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In 2000, Ait-Sahalia and Lo have argued that Economic VaR (E-VaR) calculated under option market implied risk neutral density (RND) is a more relevant measure of risk than historically based VaR. As industry practice requires VaR at high confidence level of 99%, Extreme Economic Value at Risk (EE-VaR) based on the Generalized Extreme Value (GEV) distribution has been proposed as a new risk measure. This follows from a GEV option pricing model developed by Markose and Alentorn in 2005 which shows that the GEV implied RND can accurately capture negative skewness and fat tails, with the latter explicitly determined by the market implied tail index. Here, the term structure of the GEV based RNDs is estimated which permits the calibration of an empirical scaling law for EE-VaR, and thus, obtain daily EE-VaR for any time horizon. Backtesting results for the FTSE 100 index from 1997 to 2003, show that EE-VaR has fewer violations than historical VaR. Further, there are substantial savings in risk capital with EE-VaR at 99% as compared to historical VaR corrected by a factor of 3 to satisfy the violation bound. The efficiency of EE-VaR arises because an implied VaR estimate responds quickly to market events and in some cases even anticipates them. In contrast, historical VaR reflects extreme losses in the past for longer.
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Centre for
Finance and
Amadeo Alentorn
Sheri Markose
Generalized Extreme Value
Distribution and Extreme
Economic Value at Risk
October 2007
Generalized Extreme Value Distribution and
Extreme Economic Value at Risk (EE-VaR)
Amadeo Alentornand Sheri Markose
September, 2007
Ait-Sahalia and Lo (2000) and Panigirtzoglou and Skiadopoulos (2004)
have argued that Economic VaR (E-VaR), calculated under the option
market implied risk neutral density is a more relevant measure of risk
than historically based VaR. As industry practice requires VaR at high
confidence level of 99%, we propose Extreme Economic Value at Risk
(EE-VaR) as a new risk measure, based on the Generalized Extreme Value
(GEV) distribution. Markose and Alentorn (2005) have developed a GEV
option pricing model and shown that the GEV implied RND can accu-
rately capture negative skewness and fat tails, with the latter explicitly
determined by the market implied tail index. Here, we estimate the term
structure of the GEV based RNDs, which allows us to calibrate an em-
pirical scaling law for EE-VaR, and thus, obtain daily EE-VaR for any
time horizon. Backtesting results for the FTSE 100 index from 1997 to
2003, show that EE-VaR has fewer violations than historical VaR. Fur-
ther, there are substantial savings in risk capital with EE-VaR at 99%
as compared to historical VaR corrected by a factor of 3 to satisfy the
violation bound. The efficiency of EE-VaR arises because an implied VaR
estimate responds quickly to market events and in some cases even antici-
pates them. In contrast, historical VaR reflects extreme losses in the past
for longer.
Centre for Computational Finance and Economic Agents (CCFEA), Old Mutual Asset
Managers (UK) Ltd
Centre for Computational Finance and Economic Agents (CCFEA),Department of Eco-
nomics, University of Essex. Corresponding author: Sheri Markose. Email address: We are grateful for comments from two anonymous referees which have
improved the paper. We acknowledge useful discussions with Thomas Lux, Olaf Menkens,
Christian Schlag, Christoph Schleicher, Radu Tunaru and participants at the FMA 2005 at
Siena and CEF 2005 in Washington DC.
JEL classification: G10, G14.
Keywords: Economic Value-at-Risk, EE-VaR, empirical scaling law,
term structure of implied RNDs.
1 Introduction
Value-at-Risk (VaR) has become the most popular measure for risk man-
agement. Value-at- Risk, denoted by VaR (q, k), is an estimate, for a given
confidence level q, of the maximum that can be lost from a portfolio over a given
time horizon k. An alternative measure of risk is the Economic VaR (E-VaR)
proposed by Ait-Sahalia and Lo (2000) and calculated under the option-implied
risk neutral density. It has been argued that E-VaR is a more general measure of
risk, since it incorporates investor risk preferences, demand–supply effects, and
market implied probabilities of losses or gains, Panigirtzoglou and Skiadopoulos
(2004). E-VaR can be seen as a forward looking measure to quantify market
sentiment about the future course of financial asset prices, whereas historical
or statistically based VaR (S-VaR) is backward looking, based on the historical
data. With the development in 1993 of the traded option implied VIX index
for the SP-500 returns volatility over a 30 calendar day horizon, the so called
investor fear gauge”, a significant move toward the use of a market implied
rather than a historical measure of risk in practical aspects of risk management
has occurred. Policy makers such as the Bank of England use traded option
implied risk neutral density, volatility and quantile measures to gauge market
sentiment regarding future asset prices. 1
Given the industry standard for 10 day VaR at high confidence levels of 99%,
it is important to correctly model the distribution of the extreme values of asset
returns, as it is well known that the probability distributions of asset returns are
not Gaussian especially at short time horizons (see, Cont, 2001). In the man-
agement of risk, the modelling of asymmetries and the asymptotic behaviour of
the tails of the distribution of losses is important. Extreme value theory is a
robust framework to analyse the tail behaviour of distributions. Extreme value
theory has been applied extensively in hydrology, climatology and also in the
1For the VIX index see and for the Bank of England op-
tion traded implied probability density functions, volatility and quantile measures see, . In particular, market risk premia for a
given holding period is estimated as payoffs from volatility swaps which effectively take the
difference between realized volatility and the option implied volatility.
insurance industry (see, Embrechts et. al. 1997). Despite early work by Man-
delbrot (1963) on the possibility of fat tails in financial data and evidence on the
inapplicability of the assumption of log normality in option pricing, a system-
atic study of extreme value theory for financial modelling and risk management
has only begun recently. Embrechts et. al. (1997) is a comprehensive source
on extreme value theory and applications.2Dacorogna et. al. (2001) develop
a VaR estimate based on the extreme value Pareto distribution for the tails of
the distribution which is then empirically estimated from high frequency data
using a bootstrap method for the Hill estimator.
In this paper we propose Extreme Economic Value at Risk (EE-VaR) as a
new risk measure, which is calculated from an implied risk neutral density that
is based on the Generalized Extreme Value (GEV) distribution. It has been
shown in Markose and Alentorn (2005) that the GEV option pricing model
not only accurately captures the negative skewness and higher kurtosis of the
implied risk neutral density (RND), but it also delivers the market implied tail
index that governs the tail shape. It is important to note that the GEV does not
pose a priori restrictions on the tail shape as the GEV distribution encompasses
the thin and short tailed class of the Gumbel and Weibull , respectively, along
with the fat tailed Fréchet.3Indeed, one of the main findings from Alentorn and
Markose (2006) and Alentorn (2007) is that the daily implied tail shape parame-
ter estimated without maturity effects from the GEV RND model indicates that
market perception of fat tailed behaviour of extreme events is interspersed with
thin and short tailed Gumbel and Weibull values.4Hence, the assumption of the
GEV parametric model for the RND overcomes problems, associated with the
estimation of the risk neutral density function to flexibly include extreme values
and fat tails, which are often encountered with many non-parametric methods
and with the use of parametric models such as the Gaussian.5In this paper we
2Embrechts (1999, 2000) considers the potential and limitations of extreme value theory for
risk management. Without being exhaustive here, De Haan et. al. (1994) and Danielsson and
de Vries (1997) study quantile estimation. Bali (2003) uses the GEV distribution to model
the empirical distribution of returns. Mc Neil (1999) gives an extensive overview of extreme
value theory for risk management. See also Dowd (2002, pp.272-284).
