{T}-K_{var}$
$\displaystyle=\sum_{i=1}^{252\times
T}\ln\left(\frac{P_{i}}{P_{i-1}}\right)^{2}-K_{var}.$

The main issue is how to compute the element $imp_{t}^{T}$. Since, as we have
seen in the previous Section, the pricing using the SABR formula is a good
approximation of the market price, we will use that approach, specifically

$imp_{t}^{T}=\frac{\alpha^{2}}{\nu^{2}(T-t)}\left(e^{\nu^{2}(T-t)}-1\right),$
(40)

where $\alpha$ and $\nu$ are the parameters of the SABR model computed at the
beginning of the contract. The dynamic of the Mark to Market price can be seen
in the following Figure:

Figure 9: Mark to Market value of a $1$ year Variance Swap on the SPX index.
The starting day is July 1st, 2011.

The Mark to Market value in Figure 9 is scaled by a factor
$\frac{1}{K_{var}}$. This choice, suggested in [14], is done in order to have
the vega of the Mark to Market value constant.
As we can see, if we did not consider all the behavior of the Mark to Market
price from the beginning to maturity, we would have considered only the
initial value $0$ and the final value $0.0332$. Meanwhile, the evolution of
the price has been consistently different. This suggests that keeping the
contract up to maturity is not always a good idea, since, for example, selling
it in November, would have given a much better return to the strategy. Even if
we do not consider the possibility to buy or sell the contract before
maturity, it is important to consider what could have been the value of the
contract, in the view of P&L calculations.

###  2.7 Back test on a long strategy

In this section we will focus on a simple strategy, that we will apply for all
the maturities considered, i.e., $1$ month, $3$ months, $6$ months and $1$
year. The idea behind this strategy is that we enter in a new contract every
month, and we keep every derivative up to maturity. We are keeping the
variance swaps until maturity, and at the end of the contract we receive the
difference between the realized variance and the strike fixed at the beginning
of the contract.
In the case of $1$ month Variance Swaps we are keeping only one derivative in
our portfolio at every time $t$, while, for example in the case of $3$ months
Variance Swaps, we start with one derivative, then after one month we add a
second Variance Swap, and at the second month we add the third Variance Swap.
At the third month the first Variance Swap is expired, and we can add a new
one. In this way, we have always at maximum $3$ Variance Swaps in the
portfolio, in the case of $3$ months Variance Swaps.
Since the longest maturity considered is $12$ months, the maximum number of
Variance Swaps in the portfolio would be $12$. This is done from July 2011 to
June 2016.

Figure 10: Profit and Loss of the long strategy using $1$ month Variance Swap
on the SPX index. The starting day is July 1st, 2011. Figure 11: Profit and
Loss of the long strategy using $1$ year Variance Swap on the SPX index. The
starting day is July 1st, 2011.

Figures 10 and Figure 11 show similar trend but different peculiarities. In
fact, the spikes presents in the $1$ month strategy are more important then
the ones of the $1$ year strategy. The $1$ year strategy presents a more
interesting trend, with less volatility with respect to the monthly case. This
can also be shown in Table 3.

The final Sharpe ratios are calculated with the following formula: we call
$cumMtM_{t}$ the cumulative Mark to Market value of the strategy, i.e., the
sum of the Mark to Market values of the Variance Swaps which are in the
portfolio at time $t$, so we are considering a maximum of $12$ Variance Swaps.
Let $retMtM_{t}$ the returns of $cumMtM_{t}$,

$retMtM_{t}=cumMtM_{t}-cumMtM_{t-1}.$

The Sharpe ratios of this strategy are then computed as the mean of the time
series $retMtM_{t}$ divided by its standard deviation.
This computation has be done for all the maturities and for five different
assets, two equities and three FXs.

| 1 month | 3 months | 6 months | 1 year
---|---|---|---|---
SPX | -0.31 | -0.77 | -1.10 | -1.34
EUROSTOXX | -1.01 | -0.13 | -1.03 | -
EURUSD | -0.09 | -0.48 | -0.44 | -0.87
GBPUSD | -0.48 | -0.51 | -0.57 | -0.31
USDJPY | -0.26 | -0.31 | -0.34 | -0.68
Table 3: Sharpe Ratio of the only long strategies on different assets, using
Variance Swaps with different maturities.

