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Guo, 1996a, 1996b; Jorion, 1995, 1996; Martens and Zein, 2004; Pong,
Shackleton, Taylor and Xu, 2002; Szakmary, Ors, Kim and Davidson,
2002; Xu and Taylor, 1995). Some studies ¬nd combined forecast is the
best choice (see Dunis, Law and Chauvin, 2000; Taylor and Xu, 1997).
Two studies ¬nd high-frequency intraday data can produce more ac-
curate time series forecast than implied. Fung and Hsieh (1991) ¬nd
one-day-ahead time series forecast from a long-lag autoregressive model
¬tted to 15-minutes returns is better than implied volatility. Li (2002)
¬nds the ARFIMA model outperformed implied in long-horizon fore-
casts while implied volatility dominates over shorter horizons. Implied
volatility forecasts were found to produce higher R 2 than other long
memory models, such as the Log-ARFIMA model in Martens and Zein
(2004) and Pong, Shackleton, Taylor and Xu (2004). All these long mem-
ory forecasting models are more recent and are built on volatility com-
piled from high-frequency intraday returns, while the implied volatility
remains to be constructed from less frequent daily option prices.

11.3.4 Other assets
The forecasting power of implied volatility from interest rate options was
tested in Edey and Elliot (1992), Fung and Hsieh (1991) and Amin and
Ng (1997). Interest rate option models are very different from other
option pricing models because of the need to price the whole term
structure of interest rate derivatives consistently all at the same time in
order to rule out arbitrage opportunities. Trading in interest rate instru-
ments is highly liquid as trading friction and execution cost are negli-
gible. Practitioners are more concerned about the term structure ¬t than
the time series ¬t, as millions of pounds of arbitrage pro¬ts could change
hands instantly if there is any inconsistency in contemporaneous prices.
Earlier studies such as Edey and Elliot (1992) and Fung and Hsieh
(1991) use the Black model (a modi¬ed version of Black“Scholes) that
prices each interest rate option without cross-referencing to prices of
other interest rate derivatives. The single factor Heath“Jarrow“Morton
model, used in Amin and Ng (1997) and ¬tted to short rate only, works
in the same way, although the authors have added different constraints
to the short-rate dynamics as the main focus of their paper is to compare
different variants of short-rate dynamics. Despite the complications, all
three studies ¬nd signi¬cant forecasting power is implied of interest rate
(futures) options. Amin and Ng (1997) in particular report an R 2 of 21%
128 Forecasting Financial Market Volatility


for 20-day-ahead volatility forecasts, and volatility historical average
adds only a few percentage points to the R 2 .
Implied volatilities from options written on non¬nancial assets were
examined in Day and Lewis (1993, crude oil), Kroner, Kneafsey and
Claessens (1995, agriculture and metals), Martens and Zein (2004,
crude oil) and a recent study (Szakmary, Ors, Kim and Davidson, 2002)
that covers 35 futures options contracts across nine markets including
S&P500, interest rates, currency, energy, metals, agriculture and live-
stock futures. All four studies ¬nd implied volatility dominates time
series forecasts although Kroner, Kneafsey and Claessens (1995) ¬nd
combining GARCH and implied produces the best forecast.
12
Volatility Models in Risk
Management

The volatility models described in this book are useful for estimating
value-at-risk (VaR), a measure introduced by the Basel Committee in
1996. In many countries, it is mandatory for banks to hold a minimum
amount of capital calculated as a function of VaR. Some ¬nancial in-
stitutions other than banks also use VaR voluntarily for internal risk
management. So volatility modelling and forecasting has a very impor-
tant role in the ¬nance and banking industries. In Section 12.1, we give
a brief background of the Basel Committee and the Basel Accords. In
Section 12.2, we de¬ne VaR and explain how the VaR estimate is tested
according to regulations set out in the Basel Accords. Section 12.3 de-
scribes how volatility models can be combined with extreme value theory
to produce, hitherto the most accurate, VaR estimate. The content in this
section is largely based on McNeil and Frey (2000) in the context where
there is only one asset (or one risk factor). A multivariate extension is
possible but is still under development. Section 12.4 describes various
ways to evaluate the VaR model based on Lopez (1998).
Market risk and VaR represent only one of the many types of risk
discussed in the Basel Accords. We have speci¬cally omitted credit risk
and operational risk as volatility models have little use in predicting
these risks. Readers who are interested in risk management in a broader
context could refer to Jorion (2001) or Banks (2004).


12.1 BASEL COMMITTEE AND BASEL
ACCORDS I & II
The Basel Accords have been in place for a number of years. They set out
an international standard for minimum capital requirement among in-
ternational banks to safeguard against credit, market and operational
risks. The Bank for International Settlements (BIS) based at Basel,
Switzerland, hosts the Basel Committee who in turn set up the Basel
Accords. While Basel Committee members are all from the G10 coun-
tries and have no formal supranational supervisory authority, the Basel
130 Forecasting Financial Market Volatility


Accords have been adopted by almost all countries that have active
international banks. Many ¬nancial institutions that are not regulated
by the national Banking Acts also pay attention to the risk management
procedures set out in the Basel Accords for internal risk monitoring
purposes. The IOSCO (International Organization of Securities Com-
missions), for example, has issued several parallel papers containing
guidelines similar to the Basel Accords for the risk management of
derivative securities.
The ¬rst Basel Accord, which was released in 1988 and which became
known as the Capital Accord, established a minimum capital standard
at 8% for assets subject to credit risk:
Liquidity-weighted assets
≥ 8%. (12.1)
Risk-weighted assets
Detailed guidelines were set for deriving the denominator according to
some prede¬ned risk weights; typically a very risky loan will be given
a 100% weight. The numerator consists of bank capital weighted the
liquidity of the assets according to a list of weights published by the
Basel Committee.
In April 1995, an amendment was issued to include capital charge
for assets that are vulnerable to ˜market risk™, which is de¬ned as the
risk of loss arising from adverse changes in market prices. Speci¬cally,
capital charges are to be supplied: (i) to the current market value of
open positions (including derivative positions) in interest rate related
instruments and equities in banks™ trading books, and (ii) to banks™ total
currency and commodities position in respect of foreign exchange and
commodities risk respectively. A detailed ˜Standardised Measurement
Method™ was prescribed by the Basel Committee for calculating the
capital charge for each market risk category.
If we rearrange Equation (12.1) such that
Liquidity-weighted assets ≥ 8% — Risk-weighted assets, (12.2)
then the market risk related capital charge is added to credit risk related
˜Liquidity-weighted assets™ in the l.h.s. of Equation (12.2). This effec-
tively increases the ˜Risk-weighted assets™ in the r.h.s. by 12.5 times the
additional market risk related capital charge.
In January 1996, another amendment was made to allow banks to use
their internal proprietary model together with the VaR approach for cal-
culating market risk related risk capital. This is the area where volatility
models could play an important role because, by adopting the internal
Volatility Models in Risk Management 131


approach, the banks are given the ¬‚exibility to specify model parameters
and to take into consideration the correlation (and possible diversi¬ca-
tion) effects across as well as within broad risk factor categories. The
condition for the use of the internal model is that it is subject to regular
backtesting procedures using at least one year™s worth of historical data.
More about VaR estimation and backtesting will be provided in the next
sections.
In June 2004, Basel II was released with two added dimensions, viz.
supervisory review of an institution™s internal assessment process and
capital adequacy, and market discipline through information disclosure.
Basel II also saw the introduction of operational risk for the ¬rst time
in the calculation of risk capital to be included in the denominator in
Equation (12.1). ˜Operational risk™ is de¬ned as the risk of losses result-
ing from inadequate or failed internal processes, people and systems, or
external events. The Basel Committee admits that assessments of oper-
ational risk are imprecise and it will accept a crude approximation that
is based on applying a multiplicative factor to the bank™s gross income.


12.2 VaR AND BACKTEST
In this section, we discuss market risk related VaR only, since this is the
area where volatility models can play an important role. The computa-
tion of VaR is needed only if the bank chooses to adopt its own internal
model for calculating market risk related capital requirement.