3The Gumbel class includes the normal, exponential, gamma and log normal while the
Weibull include distributions such as the uniform and beta. Examples of fat tailed distribu-
tions that belong to the Fréchet class are Pareto, Cauchy, Student-t and mixture distributions.
4Even during extreme events, though the implied tail index results in fat tails for the GEV
RND based returns– at all times the first four moments were bounded.
5In order to estimate risks at high confidence levels, such as 99% - many non-parametric
methods for RND estimation fail to capture tail behaviour of distributions because of sparse
data for options traded at very high or very low strike prices. Hence, parametric models have
become unavoidable. This, however, replaces sampling error by model error. Markose and
will focus on estimating the term structure of the GEV based implied RNDs,
which allows us to calibrate an empirical scaling law for EE-VaR at different
confidence levels, and thus, to obtain the daily EE-VaR for any time horizon,
without having to employ the widely used but incorrect square root of time
scaling rule.
There is a vast literature on the analysis of information implied from option
markets. One of the areas that has received the most attention is the study of
the implied volatility surfaces, such as in Day and Craig (1988), Ncube (1996),
Dumas, Fleming and Whalley (1998) and others. The great majority of studies
of implied distributions have focused on the analysis of the distributions at a
single point in time for event studies, such as Bates (1991) for the study of the
1987 crash, Gemmill and Saflekos (2000) for the study of British elections, and
Melick and Thomas (1996) for the analysis of oil prices during the Gulf war
crisis. Starting with the study of the day to day dynamics of implied volatility
surfaces (see Cont and Da Fonseca (2002)), recently, Clews et. al. (2001) and
Panigirtzoglou and Skiadopoulos (2004) have developed a framework for the
analysis of dynamics of implied RND functions.
A problem encountered when looking at the daily dynamics of RNDs, or
RND implied measures such as volatility6or their associated quantile values, the
E-VaR, is the time to maturity and the contract switch effects (see, Melick and
Thomas, 1998). RNDs are usually constructed using the options with shortest
time to maturity. Since options have a fixed expiry date, this means that both
the time horizon of the RND and the holding period of the underlying asset
change with time to maturity. The degree of uncertainty decreases as the expiry
date approaches. Uncertainty jumps up again when the option with the shortest
time to maturity expires, and we switch to options with the next expiration
date. For instance, given that options on the FTSE 100 index expire on the
third Friday of the expiry month, the jump would occur on the third Monday
of the expiry month. Note also that option prices with less than 5 working days
to maturity are usually excluded. Thus, the problem associated with obtaining
constant horizon RNDs and option implied values for VaR or volatility for the
underlying assets from traded options is non-trivial. Clearly, the use of E-VaR
Alentorn (2005) have argued that as the GEV distribution encompasses the 3 main classes
of tail behaviour, it mitigates model error and further there is parsimony in the number of
parameters necessary to define the distribution.
6Alentorn and Markose (2006) give an extensive survey of the studies done on removing
maturity effects on implied volatility and higher moments of the RND. Here, we focus on the
quantile values, E-VaR.
for risk management is feasible only if it can be calculated and reported daily
for a constant time horizon or holding period that is required.
With regard to the traded option implied E-VaR, to our knowledge, there
are only three previous studies that have carried out an empirical analysis of
E-VaR and two of these study daily constant horizon E-VaR. Ait-Sahalia and
Lo (2000) estimated the E-VaR for a 126 day horizon. Clews et al. (2001)
have suggested a semi-parametric methodology that can remove maturity effects
in the construction of constant horizon RNDs. The methodology consists of
interpolating the Black-Scholes implied volatility surface in delta space at a
given time horizon, and then deriving the implied RND by calculating the second
derivative of the call pricing function, using the Breeden and Litzenberger (1978)
result. This methodology is used by the Monetary Instruments and Markets
Division at the Bank of England to report daily E-VaR values for the FTSE
100 index at confidence levels ranging from 5% to 95% for the FTSE 100, for
a 3 month constant horizon RND. However, with this methodology, it is not
possible to construct a constant time horizon implied RND for a time horizon
shorter than the shortest maturity available, given that the implied volatility
surface in delta space is non-linear. Panigirtzoglou and Skiadopoulos (2004)
looked at the E-VaR calculated at 95% confidence level for constant horizons of
1, 3 and 6 months for every 14 days during the year 2001. However, the problem
of reporting daily E-VaR at short constant horizons such as 10 days remains and
typically semi-parametric methods for RND extraction fail to report E-VaR at
99% confidence level.
In this paper, we focus on obtaining a daily estimate of a constant time
horizon GEV based E-VaR using a discrete term structure of RNDs. In Sec-
tion 2, the new methodology we propose proceeds by first constructing a daily
discrete term structure of implied RNDs, using option prices of all maturities
available and a cross section of strikes for each maturity. Hence, there is a
RND for each maturity available for traded options in a given day. Assuming
the parametric GEV model for the RND, we calculate the EE-VaR at different
confidence levels as the quantile values for the RND for each available maturity.
We exploit the linear behaviour of quantile values vis-à-vis the holding period,
k, in the log-log scale to derive an empirical scaling law for different confidence
levels, q.7One of the advantages of this linear relationship is that it allows us
7The empirical evidence for the scaling parameter b in the relationship, V aR(q, k ) =
V aR(q, 1)kb, which is linear in logs has been studied by Hauksson (2001), Menkens (2004)
and Provizionatou et. al.(2005) in the context of historical VaR. Also, Dacorogna et. al.
to both interpolate and extrapolate from the available maturities and obtain
daily E-VaR values for any constant horizon from 1 day to m days and can be
used regardless of the method for extracting the discrete RNDs. To test the
robustness of our methodology we use the daily 90 day E-VaR reported by the
Bank of England for the 95% confidence level to compare the performance of
the GEV implied EE-VaR and also E-VaRs obtained from parametric RNDs
for the Black Scholes and the Mixture of two Lognormals. We then proceed
to report a 10 day EE-VaR which is easily done with our method regardless
of the time horizon of the closest maturity option contracts. We analyse the
performance of EE-VaR for different confidence levels, different time horizons,
and for a large dataset, and compare it with the performance of historical VaR
and the Black-Scholes E-VaR. In this paper we perform an in depth analysis of
the daily EE-VaR performance for over 7 years, using daily closing index option
prices on the FTSE 100 from 1997 to 2003. This is the first paper to do this
and the empirical implementation and results are reported in Sections 3 and 4.