###  2.8 Stationarity test on implied volatilities and Variance swaps

In this section we will be interested in highlighting mean reverting features
appearing on cross-asset derivatives. A full cross-sectional analysis,
focusing on long/short trades, will come in handy for magnifying the mean-
reversion effects already appearing on single assets: for this, we will focus
on Equity and FX spaces, although other asset classes could be easily taken
into account as well. Mean reversion is an important characteristic of a time
series. From a strategic point of view, knowing the mean reversion time of a
price series can give suggestions on when to buy / sell, in order to amplify
the final return of the strategy. In order to study mean reversion of a time
series, we introduce the basic concepts.
An Autoregressive $1$ model, also known as $AR(1)$ specifies that the variable
$y_{t}$ depends linearly from its previous value and on a stochastic term:

$y_{t}-m=A(y_{t-1}-m)+\varepsilon_{t}.$ (41)

The process is stationary if $|A|<1$, and in this case it is possible to
compute the mean of the process, that is $\mathbb{E}\left[y_{t}\right]=m$. A
common way to quantify the speed of mean reversion of an autoregressive
process is the half-life, which is defined as the working days needed so that
the magnitude of the forecast become half of that of the forecast origin.
Since, given a starting value $y_{k}$, the forecast value after $n$ working
days is $\hat{y}_{n+k}=A^{k}y_{n}$, we have that
$A^{k}y_{n}=\displaystyle\frac{1}{2}y_{n}$, so that the half life can be
computed as

$k=\frac{\log\left(\frac{1}{2}\right)}{\log\left(|A|\right)}.$ (42)

If the parameter $A$ is close to $0$, then the half life of the process is
really small, and it is possible to exploit this property from an allocation
point of view.

####  2.8.1 Correlation between implied volatilities and Variance Swap

Since Variance Swap are not really liquid on all asset classes, and due to the
fact that the standard version of Bloomberg does not display the time series
of Variance Swaps, it is useful to find a good proxy for the price of Variance
Swaps, that can give the same behavior of the time series. Following the
formulas of the stochastic volatility models, especially in the SABR case, we
can see that the prices are always given by the parameter $\sigma_{0}$
multiplied by a corrective factor. So a good idea would be to study the
correlation between the time series of the ATM volatility and of the Variance
Swap.

Figure 12: Time series of the square root of the $1$ year Variance Swap and
$1$ year At The Money Volatility on the SPX index. The Variance Swap prices
are obtained using the replication formula.

Motivated by the result in Figure 12, we analize the correlation between
Variance Swap prices and ATM volatility:

| 1 m | 3 m | 6 m | 12 m | 18 m
---|---|---|---|---|---
SPX | 99.82% | 99.74% | 99.79% | 99.94% | 99.89%
Eurostoxx | 99.15% | 99.38% | 99.12% | 98.97% | 98.25%
UKX | 99.52% | 99.51% | 99.41% | 99.17% | 98.67%
Nikkei | 98.24% | 98.00% | 97.32% | 97.38% | 97.36%
Table 4: Correlation between the time series of the square root of the
Variance Swap and the At The Money Volatility.

As we can see in Table 4, it is natural to study directly the behavior of the
ATM Volatility, since it will give all the properties needed from the
Autoregressive model, due to the fact that the correlation between ATM
volatility and (the square root of) the price of Variance Swaps is always
greater than $97\%$, on all assets and all maturities.

