GARCH deviations to are not in¬‚uenced

by ˜outliers™.

GARCH-t proxy ˜actual

(ranked) vol.™ Performance of

GARCH-t is

consistently much

worse. Same results

for all four stock

markets

Continued

Data Data Forecasting Forecasting Evaluation

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

41. Franses and Stock indices 1986“94 W QGARCH 1 week ahead MedSE QGARCH is best if

Van Dijk (Germany, RW GARCH estimated from data has no

(1996) Netherlands, GJR rolling 4 years. extremes. RW is

Spain, Italy, (ranked) Use weekly best when 87™s crash

Sweden) squared is included. GJR

deviations to cannot be

proxy ˜actual recommended.

vol.™ Results are likely to

be in¬‚uenced by

MedSE that penalize

nonsymmetry.

Brailsford and Faff

(1996) support GJR

as best model

although it

underpredicts over

70% of the time

42. Frennberg and VW Swedish In: M 1 month ahead MAPE, R 2 is S: seasonality

AR12 (ABS)-S RW,

Hansson stock market 1919“1976 estimated from 2“7% in ¬rst adjusted. RW model

ImpliedBS ATM Call

(1996) returns Out: 1977“82, (option maturity recursively period and seems to perform

1983“90 closest to 1 month) re-estimated 11“24% in remarkably well in

Index option Jan87“Dec90 GARCH-S, ARCH-S expanding second, more such a small stock

(European (ranked) sample. Use volatile period. market where

style) daily ret. to returns exhibit

H0 : ±implied = 0

Models that are not

compile strong seasonality.

and βimplied = 1

adj. for seasonality

monthly vol., cannot be Option was

did not perform as

adjusted for rejected with introduced in 86 and

well

autocorrelation robust SE covered 87™s crash;

outperformed by

RW. ARCH/

GARCH

did not perform as

well in the more

volatile second

period

43. Fung, Lie and £/$, C$/$, 1/84“2/87 D Option RMSE, MAE of Each day, 5 options

ImpliedOTM>ATM

Moreno FFr$, DM/$, (pre-crash) maturity; overlapping were studied; 1

Impliedvega, elasticity

(1990) ¥/$ & SrFr/$ overlapping forecasts ATM, 2 just in and 2

Impliedequal weight

options on periods. Use just out. De¬ne

HIS40 days , ImpliedITM

PHLX (ranked, all implied sample SD of ATM as S = X,

are from calls) daily returns OTM marginally

over option outperformed ATM.

maturity to Mixed together

proxy ˜actual implied of different

vol.™ contract months

Continued

Data Data Forecasting Forecasting Evaluation

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

44. Fung and S&P500, DM$ 3/83“7/89 D (15 min) RV-AR(n) 1 day ahead. RMSE and MAE RV: Realized vol.

Hsieh (1991) US T-bond (DM/$ Use 15-min from 15-min returns.

ImpliedBAW NTM Call/Put of log σ

futures from RV, RW (C-t-C) data to AR(n):

Futures and 26 Feb 85) HL construct autoregressive lags

futures options (ranked, some of the ˜actual vol.™ of order n. RW

differences are (C-t-C): random

small) walk forecast based

on close to close

returns. HL:

Parkinson™s daily

high-low method.

Impact of 1987

crash does not

appear to be drastic

possibly due to

taking log. In

general,

high-frequency data

improves

forecasting power

greatly

45. Gemmill 13 UK stocks May78“Jul83 M 13“21 non- ME, RMSE, Adding HIS

ImpliedITM

(1986) LTOM options. overlapping MAE aggregated increases R 2 from

ImpliedATM, vega WLS

option maturity across stocks and 12% to 15%. But ex

Impliedequal, OTM, elasticity

Stock price Jan 78“Nov83 D

(each average time. R 2 are ante combined

HIS20 Weeks

(ranked, all implied are 19 weeks). Use 6“12% (pooled) forecast from HIS

from calls) sample SD of and 40% (panel and ImpliedITM

weekly returns with ¬rm speci¬c turned out to be

over option intercepts). All worse than

maturity to individual forecasts.

± > 0, β < 1

proxy ˜actual Suffered small

vol.™ sample and

nonsynchroneity

problems and

omitted dividends

Continued

Data Data Forecasting Forecasting Evaluation

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

46. Gray (1996) US 1m T-Bill 1/70“4/94 W RSGARCH with time 1 week ahead R 2 calculated Volatility follows

varying probability (model not without constant GARCH and CIR

GARCH re-estimated). term, is 4 to 8% square root process.

Constant variance Use weekly for RSGARCH, Interest rate rise

(ranked) squared negative for increases probability

deviation to some CV and of switching into

proxy volatility GARCH. high-volatility

Comparable regime

RMSE and MAE Low-volatility

between persistence and

GARCH and strong rate level

RSGARCH mean reversion at

high-volatility state.

At low-volatility

state, rate appears

random walk and

volatility is highly

persistent

47. Guo (1996a) PHLX US$/¥ Jan91“Mar93 D Information not Regression with Use mid of bid“ask

ImpliedHeston

options available robust SE. No option price to limit

ImpliedHW

information on ˜bounce™ effect.

ImpliedBS

GARCH R 2 and forecast Eliminate

biasedness ˜nonsynchroneity™

HIS60

(ranked) by using

simultaneous

exchange rate and

option price. HIS

and GARCH contain

no incremental

information.

ImpliedHeston and

ImpliedHW are

comparable and are

marginally better

than ImpliedBS .

Only have access to

abstract

Continued

Data Data Forecasting Forecasting Evaluation

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

48. Guo (1996b) PHLX US$/¥, Jan86“Feb93 Tick 60 days ahead. US$/DM R 2 is 4, Conclusion same as

ImpliedHW (WLS, 0.8

US$/DM Use sample 3, 1% for the Guo (1996a). Use

< S/X < 1.2, 20 <

options variance of three methods. Barone-

T < 60 days)

GARCH (1, 1) daily returns to (9, 4, 1% for Adesi/Whaley

Spot rate D

proxy actual US$/¥) All approximation for

HIS60 days

(ranked) volatility forecasts are American options.

biased No risk premium for

volatility variance

± > 0, β < 1

with robust SE risk. GARCH has no

incremental

information. Visual

inspection of ¬gures

suggests implied

forecasts lagged

actual

49. Hamid (1998) S&P500 futures 3/83“6/93 D 13 schemes (including Non- RMSE, MAE Implied is better

options HIS, implied cross- overlapping 15, than historical and

strike average and 35 and 55 days cross-strike

intertemporal ahead averaging is better

averages) than intertemporal

(ranked, see comment) averaging (except

during very

turbulent periods)

50. Hamilton and Excess stock 1/65“6/93 M Bivariate RSARCH 1 month ahead. MAE Found economic

Lin (1996) returns Univariate RSARCH Use squared recessions drive

(S&P500 minus GARCH+L monthly ¬‚uctuations in stock

T-Bill) & Ind. ARCH+L residual returns returns volatility. ˜L™

Production AR(1) to proxy denotes leverage

(ranked) volatility effect. RS model

outperformed

ARCH/GARCH+L

51. Hamilton and NYSE VW 3/7/62“ W RSARCH+L 1, 4 and 8 MSE, MAE, Allowing up to

Susmel stock index 29/12/87 GARCH+L weeks ahead. MSLE, MALE. 4 regimes with t

(1994) ARCH+L Use squared Errors calculated distribution.