Backtesting results, based on the FTSE 100 index from 1997 to 2003, show that
EE-VaR has fewer violations than historical VaR. Note that statistical VaR is
done for a 1 day return and then scaled by the square root of time rule. The 10
day S-VaR when corrected by a multiplication factor of 3, to satisfy the viola-
tion bound, requires substantially more risk capital than EE-VaR. This saving
in risk capital with EE-VaR at high confidence levels of 99% arises because an
implied VaR estimate responds quickly to market events and in some cases even
anticipate them. In contrast, VaR estimates based on historical data reflect
extreme losses in the past for longer.
2 Model and Methodology
2.1 Extraction of GEV based RND from option prices
A large number of methods have been proposed for extracting implied distri-
butions from option prices since the seminal work of Breeden and Litzenberger
(1978), (see Jackwerth (1999) for an extensive survey). In this paper we use the
methodology proposed by Markose and Alentorn (2005) based on the General-
ized Extreme Value (GEV) distribution.
Let Stdenote the underlying asset price at time t. The European call option
(2001) derived an extreme value based VaR scaling law for high frequency forex data. Here,
we investigate the scaling relationship for implied VaR, rather than for historical VaR.
Ctis written on this asset with strike Kand maturity T. We assume the
interest rate ris constant. Following the Harrison and Pliska (1981) result on
the arbitrage free European call option price, there exists a risk neutral density
(RND) function, g(ST), such that the equilibrium call option price can bewritten
Ct(K) = EQ
t[er(Tt)max (STK, 0)]
Also, the following martingale condition holds for the stock price
Here EQ
t[.]is the risk-neutral expectation operator, conditional on all infor-
mation available at time t, and g(ST)is the risk-neutral density function of the
underlying at maturity. Note that the GEV option pricing model in Markose
and Alentorn (2005) is based on the assumption that negative returns, LT, as
defined in equation (3) below, follow a GEV distribution:
= 1 ST
The GEV distribution, in the form in von Mises (1936) (see, Reiss and
Thomas, 2001, p. 16-17) which incorporates a location parameter µ, a scale
parameter σ, and a tail shape parameter ξ, is defined by:
Fξ,µ,σ (x) = exp 1 + ξ
σ(xµ)1!, ξ 6= 0,(4)
1 + ξ(xµ)
F0,µ,σ (x) = exp exp xµ
σ, ξ = 0.(5)
The tail shape parameter ξ= 0 yields thin tailed distributions with the so
called tail index 1=αbeing equal to infinity, implying that all moments
of this class of distributions exist. When ξ < 0the GEV distribution class is
Weibull. The fat tailed Fréchet distributions arise when ξ > 0and note ξ > 25
is sufficient to imply infinite kurtosis.
The RND function g(ST)in (1) for the underlying asset price given that
LTis assumed to satisfy the GEV density function (see, Reiss and Thomas, p.
16-17) is given by8:
g(ST) = 1
Stσ1 + ξ(LTµ)
exp 1 + ξ(LTµ)
1 + ξ
σ(LTµ) = 1 + ξ
Note if the above condition in (7) is not satisfied, the GEV density function is
not defined on the real line. When ξ > 0and the distribution for Ltis fat tailed,
condition (7) implies that the GEV density function for the price is truncated
on the right, that is, the probability that the price will rise above this truncation
value is zero. On the other hand, when the ξ < 0and Ltis Weibull class, the
GEV density function for STis truncated on the left implying that the price
will not fall below the truncation value. Markose and Alentorn (2005) find that
while this did affect the limits of integration for the option price equation in
(1), the closed form solution for the call (and put) option for all cases of ξ6= 0
is identical. Omitting the proof , which can be found in Markose and Alentorn
(2005) the closed form GEV RND based call option price is given by
Ct(K) = er(Tt){Stσ
ξΓ(1 ξ, H 1)
(St(1 µ+σ
where H= 1 + ξ
Stµand Γ(1 ξ, H 1) = ´
H1zξezdz is the
incomplete Gamma function.
The structural GEV parameters ξ,µand σcan be estimated by minimizing
the sum of squared errors (SSE) between the analytical solution of the GEV
option pricing equations in (8) and the observed traded option prices with strikes
Ki, as given in (9) below:
SSE(t) = min
ξ,µ,σ (N
i=1 Ct(Ki)f
8Note the relationship between the density function for Lt,f(Lt), and that for the under-
lying, g(ST), is given by the general formula g(ST) = f(LT)|∂LT
For purposes of comparison, we use the above method to back out the re-
spective implied parameters for the Black-Scholes model and also the RND from
the Mixture of two Lognormals (MLN) first constructed by Ritchey (1990).
At the estimation stage, we use the data on the index futures contract with
the same maturity as the options and as the futures price at maturity yields,
FT=ST, the no arbitrage martingale condition in (2) enables us to substitute
out EQ(ST)by using Ft,T =EQ(ST), This also vitiates the need for data on the
dividend yield rate. The optimization problem in (10), was performed using the
non-linear least squares algorithm from the Optimization toolbox in MatLab.9
2.2 EE-VaR calculation from GEV RND
The quantile for the GEV distribution i.e. the VaR value associated with
a given confidence level q, is given as a function of the three GEV parameters
(see, Dowd 2004: pp. 274):
V aR =µσ
ξh1(log (q))ξi, ξ 6= 0,(10)
V aR =µσlog [log (1/q)] , ξ = 0.(11)
On substituting the implied GEV parameters from daily traded option prices
for a given maturity horizon, the extreme economic value at risk (EE-VaR) is
calculated from (10) and (11).
The results obtained using EE-VaR will be compared with E-VaR values
under the Gaussian assumptions of the Black-Scholes model and that of the
mixture of two lognormals. The quantile of the normal distribution is used
to calculate the E-VaR values for the Gaussian case using the Black-Scholes
implied volatility. The MLN method models the RND as a weighted sum of two
lognormals, and is given by:
f(ST) = ph(ST|µ1T, σ1T) + (1 p)h(ST|µ2, σ2T).(12)
The MLN RND has been extensively used in the literature, given that it is
9For a more detailed analysis of the estimation results, including time series of implied
parameters, pricing performance and comparison of results of the GEV model with other
parametric models, can be found in Alentorn and Markose (2006) and Alentorn (2007). As
already noted in the Introduction, the daily implied tail shape parameters ξ, for the sample
period ranged between -0.2 and +0.22.
very flexible, and allows the modelling of different levels of skewness, as well as
bimodal densities. However, compared to the GEV RND it has are five unknown
parameters θ={µ1, µ2, σ1, σ2, p}, the means of each lognormal function µ1and
µ2, the standard deviations σ1and σ2, and the weighting coefficient p. We
obtain the set of implied parameters ˆ
θby the method in (9). Then, E-VaR is
calculated as the quantile of the MLN density, which consists of a weighted sum
of the two inverse cumulative distribution functions, H , and given by:
EV aR(q, k) = ˆpH1(q|ˆµ1,ˆσ1, T ) + (1 ˆp)H1(q|ˆµ2,ˆσ2, T ).(13)
Some authors, such as Shiratsuka (2001) and Melick (1999), argue that the
values for the higher quantiles of implied RNDs are very sensitive to the choice
of RND estimation technique, since the range of strike prices that are actually
traded is very limited and the tails of the estimated implied RND vary depending
on the procedure employed. Table 1 below shows the percentage number of days
between 1997 and 2003 with traded put options with strike below each of the
confidence levels.