####  2.8.2 Stationarity test on implied volatilities: the Dickey - Fuller
test

Stationarity is a statistical concept which refers to the property of a
financial variable to oscillate in a range around a pre-specified value. If
that is the case, one can set up a mean reversion strategy for trading the
dislocations of an asset from its convergence value.
The concept of stationarity is directly related to that of unit roots in the
context of autoregressive models. A unit-root essentially mimics a Brownian
motion-type of behavior, whereby the latest price encompasses all the relevant
information, so that there is no corrective force towards any other pre-
determined value than the latest value itself.
Stationarity of financial assets can be assessed by performing a Dickey Fuller
test. The test investigates the unit root property of the time series in the
framework of autoregressive models, by testing a set of autocorrelation (ACF)
functions of the time series.
The null hypothesis of the test is that a unit root is present in the time
series and that the asset is non-stationary. The result of the test is a
p-value related to the probability of rejecting the null hypothesis when the
latter is in fact true. In practice one fixes a threshold or significance
level (usually at 5%) and rejects the null-hypothesis if the p-value is below
the threshold: essentially, when the p-value of the Dickey-Fuller test applied
on an asset is below the threshold, we reject the hypothesis that that asset
is non-stationary, which supports the suggestion that the asset might be in
fact stationary. When the p-value is above the threshold, we simply do not
have enough elements for rejecting the null hypothesis of non-stationarity.
The test can be applied to any financial asset; in this piece we have applied
to a set of spot, return and volatility time series. For numerical tests, we
have relied on the adf.test function from the tseries R package.

| 1m | 3m | 6m | 12m | 18m
---|---|---|---|---|---
SPX | 0.08 | 0.25 | 0.36 | 0.43 | 0.46
Eurostoxx | 0.01 | 0.05 | 0.10 | 0.21 | 0.27
Footsie | 0.01 | 0.03 | 0.05 | 0.06 | 0.06
Nikkei | 0.01 | 0.04 | 0.10 | 0.04 | 0.17
Swiss | 0.01 | 0.07 | 0.21 | 0.34 | 0.39
Canada | 0.03 | 0.17 | 0.34 | 0.40 | 0.40
Australia | 0.09 | 0.27 | 0.26 | 0.46 | 0.36
Norway | 0.03 | 0.03 | 0.40 | 0.34 | 0.28
Sweden | 0.02 | 0.09 | 0.15 | 0.11 | 0.10
EURUSD | 0.02 | 0.13 | 0.28 | 0.39 | 0.67
GBPUSD | 0.01 | 0.09 | 0.23 | 0.28 | 0.64
USDJPY | 0.01 | 0.01 | 0.03 | 0.06 | 0.43
USDCHF | 0.01 | 0.02 | 0.08 | 0.19 | 0.50
USDCAD | 0.03 | 0.14 | 0.25 | 0.31 | 0.46
AUDUSD | 0.01 | 0.06 | 0.10 | 0.26 | 0.55
NZDUSD | 0.01 | 0.05 | 0.18 | 0.38 | 0.56
USDNOK | 0.03 | 0.12 | 0.28 | 0.37 | 0.62
USDSEK | 0.05 | 0.22 | 0.33 | 0.44 | 0.67
Table 5: p values of the Dickey Fuller test on the At The Money Volatility
series, for different assets and different maturities.

As we can see in Table 5, for maturities larger than $6$ months, the Dickey
Fuller test "shows" that there is no stationarity of the process, and so the
time series are not mean reverting. Since the most liquid Variance Swaps are
the ones with $1$ year maturity, we understand that trading on single assets
is not a good idea, so it would be interesting to study the possible behavior
of combination of Variance Swaps.

###  2.9 Spread on Variance Swaps

We now want to test whether spreads of volatilities might exhibit better
stationarity properties than for the single vol assets. We focus on $1$ year
volatilities.
We will apply the Dickey Fuller test on the spreads of the $1$ year vol: here
for simplicity we will consider the case where the spread is

$\Delta^{1,2}_{t}=y_{t}^{1}-y_{t}^{2},$

i.e., with a relative $\beta$ of $1$. A more sophisticated choice could be to
use a different $\beta$ that could permit to balance the weights of the two
Variance Swap / ATM volatilities series. This is very important in the cross
asset scenario: in fact, different asset classes present different ATM
volatility levels, and then different Variance Swap levels. Equity
volatilities are higher than FX volatilities, and choosing a simple spread
with $\beta=1$ would keep the properties of the equity series prominent with
respect to the FX series. Nevertheless, in order to simplify the computation
and the methodologies, we will keep $\beta=1$.

