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