Table 1: Percentage number of days with put option prices with strikes below
each of the confidence levels FTSE-100 Traded Options (1997-2003)
Confidenc level Percentage number of days
70% 94%
80% 86%
90% 68%
95% 51%
99% 22%
Hence, we will also compare the quantile values obtained from the parametric
RND models with those at the highest confidence level of 95% reported by the
Bank of England which uses the semi-parametric RND method discussed earlier.
3 Data description
The data used in this study are the daily settlement prices of the FTSE
100 index call and put options published by the London International Financial
Futures and Options Exchange (LIFFE). These settlement prices are based on
quotes and transactions during the day and are used to mark options and futures
positions to market. Options are listed at expiry dates for the nearest three
months and for the nearest March, June, September and December. FTSE 100
options expire on the third Friday of the expiry month. The FTSE 100 option
strikes are in intervals of 50 or 100 points depending on time-to-expiry, and the
minimum tick size is 0.5.
The period of study was from 1997 to 2003, so there were 28 expiration
dates (7 years with 4 contracts per year). This period includes some events,
such as the Asian crisis, the LTCM crisis and the 9/11 attacks, which resulted
in a sudden fall of the underlying FTSE 100 index, and will be useful to analyze
the performance of the methods under extreme events. The average number of
maturities available with more than 3 options traded in our sample (1997-2003)
is displayed in Table 2 below . In average across all years, we have 5.33 different
maturities each day.
Table 2: Average number of maturities available FTSE-100 Traded Options
Year Average number of maturities available
1997 3.96
1998 4.57
1999 5.19
2000 5.49
2001 5.84
2002 6.19
2003 6.09
Average 5.33
The LIFFE exchange quotes settlement prices for a wide range of options,
even though some of them may have not been traded on a given day. In this
study we only consider prices of traded options, that is, options that have a non-
zero volume. The data were also filtered to exclude days when the cross-sections
of options had less than three option strikes, since a minimum of three strikes
is required to estimate the three parameters of the GEV model.10 Also, options
whose prices were quoted as zero or that had less than 5 days to expiry were
eliminated. Finally, option prices were checked for violations of the monotonicity
10The number of option prices needed to extract the RND must be at least equal than the
number of degrees of freedom for the parametric method used. The number of degrees of
freedom is equal to the number of parameters that need to be estimated minus the number
of constraints. For example, the GEV model has three parameters while the mixture of
lognormals have five parameters.
11Monotonicity requires that the call (put) prices are strictly decreasing (increasing) with
respect to the exercise price.
The risk-free rates used are the British Bankers Association’s 11 a.m. fixings
of the 3-month Short Sterling London InterBank Offer Rate (LIBOR) rates from
the website Even though the 3-month LIBOR market does not
provide a maturity-matched interest rate, it has the advantages of liquidity and
of approximating the actual market borrowing and lending rates faced by option
market participants (Bliss and Panigirtzoglou, 2004).
4 Empirical Modelling and Results
4.1 Term structure of RNDs
To calculate the EE-VaR, ideally, one would use a RND implied by options
with time to maturity exactly equal to the time horizon we are interested. That
is, to calculate the 10 day EE-VaR we would use prices from options that mature
in 10 days to obtain an implied RND, and calculate the quantile of that density
at the confidence level required. However, in practice, we only have options that
expire every month during the next three months, and also, options that expire
in March, June, September and December. In the original study of Markose
and Alentorn (2005), at each trading day, only the RND implied by the closest
to maturity contracts for which futures contracts were available (March, June,
September and December) was extracted. Here, we propose, on a daily basis,
the extraction of an RND for each of the maturities with a sufficient number of
traded option prices. Then, using this discrete set of RNDs, each with a different
maturity, we can construct what we call a term structure of implied RNDs. This
term structure can be visualized as a 3 dimensional chart that displays, for a
given day, how the implied RNDs vary across different maturities. For purposes
of illustration, Figure 1 below displays the implied RND term structure for a
typical day, 21 August 2001, using the GEV model. Note from Figure 1 that the
main feature of the term structure, which is independent of the RND extraction
method used, is that the peakedness of the RNDs decreases as the time horizon
increases. This term structure of implied RNDs will be used in the following
section to obtain constant time horizon E-VaRs.
Table 3 below displays the actual EE-VaR values. As one would expect, the
EE-VaR values increase both with confidence level and with time horizon. Also,
note how the number of options prices available decreases as time to maturity
increases, that is, the options with the closest to maturity dates are the ones
that have the widest range of traded strikes.
Figure 1: Term Structure of GEV based implied RNDs on 21 August 01
Table 3: EE-VaR values for each available maturity and at different confidence
levels on 21 August 01
Expiry Days to Number EE VaR
month maturity options 70% 80% 90% 95% 99%
Sep-01 31 44 2.4% 4.4% 7.4% 10.1% 15.6%
Oct-01 59 31 3.1% 5.9% 10.2% 14.0% 21.7%
Nov-01 87 13 3.7% 7.2% 12.7% 17.5% 27.4%
Dec-01 122 16 4.2% 8.5% 15.0% 20.8% 32.6%
Mar-02 213 13 5.7% 11.4% 20.1% 27.7% 42.8%
Jun-02 304 10 6.9% 13.7% 23.8% 32.4% 49.0%
4.2 Empirical Scaling of EE-VaR
One of the requirements of the Basel accord is that banks should report the
daily 10 day VaR at 99% confidence level of their portfolios. However, there are
some difficulties with estimating the 10 day VaR, due to the need for a long time
series in order to compute the 10 day returns, and then, calculate the quantiles
of their distribution. In practice, the square root of time scaling rule is widely
used to scale up the 1 day VaR to the 10 day VaR. This scaling rule is only
appropriate for time series that have Gaussian properties, but it has been well
established in the literature for a long time (see, Fama (1965) and Mandelbrot
(1967)), that financial data is non-Gaussian. Following the wide spread use
of VaR as a risk measure and reporting requirement, there have been several
recent studies that looked at the problem of scaling VaR, such as McNeil and
Frey (2000), Hauksson et. al. (2001), Kaufmann and Patie (2003), Danielsson
and Zigrand (2004), Menkens (2004) and Provizionatou et al (2005).