12.2.1 VaR
˜Value-at-risk™ (VaR) is de¬ned as the 1% quantile of the lower tail
distribution of the trading book held over a 10-day period.1 The capital
charge will then be the higher of the previous day™s VaR or three times the
average daily VaR of the preceding 60 business days. The multiplicative
factor of three was used as a cushion for cumulative losses arising from
adverse market conditions and to account for potential weakness in the
modelling process.2 Given that today™s portfolio value is known, the
prediction of losses over a 10-day period amounts to predicting the rate
of change (or portfolio returns) over the 10-day period (Figure 12.1).

1
A separate VaR will be calculated for each risk factor. So there will be separate VaR for interest rate related
instruments, equity, foreign exchange risk and commodities risk. If correlations among the four risk factors are
not considered, then the total VaR will be the sum of the four VaR estimates.
2
While one may argue that such a multiplicative factor is completely arbitrary, it is nevertheless mandatory.
132 Forecasting Financial Market Volatility




Figure 12.1 Returns distribution and VaR


The VaR estimate for tomorrow™s trading position is then calculated as
today™s portfolio value times the 1% quantile value if it is a negative
return. (A positive return will not attract any risk capital.)
The discovery of stochastic volatility has led to the common practice of
modelling returns distribution conditioned on volatility level at a speci¬c
point in time. Volatility dynamic has been extensively studied since the
seminal work of Engle (1982). It is now well known that a volatile period
in the ¬nancial markets tends to be followed by another volatile period,
whereas a tranquil period tends to be followed by another tranquil period.
VaR as de¬ned by the Basel Committee is a short-term forecast. Hence a
good VaR model should fully exploit the dynamic of volatility structure.


12.2.2 Backtest
For banks who decide to use their own internal models, they have to
perform a backtest procedure at least at a quarterly interval. The back-
test procedure involves comparing the bank™s daily pro¬ts and losses
with model-generated VaR measures in order to gauge the quality and
accuracy of their risk measurement systems. Speci¬cally, this is done by
counting from the record of the last 12 months (or 250 trading days) the
number of times when actual losses are greater than the predicted risk
measure. The proportion actually covered can then be checked to see if
it is consistent with a 99% level of con¬dence.
In the 1996 backtest document, the Basel Committee was unclear
about whether the exceptional losses should take into account fee income
and changes in portfolio position. Hence the long discussion about the
choice of 10-day, 1-day or intraday intervals for calculating exceptional
Volatility Models in Risk Management 133


losses, and the discussion of whether actual or simulated trading results
should be tested. Before we discuss these two issues, it is important
to note that (i) actual VaR violations could be due to an inadequate
(volatility) model or a bad decision (e.g. a decision to change portfolio
composition at the wrong time); (ii) ¬nancial market volatility often does
not obey the scaling law, i.e. variance of 10-day return is not equal to

1-day variance times 10.
To test model adequacy, the backtest should be based on a simulated
portfolio assuming that the bank has been holding the same portfolio for
the last 12 months. This will help to separate a bad model from a bad
decision. Given that the VaR used for calculating the capital requirement
is for a 10-day holding period, the backtest should also be performed
using a 10-day window to accumulate pro¬ts/losses. The current rules,
which require the VaR test to be calculated for a 1-day pro¬ts/losses,
fail to recognize the volatility dynamics over the 10-day period is vastly
different from the 1-day volatility dynamic.
The Basel Committee recommends that backtest also be conducted on
actual trading outcomes in addition to the simulated portfolio position.
Speci¬cally, it recommends a comprehensive approach that involves a
detailed attribution of income by source, including fees, spreads, market
movements and intraday trading results. This is very useful for uncov-
ering risks that are not captured in the volatility model.

12.2.3 The three-zone approach to backtest evaluation
The Basel Committee then speci¬es a three-zone approach to evaluate
the outcome of the backtest. To understand the rationale of the three-zone
approach, it is important to recognize that all appropriately implemented
control systems are subject to random errors. On the other hand, there
are cases where the control system is a bad one and yet there are no
failures. The objective of the backtest is to distinguish the two situations
which are known as type I (rejecting a system when it is working) and
type II (accepting a bad system) errors respectively.
Given that the con¬dence level set for the VaR measure is 99%, there
is a 1% chance of exceptional losses that are tolerated. For 250 trad-
ing days, this translates into 2.5 occurrences where the VaR estimate
will be violated. Figure 12.2 shows the outcome of simulations involv-
ing a system with 99% coverage and 95% coverage as reported in the
Basel document (January 1996, Table 1). We can see that the number
of exceptions under the true 99% coverage can range from 0 to 9, with
134 Forecasting Financial Market Volatility




Figure 12.2 Type I and II backtest errors


most of the instances centred around 2. If the true coverage is 95%, we
might get four or more exceptions with the mean centred around 12. If
only these two coverages are possible then one may conclude that if the
number of exceptions is four or below, then the 99% coverage is true. If
there are ten or more exceptions it is more likely that the 99% coverage
is not true. If the number of exceptions is between ¬ve and nine, then it
is possible that the true coverage might be from either the 99% or the
95% population and it is impossible to make a conclusion.
The dif¬culty one faces in practice is that there could potentially be an
endless range of possible coverages (i.e. 98%, 97%, 96%, . . . , etc). So
while we are certain about the range of type I errors (because we know
that our objective is to have a 99% coverage), we cannot be precise about
the possible range of type II errors (because we do not know the true
coverage). The three-zone approach and the recommended increase in
scaling factors (see Table 12.1) is the Basel Committee™s effort to seek
a compromise in view of this statistical uncertainty. From Table 12.1,
if the backtest reveals that in the last trading year there are 10 or more
VaR violations, for example, then the capital charge will be four times
VaR, instead of three times VaR.
Volatility Models in Risk Management 135


Table 12.1 Three-zone approach to internal backtesting of VaR

No. of Increase in Cumulative
Zones exceptions scaling factor probability

Green 0“4 0 8.11 to 89.22%
+0.40
5 95.88%
+0.50
6 98.63%
+0.65
Yellow 7 99.60%
+0.75
8 99.89%
+0.85
9 99.97%
+1
Red 10 or more 99.99%

Note: The table is based on a sample of 250 observations. The cumulative
probability is the probability of obtaining a given number of or fewer ex-
ceptions in a sample of 250 observations when the true coverage is 99%.
(Source: Basel 1996 Amendment, Table 2).


When the backtest signals a red zone, the national supervisory body
will investigate the reasons why the bank™s internal model produced such
a large number of misses, and may demand the bank to begin working
on improving its model immediately.


12.3 EXTREME VALUE THEORY AND
VaR ESTIMATION
The extreme value approach to VaR estimation is a response to the ¬nd-
ing that standardized residuals of many volatility models have longer
tails than the normal distribution. This means that a VaR estimate pro-
duced from the standard volatility model without further adjustment will
underestimate the 1% quantile. Using GARCH-t, where the standard-
ized residuals are assumed to follow a Student-t distribution, partially
alleviates the problem, but GARCH-t is inadequate when the left tail
and the right tail are not symmetrical. The EVT-GARCH method pro-
posed by McNeil and Frey (2000) is to model conditional volatility and
marginal distribution of the left tail separately. We need to model only
the left tail since the right tail is not relevant as far as VaR computation
is concerned.
Tail event is by de¬nition rare and a long history of data is required to
uncover the tail structure. For example, one would not just look into the
last week™s or the last year™s data to model and forecast the next earth-
quake or the next volcano eruption. The extreme value theory (EVT) is
best suited for studying rare extreme events of this kind where sound
136 Forecasting Financial Market Volatility


statistical theory for maximal has been well established. There is a prob-
lem, however, in using a long history of data to produce a short-term
forecast. The model is not sensitive enough to current market condition.
Moreover, one important assumption of EVT is that the tail events are
independent and identically distributed (iid). This assumption is likely
to be violated because of stochastic volatility and volatility persistence
in particular. Volatility persistence suggests one tail event is likely to
be followed by another tail event. One way to overcome this violation
of the iid assumption is to ¬lter the data with a volatility model and
study the volatility ¬ltered iid residuals using EVT. This is exactly what
McNeil and Frey (2000) proposed. In order to produce a VaR estimate in
the original return scale to conform with the Basel requirement, we then
trace back the steps by ¬rst producing the 1% quantile estimate for the
volatility ¬ltered return residuals, and convert the 1% quantile estimate
to the original return scale using the conditional volatility forecast for
the next day.