(ranked) weekly residual from variance RSARCH with

returns to proxy and log variance leverage (L)

volatility provides best

forecast. Student-t is

preferred to GED

and Gaussian

52. Harvey and S&P100 (OEX) Oct85“Jul89 D 1 day ahead R 2 is 15% for Implied volatility

ImpliedATM calls+puts

Whaley (American binomial, implied for use calls and 4% for changes are

(1992) in pricing next puts (excluding statistically

shortest maturity >

15 days) day option 1987 crash) predictable, but

(predict changes in market was ef¬cient,

implied) as simulated

transactions (NTM

call and put and

delta hedged using

futures) did not

produce pro¬t

Continued

Data Data Forecasting Forecasting Evaluation

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

53. Heynen and 7 stock indices 1/1/80“ D SV(?) Non- MedSE SV appears to

Kat (1994) and 5 exchange 31/12/92 EGARCH overlapping 5, dominate in index

rates GARCH 10, 15, 20, 25, but produces errors

In: 80“87 RW 50, 75, 100 that are 10 times

Out: 88“92 (ranked, see also days horizon larger than

comment) with constant (E)GARCH in

(87™s crash

update of exchange rate. The

included in

parameters impact of 87™s crash

in-sample)

estimates. Use is unclear. Conclude

sample standard that volatility model

deviations of forecasting

daily returns to performance

proxy ˜actual depends on the asset

vol.™ class

54. Hol and S&P100 (VXO) 2/1/86“ D SIV 1, 2, 5, 10, 15 R 2 ranges SVX is SV with

Koopman 29/6/2001 SVX+ and 20 days between 17 and impliedVXO as an

(2002) SV ahead. Use 33%, MSE, exogenous variable

Out:

(ranked) 10-min returns MedSE, MAE. while SVX+ is SVX

Jan97“Jun01

to construct with persistence

± and β not

˜actual vol.™ reported. All adjustment. SIV is

forecasts stochastic implied

underestimate with persistence

actuals parameter set equal

to zero

55. Hwang and LIFFE stock 23/3/92“ D Log-ARFIMA-RV Forecast

1, 5, 10, 20, . . . , MAE, MFE

Satchell options 7/10/96 Scaled truncated 90, 100, 120 impliedATM BS of

(1998) Detrended days ahead IV shortest maturity

240 daily out-

Unscaled truncated estimated from option (with at 15

of-sample

a rolling sample trading days to

MAopt n=20 -IV

forecasts.

of 778 daily maturity). Build MA

Adj MAopt n=20 -RV

GARCH-RV observations. in IV and ARIMA

(ranked, forecast Different on log (IV). Error

implied) estimation statistics for all

intervals were forecasts are close

tested for except those for

robustness GARCH forecasts.

The scaling in

Log-ARFIMA-RV

is to adjust for

Jensen inequality

56. Jorion (1995) DM/$, ¥/$, 1/85“2/92 D 1 day ahead and R 2 is 5% (1-day) Implied is superior

ImpliedATM BS call+put

SrFr/$ futures 7/86“2/92 option maturity. or 10“15% to the historical

GARCH (1, 1), MA20

options on 3/85“2/92 (ranked) Use squared (option methods and least

CME returns and maturity). With biased. MA and

aggregate of robust SE, GARCH provide

square returns only marginal

±implied > 0 and

to proxy actual incremental

βimplied < 1 for

volatility long horizon and information

is unbiased for

1-day forecasts

Continued

Data Data Forecasting Forecasting Evaluation

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

57. Jorion (1996) DM/$ futures Jan85“Feb92 D 1 day ahead, R 2 about 5%. R 2 increases from

ImpliedBlack, ATM

options on GARCH(1, 1) use daily 5% to 19% when

H0 : ±implied = 0,

CME (ranked) squared to unexpected trading

βimplied = 1

proxy actual cannot be volume is included.

volatility rejected with Implied volatility

robust SE subsumed

information in

GARCH forecast,

expected futures

trading volume and

bid“ask spread

58. Karolyi 74 stock 13/1/84“ M 20 days ahead MSE Bayesian adjustment

Bayesian impliedCall

(1993) options 11/12/85 volatility to implied to

ImpliedCall

incorporate

HIS20,60

(Predict option price cross-sectional

not ˜actual vol.™) information such as

¬rm size, leverage

and trading volume

useful in predicting

next period option

price

59. Klaassen US$/£, 3/1/78“ D RSGARCH 1 and 10 days MSE of variance, GARCH(1, 1)

(1998) US$/DM and 23/7/97 RSARCH ahead. Use regression forecasts are more

US$/¥ GARCH(1, 1) mean adjusted though R 2 is not variable than RS

Out: 20/10/87“

(ranked) 1- and 10-day reported models. RS provides

23/7/97

return squares statistically

to proxy actual signi¬cant

volatility improvement in

forecasting volatility

for US$/DM but not

the other exchange

rates

60. Kroner, Futures options Jan87“Dec90 D 225 calendar MSE, ME GR: Granger and

GR > COMB

Kneafsey and on cocoa, (kept last 40 days (160 Ramanathan

ImpliedBAW Call

Claessens cotton, corn, observations working days) (1984)™s regression

(WLS > AVG

(1995) gold, silver, for out-of- ahead, which is weighted combined

> ATM)

sugar, wheat sample longer than forecast, COMB: lag

HIS7 weeks > GARCH

forecast) (ranked) average implied in GARCH

conditional variance

Futures prices Jan87“ Jul91

equation. Combined

method is best

suggests option

market inef¬ciency

Continued

Data Data Forecasting Forecasting Evaluation

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

61. Lamoureux Stock options for 19/4/82“ D 90 to 180 days ME, MAE, Implied volatility is

ImpliedHull-White NTM Call

and 10 non-dividend- 31/3/84 (intermediate term to matching option RMSE. best but biased. HIS

Lastrapes paying stocks maturity, WLS) maturity estimated Average provides incremental

(1993) (CBOE) using rolling 300 implied is info. to implied and

HISupdated expanding estimate

GARCH observations and lower than has the lowest RMSE.

