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Centre for

Computational

Finance and

Economic

Agents

Working

Paper

Series

www.ccfea.net

WP013-07

Amadeo Alentorn

Sheri Markose

Generalized Extreme Value

Distribution and Extreme

Economic Value at Risk

(EE-VaR)

October 2007

Generalized Extreme Value Distribution and

Extreme Economic Value at Risk (EE-VaR)

Amadeo Alentorn∗and Sheri Markose†

September, 2007

Abstract

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:

scher@essex.ac.uk 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.

1

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 www.cboe.com/micro/vix/ and for the Bank of England op-

tion traded implied probability density functions, volatility and quantile measures see,

www.bankofengland.co.uk/statistics/impliedpdfs/ . 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.

2

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

3

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.

4

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.

5

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.

6

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

as:

Ct(K) = EQ

t[e−r(T−t)max (ST−K, 0)]

=e−r(T−t)ˆ∞

K

(ST−K)g(ST)dST.(1)

Also, the following martingale condition holds for the stock price

St=e−r(T−t)EQ

t[ST].(2)

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:

LT=−RT=−ST−St

St

= 1 −ST

St

.(3)

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(x−µ)−1/ξ!, ξ 6= 0,(4)

with

1 + ξ(x−µ)

σ>0,

and

F0,µ,σ (x) = exp −exp x−µ

σ, ξ = 0.(5)

The tail shape parameter ξ= 0 yields thin tailed distributions with the so

called tail index 1/ξ =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

7

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(LT−µ)

σ−1−1/ξ

exp −1 + ξ(LT−µ)

σ−1/ξ!,(6)

with

1 + ξ

σ(LT−µ) = 1 + ξ

σ1−ST

St−µ>0.(7)

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) = e−r(T−t){−Stσ

ξΓ(1 −ξ, H −1/ξ)

−(St(1 −µ+σ

ξ)−K)(−e−H−1/ξ )},(8)

where H= 1 + ξ

σ1−K

St−µand Γ(1 −ξ, H −1/ξ) = ´∞

H−1/ξ z−ξe−zdz 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

X

i=1 Ct(Ki)−f

Ct((Ki)2).(9)

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

∂ST|=f(LT)1

St.

8

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 =µ−σ

ξh1−(−log (q))−ξi, ξ 6= 0,(10)

and

V aR =µ−σlog [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, σ1√T) + (1 −p)h(ST|µ2, σ2√T).(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.

9

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:

E−V aR(q, k) = ˆpH−1(q|ˆµ1,ˆσ1, T ) + (1 −ˆp)H−1(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

10

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

(1997-2003)

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

condition.11

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.

11

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 www.bba.org.uk. 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.

12

Figure 1: Term Structure of GEV based implied RNDs on 21 August 01

0

100

200

300

−0.5

0

0.5

0

1

2

3

4

5

6

7

Time horizon (days)

RNDs and EE−VaRs for 21−Aug−01

Negative returns

Probability density

RND

99%

95%

90%

80%

70%

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

13

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).

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

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

log( Days )

log( EE−VaR )

99%

95%

90%

80%

70%

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

15

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(yi−ˆyi)2

PTN

i=T1wi(yi−yi)2, with

TN

X

i=T1

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

R2≤90% 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%.

16

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

17

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%

(0.12)†level

0.48 -2.04

80% (51.1%)# (0.23)†0.9%

(0.09)†

0.51 -1.84

90% (52.4%)# (0.23)†1.4%

(0.09)†

0.53 -1.73

95% (52.1%)# (0.23)†1.9%

(0.09)†

0.56 -1.57

99% (52.6%)# (0.25)†2.7%

(0.11)†-1.88

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.

18

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.

19

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

20

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-

els.

21

Table 9: Percentage violations of EE-VaR

Horizon Conﬁ- level

dence

(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

dence

(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

dence

(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

22

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

23

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.

24

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

25

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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