12.3.1 The model
Following the GARCH literature, let us write the return process as
rt = µ + z t h t . (12.3)
In the process above, we assume there is no serial correlation in daily
return. (Otherwise, an AR(1) or an MA(1) term could be added to the
r.h.s. of (12.3).) Equation (12.3) is estimated with some appropriate
speci¬cation for the volatility process, h t (see Chapter 4 for details). For
stock market returns, h t typically follows an EGARCH(1,1) or a GJR-
GARCH(1,1) process. The GARCH model in (12.3) is estimated using
quasi-maximum likelihood with a Gaussian likelihood function, even
though we know that z t is not normally distributed. QME estimators are
unbiased and since the standardized residuals will be modelled using an
extreme value distribution, such a procedure is deemed appropriate.
The standardized residuals z t are obtained by rearranging (12.3)
rt ’ µ
zt = √ .
ht
Since our main concern is about losses, we could multiply z t by ’1 so
that we are always working with positive values for convenience. The
z t variable is then ranked in descending order, such that z (1) ≥ z (2) ≥
· · · ≥ z (n) where n is the number of observations.
Volatility Models in Risk Management 137


The next stage involves estimating the generalized Pareto distribu-
tion (GPD) to all z that are greater than a high threshold u. The GPD
distribution has density function
± ’1/ξ

 1 ’ 1 + ξ (z ’ u) for ξ = 0
β
f (z) = .


1 ’ e’(z’u)/β for ξ = 0
The parameter ξ is called the tail index and β is the scale parameter.
The estimation of the model parameters (the tail index ξ in particular)
and the choice of the threshold u are not independent processes. As
u becomes larger, there will be fewer and fewer observations being
included in the GPD estimation. This makes the estimation of ξ very
unstable with a large standard error. But, as u decreases, the chance of an
observation that does not belong to the tail distribution being included in
the GPD estimation increases. This increases the risk of the ξ estimate
being biased. The usual advice is to estimate ξ (and β) at different
levels of u. Then starting from the highest value of u, a lower value of
u is preferred unless there is a change in the level of ξ estimate, which
indicates there may be a possible bias caused by the inclusion of too
many observations in the GPD estimation.
Once the parameters ξ and β are estimated, the 1% quantile is obtained
by inverting the cumulative density function
’1/ξ
z’u
k
F (z) = 1 ’ 1+ξ ,
β
n
’ξ
β 1’q
zq = u + ’1 ,
ξ k/n

where q = 0.01 and k is the number of z exceeding the threshold u.
We are now ready to calculate the VaR estimate using (12.3) and the
volatility forecast for the next day:

VaRt+1 = current position — µ ’ z q h t+1 .


12.3.2 10-day VaR
It is well-known that the variance of a Gaussian variable follows a simple
scaling law and the Basel Committee, in its 1996 Amendment, states that
138 Forecasting Financial Market Volatility



it will accept a simple T scaling of 1-day VaR for deriving the 10-day
VaR required for calculating the market risk related risk capital.
The stylized facts of ¬nancial market volatility and research ¬ndings
have repeatedly shown that a 10-day VaR is not likely to be the same as

10 — 1-day VaR. First, the dynamic of a stationary volatility process
suggests that if the current level of volatility is higher than unconditional
volatility, the subsequent daily volatility forecasts will decline and con-
verge to unconditional volatility, and vice versa for the case where the
initial volatility is lower than the unconditional volatility. The rate of
convergence depends on the degree of volatility persistence. In the case
where initial volatility is higher than unconditional volatility, the scal-

ing factor will be less than 10. In the case where initial volatility is
lower than unconditional volatility, the scaling factor will be more than

10. In practice, due to volatility asymmetry and other predictive vari-
ables that might be included in the volatility model, it is always best
to calculate h t+1 , h t+2 , · · · , h t+10 separately. The 10-day VaR is then
produced using the 10-day volatilty estimate calculated from the sum of
h t+1 , h t+2 , · · · , h t+10 .
Secondly, ¬nancial asset returns are not normally distributed.
Danielsson and deVries (1997) show that the scaling parameter for
quantile derived using the EVT method increases at the approximate
rate of T ξ , which is typically less than the square-root-of-time adjust-
ment. For a typical value of ξ (= 0.25) , T ξ = 1.778, which is less than
100.5 (= 3.16). McNeil and Frey (2000) on the other hand dispute this
¬nding and claim the exponent to be greater than 0.5. The scaling fac-
tor of 100.5 produced far too many VaR violations in the backtest of ¬ve
¬nancial series, except for returns on gold. In view of the con¬‚icting em-
pirical ¬ndings, one possible solution is to build models using 10-day
returns data. This again highlights the dif¬culty due to the inconsistency
in the rule applies to VaR for calculating risk capital and that applies to
VaR for backtesting.

12.3.3 Multivariate analysis
The VaR computation described above is useful for the single asset case
and cases where there is only one risk factor. The cases for multi-asset
and multi-risk-factor are a lot more complex which require multivariate
extreme theories and a better understanding of the dependence struc-
ture between the variables of interest. Much research in this area is
still ongoing. But it is safe to say that correlation coef¬cient, the key
Volatility Models in Risk Management 139


measure used in portfolio diversi¬cation, can produce very misleading
information about the dependence structure of extreme events in ¬nan-
cial markets (Poon, Rockinger and Tawn, 2004). The VaR of a portfolio
is not a simple function of the weighted sum of the VaR of the individ-
ual assets. Detailed coverage of multivariate extreme value theories and
applications is beyond the scope of this book. The simplest solution we
could offer here is to treat portfolio returns as a univariate variable and
apply the procedures above. Such an approach does not provide insight
about the tail relationship between assets and that between risk factors,
but it will at least produce a sensible estimate of portfolio VaR.


12.4 EVALUATION OF VaR MODELS
In practice, there will be many different models for calculating VaR,
many of which will satisfy Basel™s backtest requirement. The important
questions are ˜Which model should one use?™ and ˜If there are excep-
tions, how do we know if the model is malfunctioning?™. Lopez (1998)
proposes two statistical tests and a supplementary evaluation that is based
on the user specifying a loss function.
The ¬rst statistical test involves modelling the number of exceptions
as independent draws from a binomial distribution with a probability
of occurrence equal to 1%. Let x be the actual number of exceptions
observed for a sample of 250 trading outcomes. The probability of ob-
serving x exceptions from a 99 % coverage is

Pr (x) = C x — 0.01x — 0.99250’x .
250


The likelihood ratio statistic for testing if the actual unconditional cov-
erage ± = x/250 = 0.01 is

LRuc = 2 log ± x — (1 ’ ±)250’x ’ log 0.01x — 0.99250’x .