In this study we are faced with a similar problem, but instead of having to
scale up the 1 day E-VaR, we need to scale down from the maturities available,
to 10 day and 1 day E-VaR. Without resorting to scaling, we would only be
able to calculate the 10 day VaR for only one day each month, the day when
there are exactly 10 days to maturity for the closest to maturity contract (in
the case of FTSE 100 data, it would be around the first Friday of each month,
since contracts mature in the third Friday of the month). Following a similar
approach as in Hauksson et. al. (2001) and Menkens (2004), we have identified
an empirical scaling law for EE-VaR against time horizon that is linear in a
log-log scale.
log (EE V aR (k, q)) = b(q) log (k) + c(q),(14)
where kis the number of days, c(k)is the 1-day EE-VaR value (given that
log(1) = 0), and the slope b(q)is the EE-VaR scaling parameter for a given
confidence level q. Once we estimate the parameters b(q)and c(q)for a given
day and for a given confidence level q, we can obtain the k-day EE-VaR value
as follows:
EE V aR (k, q) = 10ˆ
b(q) log(k)+ˆc(q).(15)
Figure 2 below displays the EE-VaR values obtained from the RNDs in Figure
1 above, using the linear regression line from equation (14).
Figure 2: log-log plot for 21 August 01, with the estimated linear scaling rule
for each confidence level.
1.4 1.6 1.8 2 2.2 2.4 2.6
log( Days )
log( EE−VaR )
Table 4: Regression coefficients b and c for 21 August 2001.
Confidence level b c
70% 0.47 -2.18
80% 0.50 -2.09
90% 0.51 -1.91
95% 0.52 -1.82
99% 0.51 -1.73
Average 0.50 -1.95
While we report the average of the full set of daily scaling coefficients, b,
implied from the term structure of EE-VaR for the sample period, what Table
4 indicates is how the E-VaR based scaling coefficients differ from scaling in
historical VaR. Hauksson et. al. (2001) report these to be around .43 (though
it is not clear what confidence level this is for) while Provitinatou et. al report
scaling coefficients which range from .47 to .45 for the 70% and 99% confidence
levels, respectively. As will be seen, in general the market implied VaR scales
more vigorously with time at higher quantiles. However, the size of the scaling
coefficients in Tables 4 and 5 should not be confused with implying unbounded
second and higher moments of the RND functions as the implied tail parameters
ξat all times for the sample period showed that up to 4 moments exist.
4.3 Improving the estimation of the linear scaling law by
using WLS
The linear regression estimated to obtain the time scaling for EE-VaR can
be affected by EE-VaR values calculated from an RND constructed from very
few option prices. The EE-VaR estimates in such cases will have very wide
confidence intervals. As an example, take the data and regression for 12 Nov
97, shown in Figure 3 below. The R2of the OLS regression was 64.8%, a very
poor fit. The EE-VaR value furthest away from maturity was obtained from
an RND estimated using only 4 option prices, and thus the confidence intervals
of the EE-VaR estimate are much wider than the EE-VaR values obtained for
closer maturities, which are based on RNDs extracted using around 25 contracts.
One method to solve this issue is to use a Weighted Linear Squares (WLS)
regression, using the number of option prices available at each maturity relative
to total prices as weights for the EE-VaR values.
W eighted R2= 1 PTN
i=T1wi(yiyi)2, with
wi= 1,(16)
wi=N umberOf P riceAtM aturityi
T otalN umberOf P rices (17)
Table 5: Average R2for different quantiles, and number of days with different
ranges of R2
Confidence level 70% 80% 90% 95% 99%
Average R287.9% 97.9% 98.8% 98.7% 97.9%
Number of days with R2>99% 429 1152 1410 1320 901
99% R2>90% 860 510 282 375 742
R290% 444 71 41 38 90
Table 6 below shows the average weighted R2at each confidence level. Note
how the fitting performance increases with confidence level, while it is lowest at
96.8% for the lowest quantile of 70%.
Figure 3: Example of linear regression using OLS vs. WLS for a day when there
are some maturities with very few option prices available.
Table 6: Average weighted R2at each quantile
Confidence level 70% 80% 90% 95% 99%
Weighted R296.8% 99.4% 99.6% 99.5% 99.3%
4.4 Full Regression results for scaling in EE-VaR
The average regression coefficients b and c across the 1733 days in our sample
period are displayed in Table 7 below. We can see that the slope bincreases
with the confidence interval, and that intercept, i.e. the 1 day EE-VaR also
increases with confidence level, as one would expect. The standard deviation of
the estimates at different confidence levels is fairly constant. We also report the
percentage number of days where bwas found to be statistically significantly
different from one half.12 On average, irrespective of the confidence level, we
12Newey-West (1987) heterosckedasticity and autocorrelation consistent standard error was
found that in around 50% of the days the scaling was significantly different than
0.5, the scaling implied by the square root of time rule.
Table 7: Average regression coefficients b,c and exp(c) across the 1733 days in
the sample, for the GEV case. #: The percentage number of days where b was
statistically significantly different from 0.5. : Standard deviation of b
Confidence level q b(q) c EVaR(1,q) = exp(c)
0.41 -2.24
70% (51.3%)# (0.26)0.6%
0.48 -2.04
80% (51.1%)# (0.23)0.9%
0.51 -1.84
90% (52.4%)# (0.23)1.4%
0.53 -1.73
95% (52.1%)# (0.23)1.9%
0.56 -1.57
99% (52.6%)# (0.25)2.7%
Figure 4 displays the time series of the b estimates (the slope of the scaling
law) for the GEV case at the 95% confidence level. Even though the average
value of b at this confidence level is 0.53 (see Table 7), it appears to be time
varying and takes values that range from 0.2 to 1.
used to test the null hypothesis H0 :b= 0.5. We employed this methodology when testing the
statistical significance of the estimated slope b, because E-VaR estimates are for overlapping
horizons, and therefore are auto-correlated. The Newey-West lag adjustment used was n - 1.
Figure 4: Time series of b estimates for the GEV model at 95% confidence
4.5 Comparison of EE-VaR with other models for E-VaR
Figure 5 below displays the time series of 90 day EVaR estimates at the 95%
confidence level for each of the three parametric methods for RND extraction
(combined with their respective empirical scaling regression results) together
with the estimates from the Bank of England non-parametric method. The 90
day FTSE returns are also displayed in Figure 5.
Table 8 below shows the sample mean and standard deviation of each of the
four EVaR time series. If we use the BoE values as the benchmark, we can
see that on average, the mixture of lognormals method overestimates EVaR,
while the Black-Scholes method underestimates it. Among the three parametric
methods, the GEV method yields the time series of EVaRs closest to the BoE
one. This can be seen both from Table 8 and also in the Figure 5, where the
BoE time series practically overlaps the GEV E-VaR time series. This confirms
that the GEV based VaR calculation is equivalent to the semi-parametric one
used by the Bank of England but also has the added advantage, as we will see
in the next section, of being capable of providing 10 day E-VaRs at the industry
standard 99% confidence level.