(ranked, based on expanding sample. actual for all When all three

regression result) Use sample stocks. R 2 on forecasts are included;

variance of daily variance varies ± > 0,

returns to proxy between 3 and 1 > βimplied > 0,

˜actual vol.™ 84% across βGARCH = 0,

stocks and βHIS < 0 with robust

models SE. Plausible

explanations include

option traders

overreact to recent

volatility shocks, and

volatility risk

premium is nonzero

and time-varying

62. Latane and 24 stock options 5/10/73“ W In-sample forecast Cross-section Used European model on

Implied vega weighted

Rendleman from CBOE 28/6/74 and forecast that correlation American options and

HIS4 years

omitted dividends.

(1976) (ranked) extend partially into between

volatility ˜Actual™ is more

the future. Use

weekly and monthly estimates for 38 correlated (0.686) with

returns to calculate weeks and a ˜Implied™ than HIS

actual volatility of 2-year period volatility (0.463) Highest

various horizons correlation is that

between implied and

actual standard

deviations which were

calculated partially into

the future

63. Lee (1991) $/DM, $/£, $/¥, 7/3/73“ W Kernel (Gaussian, 1 week ahead (451 RMSE, MAE. It Nonlinear models are, in

$/FFr, $/C$ 4/10/89 (Wed, truncated) observations in is not clear how general, better than linear

(Fed. Res. 12pm) Index (combining sample and 414 actual volatility GARCH. Kernel method

Out: 21

Bulletin) ARMA and observations was estimated is best with MAE. But

Oct81

GARCH) out-of-sample) most of the RMSE and

“11

EGARCH (1, 1) MAE are very close.

Oct89.

GARCH (1, 1) Over 30 kernel models

IGARCH with were ¬tted, but only

trend those with smallest

(rank changes see RMSE and MAE were

comment for reported. It is not clear

general how the nonlinear

assessment) equivalence was

constructed. Multi-step

forecast results were

mentioned but not shown

Data Data Forecasting Forecasting Evaluation

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

64. Li (2002) $/DM, $/£, $/¥ 3/12/86“ Tick 1, 2, 3 and 6 months MAE. R 2 ranges Both forecasts have

ImpliedGK OTC ATM

30/12/99 (5 min) ahead. Parameters 0.3“51% (implied), incremental

ARFIMArealised

OTC ATM

In: 12/8/86“ (implied better at not re-estimated. 7.3“47% (LM), information

options

11/5/95 shorter horizon Use 5-min returns 16“53% especially at long

$/£, $/¥

D and ARFIMA to construct ˜actual (encompass). For horizon. Forcing:

19/6/94“

$/DM

D better at long vol.™ both models, H0 :

13/6/99 ± = 0, β = 1

19/6/94“ horizon) produce

± = 0, β = 1 are

30/12/98 rejected and low/negative R 2

(especially for long

typically β < 1

with robust SE horizon). Model

realized standard

deviation as

ARFIMA without

log transformation

and with no

constant, which is

awkward as a

theoretical model

for volatility

65. Lopez C$/US$, 1980“1995 D SV-AR(1)-normal 1 day ahead and MSE, MAE, LL is the logarithmic

(2001) DM/US$, GARCH-gev probability forecasts LL, HMSE, loss function from Pagan

¥/US$, US$/£ EWMA-normal for four ˜economic GMLE and QPS and Schwert (1990),

In: 1980“1993

events™, viz. cdf of (quadratic HMSE is the

Out: GARCH-normal, -t

speci¬c regions. Use probability heteroscedasticity-adj.

1994“1995 EWMA-t

daily squared residuals scores) MSE from Bollerslev

AR(10)-Sq, -Abs

to proxy volatility. Use and Ghysels (1996) and

Constant

empirical distribution GMLE is the Gaussian

(approx. rank, see

to derive cdf quasi-ML function from

comments)

Bollerslev, Engle and

Nelson (1994). Forecasts

from all models are

indistinguishable. QPS

favours SV-n, GARCH-g

and EWMA-n

RMSE,

66. Loudon, FT All Share Jan71“Oct97 D EGARCH, GJR, Parameters estimated TS-GARCH is an

regression on

Watt and TS-GARCH, in period 1 (or 2) used absolute return version

Sub-periods:

log volatility

Yadav TGARCH to produce conditional of GARCH. All

Jan71“Dec80

(2000) NGARCH, variances in period 2 and a list of GARCH speci¬cations

Jan81“Dec90

diagnostics. R 2

VGARCH, (or 3). Use GARCH have comparable

Jan91“Oct97

GARCH, squared residuals as is about 4% in performance though

MGARCH ˜actual™ volatility period 2 and 5% nonlinear, asymmetric

(no clear rank, in period 3 versions seem to fare

forecast better. Multiplicative

GARCH vol.) GARCH appears worst,

followed by NGARCH

and VGARCH (Engle

and Ng 1993)

Continued

Data Data Forecasting Forecasting Evaluation

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

67. Martens S&P500 Jan94“ Tick Heteroscedasticity Scaled down one large

ImpliedBAW VXOstyle Non-overlapping 1, 5,

and futures, Dec2000 Log-ARFIMA 10, 20, 30 and 40 days adjusted RMSE. oil price.

Zein ¥/US$ futures, Jan96“ GARCH ahead. 500 daily R 2 ranges Log-ARFIMA

(2004) Dec2000 (ranked, see observations in 25“52% truncated at lag 100.

Crude Jun93“ comment also) in-sample which (implied), Based on R 2 , Implied

oil futures Dec2000 expands on each 15“48% (LM) outperforms GARCH

iteration across assets and in every case, and

horizons. Both beats Log-ARFIMA in

models provide ¥/US$ and crude oil.

incremental info. Implied has larger

to encompassing HRMSE than

regr. Log-ARFIMA in most

cases. Dif¬cult to

comment on implied™s

biasedness from

information presented

68. McKenzie 21 A$ bilateral Various length D Square vs. power 1 day ahead absolute RMS, ME, MAE. The optimal power is

(1999) exchange rates from 1/1/86 or transformation returns Regressions closer to 1 suggesting

4/11/92 to (ARCH models suggest all ARCH squared return is not

31/10/95 with various lags. forecasts are the best speci¬cation

See comment for biased. No R 2 in ARCH type model

rank) was reported for forecasting purpose

ME, MAE, CGARCH is the

69. McMillan, FTSE100 Jan84“Jul96 D, W, RW, MA, ES, EWMA j = 1 day, 1 week and

RMSE for component GARCH

Speight and FT All Share Jan69“Jul96 M GARCH, 1 month ahead based

on the three data symmetry loss model. Actual

Gwilym TGARCH,

function. volatility is proxied by

(2000) EGARCH, frequencies. Use j

Out: period squared returns MME(U) and mean adjusted squared

CGARCH

1996“1996 for to proxy actual MME(O), mean returns, which is likely

HIS, regression,

both series. (ranked) volatility mixed error that to be extremely noisy.

penalize Evaluation conducted

under/over on variance, hence

predictions forecast error statistics

are very close for most

models. RW, MA, ES

dominate at low

frequency and when

crash is included.