The LRuc test statistic has an asymptotic χ 2 distribution with one degree
of freedom.
The second test makes use of the fact that VaR is the interval forecast
of the lower 1% tail of the one-step-ahead conditional distribution of re-
turns. So given a set of VaRt , the indicator variable It+1 is constructed as

for rt+1 ¤ VaRt
1
It+1 = .
for rt+1 > VaRt
0
140 Forecasting Financial Market Volatility


If VaRt provides correct conditional coverage, It+1 must equal un-
conditional coverage, and It+1 must be serially independent. The LRcc
test is a joint test of these two properties. The relevant test statistic is
LRcc = LRuc + LRind ,
which has an asymptotic χ 2 distribution with two degrees of freedom.
The LRind statistic is the likelihood ratio statistic for the null hypothesis
of serial independence against ¬rst-order serial dependence.
The LRuc test and the LRcc test are formal statistical tests for the
distribution of VaR exceptions. It is useful to supplement these formal
tests with some numerical scores that are based on the loss function of
the decision maker. The loss fuction is speci¬ed as the cost of various
outcomes below:
f (rt+1 , VaRt ) for rt+1 ¤ VaRt
Ct+1 = .
g (rt+1 , VaRt ) for rt+1 > VaRt
Since this is a cost function and the prevention of VaR exception is of
paramount importance, f (x, y) ≥ g (x, y) for a given y. The best VaR
model is one that provides the smallest total cost, Ct+1 .
There are many ways to specify f and g depending on the con-
cern of the decision maker. For example, for the regulator, the concern
is principally about VaR exception where rt+1 ¤ VaRt and not when
rt+1 > VaRt . So the simplest speci¬cation for f and g will be f = 1
and g = 0 as follows:
1 for rt+1 ¤ VaRt
Ct+1 = .
0 for rt+1 > VaRt
If the exception as well as the magnitude of the exception are both
important, one could have
1 + (rt+1 ’ VaRt )2 for rt+1 ¤ VaRt
Ct+1 = .
for rt+1 > VaRt
0
The expected shortfall proposed by Artzner, Delbaen, Eber and Heath
(1997, 1999) is similar in that the magnitude of loss above VaR is
weighted by the probability of occurrence. This is equivalent to
|rt+1 ’ VaRt | for rt+1 ¤ VaRt
Ct+1 = .
for rt+1 > VaRt
0
For banks who implement the VaR model and has to set aside capital
reserves, g = 0 is not appropriate because liquid assets do not provide
Volatility Models in Risk Management 141


good returns. So one cost function that will take into account the oppor-
tunity cost of money is
|rt+1 ’ VaRt |γ for rt+1 ¤ VaRt
Ct+1 = ,
|rt+1 ’ VaRt | — i for rt+1 > VaRt
where γ re¬‚ects the seriousness of large exception and i is a function of
interest rate.
13
VIX and Recent Changes in VIX

The volatility index (VIX) compiled by the Chicago Board of Option
Exchange has always been shown to capture ¬nancial turmoil and pro-
duce good forecast of S&P100 volatility (Fleming, Ostdiek and Whaley,
1995; Ederington and Guan, 2000a; Blair, Poon and Taylor, 2001; Hol
and Koopman, 2002). It is compiled on a real-time basis aiming to re-
¬‚ect the volatility over the next 30 calendar days. In September 2003,
the CBOE revised the way in which VIX is calculated and in March
2004 it started futures trading on VIX. This is to be followed by options
on VIX and another derivative product may be variance swap. The old
version of VIX, now renamed as VXO, continued to be calculated and
released during the transition period.


13.1 NEW DEFINITION FOR VIX
There are three important differences between VIX and VXO:

(i) The new VIX uses information from out-of-the-money call and put
options of a wide range of strike prices, whereas VXO uses eight
at- and near-the-money options.
(ii) The new VIX is model-free whereas VXO is a weighted average of
Black“Scholes implied volatility.
(iii) The new VIX is based on S&P500 index options whereas VXO is
based on S&P100 index options.

The VIX is calculated as the aggregate value of a weighted strip of
options using the formula below:

2
2 Ki r T 1 F
σvi x = e Q (K i ) ’ ’1 ,
2
(13.1)
K i2
T T K0
i
F = K 0 + er T (c0 ’ p0 ) , (13.2)
K i+1 + K i’1
Ki = , (13.3)
2
144 Forecasting Financial Market Volatility


where r is the continuously compounded risk-free interest rate to expi-
ration, T is the time to expiration (in minutes!), F is the forward price of
the index calculated using put“call parity in (13.2), K 0 is the ¬rst strike
just below F, K i is the strike price of ith out-of-the-money options (i.e.
call if K i > F and put if K i < F), Q (K i ) is the midpoint of the bid“ask
spread for option at strike price K i , K i in (13.3) is the interval between
strike prices. If i is the lowest (or highest) strike, then K i = K i+1 ’ K i
(or K i = K i ’ K i’1 ).
Equations (13.1) to (13.3) are applied to two sets of options contracts
for the near term T1 and the next near term T2 to derive a constant 30-day
volatility index VIX:


N T2 ’ N30 2 N30 ’ N T1 N365
VIX = 100 — T1 σ1 + T2 σ2 — ,
2
N T2 ’ N T1 N T2 ’ N T1 N30

where N„ is the number of minutes (N30 = 30 — 1400 = 43,200 and
N365 = 365 — 1400 = 525,600).


13.2 WHAT IS THE VXO?
VXO, the predecessor of VIX, was released in 1993 and replaced by
the new VIX in September 2003. VXO is an implied volatility com-
posite compiled from eight options written on the S&P100. It is con-
structed in such a way that it is at-the-money (by combining just-in-
and just-out-of-the-money options) and has a constant 28 calendar days
to expiry (by combining the ¬rst nearby and second nearby options
around the targeted 28 calendar days to maturity). Eight option prices are
used, including four calls and four puts, to reduce any pricing bias and
measurement errors caused by staleness in the recorded index level.
Since options written on S&P100 are American-style, a cash-dividend
adjusted binomial model was used to capture the effect of early ex-
ercise. The mid bid“ask option price is used instead of traded price
because transaction prices are subject to bid“ask bounce. (See Whaley
(1993) and Fleming, Ostdiek and Whaley (1995) for further details.)
Owing to the calendar day adjustment, VIX is about 1.2 times (i.e.

365/252) greater than historical volatility computed using trading-day
data.
60
VIX for the period 2 Jan 1990 to 28 June 2004 VIX vs. VXO
2 Jan 1990 to 28 June 2004
50
VIX
40
40
30
35
20

10 30

0
25
90 90 91 92 93 94 95 96 97 98 99 00 01 02 03
19 19 19 19 19 19 19 19 19 19 19 20 20 20 20
1/ 2/ 2/ 2/ 2/ 2/ 2/ 2/ 1/ 1/ 1/ 1/ 1/ 1/ 1/
/0 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1
02 27 24 18 15 12 07 03 28 25 23 17 21 19 17 20


15
60
VXO for the period 2 Jan 1990 to 28 June 2004
50
10
10 15 20 25 30 35 40
40
VXO
30

20 VXO VIX
Mean 21.057 20.079
10
Std Dev 7.277 6.385
Kurtosis 0.738 0.669
0
Skewness 0.889 0.870
0 0 1 2 3 4 5 6 7 8 9 0 1 2 3
99 99 99 99 99 99 99 99 99 99 99 00 00 00 00 Minimum 9.040 9.310
1 1 1 1 1 1 1 1 1 1 1 2 2 2 2
1/ 2/ 2/ 2/ 2/ 2/ 2/ 2/ 1/ 1/ 1/ 1/ 1/ 1/ 1/
/0 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 /1 Maximum 50.480 45.740
02 27 24 18 15 12 07 03 28 25 23 17 21 19 17

Figure 13.1 Chicago Board of Options Exchange volatility indices
146 Forecasting Financial Market Volatility


13.3 REASON FOR THE CHANGE
There are many reasons for the change; if nothing else the new volatility
index is hedgeable and the old one is not. The new VIX can be replicated
with a static portfolio of S&P500 options or S&P500 futures. Hence, it
allows hedging and, more importantly, the corrective arbitrage options of
VIX derivatives if prices are not correct. The CBOE argues that the new
VIX re¬‚ect information in a broader range of options rather than just the
few at-the-money options. More importantly, the new VIX is aiming to
capture the information in the volatility skew. It is linked to the broader-
based S&P500 index instead of the S&P100 index. The S&P500 is the
primary index for most portfolio benchmarking so derivative products
that are more closely linked to S&P500 will facilitate risk management.
Although the two volatility indices are compiled very differently, their
statistical properties are very similar. Figures 13.1(a) and 13.1(b) show
the time series plots of VIX and VXO over the period 2 January 1990
to 28 June 2004, and Figure 13.1(c) provides a scatterplot showing the
relationship between the two. The new VIX has a smaller mean and is
more stable than the old VXO. There is no doubt that researchers are
already investigating the new index and all the issues that it has brought
about, such as the pricing and hedging of derivatives written on the
new VIX.
14
Where Next?