Figure 5: BoE EVaR vs. GEV EVaR vs. Mixture lognormals for 90 days at
95% confidence level
Table 8: BoE EVaR vs. GEV EVaR vs. Mixture lognormals for 90 days at 95%
confidence level
Method Sample mean Sample standard deviation
BoE method 0.217 5.9%
GEV model 0.212 6.0%
Mixture lognormals 0.259 8.5%
Black Scholes 0.147 4.7%
4.6 EE-VaR in Risk Management : Backtesting results
The performance of a VaR methodology is usually assessed in terms of its
backtesting performance. Here we compare the backtesting performance of EE-
VaR against statistical VaR (S-VaR) estimated using the historical method and
scaled using the square root of time, and also against the E-VaR estimated
under the Black-Scholes assumptions.
The historical 1 day VaR at, say 99% confidence level, is obtained by taking
the third highest loss in the time window of the previous 250 trading days, that
is, in the previous year. Then, to obtain the 10 day VaR , the most commonly
used technique is to scale up the 1 day VaR using the square root of time
rule. The percentage number of violations at each confidence level q needs to
be below the benchmark (1-q). For example, at the 99% confidence level, the
percentage of days where the predicted VaR is exceeded by the market should
be 1%. If there are fewer violations than the benchmark 1%, it means that
the VaR estimate is too conservative, and it could impose an excessive capital
requirement for the banks. On the other hand, if there are more violations than
the benchmark 1%, the VaR estimates are too low, and the capital set aside by
the bank would be insufficient.
We use our method to calculate the GEV and Black-Scholes based E-VaR
values at constant time horizons of 1, 10, 30, 60, and 90 days. Tables 9, 10
and 11 show the results in terms of percentage number of violations for EE-
VaR, S-VaR, and Black-Scholes E-VaR, respectively. The values highlighted
in bold indicate that the benchmark percentage number of violations has been
exceeded. On the other hand, the non highlighted values indicate that the
benchmark percentage number of violations at the given confidence level has
not been exceeded. The last row of each table displays the average percentage
number of violations across maturities for each confidence level. In the row
labelled “Benchmark” we can see the target percentage number of observations
at each confidence level.
We can see that of all the three methods, EE-VaR yields the least number
of cases where the benchmark percentage violations is exceeded. However, it
exceeds the benchmark at all confidence levels for the 1 day horizon. The S-VaR
also exceeds the benchmark at all confidence levels for the 1 day horizon, but
additionally, it exceeds it for the 10 day horizon at 97% and 99% confidence
levels, and for the 30 day horizon at almost all confidence levels. The Black-
Scholes based E-VaR is the worst of all methods, exceeding the benchmark in
14 out of the 25 cases.
Looking at the averages across horizons for the three methods, we see that
EE-VaR and S-VaR yield similar results at the higher quantiles (98% and 99%),
but EE-VaR appears further away from the benchmark than S-VaR at the other
confidence levels, indicating that it may be too conservative. On average, the
Black-Scholes based E-VaR exceeds the benchmark percentage number of viola-
tions at all but the lowest confidence level, which indicates that it substantially
underestimates the probability of downward movements at high confidence lev-
Table 9: Percentage violations of EE-VaR
Horizon Confi- level
(days) 95% 96% 97% 98% 99%
Benchmark 5% 4% 3% 2% 1%
1 6.2% 5.5% 4.4% 3.8% 3.1%
10 2.4% 1.8% 1.5% 1.3% 0.6%
30 2.8% 2.0% 1.3% 0.5% 0.1%
60 2.5% 2.2% 1.3% 0.7% 0.0%
90 2.9% 2.3% 1.5% 0.8% 0.0%
Average 3.4% 2.8% 2.0% 1.4% 0.8%
Table 10: Percentage violations of Statistical VaR (S-VaR)
Horizon Confi- level
(days) 95% 96% 97% 98% 99%
Benchmark 5% 4% 3% 2% 1%
1 5.9% 4.7% 3.8% 2.5% 1.5%
10 4.6% 4.0% 3.1% 1.7% 1.4%
30 5.3% 4.2% 3.3% 2.1% 0.8%
60 4.3% 3.6% 2.8% 1.1% 0.3%
90 4.5% 2.8% 1.9% 0.7% 0.0%
Average 4.9% 3.9% 3.0% 1.6% 0.8%
Table 11: Percentage violations of Black-Scholes based E-VaR
Horizon Confi- level
(days) 95% 96% 97% 98% 99%
Benchmark 5% 4% 3% 2% 1%
1 6.8% 5.9% 5.1% 4.4% 3.5%
10 3.6% 3.2% 2.3% 2.0% 1.0%
30 4.0% 3.8% 3.3% 2.4% 1.3%
60 3.9% 3.3% 2.9% 2.4% 1.7%
90 4.6% 4.1% 3.7% 3.0% 2.0%
Average 4.6% 4.1% 3.5% 2.8% 1.9%
Figure 6 below shows the time series of 10 day FTSE returns, 10 day EE-VaR
and 10 day S-VaR, both at 99% confidence level. We have chosen to plot the
VaR of these particular set of (q , k)values as the 10 day VaR at 99% confidence
level is one of most relevant VaR measures for practitioners, given the regulatory
reporting requirements. We can see how the S-VaR is violated more times (25
times, or 1.4% of the time) than the EE-VaR (10 times, or 0.6%) by the 10 day
FTSE return.
Figure 6: Time series of the 10 day FTSE returns, 10 day EE-VaR and 10 day
S-VaR, both at a 99% confidence level.
4.7 E-VaR vs. Historic VaR x3
The Basle Committee on Banking Supervision explains in the “Overview
of the Amendment to the Capital Accord to Incorporate Market Risks” (1996)
that the multiplication factor, ranging from 3 to 4 depending on the backtesting
results of a bank’s internal model, is needed to translate the daily value-at-risk
estimate into a capital charge that provides a sufficient cushion for cumulative
losses arising from adverse market conditions over an extended period of time.
But it is also designed to account for potential weaknesses in the modelling
process. Such weaknesses exist because:
Market price movements often display patterns (such as "fat tails") that
differ from the statistical simplifications used in modelling (such as the
assumption of a "normal distribution").
The past is not always a good approximation of the future (for example
volatilities and correlations can change abruptly).
VaR estimates are typically based on end-of-day positions and generally
do not take account of intra-day trading risk.
Models cannot adequately capture event risk arising from exceptional mar-
ket circumstances.
It is interesting to note from Figure 7 that the E-VaR values in periods of market
turbulence (Asian Crisis, LTCM crisis and 9/11) are similar to the historical S-
VaR values when multiplied by a factor of 3. Thus, the multiplication factor 3
for a 10 day S-VaR appears to justify the reasoning that it will cover extreme
events. In contrast, as we are modelling extreme events explicitly under EE-VaR
such ad hoc multiplication factors are unnecessary.
Figure 7: 10 day at 99% E-VaR vs. Statistical VaR multiplied by a factor of 3
4.8 Capital requirements
The average capital requirement based on the EE-VaR, S-VaR and S-VaRx3
for the 10 day VaR at different confidence level is displayed in Table 12 below.