| SPX | Eurostoxx | Footsie | Nikkei | Swiss | Canada | Australia | Norway | Sweden
---|---|---|---|---|---|---|---|---|---
SPX | NA | 0.01 | 0.01 | 0.01 | 0.12 | 0.01 | 0.01 | 0.32 | 0.01
Eurostoxx | 0.01 | NA | 0.02 | 0.01 | 0.08 | 0.01 | 0.01 | 0.31 | 0.01
Footsie | 0.01 | 0.02 | NA | 0.01 | 0.01 | 0.03 | 0.01 | 0.04 | 0.01
Nikkei | 0.01 | 0.01 | 0.01 | NA | 0.01 | 0.01 | 0.01 | 0.01 | 0.01
Swiss | 0.12 | 0.08 | 0.01 | 0.01 | NA | 0.03 | 0.01 | 0.27 | 0.01
Canada | 0.01 | 0.01 | 0.03 | 0.01 | 0.03 | NA | 0.01 | 0.08 | 0.01
Australia | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | NA | 0.19 | 0.01
Norway | 0.32 | 0.31 | 0.04 | 0.01 | 0.27 | 0.08 | 0.19 | NA | 0.01
Sweden | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | NA
Table 6: p values of the Dickey Fuller test on spreads of $1$ year At The Money Volatility, Equity space. | EURUSD | GBPUSD | USDJPY | USDCHF | USDCAD | AUDUSD | NZDUSD | USDNOK | USDSEK
---|---|---|---|---|---|---|---|---|---
EURUSD | NA | 0.21 | 0.01 | 0.02 | 0.34 | 0.01 | 0.06 | 0.19 | 0.49
GBPUSD | 0.21 | NA | 0.01 | 0.19 | 0.02 | 0.01 | 0.01 | 0.01 | 0.30
USDJPY | 0.01 | 0.01 | NA | 0.01 | 0.03 | 0.05 | 0.04 | 0.01 | 0.03
USDCHF | 0.02 | 0.19 | 0.01 | NA | 0.36 | 0.04 | 0.07 | 0.16 | 0.32
USDCAD | 0.34 | 0.02 | 0.03 | 0.36 | NA | 0.02 | 0.02 | 0.03 | 0.12
AUDUSD | 0.01 | 0.01 | 0.05 | 0.04 | 0.02 | NA | 0.01 | 0.01 | 0.01
NZDUSD | 0.06 | 0.01 | 0.04 | 0.07 | 0.02 | 0.01 | NA | 0.01 | 0.01
USDNOK | 0.19 | 0.01 | 0.01 | 0.16 | 0.03 | 0.01 | 0.01 | NA | 0.08
USDSEK | 0.49 | 0.30 | 0.03 | 0.32 | 0.12 | 0.01 | 0.01 | 0.08 | NA
Table 7: p values of the Dickey Fuller test on spreads of $1$ year At The Money Volatility, FX space. | EURUSD | GBPUSD | USDJPY | USDCHF | USDCAD | AUDUSD | NZDUSD | USDNOK | USDSEK
---|---|---|---|---|---|---|---|---|---
SPX | 0.17 | 0.16 | 0.20 | 0.25 | 0.10 | 0.05 | 0.10 | 0.10 | 0.09
Eurostoxx | 0.01 | 0.01 | 0.04 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01
Footsie | 0.02 | 0.01 | 0.10 | 0.04 | 0.01 | 0.02 | 0.02 | 0.02 | 0.04
Nikkei | 0.01 | 0.01 | 0.02 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01
Swiss | 0.09 | 0.04 | 0.19 | 0.15 | 0.02 | 0.03 | 0.03 | 0.05 | 0.09
Canada | 0.17 | 0.14 | 0.26 | 0.24 | 0.13 | 0.09 | 0.11 | 0.12 | 0.12
Australia | 0.17 | 0.09 | 0.24 | 0.21 | 0.06 | 0.04 | 0.07 | 0.09 | 0.13
Norway | 0.23 | 0.17 | 0.26 | 0.25 | 0.27 | 0.27 | 0.17 | 0.25 | 0.27
Sweden | 0.01 | 0.01 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01
Table 8: p values of the Dickey Fuller test on spreads of $1$ year At The
Money Volatility, Cross - Asset space.