Performances of

GARCH models are

similar though not as

good

Continued

Data Data Forecasting Forecasting Evaluation

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

70. Noh, Engle S&P500 index Oct85“Feb92 D GARCH adj. for Option maturity. Equate Regression with

and options weekend and Based on 1000 days forecastability with call+put implieds,

Kane hols rolling period pro¬tability under daily dummies and

(1994) estimation the assumption of previous day returns to

ImpliedBS

weighted by an inef¬cient option predict next day

trading volume market implied and option

(ranked, predict prices. Straddle

option price strategy is not vega

not ˜actual neutral even though it

vol.™) might be delta neutral

assuming market is

complete. It is possible

that pro¬t is due to

now well-documented

post 87™s crash higher

option premium

M EGARCH(1, 2) 1 month ahead. Use

71. Pagan and US stock market 1834“1937 R 2 is 7“11% for The nonparametric

squared residual

Schw- Out: 1900“ GARCH(1, 2) 1900“25 and 8% models fared worse

2-step conditional monthly returns to

ert 1925 (low for 1926“37. than the parametric

(1990) volatility), variance proxy actual Compared with R 2 models. EGARCH

volatility

1926“1937 RS-AR(m) for variance, R 2 for came out best because

(high Kernel (1 lag) log variance is of the ability to capture

volatility) Fourier (1 or 2 smaller in 1900“25 volatility asymmetry.

lags) and larger in Some prediction bias

(ranked) 1926“37 was documented

72. Pong, US$/£ In: Jul87“ 1 month and 3 ME, MSE, Implied, ARMA and

5-, 30-min ImpliedATM, OTC

Shackleton, Dec93 (bias adj. months ahead at regression. R 2 ARFIMA have similar

quote

Taylor and Out: Jan94“ using rolling 1-month interval ranges between performance.

Xu (2004) Dec98 regr. on last 22 and 39% GARCH(1, 1) clearly

5 years monthly (1-month) and inferior. Best

data) 6 and 21% combination is

Log-ARMA(2, 1) (3-month) Implied + ARMA

Log-ARFIMA (2, 1). Log-AR(FI)MA

(1, d, 1) forecasts adjusted for

GARCH(1, 1) Jensen inequality.

Dif¬cult to comment

(ranked)

on implied™s

biasedness from

information presented

73. Poteshman S&P500 1Jun88“ D Option maturity BS R 2 is over

ImpliedHeston F test for H0 : ±BS = 0,

(2000) (SPX) & 29Aug97 (about 3.5 to 4 50%. Heston

ImpliedBS (both βBS = 1 are rejected

futures implieds are weeks, non- implied produced though t-test supports

Heston futures

from WLS of all overlapping). Use similar R 2 but H0 on individual

estimation: Tick

5-min futures very close to coef¬cients. Show

options <7

1Jun93“

inferred index being unbiased biasedness is not

months but >6

29Aug97

calendar days) return to proxy caused by bid“ask

˜actual vol.™

HIS1, 2, 3, 6 months spread. Using in σ ,

S&P500 7Jun62“May93 M (ranked) high-frequency

realized vol., and

Heston model, all help

to reduce implied

biasedness

Continued

Data Data Forecasting Forecasting Evaluation

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

74. Randolph S&P500 2/1/1986“ Daily Non-overlapping 20 ME, RMSE, MAE, Mean reversion model

MRMATM HIS

and Najand futures options 31/12/88 opening days ahead, re- MAPE (MRM) sets drift rate of

MRMATM implied

(1991) (crash included) Tick GARCH(1, 1) estimated using volatility to follow a

expanding sample. mean reverting process

HIS20 day

ATM calls only In: First 80

ImpliedBlack taking impliedATM (or

observations

HIS) as the previous day

(ranked though the

vol. Argue that GARCH

error statistics

did not work as well

are close)

because it tends to

provide a persistent

forecast, which is valid

only in period when

changes in vol. are small

75. Schmalensee 6 CBOE stock 29/4/74“23/5/75 W 1 week ahead. Statistical tests Find implied rises when

ImpliedBS call

and Trippi options (simple average ˜Actual™ proxied by reject the stock price falls, negative

56 weekly

(1978) of all strikes and weekly range and hypothesis that IV serial correlation in

observations

all maturities) average price responds positively changes of IV and a

(Forecast implied deviation to current volatility tendency for IV of

not actual different stocks to move

volatility) together. Argue that IV

might correspond better

with future volatility

76. Scott and DM/$, £/$, 14/3/83/“ Daily Non-overlapping MSE, R 2 ranges Simple B-S forecasts

ImpliedGk (vega,

Tucker C$/$, ¥/$ & 13/3/87 closing Inferred ATM, option maturity: 3, 6 from 42 to 49%. just as well as

(1989) SrFr/$ (pre-crash) tick NTM) and 9 months. Use In all cases, sophisticated CEV

American sample SD of daily model. Claimed

ImpliedCEV ± > 0, β < 1.

options on (similar rank) returns to proxy HIS has no omission of early

PHLX ˜actual vol.™ incremental info. exercise is not

content important. Weighting

scheme does not matter.

Forecasts for different

currencies were mixed

together

77. Sill (1993) S&P500 1959“1992 M HIS with exo 1 month ahead R 2 increase from Volatility is higher in

variables 1% to 10% when recessions than in

HIS additional expansions, and the

(see comment) variables were spread between

added commercial-paper and

T-Bill rates predict

stock market volatility

78. Szakmary, Futures options Various dates D Overlapping option R 2 smaller for

ImpliedBk, NTM HIS30 and GARCH

Ors, Kim on S&P500, 9 between maturity, shortest but ¬nancial have little or no

2Calls + 2Puts eq al weight

and interest rates, 5 Jan 83 and (23“28%), higher incremental

HIS30 , >10 days. Use sample

Davidson currency, 4 May 2001 GARCH SD of daily returns for metal and information content.

(2002) energy, 3 (ranked) over forecast horizon agricult. ±implied > 0 for 24

metals, 10 to proxy ˜actual vol.™ (30“37%), highest cases (or 69%), all 35

agriculture, 3 for livestock and cases βimplied < 1 with

livestock energy (47“58%) robust SE

Continued

Data Data Forecasting Forecasting Evaluation

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

79. Taylor JW DAX, 6/1/88“30/8/95 W STES (E, AE, 1 week ahead using ME, MAE, Models estimated based

(2004) S&P500, (equally split EAE) a moving window RMSE, R 2 on minimizing in-sample

Hang Seng, between in- and GJR of 200 weekly (about 30% for forecast errors instead of

FTSE 100, out-) (+Smoothed returns. Use daily HK and Japan ML. STES-EAE (smooth

Amsterdam variations) squared residual and 6% for US) transition exponential

EOE, Nikkei, GARCH returns to construct smoothing with return

Singapore All weekly ˜actual™ and absolute return as

MA20 weeks,

Share Riskmetrics volatility transition variables)

(ranked) produced consistently

better performance for

1-step-ahead forecasts

Represent one of the

80. Taylor SJ 15 US stocks Jan66“Dec76 D EWMA 1 and 10 days ahead Relative MSE

earliest studies in ARCH

(1986) FT30 Jul75“Aug82 Log-AR(1) absolute returns.