Here, the capital requirement is calculated as a percentage of the value of a
portfolio that replicates the FTSE 100 index. What is very clear is that EE-VaR
when compared to S-VaRx3 shows substantial savings in risk capital, needing
only on average 1.7% more than S-VaR to give cover for extreme events. S-
VaRx3 needs over 2.36 times as much capital for risk cover at 99% level. Figure
7 clearly gives the main draw back of historically derived VaR estimates where
the impact of large losses in the past result in high VaR for some 250 days at
a time. The EE-VaR estimates are more adept at incorporating market data
information contemporaneously.
Table 12: Average daily capital requirement based on a 10 day horizon at dif-
ferent confidence levels, with standard deviations in brackets.
Confi- level Aver-
dence age
95% 96% 97% 98% 99%
S- 6.5% 7.3% 7.8% 9.1% 10.1% 8.2%
VaR (1.9%) (2.3%) (2.4%) (2.9%) (2.8%) (2.5%)
S-VaR 19.5% 21.9% 23.4% 27.3% 30.3% 24.48%
×3(5.7%) (6.9%) (7.2%) (8.7%) (8.4%) (7.38%)
EE- 8.1% 8.7% 9.6% 10.8% 12.7% 9.9%
VaR (3.1%) (3.3%) (3.7%) (4.2%) (5.0%) (3.9%)
5 Conclusions
We propose a new risk measure, Extreme Economic Value at Risk (EE-
VaR), which is calculated from an implied risk neutral density that is based on
the Generalized Extreme Value (GEV) distribution. In order to overcome the
problem of maturity effect, arising from the fixed expiration of options, we have
developed a new methodology to estimate a constant time horizon EE-VaR by
deriving an empirical scaling law in the quantile space based on a term structure
of RNDs. Remarkably, the Bank of England semi-parametric method for RND
extraction and the constant horizon implied quantile values estimated daily for a
90 day horizon coincides closely with the EE-VaR values at 95% confidence level
showing that GEV model is flexible enough to avoid model error displayed by the
Black-Scholes and Mixture of Lognormal models. The main difference between
the Bank’s method and the one that relies on an empirical linear scaling law
of the E-VaR based on a daily term structure of the GEV RND is that shorter
than 1 month E-VaRs and in particular daily 10 day E-VaR can be reported
in our framework. This generally remains problematic in the RND extraction
method based on the implied volatility surface in delta space as it is non-linear
in time to maturity and also it cannot reliably report E-VaR for high confidence
levels of 99%.
Based on the backtesting and capital requirement results, it is clear there
is a trade-off between the frequency of benchmark violations of the VaR value,
and the amount of capital required. The 10 day EE-VaR gives fewer cases of
benchmark violations, but yields higher capital requirements compared to the
10 day S-VaR. However, when the latter is corrected by a multiplication factor
of 3, to satisfy the violation bound, the risk capital needed is more that 2.3
times as much as EE-VaR for the same cover at extreme events. This saving
in risk capital with EE-VaR at high confidence levels of 99% arises because an
implied VaR estimate responds quickly to market events and in some cases even
anticipate them.
While the power of such a market implied risk measure is clear, both as
an additional tool for risk management to estimate the likelihood of extreme
outcomes and for maintaining adequate risk cover, the EE-VaR needs further
testing against other market implied parametric models as well as S-VaR meth-
ods. This will be undertaken in future work.
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... An accurate estimation of volatility is also essential for the assessment of Value-at-Risk (VaR). Among others, Giot (2005) uses the old VIX index (VXO) to calculate the VaR for the S&P 100 portfolio; Alentorn and Markose (2008) report that the VaR calculated by option implied volatility outperforms the VaR calculated by means of historical simulation, and the option implied VaR is able to respond quickly to market shocks, even in some cases anticipating such shocks. Volatility forecasts play a major role in portfolio strategies (see e.g. ...
Corridor implied volatility is obtained from model-free implied volatility by truncating the integration domain between two barriers. Empirical evidence on volatility forecasting in various markets points to the utility of trimming the risk-neutral distribution of the underlying stock price, in order to obtain unbiased measures of future realized volatility. The aim of this paper is to investigate the optimal corridor of strike prices for volatility forecasting in the Italian market, by analyzing numerous symmetric and asymmetric corridors in a dataset for the years 2005–2010 spanning both a relatively calm period and a period of turmoil. The results indicate that put prices, providing information on the probability of a downturn of the underlying asset, provide the best indication of future realized volatility, particularly in a period of turmoil.
... 998) analyses the empirical scaling law on volatility at different time scales. He defines the standard deviation in high frequency data as the average of absolute logarithmic price changes which is the same as Müller,Dacorogna, Olsen, Pictet, and Morgenegg (1990). The authors also find a power law relationship between volatility and time interval.Alentorn and Markose (2008) find two empirical scaling laws for Economic VaR (E-VaR) and implied volatility. By using the scaling law, they can remove the maturity dependence to compute E-VaR values at any time horizon. ...
... Interested readers are referred to recent literature on Pareto tail estimation (e.g. [9]), VaR and VaR scaling (see [1] and [14], among others). Figure 1: Backtesting VaR and CVaR estimates of FTSE100 at level 1%. ...
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We discuss asymptotic scaling rules for VaR and CVaR in the context of distributions with Pareto style tails. These relationships are easily turned into semiparametric VaR and CVaR estimates with appealing backtesting properties.
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No cenário econômico, estudar o comportamento de preços de produtos, commodities ou indicadores torna possível a realização de previsões, permitindo elaboração de projeções de risco com maior precisão e, quando ocorrem eventos extremos desses preços, perdas, ou até mesmo falências podem ocorrer. Nesse sentido, a Teoria de Valores Extremos (TVE) trata de maneira adequada tais fenômenos. Foram utilizados os dados econômicos do CEPEA - ESALQ compreendida entre o período de 1997 a 2020, organizados em séries de máximos mensais e, para cada série, as distribuições Gumbel, Generalizada de Valores Extremos (GVE) e suas versões não-estacionárias foram consideradas. Pôde-se constatar que as distribuições Gumbel e GVE se ajustaram em todos os meses e os indicadores de qualidade de ajuste atestam que a distribuição Gumbel é a mais adequada em todos os meses. Nos meses de abril a outubro existe uma ligeira menor probabilidade dos preços serem superados e nos meses de novembro a fevereiro são os meses com maior probabilidade de ocorrência de altos preços de boi gordo. Pelo teste de Mann-Kendall constatou-se tendência em todas as séries, a qual foi incorporada na distribuição Gumbel não-estacionária, e o teste de razão de verossimilhanças e AIC mostraram-se favoráveis quanto à modelagem da tendência. Palavras-chave: planejamento rural; economia aplicada; distribuição Gumbel; valor de retorno; estatística aplicada. Probabilistic modeling of maximum commodity fat ox prices for the state of São Paulo ABSTRACT: In the economic scenario, studying the behavior of prices of products, commodities or indicators makes it possible to make forecasts, allowing the elaboration of risk projections with greater precision and, when extreme events of these prices occur, losses or even bankruptcies can occur. In this sense, the Extreme Value Theory (EVT) is more suitable such phenomena. Economic data from CEPEA - ESALQ covering the period from 1997 to 2020 were used, organized in series of monthly maximums and, for each series, the Gumbel, Generalized Extreme Values (GVE) distributions and their non-stationary versions were considered. It could be seen that the Gumbel and GVE distributions fit in every month and the goodness-of-fit attest that the Gumbel distribution is the most suitable in every month. In the months of April to October there is a slight lower probability of prices being exceeded and in the months of November to February are the months with the highest probability of occurrence of high fat ox prices. The Mann-Kendall test was used for testing the trend in all series, which was incorporated into the non-stationary Gumbel distribution, and the likelihood ratio test and AIC were favorable in terms of trend modeling. Keywords: rural planning; applied economics; gumbel distribution; return value; applied statistics.