It is easy to check that spreads exhibit much stronger stationarity
properties, in that most of the spreads now pass the stationarity test. The
results above suggest that taking a position on spreads of cross-asset
volatilities should permit magnifying the mean reversion strategies already
evident for the volatility variables.

####  2.9.1 A case study

Motivated by Tables 6, 7 and 8, we study the spread between SPX and Eurostoxx
and between SPX and Footsie. This two spreads show mean reversion from the
Dickey Fuller test.
We compute the half life of the two spreads, and we compare them with the half
life of the single assets:

SPX | Eurostoxx | Footsie | SPX - Eurostoxx | SPX - Footsie
---|---|---|---|---
138.89 | 26.84 | 38.15 | 2.61 | 2.16
Table 9: Half lives in days of $12$ months At The Money Volatilities and their
spreads Figure 13: Time series of the $12$ months At The Money Volatility of
the SPX index, from July 1, 2009 to July, 1 2016.

The time series of the $12$ ATM Volatility does not present mean reversion
properties, as can we see in Figure 13 and in Tables 5 and 9. Moreover, it
presents peaks in correspondence of periods of stress, as 2011 and 2013. The
use of spreads, in addition of the already shown property of magnification of
the mean return properties, will remove the peaks of the time series.

Figure 14: Time series of the $12$ months At The Money Volatility spreads
between the SPX and the Eurostoxx indexes, from July 1, 2009 to July, 1 2016.
Figure 15: Time series of the $12$ months At The Money Volatility spreads
between the SPX and the Footsie indexes, from July 1, 2009 to July, 1 2016.

Even if the mean values are low, Figures 14 and 15 show that it is possible to
remove the spikes in correspondence of high volatility periods, and it is
easily seen that the mean reversion properties are magnified, since we can see
the almost periodic behavior of the spreads time series. This observation give
now a good hint on how to possible trade this couples of Variance Swaps, in
order to exploit all the mean reversion properties. This would not have been
possible if we had focus on only one asset, since the mean reverting trading
strategies would have not worked.

###  2.10 Final analysis

As we have seen in the previous sections, there are two possible ways to
implement investment strategies based on variance swap: study the difference
between the implied and statistic parameters, and exploit the mean reverting
characteristics of the spreads.

####  2.10.1 Risk premium on Variance Swaps

The market premium on a vanilla option, say a Call option, is given by the
difference of the market price and the price given by the Black Scholes
formula where the volatility is given by a statistical estimator. Since the
Black Scholes price is a bijective function of the volatility, the market
premium is directly given by the difference between the implied volatility and
the statistical volatility.
This relation is easy to understand and to implement in the case of plain
vanilla options since, even if the Black Scholes model is not a good model, it
is a common assumption that implied volatility in the Black Scholes formulas
is "the wrong number to put in the wrong formula to obtain the right price".
Things becomes more complicated with more exotic derivatives, in particular
with Variance Swap, as in our case. We can assume that the SABR formula (28)
is the equivalent of the Black Scholes formula. In this case, the "wrong"
numbers to put in the formula would be the calibrated values of the observed
variance $\alpha$ and the volvol parameter $\nu$.
$\alpha$, if not calibrated, can be easily estimated by the ATM implied
volatility of $1$ week plain vanilla options, while the volvol parameter can
be obtained inverting formula (28), if the series of the Variance Swap prices
is available.
This has to be compared with the statistical variance and the statistical
volvol. This elements are usually obtained using an EWMA with $\lambda=0.94$
from, respectively, the stock price series and the (estimated) volatility
series. The market premium of a variance swap becomes, under this assumptions,
the sum of the premium given by the volatility and the premium given by the
volvol parameter. The advantage of the strategies using Variance Swap is that,
even if we add more complexity to the pricing and hedging procedures, we add a
component in the market premium, so even when the volatility premium is low
(so we do not have signals on whether to buy or sell plain vanilla options),
there is still the possibility to exploit the volvol premium.