Various length ARMACH-Abs 2/3 of sample used class forecasts. The issue

6 metal

Nov74“Sep82 ARMACH-Sq in estimation. Use of volatility stationarity

£/$

5 agricultural Various length HIS daily absolute is not important when

futures (ranked) returns deviation as forecast over short

4 interest rate ˜actual vol.™ horizon. Nonstationary

Various length ARMACH-Sq is

futures series (e.g. EWMA) has

similar to

the advantage of having

GARCH

fewer parameter

estimates and forecasts

respond to variance

change fairly quickly

81. Taylor SJ DM/$ futures 1977“83 D High, low and 1, 5, 10 and 20 days RMSE Best model is a weighted

(1987) closing prices ahead. Estimation average of present and

(see comment) period, 5 years past high, low and

closing prices with

adjustments for weekend

and holiday effects

82. Taylor SJ DM/$ 1/10/92“30/9/93 1 hour ahead MAE and MSE on 5-min return has

Quote Implied + ARCH

and Xu In: 9 months combined estimated from 9 std deviation and information incremental

(1997) Out: 3 months Implied, months in-sample variance to daily implied when

ARCH period. Use 5-min forecasting hourly

returns to proxy volatility

HIS9 months

DM/$ options ˜actual vol.™

D HISlast hour realised vol Friday macro news ARCH model includes

on PHLX (ranked) seasonal factors with hourly and 5-min

See comment for have no impact on returns in the last hour

details on implied forecast accuracy plus 120 hour/day/week

and ARCH seasonal factors. Implied

derived from NTM

shortest maturity (>9

calendar days) Call+Put

using BAW

Continued

Data Data Forecasting Forecasting Evaluation

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

83. Tse (1991) Topix Nikkei In: 1986“1987 D EWMA 25 days ahead ME, RMSE, MAE, Use dummies in mean

Stock Average Out: 88“89 HIS estimated from MAPE of variance equation to control for

ARCH, GARCH rolling 300 of 21 non- 1987 crash.

(ranked) observations overlapping 25-day Nonnormality provides

periods a better ¬t but a poorer

forecast. ARCH/GARCH

models are slow to

react to abrupt change

in volatility. EWMA

adjust to changes very

quickly

84. Tse and Singapore, 5 19/3/75 to D EWMA 25 days ahead RMSE, MAE EWMA is superior,

Tung (1992) VW market & 25/10/88 HIS estimated from GARCH worst.

industry GARCH rolling 425 Absolute returns > 7%

indices (ranked) observations are truncated. Sign of

nonstationarity. Some

GARCH

nonconvergence

85. Vasilellis Stock options In: W 3 months ahead. RMSE Implied: 5-day average

Combine (Implied +

and Meade 12 UK stocks 28/3/86“27/6/86 GARCH) Use sample SD of dominates 1-day

(1996) (LIFFE) Implied (various, see daily returns to implied vol. Weighting

In2 (for combined

comment) proxy ˜actual vol.™ scheme: max vega >

forecast):

GARCH vega weighted >

28/6/86’25/3/88

EWMA elasticity weighted >

Out: 6/7/88’ max elasticity with ˜>™

HIS3 months

21/9/91 (ranked, results indicates better

not sensitive to forecasting

basis use to performance.

combine) Adjustment for div.

and early exercise:

Rubinstein > Roll >

constant yield. Crash

period might have

disadvantaged time

series methods

86. Vilasuso C$/$, FFr/$, In: 13/3/79“ D FIGARCH 1, 5 and 10 days MSE, MAE, and Signi¬cantly better

(2002) DM/$, ¥/$, 31/12/97 GARCH, ahead. Used daily Diebold- forecasting

£/$ IGARCH squared returns to Mariano™s test for performance from

Out: 1/1/98“ (ranked, proxy actual sig. difference FIGARCH. Built

31/12/99 GARCH volatility FIARMA (with a

marginally better constant term) on

than IGARCH) conditional variance

without taking log.

Truncated at lag 250

Continued

Data Data Forecasting Forecasting Evaluation

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

87. Walsh and Australian 1Jan 93“ 5-min to form EWMA GARCH 1 hour, 1 day and 1 MSE, RMSE, Index with larger

Tsou (1998) indices: 31 Dec95 H, D and W (not for weekly week ahead MAE, MAPE number of stock is

VW20, VW50 returns returns) estimated from a easier to forecast due to

In: 1 year

& VW300 HIS, IEV 1-year rolling diversi¬cation, but gets

Out: 2 years (improved sample. Use square harder as sampling

extreme-value of price changes interval becomes

method) (non-cumulative) as shorter due to problem

(ranked) ˜actual vol.™ of nonsynchronous

trading. None of the

GARCH estimations

converged for the

weekly series, probably

too few observations

88. Wei and SrFr/$, 2/83“1/90 M Use European formula

Implied GK ATM call Non-overlapping 1 R 2 30%(£), 17%

Frankel DM/$, ¥/$, (shortest month ahead. Use (DM), 3%(SrFr), for American style

(1991) £/$ options maturity) option. Also suffers

sample SD of daily 0%(¥). ± > 0,

(PHLX) from nonsynchronicity

exchange rate return β < 1 (except that

problem. Other tests

to proxy ˜actual vol.™ for £/$, ± > 0,

reveal that implied

β = 1) with

D

Spot rates heterosced. tends to overpredict

consistent SE high vol. and

underpredict low vol.

Forecast/implied could

be made more accurate

by placing more weight

on long-run average

89. West and C$/$, FFr/$, 14/3/73“ W GARCH(1, 1) RMSE and Some GARCH forecasts

j = 1, 12, 24 weeks

Cho (1995) DM/$, ¥$/$, 20/9/89 IGARCH (1, 1) estimated from regression test on mean revert to

£/$ AR(12) in absolute rolling 432 weeks. variance, R 2 varies unconditional variance in

In: 14/3/73“

AR(12) in squares Use j period squared from 0.1% to 4.5% 12 to 24 weeks. It is

17/6/81

Homoscedastic returns to proxy dif¬cult to choose

Out: 24/6/81“ Gaussian kernel actual volatility between models.