The aim of this study is to compare the performance of robust estimators in the presence of explanatory variables with Generalized Extreme Value (GEV) distributions in the logistic regression model. Existence of extreme values in the logistic regression model negatively affects the bias and effectiveness of classical Maximum Likelihood (ML) estimators. For this reason, robust estimators that are less sensitive to extreme values have been developed. Random variables with extreme values may be fit in one of specific distributions. In study, the GEV distribution family was examined and five robust estimators were compared for the Fréchet, Gumbel and Weibull distributions. To the simulation results, the CUBIF estimator is prominent according to both bias and efficiency criteria for small samples. In medium and large samples, while the MALLOWS estimator has the minimum bias, the CUBIF estimator has the best efficiency. The same results apply for different contamination ratios and different scale parameter values of the distributions. Simulation findings were supported by a meteorological real data application.
In this study, we use the S&P 500 options prices to derive various tail risk indexes. We then decompose the option‐implied tail risk indexes into the conditional tail risk of stock returns and equity tail risk premia. We examine the predictive power of the conditional tail risks and equity tail risk premia for various stock portfolio returns. The results demonstrate that the tail risk indicators possess additional predictive power for stock returns in the presence of extant risk indicators and other return predictor variables.
This paper develops a new multiple time scale-based empirical framework for market risk estimates and forecasts. Ultra-high frequency data are used in the empirical analysis to estimate the parameters of empirical scaling laws which gives a better understanding of the dynamic nature of the market. A comparison of the new approach with the popular Value-at-Risk and expected tail loss measures with respect to their risk forecasts during the crisis period in 2008 is presented. The empirical results show the outperformance of the new scaling law method which turns out to be more accurate and flexible due to the scale invariance. The scaling law method promotes the use of massive real data in developing risk measurement and forecasting models.
Pension funds have adopted different management approaches to overcome the arising difficulties to maintain a solid financial status. Among these, there is the adoption of an indexation policy which is conditional on the solvability of the fund. Pension funds recognizing conditional inflation indexation are obliged to pay an additional payoff linked to the inflation rate through some specific rule. The additional payoff normally takes the form of a contingent claim conditional to a measure of sustainability of the payoff itself; in most cases, the measure is linked to an asset/liability ratio able to capture the solvability of the fund. Therefore, a full valuation of the obligation towards funds participants cannot exclude the proper appraisal of this additional option. The option payoff is conditional to a measurement asset that is different from the reference underlying asset. This structure recalls a barrier option with different measurement and payoff asset. The paper investigates the opportunity to apply barrier option schemes in an asset/liability context to provide a full valuation of the obligation towards participants. Results derive from a simulation procedure applied by means of scenario-based analysis. Numerical results give the opportunity to state the absolute and relative value of the inflation option.
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We briefly introduce some basic facts about multivariate extreme value theory and present some new results regarding finite aggregates and multivariate extreme value distributions. Based on our results high-frequency data can considerably improve the quality of estimates of extreme movements in financial markets. Secondly, we present an empirical exploration of what the tails really look like for four foreign exchange rates sampled at varying frequencies. Both temporal and spatial dependence is considered. In particular we estimate the spectral measure, which along with the tail index, completely determines the extreme value distribution. Lastly, we apply our results to the problem of portfolio optimization or risk minimization. We analyse how the expected shortfall and value-at-risk scale with the time horizon and find that this scaling is not by a factor of the square root of time as is frequently used, but by a different power of time. We show that the accuracy of risk estimation can be drastically improved by using hourly or bihourly data.
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We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.
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The financial industry, including banking and insurance, is undergoing major changes. The (re)insurance industry is increasingly exposed to catastrophic losses for which the requested cover is only just available. An increasing complexity of financial instruments calls for sophisticated risk management tools. The securitization of risk and alternative risk transfer highlight the convergence of finance and insurance at the product level. Extreme value theory plays an important methodological role within risk management for insurance, reinsurance, and finance.
Many financial applications, such as risk analysis and derivatives pricing, depend on time scaling of risk. A common method for this purpose, though only correct when returns are iid normal, is the square–root–of–time rule where an estimated quantile of a return distribution is scaled to a lower frequency by the square-root of the time horizon. The aim of this paper is to examine time scaling of risk when returns follow a jump diffusion process. It is argued that a jump diffusion is well-suited for the modeling of systemic risk, which is the raison d’etre of the Basel capital adequacy proposals. We demonstrate that the square–root–of–time rule leads to a systematic underestimation of risk, whereby the degree of underestimation worsens with the time horizon, the jump intensity and the confidence level. As a result, even if the square–root–of–time rule has widespread applications in the Basel Accords, it fails to address the objective of the Accords.
The paper motivates the use of the statistical extreme value theory for the problem of portfolio selection in economics, both theoretically and empirically. It is shown that the conventional safety first criterion developed by Roy can be successfully improved upon by exploiting the fat tail property of asset returns. Extreme value theory is seen to provide a better bound than the Chebyshev bound. In the empirical application we calculate minimum threshold return levels given very low exceedance probabilities for bond and equity investors. A proof of a new quantile estimator is obtained in the appendix. The data cover at least a half-century of returns and allow for evaluation of investment risks in the long run.
The paper performs an empirical estimation of time-varying volatility using OLS regression. Error Components, and Dummy Variable models, by regressing the implied volatility on time to maturity, the strike price and a dummy. Both the daily OLS equations and the panel data model provide more accurate estimates of Black and Scholes option prices than the bench-mark standard deviation of log returns. FT-SE 100 Index European options are used for empirical analysis.
This paper presents a new approach to modeling the dynamics of implied distributions. First, we obtain a parsimonious description of the dynamics of the S&P 500 implied cumulative distribution functions by applying principal components analysis. Subsequently, we develop new arbitrage-free Monte Carlo simulation methods that model the evolution of the whole distribution through time as a diffusion process. Our approach generalizes the conventional approaches of modeling only the first two moments as diffusion processes, and it has important implications for “smile-consistent” option pricing and for risk management. The out-of-sample performance within a Value-at-Risk framework is examined.