The volatility forecasting literature is still very active. Many more new
results are expected in the near future. There are several areas where
future research could seek to make improvements. First is the issue
about forecast evaluation and combining forecasts of different models.
It would be useful if statistical tests were conducted to test whether the
forecast errors from Model A are signi¬cantly smaller, in some sense,
than those from Model B, and so on for all pairs. Even if Model A is found
to be better than all the other models, the conclusion is NOT that one
should henceforth forecast volatility with Model A and ignore the other
models as it is very likely that a linear combination of all the forecasts
might be superior. To ¬nd the weights one can either run a regression
of empirical volatility (the quantity being forecast) on the individual
forecasts, or as an approximation just use equal weights. Testing the
effectiveness of a composite forecast is just as important as testing the
superiority of the individual models, but this has not been done very
often or across different data sets.
A mere plot of any measure of volatility against time will show the
familiar ˜volatility clustering™ which indicates some degree of forecast-
ability. The biggest challenge lies in predicting changes in volatility. If
implied volatility is agreed to be the best performing forecast, on average,
this is in agreement with the general forecast theory, which emphasizes
the use of a wider information set than just the past of the process
being forecast. Implied volatility uses option prices and so potentially
the information set is richer. What needs further consideration is if all
of its information is now being extracted and if it could still be widened
to further improve forecast accuracy especially that of long horizon
forecasts. To achieve this we need to understand better the cause of
volatility (both historical and implied). Such an understanding will help
to improve time series methods, which are the only viable methods when
options, or market-based forecast, are not available.
Closely related to the above is the need to understand the source of
volatility persistence and the volume-volatility research appears to be
promising in providing a framework in which volatility persistence may
148 Forecasting Financial Market Volatility


be closely scrutinized. The mixture of distribution hypothesis (MDH)
proposed by Clark (1973), the link between volume-volatility and mar-
ket trading mechanism in Tauchen and Pitts (1983), and the empirical
¬ndings of the volume-volatility relationship surveyed in Karpoff (1987)
are useful starting points. Given that Lamoureux and Lastrapes (1990)
¬nd volume to be strongly signi¬cant when it is inserted into the ARCH
variance process, while returns shocks become insigni¬cant, and that
Gallant, Rossi and Tauchen (1993) ¬nd conditioning on lagged vol-
ume substantially attenuates the ˜leverage™ effect, the volume-volatility
research may lead to a new and better way for modelling returns distri-
butions. To this end, Andersen (1996) puts forward a generalized frame-
work for the MDH where the joint dynamics of returns and volume are
estimated, and reports a signi¬cant reduction in the estimated volatility
persistence. Such a model may be useful for analysing the economic
factors behind the observed volatility clustering in returns, but such a
line of research has not yet been pursued vigorously.
There are many old issues that have been around for a long time.
These include consistent forecasts of interest rate volatilities that sat-
is¬es the no-arbitrage relationship across all interest rate instruments,
more tests on the use of absolute returns models in comparison with
squared returns models in forecasting volatility, a multivariate approach
to volatility forecasting where cross-correlation and volatility spillover
may be accommodated, etc.
There are many new adventures that are currently under way as well.1
These include the realized volatility approach, noticeably driven by
Andersen, Bollerslev, Diebold and various co-authors, the estimation
and forecast of volatility risk premium, the use of spot and option price
data simultaneously (e.g. Chernov and Ghysels, 2000), and the use of
Bayesian and other methods to estimate stochastic volatility models (e.g.
Jones, 2001), etc.
It is dif¬cult to envisage in which direction volatility forecasting re-
search will ¬‚ourish in the next ¬ve years. If, within the next ¬ve years,
we can cut the forecast error by half and remove the option pricing bias in
ex ante forecast, this will be a very good achievement indeed. Producing
by then forecasts of large events will mark an important milestone.

1
We thank a referee for these suggestions.
Appendix

Data Data Forecasting Forecasting Evaluation
Author(s) Asset(s) period frequency methods and rank horizon and R-squared Comments

ME, RMSE, GARCH is least biased and
1. Akigray CRSP VW Jan63“Dec86 D GARCH(1,1) 20 days ahead
MAE, MAPE produced best forecast
(1989) & EW (pre-crash) ARCH(2) estimated from
especially in periods of
indices split into EWMA rolling 4 years
high volatility and when
4 subperiods HIS data. Daily
changes in volatility persist.
of 6 years (ranked) returns used to
construct Heteroscedasticity is less
each
˜actual vol.™; strong in low-frequency
adjusted for data and monthly returns
serial are approximately Normal
correlation
2. Alford and 6879 stocks 12/66“6/87 W, M ˜Shrinkage™ forecast 5 years starting MedE, MedAE To predict 5-year monthly
Boatman listed in NYSE/ (HIS adjusted from 6 months volatility one should use
(1995) ASE & towards after ¬rm™s 5 year™s worth of weekly or
NASDAQ comparable ¬scal year monthly data. Adjusting
¬rms) historical forecast using
HIS industry and size produced
Median HIS vol. of best forecast
˜comparable™ ¬rm
(ranked)
Continued
Data Data Forecasting Forecasting Evaluation
Author(s) Asset(s) period frequency methods and rank horizon and R-squared Comments

3. Amin and 3M 1/1/88“ D 20 days ahead R 2 is 21% for Interest rate models that
ImpliedAmerican All Call+Put
Ng (1997) Eurodollar 1/11/92 (WLS, 5 variants of (1 day ahead implied and 24% incorporate volatility term
futures & the HJM model) forecast for combined. structure (e.g. Vasicek)
futures HIS produced from perform best. Interaction
H0 : ±implied = 0,
options (ranked) in-sample with term capturing rate level
βimplied = 1
lag implied in cannot be and volatility contribute
GARCH/GJR not rejected with additional forecasting
discussed here) robust SE power
4. Andersen DM/$, ¥/$ In: D (5 min) GARCH(1, 1) 1 day ahead, use R 2 is 5 to 10% R 2 increases monotonically
and 1/10/87“ 5-min returns to for daily squared with sample frequency
Bollerslev 30/9/92 construct ˜actual returns, 50% for
(1998) Out: vol.™ 5-min square
1/10/92“ returns
30/9/93
1/12/86“ Tick VAR-RV, AR-RV, 1 and 10 days 1-day-ahead R 2
5. Andersen, ¥/US$, RV is realized volatility, D
30/6/99 (30 min) FIEGARCH-RV ahead. ˜Actual
Bollerslev, DM/US$ ranges between is daily return, and ABS is
GARCH-D, RM-D, vol.™ derived from
Diebold Reuters In: 27 and 40% daily absolute return. VAR
and Labys FXFX 1/12/86“ FIEGARCH-D 30-min returns (1-day-ahead) allows all series to share
VAR-ABS
(2001) quotes 1/12/96, and 20 and 33% the same fractional
10 years (ranked) (10-days-ahead). integrated order and
Out: cross-series linkages.
2/12/96“ Forecast improvement is
30/6/99, largely due to the use of
2.5 years high-frequency data (and
realised volatility) instead
of the model(s)
6. Andersen, DM/US$ 1/12/86“ 5 min GARCH(1, 1) at 1, 5 and 20 RMSE, MAE, HRMSE and
Bollerslev Reuters quotes 30/11/96 5-min, 10-min, 1-hr, days ahead, use HRMSE, HMAE are
and Lange 8-hr, 1-day, 5-day, 5-min returns to HMAE, LL heteroscedasticity-
In: 1/10/87“
(1999) 20-day interval construct adjusted error
30/9/92
˜actual vol.™ statistics; LL is the
logarithmic loss
function.
High-frequency
returns and
high-frequency
GARCH(1, 1)
models improve
forecast accuracy.
But, for sampling
frequencies shorter
than 1 hour, the
theoretical results
and forecast
improvement break
down
7. Bali (2000) 3-, 6-, 8/1/54“ W NGARCH 1 week ahead. R 2 increases CKLS: Chan,
12-month 25/12/98 GJR, TGARCH Use weekly from 2% to 60% Karolyi, Longstaff
T-Bill rates AGARCH, QGARCH interest rate by allowing for and Sanders (1992)
TSGARCH absolute change asymmetries,
GARCH to proxy ˜actual level effect and
VGARCH vol.™ changing
Constant vol. (CKLS) volatility
(ranked, forecast both
level and volatility)
Continued
Data Data Forecasting Forecasting Evaluation
Author(s) Asset(s) period frequency methods and rank horizon and R-squared Comments