12/4/89 (no clear rank) Nonparametric method

came out worst though

statistical tests for do not

reject null of no

signi¬cance difference in

most cases

90. Wiggins S&P500 4/82“12/89 D ARMA model with 1 week ahead and 1 Bias test, ef¬ciency Modi¬ed Parkinson

(1992) futures 2 types of month ahead. test, regression approach is least biased.

estimators: Compute actual C-t-C estimator is three

1. Parkinson/Garmen- volatility from daily times less ef¬cient than

Klass extreme observations EV estimators. Parkinson

value estimators estimator is also better

2. Close-to-close than C-t-C at forecasting.

estimator 87™s crash period

(ranked) excluded from analysis

Continued

Data Data Forecasting Forecasting Evaluation

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

91. Xu and £/$, DM/$, ¥/$ In: Jan 85“Oct89 D ME, MAE, Implied works best and is

ImpliedBAW NTM TS or Non-overlapping 4

Out:

Taylor & SrFr/$ weeks ahead, RMSE, When unbiased. Other forecasts

short

(1995) PHLX options 18Oct89“4Feb92 have no incremental

GARCHNormal or GED estimated from a ±implied is set

rolling sample of equal to 0, information. GARCH

HIS4 weeks

Corresponding

(ranked) 250 weeks daily forecast performance not

βimplied = 1

futures rates

data. Use cannot be sensitive to distributional

cumulative daily rejected assumption about

squared returns to returns. The choice of

proxy ˜actual vol.™ implied predictor (term

structure, TS, or short

maturity) does not affect

results

92. Yu (2002) NZSE40 Jan80“Dec98 D SV (of log variance) 1 month ahead RMSE, MAE, Range of the evaluation

GARCH (3, 2), estimated from Theil-U and measures for most

In: 1980“1993

GARCH (1, 1) previous 180 to 228 LINEX on models is very narrow.

Out: 1994“1998

months of daily variance Within this narrow

HIS, MA5 yr or 10 yr

data. Use aggregate range, SV ranked ¬rst,

ES and EWMA

of daily squared performance of GARCH

(monthly revision)

returns to construct was sensitive to

Regressioniag-1

actual monthly evaluation measure;

ARCH(9), RW,

volatility regression and EWMA

(ranked)

methods did not perform

well. Worst performance

from ARCH(9) and RW.

Volatile periods (Oct 87

and Oct 97) included in

in- and out-of-samples

93. Zumbach USD/CHF, 1/1/89“ H LM-ARCH 1 day ahead RMSE. Realized LM-ARCH, aggregates

(2002) USD/JPY 1/7/2000 F-GARCH estimated from volatility measured high-frequency squared

GARCH previous 5.5 years using hourly returns returns with a set of

And their integrated power law weights, is the

counterparts best though difference is

(ranked) small. All integrated

versions are more stable

across time

Ranked: models appear in the order of forecasting performance; best performing model at the top. If two weighting schemes or two forecasting models appear

at both sides of ˜>™, it means the l.h.s. is better than the r.h.s. in terms of forecasting performance. SE: Standard error. ATM: At the money. NTM: Near the

money. OTM: Out of the money. WLS: an implied volatility weighting scheme used in Whaley (1982) designed to minimize the pricing errors of a collection of

options. In some cases the pricing errors are multiplied by trading volume or vega to give ATM implied a greater weight. HIS: Historical volatility constructed

based on past variance/standard deviation. VXO: Chicago Board of Option Exchange™s volatility index derived from S&P100 options. VXO was renamed VXO

in September 2003. The current VXO is compiled using a model-free implied volatility estimate. All the research papers reviewed have used VXO (i.e. the old

VIX.) RS: Regime switching. BS: Black“Scholes. BK: Black model for pricing futures option. BAW: Barone-Adesi and Whaley American option pricing formula.

HW: Hull and White option pricing model with stochastic volatility. FW: Fleming and Whaley (1994) modi¬ed binomial method that takes into account wildcard

option. GK: Garman and Kohlhagan model for pricing European currency option. HJM: Heath, Jarrow and Morton (1992) forward rate model for interest rates.

References

Aggarwal, R., C. Inclan and R. Leal (1999) Volatility in emerging stock markets, Journal

of Financial and Quantitative Analysis, 34, 1, 33“55.

Ait-Sahalia, Y., P. A. Mykland and L. Zhang (2003) How often to sample a continuous-

time process in the presence of market microstructure noise, Working paper, Univer-

sity of Princeton.

Akgiray, V. (1989) Conditional heteroskedasticity in time series of stock returns: evi-

dence and forecasts, Journal of Business, 62, 55“80.

Alexander, C. (2001), Market Models: A Guide to Financial Data Analysis, John Wiley

& Sons Ltd, Chichester.

Alford, A.W., and J.R. Boatsman (1995) Predicting long-term stock return volatility:

Implications for accounting and valuation of equity derivatives, Accounting Review,

70, 4, 599“618.

Alizadeh, S., M.W. Brandt and F.X. Diebold (2002) Range-based estimation of stochastic

volatility models, Journal of Finance, 57, 3, 1047“1092.

Amin, K., and V. Ng (1997) Inferring future volatility from the information in implied

volatility in Eurodollar options: A new approach, Review of Financial Studies, 10,

333“367.

Andersen, T.G. (1996) Return volatility and trading volume: An information ¬‚ow inter-

pretation of stochastic volatility, Journal of Finance, 51, 1, 169“204.

Andersen, T.G., L. Benzoni and J. Lund (2002) An empirical investigation of continuous

time equity return models, Journal of Finance, 57, 1239“1284.

Andersen, T., and T. Bollerslev (1998) Answering the skeptics: Yes, standard volatility

models do provide accurate forecasts, International Economic Review, 39, 4, 885“

905.

Andersen, T.G., T. Bollerslev, F.X. Diebold and H. Ebens (2001) The distribution of

realized stock return volatility, Journal of Financial Economics, 61, 1, 43“76.

Andersen, T.G., T. Bollerslev, F.X. Diebold and P. Labys (2001) The distribution of

realized exchange rate volatility, Journal of American Statistical Association, 96,

453, 42“57.

Andersen, T.G., T. Bollerslev and S. Lange (1999) Forecasting ¬nancial market volatility:

Sample frequency vis-` -vis forecast horizon, Journal of Empirical Finance, 6, 5,

a

457“477.

Andersen, T.G., T. Bollerslev, F.X. Diebold and P. Labys, 2003, Modeling and forecasting

realized volatility, Econometrica, 71, 2, 529“626.

202 References

Andersen, T.G., and B.E. Sorensen (1997) GMM and QML asymptotic standard

deviations in stochastic volatility models, Journal of Econometrics, 76, 397“

403.

Artzner, P., F. Delbaen, J. Eber and D. Heath (1997) Thinking coherently, RISK Maga-

zine, 10, 11, 68“71.

Artzner, P., F. Delbaen, J. Eber and D. Heath (1999) Coherent measures of risk, Mathe-

matical Finance, 9, 3, 203“228.

Baillie, R.T., and T. Bollerslev (1989) The message in daily exchange rates: a conditonal-

variance tale, Journal of Business and Economic Statistics, 7, 3, 297“305.

Baillie, R.T., T. Bollerslev and H.O. Mikkelsen (1996) Fractionally integrated general-

ized autoregressive conditional heteroskedasticity, Journal of Econometrics, 74, 1,

3“30.