####  2.10.2 Mean reversion on spreads of Variance Swaps

Mean reversion properties of Variance Swaps have been studied in the previous
section: a mean reverting process can give trading signals on when to buy and
when to sell an asset or a derivative. When the price is high and we know
that, according to the model, the price is lowering, then it is a good idea to
sell, because it would not be possible to extract more value from the carry.
Similarly, when the price is low and the models suggests that the series is
reverting to the mean, then it is a good choice to buy, since the wait would
only make the price of the asset / derivative higher.
In case of derivatives, it use useful to study the half life of the process.
In the case of derivatives, if the half life is comparable to the maturity of
the option, than this strategy of buying / selling would not be applicable,
because the time taken by the process to return to his mean would overcome the
maturity date.
Motivated by this theoretical and empirical observation, the spread of
Variance Swaps is a great way to highlight mean reverting characteristic and
to implement trading strategies. In fact, as already seen, the series of the
spread is much more mean reverting, with the half life usually under five
trading days. Even if we did not consider the presence of trading costs, the
possibility of obtain value from the strategy on a weekly basis is interesting
and suggests full attention.
In this work we did not implement any trading strategy. This is due to the
fact that the scope of the Research team is to suggest new technologies and
new trading ideas to the investors and to the trading desks. The choice of the
derivatives to trade, the asset class, the most interesting strategy, is not
the job of the research department, so there will not be any market
implementation in this report.

bla

## References

  * [1] Engle, R. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation.  Econometrica 50 (1982), 987–1007.
  * [2] Bollerslev, T. Generalized autoregressive conditional heteroskedaticity.  Journal of Econometrics 31 (1986), 307–327.
  * [3] Glosten, L. Jagannathan, R., and D. Runkle On the relation between the expected value and the volatility of the nominal excess return on stocks.  Journal of Finance 48, (1993), 1779–1801.
  * [4] Nelson, D.B. Conditional heteroskedaticity in asset returns: A new approach.  Econometrica 59, (1991), 347–370.
  * [5] Parkinson, M. The extreme value method for estimating the variance of the rate of return.  Journal of Business 53, (1980), 61–65.
  * [6] Garman, M.B. and Klass, M.J. On the estimation of security price volatilities from historical data.  Journal of Business 53, (1980), 67–78.
  * [7] Rogers, L. C. G. and Satchell, S.E.  Estimating variance from high, low and closing prices.  Annals of Applied Probability 1, (1991), 504–512.
  * [8] Yang, D. and Zhang, Q.  Drift independent volatility estimation based on high, low, open and close prices.  Journal of Business 73, (2000), 477–491.
  * [9] Bernett, C. and Gil, M. Measuring Historical Volatility.  Santander Investment Securities working paper, (2013).
  * [10] Reider, R. Volatility Forecasting I: Garch models.  Time Series Analysis and Statistical Arbitrage course at New York University.
  * [11] Tsay, R. S. Analysis of Financial Time Series, Second Edition.  Wiley, (2005).
  * [12] Saichev, A., Sornette, D. and Filimonov, V. Efficient high-frequency variance estimators.  Risk Magazine, (2013)
  * [13] Demeterfi, K., Derman, E., Kamal, M. and Zou, J. More Than You Ever Wanted To Know About Volatility Swaps.  Goldman Sachs Quantitative Strategies Research Notes, (1999)
  * [14] Bossu, S., Guichard, R. and Strasser, E. Just What You Need To Know About Variance Swaps.  JP Morgan Equity Derivatives Research Notes, (2005)
  * [15] Carr, P. and Madan, D. Optimal positioning in derivative securities.  Quantitative Finance 1, (2001), 19–37
  * [16] Hagan, P. S., Kumar, D., Lesniewski, A.S., and Woodward., D.E.  Managing smile risk.  Wilmott Magazine, (2002), 84–108
  * [17] Heston, S.  A closed-form solution for options with stochastic volatility with applications to bond and currency options.  Review of Financial Studies 6, (1993), 327–343
  * [18] Stein, E. and Stein, J.  Stock Price Distributions with Stochastic Volatility: An Analytic approach.  Review of Financial Studies 4, (1991)
  * [19] Henry-Labordere, P.  A General Asymptotic Implied Volatility for Stochastic Volatility Models.  Proceedings "Petit dejeuner de la Finance", (2005)