8. Beckers 62 to 116 28/4/75“ D FBSD Over option™s MPE, MAPE. FBSD: Fisher Black™s
(1981) 21/10/77 maturity (3 Cross sectional
stocks option pricing service
ImpliedATM call, 5 days ave
options months), 10 R 2 ranges takes into account stock
Impliedvega call, 5 days ave
non-overlapping between 34 and vol. tend to move
RWlast quarter
(ranked, both implieds cycles. Use 70% across together, mean revert,
are 5-day average sample SD of models and leverage effect and
because of large daily returns over expiry cycles. implied can predict
variations in daily option maturity FBSD appears to future. ATM, based on
stock implied) to proxy ˜actual be least biased vega WLS, outperforms
vol.™ vega weighted implied,
with ± = 0, β =
and is not sensitive to ad
1. ± > 0, β < 1
for the other two hoc dividend adjustment.
implieds Incremental information
from all measures
suggests option market
inef¬ciency. Most
forecasts are upwardly
biased as actual vol. was
on a decreasing trend
50 stock 4 dates: D (from ditto Cross-sectional TISD: Single intraday
TISDvega
options 18/10/76, Tick) R 2 ranges transaction data that has
ImpliedATM call, 1 days ave
24/1/77, (ranked) between 27 and the highest vega. The
18/4/77, 72% across superiority of TISD over
18/7/77 models and implied of closing option
expiry cycles prices suggest signi¬cant
non-simultaneity and
bid“ask spread problems
SP 1/1/88“ D GARCH One step ahead. Cox MLE RMSE Consider if
9. Bera and Daily SP500,
(LE: logarithmic heteroscedasticity is
Higgins Weekly $/£, 28/5/93 W Bilinear model Reserve 90% of
$/£ M (ranked) data for error) due to bilinear in
(1997) Monthly US
Ind. Prod. 12/12/85“ estimation level. Forecasting
results show strong
28/2/91
preference for
Ind. Prod.
GARCH
1/60“3/93
10. Blair, S&P100 2/1/87“ Tick 1, 5, 10 and 20 1-day-ahead R 2 Using squared returns
ImpliedVXO
Poon and (VXO) 31/12/99 GJR days ahead is 45% for VXO, reduces R 2 to 36% for
Taylor estimated using a and 50% for both VXO and
HIS100
Out: 4/1/93“
(2001) (ranked) rolling sample of combined. VXO combined. Implied
31/2/99
1000 days. Daily is downward volatility has its own
actual volatility is biased in persistence structure.
calculated from out-of-sample GJR has no
5-min returns period incremental
information though
integrated HIS vol.
can almost match IV
forecasting power
11. Bluhm and German DAX In: 1/1/88“ D 45 calendar days, MAPE, LINEX Ranking varies a lot
ImpliedVDAX
Yu (2000) stock index 28/6/96 GARCH(-M), SV 1, 10 and 180 depend on forecast
and VDAX the Out: 1/7/66“ EWMA, EGARCH, trading days. horizons and
DAX volatility 30/6/99 GJR, ˜Actual™ is the performance measures
index HIS sum of daily
(approx. ranked) squared returns
Continued
Data Data Forecasting Forecasting Evaluation
Author(s) Asset(s) period frequency methods and rank horizon and R-squared Comments

12. Boudoukh, 3-month US 1983“1992 D EWMA, MDE 1 day ahead MSE and MDE is multivariate
Richard- T-Bill GARCH(1, 1), HIS based on 150-day regression. MDE density estimation where
son and (ranked) rolling period has the highest volatility weights depend
Whitelaw estimation. R 2 while EWMA on interest rate level and
(1997) Realized has the smallest term spread. EWMA and
volatility is the MSE MDE have comparable
daily squared performance and are
changes averaged better than HIS and
GARCH
across t + 1 to
t +5
13. Brace and Futures option 1986“87 D 20 days ahead. Adj R 2 are 20% Large ¬‚uctuations of R 2
HIS5, 20, 65 days
Hodgson on Australian (HIS), 17% (HIS from month to month.
ImpliedNTM call, 20“75 days Use daily returns
(1991) Stock Index (ranked) to calculate Results could be due to
+ implied). All
(marking to standard the dif¬culty in valuing
± > 0 and sig.
market is deviations some uni regr futures style options
needed for this coeff are sig
option) negative (for
both HIS and
implied)
D GJR, 1 month ahead. ME, MAE, Though the ranks are
14. Brailsford Australian 1/1/74“
Regr, HIS, GARCH, Models estimated RMSE, MAPE, sensitive, some models
and Faff Statex- 30/6/93
MA, EWMA, from a rolling and a collection dominate others;
(1996) Actuaries In: Jan74“
RW, ES 12-year window of asymmetric
Accumulation Dec85 MA12 > MA5 and Regr.
Out: Jan86“ (rank sensitive to error loss functions
Index for top > MA > EWMA > ES.
56 Dec93 statistics) GJR came out quite well
but is the only model that
(include 87™s
always underpredicts
crash period)
15. Brooks DJ Composite 17/11/78“ D RW, HIS, MA, ES, 1 day ahead MSE, MAE of Similar performance across
(1998) 30/12/88 EWMA, AR, GARCH, squared returns variance, % models especially when
EGARCH, GJR, using rolling overpredict. R 2 87™s crash is excluded.
Out:
Neural network 2000 is around 4% Sophisticated models such
17/10/86“
(all about the same) observations for increases to 24% as GARCH and neural net
30/12/88
estimation for pre-crash data did not dominate. Volume
did not help in forecasting
volatility
16. Canina and S&P100 15/3/83“ D 7 to 127 calendar Combined R 2 is Implied has no correlation
HIS60 calendar days
Figlewski (OEX) 28/3/87 days matching 17% with little with future volatility and
ImpliedBinomial Call
(1993) (pre-crash) (ranked) option maturity, contribution does not incorporate info.
overlapping from implied. All contained in recently
Implied in 4 maturity
forecasts with observed volatility. Results
±implied > 0,
gp, each subdivided
Hansen std error. appear to be peculiar for
βimplied < 1 with
into 8 intrinsic gp
Use sample SD robust SE pre-crash period. Time
of daily returns to horizon of ˜actual vol.™
proxy ˜actual changes day to day.
vol.™ Different level of implied
aggregation produces
similar results
17. Cao and Tsay Excess returns 1928“1989 M TAR 1 to 30 months. MSE, MAE TAR provides best
(1992) for S&P, VW EGARCH(1, 0) Estimation period forecasts for large stocks.
EW indices ARMA(1, 1) ranges from 684 EGARCH gives best
GARCH(1, 1) to 743 months long-horizon forecasts for
(ranked) Daily returns small stocks (may be due to
used to construct leverage effect). Difference
˜actual vol.™ in MAE can be as large as
38%
Continued
Data Data Forecasting Forecasting Evaluation
Author(s) Asset(s) period frequency methods and rank horizon and R-squared Comments