Baillie, R.T., T. Bollerslev and M.R. Redfearn (1993) Bear squeezes, volatility spillovers

and speculative attacks in the hyperin¬‚ation 1920s foreign exchange, Journal of

International Money and Finance, 12, 5, 511“521.

Bakshi, G., C. Cao and Z. Chen (1997) Empirical performance of alternative option

pricing models, Journal of Finance, 52, 5, 2003“2049.

Bakshi, G., and N. Kapadia (2003) Delta-hedged gains and the negative market volatility

risk premium, Review of Financial Studies, 16, 2, 527“566.

Bali, T.G. (2000) Testing the empirical performance of stochastic volatility models of

the short-term interest rate, Journal of Financial and Quantitative Analysis, 35, 2,

191“215.

Ball, C.A., and W.N. Torous (1984) The maximum likelihood estimation of security price

volatility: Theory, evidence and application to option pricing, Journal of Business,

57, 1, 97“113.

Bandi, F.M., and J.R. Russell (2004) Separating microstructure noise from volatility,

Working paper, University of Chicago.

Banks, E. (2004) Alternative risk transfer: Integrated risk management through insur-

ance, reinsurance and the capital markets, John Wiley & Sons Ltd, Chichester.

Barndorff-Nielsen, O.E., and N. Shephard (2003) Power and bipower variation with

stochastic volatility and jumps, Working paper, Oxford University.

Barone-Adesi, G., and R.E. Whaley (1987) Ef¬cient analytic approximation of American

option values, Journal of Finance, 42, 2, 301“320.

Bates, D.S. (1996) Testing option pricing models, in: Maddala, G.S., and C.R. Rao

(eds), Handbook of Statistics, vol. 14: Statistical Methods in Finance, Elsevier, North

Holland, Amsterdam, pp. 567“611.

Bates, D.S. (2000) Post-87 crash fears in S&P 500 futures options, Journal of Econo-

metrics, 94, 181“238.

Beckers, S. (1981) Standard deviations implied in option prices as predictors of future

stock price variability, Journal of Banking and Finance, 5, 363“382.

Beckers, S, (1993) Variances of security price returns based on high, low and closing

prices, Journal of Business, 56, 97“112.

Benzoni, L. (2001) Pricing options under stochastic volatility: An empirical investiga-

tion, Working paper, Carlson School of Management, Minneapolis, MN.

Bera, A.K., and M.L. Higgins (1993) ARCH models: properties, estimation and testing,

Journal of Economic Surveys, 7, 4, 305“365.

Bera, A., and M. Higgins (1997) ARCH and bilinearity as competing models for non-

linear dependence, Journal of Business and Economic Statistics, 15, 1, 43“50.

Beran, J. (1994) Statistics for Long Memory Process, John Ryland, Chapman & Hall.

Black, F. (1975) Fact and fantasy in the use of options, Financial Analysts Journal, 31,

36“41.

References 203

Black, F, (1976) Studies of stock price volatility of changes, American Statistical Asso-

ciation Journal, 177“181.

Blair, B., S.-H. Poon and S.J. Taylor (2001) Forecasting S&P 100 volatility: The incre-

mental information content of implied volatilities and high frequency index returns,

Journal of Econometrics, 105, 5“26.

Bluhm, H.H.W., and J. Yu (2000) Forecasting volatility: Evidence from the German

stock market, Working paper, University of Auckland.

Bollen, B., and B., Inder (2002) Estimating daily volatility in ¬nancial markets utilizing

intraday data, Journal of Empirical Finance, 9, 551“562.

Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity, Journal

of Econometrics, 31, 307“328.

Bollerslev, T. (1987) A conditionally heteroskedastic time series model for speculative

prices and rates of return, Review of Economics and Statistics, 69, 3, 542“547.

Bollerslev, T. (1990) Modelling the coherence in short-run nominal exchange rates:

A multivariate generalized ARCH model, Review of Economics and Statistics, 72,

498“505.

Bollerslev, T., R.Y. Chou and K.P. Kroner (1992) ARCH modeling in ¬nance: A Review

of the theory and empirical evidence, Journal of Econometrics, 52, 5“59.

Bollerslev, T., R.F. Engle and D.B. Nelson (1994) ARCH models, in: Engle, R.F.,

and D.L. McFadden (eds), Handbook of Econemetrics, Vol. IV, North Holland,

Amsterdam, pp. 2959“3038.

Bollerslev, T., R.F. Engle and J.M. Wooldridge (1988) A capital asset pricing model

with time-varying covariances, Journal of Political Economy, 96, 1, 116“131.

Bollerslev, T., and E. Ghysels (1996) Periodic autoregressive conditional heteroskedas-

ticity, Journal of Business and Economic Statistics, 14, 2, 139“151.

Bollerslev, T., and H.O. Mikkelsen (1996) Modeling and pricing long memory in stock

market volatility, Journal of Econometrics, 73, 1, 151“184.

Bollerslev, T., and H. O. Mikkelsen (1999) Long-term equity anticipation securities and

stock market volatility dynamics, Journal of Econometrics, 92, 75“99.

Boudoukh, J., M. Richardson and R.F. Whitelaw (1997) Investigation of a class of

volatility estimators, Journal of Derivatives, 4, 3, 63“71.

Brace, A., and A. Hodgson (1991) Index futures options in Australia “ An empirical

focus on volatility, Accounting and Finance, 31, 2, 13“31.

Brailsford, T.J., and R.W. Faff (1996) An evaluation of volatility forecasting techniques,

Journal of Banking and Finance, 20, 3, 419“438.

Brooks, C. (1998) Predicting stock market volatility: Can market volume help? Journal

of Forecasting, 17, 1, 59“80.

Buraschi, A., and J. C. Jackwerth (2001) The price of a smile: Hedging and spanning in

option markets, Review of Financial Studies, 14, 2, 495“527.

Canina, L., and S. Figlewski (1993) The informational content of implied volatility,

Review of Financial Studies, 6, 3, 659“681.

Cao, C.Q., and R.S. Tsay (1992) Nonlinear time-series analysis of stock volatilities,

Journal of Applied Econometrics, December, Supplement, 1, S165“S185.

Chan, K.C., G.A. Karolyi, F.A. Longstaff and A.B. Sanders (1992) An empirical com-

parision of alternative models of the short-term interest rate, Journal of Finance, 47,

3, 1209“1227.

Chernov, M. (2001) Implied volatilities as forecasts of future volatility, the market risk

premia, and returns variability, Working paper, Columbia Business School.

Chernov, M., and E. Ghysels (2000) A study towards a uni¬ed approach to the joint es-

timation of objective and risk neutral measures for the purposes of options valuation,

Journal of Financial Economics, 56, 3, 407“458.

204 References

Chiras, D., and S. Manaster (1978) The information content of option prices and a test

of market ef¬ciency, Journal of Financial Economics, 6, 213“234.