18. Chiras and All stock 23 months M Implied (weighted by 20 months ahead. Cross-sectional Implied outperformed HIS
Manaster options from from Jun73 price elasticity) Use SD of 20 R 2 of implied especially in the last 14
(1978) CBOE to Apr75 monthly returns ranges 13“50% months. Find implied
HIS20 months
(ranked) to proxy ˜actual across 23 increases and better
vol.™ months. HIS behave after dividend
adds 0“15% to adjustments and evidence
R2 of mispricing possibly due
to the use European
pricing model on
American style options
19. Christensen S&P100 Nov83“ M Non-overlapping R 2 of log var are Not adj. for dividend and
ImpliedBS ATM 1-month Call
and (OEX) May95 24 calendar (or 39% (implied), early exercise. Implied
HIS18 days
Prabhala (ranked) 18 trading) days. 32% (HIS) and dominates HIS. HIS has
Monthly
(1998) Use SD of daily 41% no additional information
expiry cycle
returns to proxy (combined). in subperiod analysis.
˜actual vol.™ Proved that results in
± < 0 (because
Canina and Figlewski
of log), β < 1
with robust SE. (1993) is due to pre-crash
Implied is more characteristics and high
biased before degree of data overlap
the crash relative to time series
length. Implied is
unbiased after controlling
for measurement errors
using impliedt’1 and
HISt’1
20. Christoffersen 4 stk indices 1/1/73“ D No model. 1 to 20 days Run tests and Equity and FX:
and Diebold 4 ex rates US 1/5/97 Markov forecastability decrease
(no rank; evaluate
rapidly from 1 to 10
(2000) 10-year volatility transition matrix
days. Bond: may extend
T-Bond forecastability (or eigenvalues
persistence) by (which is as long as 15 to 20 days.
checking interval basically Estimate bond returns
forecasts) 1st-order serial from bond yields by
coef¬cient of assuming coupon equal
the hit sequence to yield
in the run test)
21. Cumby, ¥/$, stocks 7/77 to W EGARCH 1 week ahead, R 2 varies from EGARCH is better than
Figlewski (¥, $), bonds 9/90 HIS estimation period 0.3% to 10.6%. naive in forecasting
and (¥, $) (ranked) ranges from 299 to volatility though
Hasbrouck 689 weeks R-squared is low.
(1993) Forecasting correlation
is less successful
22. Day and S&P100 Out: W 1 week ahead R 2 of variance Omit early exercise.
ImpliedBS Call (shortest
Lewis (1992) OEX option 11/11/83“ estimated from a regr. are 2.6% Effect of 87™s crash is
but > 7 days,
31/12/89 volume WLS) rolling sample of (implied) and unclear. When weekly
410 observations. 3.8% (encomp.). squared returns were
HIS1 week
Reconstructed In: 2/1/76“
S&P100 GARCH Use sample All forecasts used to proxy ˜actual
11/11/83
EGARCH variance of daily add marginal vol,™ R 2 increase and
(ranked) returns to proxy was max for HIS
info. H0 :
weekly ˜actual contrary to expectation
±implied =
vol.™ (9% compared with
0, βimplied = 1
cannot be 3.7% for implied)
rejected with
robust SE
Continued
Data Data Forecasting Forecasting Evaluation
Author(s) Asset(s) period frequency methods and rank horizon and R-squared Comments

23. Day and Crude oil 14/11/86“18/3/91 D Option maturity of ME, RMSE, Implied performed
ImpliedBinomial ATM Call
Lewis (1993) futures 4 nearby contracts, MAE. R 2 of extremely well.
HISforecast horizon
options GARCH-M (average 13.9, 32.5, variance regr. are Performance of HIS
Crude oil 8/4/83“18/3/91 EGARCH-AR(1) 50.4 and 68 trading 72% (short mat.) and GARCH are
futures coincide with (ranked) days to maturity). and 49% (long similar. EGARCH
Estimated from
Kuwait invasion by maturity). With much inferior. Bias
rolling 500
Iraq in second half adjusted and combined
robust SE ± > 0
observations
of sample for short and forecasts do not
perform as well as
± = 0 for long,
unadjusted implied.
β = 1 for all
maturity GARCH has no
incremental
information. Result
likely to be driven by
Kuwait invasion by
Iraq

24. Dimson and UK FT All 1955“89 Q ES, Regression Next quarter. Use MSE, RMSE, Recommend
Marsh Share RW, HA, MA daily returns to MAE, RMAE exponential smoothing
(1990) (ranked) construct ˜actual and regression model
vol.™ using ¬xed weights.
Find ex ante
time-varying
optimization of
weights does not work
well ex post
25. Doidge and Toronto 35 stock In: 2/8/88“31/12/91 D Combine3 1 month ahead MAE, MAPE, Combine1 equal
Wei (1998) index & European Out: 1/92“7/95 Combine2 from rolling RMSE weight for GARCH
options GARCH sample and implied forecasts.
EGARCH estimation. No Combine2 weighs
mention on GARCH and implied
HIS100 days
Combine1 how ˜actual based on their recent
vol.™ was forecast accuracy.
ImpliedBS Call+Put
derived Combine3 puts implied
(All maturities >
in GARCH conditional
7 days, volume
WLS) variance. Combine3
was estimated using
(ranked)
full sample due to
convergence problem;
so not really
out-of-sample forecast
26. Dunis, Laws DM/¥, £/DM, In: 2/1/91“27/2/98 D GARCH(1, 1) 1 and 3 months RMSE, MAE, No single model
dominates though SV
and Chauvin £/$, $/CHF, Out: 2/3/98“ AR(10)-Sq returns (21 and 63 MAPE, Theil-U,
(2000) $/DM, $/¥ 31/12/98 AR(10)-Abs returns trading days) CDC (Correct is consistently worst,
SV(1) in log form with rolling Directional and implied always
improves forecast
estimation. Change index)
HIS21 or 63 trading days
accuracy. Recommend
1- & 3-M forward Actual
equal weight combined
volatility is
ImpliedATM quotes
forecast excluding SV
Combine calculated as
Combine (except the average
SV) absolute return
(rank changes over the
across currencies forecast
and forecast horizon
horizons)
Continued
Data Data Forecasting Forecasting Evaluation
Author(s) Asset(s) period frequency methods and rank horizon and R-squared Comments

27. Ederington and S&P500 futures 1Jan88“30Apr98 D Overlapping Panel R 2 19% Information content of
ImpliedBK 16Calls 16 Puts
Guan (1999) options 10 to 35 days and individual implied across strikes
HIS40 days
˜Frown™ (ranked) matching R 2 ranges exhibit a frown shape
maturity of 6“17% (calls) with options that are
nearest to and 15“36% NTM and have
expiry option. (puts). Implied moderately high
Use SD of is biased and strikes possess largest
daily returns to inef¬cient, information content.
proxy ˜actual HIS typically adds
±implied > 0 and
vol.™ 2“3% to the R 2 and
βimplied < 1 with
robust SE nonlinear implied
terms add another
2“3%. Implied is
unbiased and ef¬cient
when measurement
error is controlled
using Impliedt’1 and
HISt’1 .
5 DJ Stocks 2/7/62“30/12/94 D GW: geometric
28. Ederington and GWMAD n = 10, 20, 40, RMSE, MAE
Guan (2000a) S&P500 3m 2/7/62“29/12/95 80 and 120 weight, MAD: mean
GWSTD ,
˜Forecasting Euro$ rate 10yr 1/1/73“20/6/97 GARCH, EGARCH days ahead absolute deviation,
volatility™ T-Bond yield 2/1/62“13/6/97 AGARCH estimated from STD: standard
DM/$ 1/1/71“30/6/97 a 1260-day deviation. Volatility
HISMAD, n, HISSTD, n
(ranked, error statistics rolling aggregated over a
window; longer period
are close; GWMAD
leads consistently parameters produces a better
forecast. Absolute
though with only re-estimated
every 40 days. returns models
small margin)
generally perform
Use daily
squared better than square
returns models
deviation to
proxy ˜actual (except GARCH >
AGARCH). As
vol.™
horizon lengthens, no
procedure dominates.
GARCH and
EGARCH estimations
were unstable at times
Continued
Data Data Forecasting Forecasting Evaluation
Author(s) Asset(s) period frequency methods and rank horizon and R-squared Comments

Overlapping RMSE, MAE, VXO: 2calls+2puts,
29. Ederington and S&P500 futures In: 4/1/88“31/12/91 D Implied:— 99%, VXO
Out:
Guan (2000b) options 10 to 35 days MAPE NTM weighted to get
HIS40 trading days
2/1/92“31/12/92 ˜*™ indicates
˜Averaging™ matching ATM. Eq4/32:
Implied: VXO > Eq4
individual
maturity of calls+puts equally
Implied: WLS >
implieds were
nearest to weighted. WLS, vega
vega > Eq32 >
corrected for
expiry option. and elasticity are other
elasticity
biasedness ¬rst
(ranked) Use SD of weighting scheme.
before
daily returns to 99% means 1% of regr.
averaging using
proxy ˜actual error used in weighting
in-sample regr.
vol.™ all implieds. Once the
on realized biasedness has been
corrected using regr.,
little is to be gained by
any averaging in such a
highly liquid S&P500
futures market
30. Ederington and S&P500 futures 1/1/83“14/9/95 D Overlapping R 2 ranges GARCH parameters
ImpliedBlack 4NTM
Guan (2002) options option maturity 22“12% from were estimated using
GARCH, HIS40 days
˜Ef¬cient (ranked) 7“90, 91“180, short to long whole sample.
predictor™ 181“365 and horizon. Post 87™s GARCH and HIS add
7“365 days crash R 2 nearly little to 7“90 day R 2 .
ahead. Use doubled. Implied When 87™s crash was
sample SD over is ef¬cient biased; excluded HIS add sig.
forecast horizon explanatory power to
±implied > 0 and
to proxy ˜actual 181“365 day forecast.
βimplied < 1 with
vol.™ robust SE When measurement
errors were controlled
using impliedt’5 and
impliedt+5 as
instrument variables
implied becomes
unbiased for the whole
period but remains
biased when crash
period was excluded
Continued
Data Data Forecasting Forecasting Evaluation
Author(s) Asset(s) period frequency methods and rank horizon and R-squared Comments

31. Edey and Elliot Futures options Futures W Option maturity Regression (see R 2 cannot be compared
ImpliedBK NTM,call
(1992) on A$ 90d-Bill, options: up to 3M. Use comment). In with other studies
ImpliedBK NTM, put
10yr bond, Stock inception to sum of (return most cases because of the way
index 12/88 (No rank, 1 call and square plus ˜actual™ is derived and
±implied > 0 and
1 put, selected based lagged squares returns
Impliedt+1 ) as βimplied < 1 with
on highest trading ˜actual vol.™ robust SE. For were added to the RHS
volume) stock index option
βimplied = 1 cannot
be rejected using
robust SE
A$/US$ options A$/US$ W Constant 1M.
option: Use sum of
12/84“12/87 weekly squared
returns to proxy
˜actual vol.™
32. Engle, Ng and 1 to 12 months Aug64“Nov85 M 1-factor ARCH 1 month ahead Model ¬t Equally weighted bill
Rothschild T-Bill returns, Univariate ARCH-M volatility and portfolio is effective in
(1990) VW index of (ranked) risk premium of predicting (i.e. in an
NYSE & AMSE 2 to 12 months expectation model)
stocks T-Bills volatility and risk
premia of individual
maturities
33. Feinstein MSE, MAE, ME. Atlanta: 5-day
S&P500 futures Jun83“Dec88 Option expiry Implied: 23 non-
(1989b) options (CME) cycle overlapping T-test indicates average of
Atlanta > average >
Just-OTM call
forecasts of 57, all ME > 0
vega > elasticity
implied using
Just- 38 and 19 days (except HIS) in
ahead. Use the post-crash exponential weights.
OTMCall > P+C > Put
period which In general Just-OTM
sample SD of
HIS20 days
(ranked, note daily returns means implied Impliedcall is the best
over the option was upwardly
pre-crash rank is
very different and maturity to biased
erratic) proxy ˜actual
vol.™
34. Ferreira French & In: W 1 week ahead. Regression, L: interest rate level,
ES, HIS26, 52, all
(1999) German Jan81“Dec89 GARCH(-L) Use daily MPE, MAPE, E: exponential.
interbank 1M Out: (E)GJR(-L) squared rate RMSPE. R 2 is French rate was very
mid-rate Jan90“Dec97 (rank varies between changes to 41% for France volatile during ERM
French and proxy weekly and 3% for crises. German rate
(ERM crises:
German rates, volatility Germany was extremely stable
Sep92“
sampling method in contrast.
Sep93)
and error statistics) Although there are
lots of differences
between the two
rates, best models
are nonparametric;
ES (French) and
simple level effect
(German). Suggest a
different approach is
needed for
forecasting interest
rate volatility
Continued
Data Data Forecasting Forecasting Evaluation
Author(s) Asset(s) period frequency methods and rank horizon and R-squared Comments

35. Figlewski S&P500 3m US 1/47“12/95 M 6, 12, 24, 36, RMSE Forecast of volatility
HIS6, 12, 24, 36, 48, 60m
(1997) T-Bill 20yr 1/47“12/95 GARCH(1, 1) for 48, 60 months. of the longest
T-Bond DM/$ 1/50“7/93 S&P and bond Use daily horizon is the most
1/71“11/95 yield. returns to accurate. HIS uses
(ranked) compute ˜actual the longest
vol.™ estimation period is
the best except for
short rate
S&P500 2/7/62“ D GARCH(1, 1) 1, 3, 6, 12, 24 RMSE GARCH is best for
3m US 29/12/95 months S&P but gave worst
HIS1, 3, 6, 12, 24, 60 months
T-Bill 2/1/62“ (S&P™s rank, reverse performance in all
20yr 29/12/95 for the others) the other markets. In
T-Bond 2/1/62“ general, as
DM/$ 29/12/95 out-of-sample
4/1/71“ horizon increases,
30/11/95 the in-sample length
should also increase
36. Figlewski and S&P500 US 1/4/71“ D 1, 3, 12 months RMSE ES works best for
His3, 12, 60 months ES
Green (1999) LIBOR 10yr 12/31/96 (rank varies) for daily data S&P (1“3 month)
T-Bond yield Out: From and short rate (all
DM/$ Feb96 three horizons). HIS
works best for bond
yield, exchange rate
and long horizon
S&P forecast. The
longer the forecast
horizon, the longer
the estimation
period
1/4/71“ M 24 and 60 For S&P, bond yield
His26, 60, all months ES
12/31/96 (ranked) months for and DM/$, it is best
Out: From monthly data to use all available
Jan92 ˜monthly™ data. 5
year™s worth of data
works best for short
rate
Continued
Data Data Forecasting Forecasting Evaluation
Author(s) Asset(s) period frequency methods and rank horizon and R-squared Comments

37. Fleming S&P100 (OEX) 10/85“4/92 D Option maturity R 2 is 29% for Implied dominates.
ImpliedFW ATM calls
(1998) (all monthly forecast All other variables
ImpliedFW ATM puts (shortest but >
observations (both implieds are 15 days, and 6% for daily related to volatility
that overlap WLS using all average 30 forecast. All such as stock
with 87™s ATM options in calendar days), returns, interest rate
±implied = 0,
crash were the last 10 minutes 1 and 28 days and parameters of
βimplied < 1 with
removed) before market ahead. Use robust SE for the GARCH do not
close) daily square last two ¬xed possess information
ARCH/GARCH return horizon forecasts incremental to that
deviations to contained in implied
HISH-L 28 days
(ranked) proxy ˜actual
vol.™
38. Fleming, S&P500, 3/1/83“ D Exponentially Daily Sharpe ratio Ef¬cient frontier of
Kirby T-Bond and 31/12/97 weighted var-cov rebalanced (portfolio return volatility timing
and Ostdiek gold futures matrix portfolio over risk) strategy plotted
(2000) above that of ¬xed
weight portfolio
39. Fleming, S&P100 (VXO) Jan86“Dec92 D, W 28 calendar (or R 2 increased VXO dominates
Implied VXO
Ostdiek and 20 trading) day. from 15% to HIS, but is biased
HIS20 days
Whaley (ranked) Use sample SD 45% when crash upward up to 580
(1995) of daily returns is excluded. basis points.
to proxy ˜actual Orthogonality test
±VXO = 0,
vol.™ rejects HIS when
βVXO < 1 with
robust SE VXO is included.
Adjust VXO
forecasts with
average forecast
errors of the last 253
days helps to correct
for biasedness while
retaining implied™s
explanatory power
MSE and MedSE Forecasting
40. Franses and Dutch, German, 1983“94 W AO-GARCH 1 week ahead
performance
Ghijsels Spanish and (GARCH adjusted estimated from
signi¬cantly
(1999) Italian stock for additive previous 4
years. Use improved when
market returns outliers using the
parameter estimates

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