Chong, Y.Y., and D.F. Hendry, (1986) Econometric evaluation of linear macro-economic

models, Review of Economics Studies, 53, 671“690.

Christensen, B.J., and N.R. Prabhala (1998) The relation between implied and realized

volatility, Journal of Financial Economics, 50, 2, 125“150.

Christie, A.A. (1982) The stochastic behaviour of common stock variances: Value,

leverage, and interest rate effect, Journal of Financial Economics, 10, 407“

432.

Christodoulakis, G.A., and S.E. Satchell (1998) Hashing GARCH: A re-assessment

of volatility forecast and perfomance, Chapter 6, pp. 168“192, in: Knight, J., and

S. Satchell (eds), Forecasting Volatility in the Financial Markets, Butterworth.

Christoffersen, P.F., and F.X. Diebold (2000) How relevant is volatility forecasting for

risk management? Review of Economics and Statistics, 82, 1, 12“22.

Clark, P. (1973) A subordinated stochastic process model with ¬nite variance for spec-

ulative prices, Econometrica, 41, 135“156.

Corradi, V. (2000) Reconsidering the continuous time limit of the GARCH(1,1) process,

Journal of Econometrics, 96, 145“153.

Cox, J.C., S.A. Ross and M. Rubinstein (1979) Option pricing: A simpli¬ed approach,

Journal of Financial Economics, 7, 229“263.

Cumby, R., S. Figlewski and J. Hasbrouck (1993) Forecasting volatilities and correlations

with EGARCH models, Journal of Derivatives, 1, 51“63.

Danielsson, J. (1994) Stochastic volatility in asset prices: Estimation with simulated

maximum likelihood, Journal of Econometrics, 64, 375“400.

Danielsson, J., and C.G. de Vries (1997) Tail index and quantile estimation with very

high frequency data, Journal of Empirical Finance, 4, 241“257.

Das, S.R., and R.K. Sundaram (1999) Of smiles and smirks: A term structure perpective,

Journal of Financial and Quantitative Analysis, 34, 2.

Davidian, M., and R.J. Carroll (1987) Variance function estimation, Journal of American

Statistical Association, 82, 1079“1091.

Day, T.E., and C.M. Lewis (1992) Stock market volatility and the information content

of stock index options, Journal of Econometrics, 52, 267“287.

Day, T.E., and C.M. Lewis (1993) Forecasting futures market volatility, Journal of

Derivatives, 1, 33“50.

Diebold, F.X. (1988) Empirical Modeling of Exchange Rate Dynamics, Springer Verlag,

New York.

Diebold, F.X., A. Hickman, A. Inoue and T. Schuermann (1998) Scale models, RISK

Magazine, 11, 104“107.

Diebold, F.X., and A. Inoue (2001) Long memory and regime switching, Journal of

Econometrics, 105, 1, 131“159.

Diebold, F.X., and J.A. Lopez (1995) Modelling volatility dynamics, in: Hoover, K.

(ed.), Macroeconomics: Developments, Tensions and Prospects, Kluwer, Dordtecht,

pp. 427“466.

Diebold, F.X., and R.S. Mariano (1995) Comparing predictive accuracy, Journal of

Business and Economic Statistics, 13, 253“263.

Dimson, E., and P. Marsh (1990) Volatility forecasting without data-snooping, Journal

of Banking and Finance, 14, 2“3, 399“421.

Ding, Z., C.W.J. Granger and R.F. Engle (1993) A long memory property of stock market

returns and a new model, Journal of Empirical Finance, 1, 83“106.

Doidge, C., and J.Z. Wei (1998) Volatility forecasting and the ef¬ciency of the Toronto

35 index options market, Canadian Journal of Administrative Science, 15, 1, 28“38.

References 205

Drost, F.C., and T.E. Nijman (1993) Temporal aggregation of GARCH process, Econo-

metrica, 61, 4, 909“927.

Duan, J. (1997) Augmented GARCH(p,q) process and its diffusion limit, Journal of

Econometrics, 79, 97“127.

Duf¬e, D., and K.J. Singleton (1993) Simulated moments estimation of Markov models

of asset prices, Econometrica, 61, 929“952.

Dunis, C.L., J. Laws and S. Chauvin (2000) The use of market data and model combi-

nation to improve forecast accuracy, Working paper, Liverpool Business School.

Durbin, J., and S.J. Koopman (2000) Time series analysis of non-Gaussian observations

based on state space models from both classical and Bayesian perpectives, Journal

of Royal Statistical Society Series, 62, 1, 3“56.

Ederington, L.H., and W. Guan (1999) The information frown in option prices, Working

paper, University of Oklahoma.

Ederington, L.H., and W. Guan (2000a) Forecasting volatility, Working paper, University

of Oklahoma.

Ederington, L.H., and W. Guan (2000b) Measuring implied volatility: Is an average

better? Working paper, University of Oklahoma.

Ederington, L.H., and W. Guan (2002) Is implied volatility an informationally ef¬cient

and effective predictor of future volatility? Journal of Risk, 4, 3.

Ederington, L.H., and J.H. Lee (2001) Intraday volatility in interest rate and foreign

exchange markets: ARCH, announcement, and seasonality effects, Journal of Futures

Markets, 21, 6, 517“552.

Edey, M., and G. Elliot (1992) Some evidence on option prices as predictors of volatility,

Oxford Bulletin of Economics & Statistics, 54, 4, 567“578.

Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the

variance of United Kingdom in¬‚ation, Econometrica, 50, 4, 987“1007.

Engle, R.F. (1993) Statistical models for ¬nancial volatility, Financial Analysts Journal,

49, 1, 72“78.

Engle, R.F., and T. Bollerslev (1986) Modelling the persistence of conditional variances,

Econometric Reviews, 5, 1“50.

Engle, R., and K.F. Kroner (1995) Multivariate simultaneous generalized ARCH, Econo-

metric Theory, 11, 122“150.

Engle, R.F., and G.J. Lee (1999) A long-run and short-run component model of stock

return volatility, in: Engle, R.F., and H. White (ed.), Cointegration, Causality and

Forecasting, Oxford University Press, Oxford, Chapter 10, pp. 475“497.

Engle, R.F., and V.K. Ng (1993) Measuring and testing the impact of news on volatility,

Journal of Finance, 48, 1749“1778.

Engle, R. F., V. Ng and M. Rothschild (1990) Asset pricing with a factor-ARCH covari-

ance structure: Empirical estimates for Treasury Bills, Journal of Econometrics, 45,

213“239.

Fair, R.C., and R.J. Shiller (1989) The informational content of ex ante forecasts, Review

of Economics and Statistics, 71, 2, 325“332.

Fair, R.C., and R.J. Shiller (1990) Comparing information in forecasts from econometric

models, American Economic Review, 80, 3, 375“380.

Feinstein, S.P. (1989a) The Black“Scholes formula is nearly linear in sigma for at-the-

money options; therefore implied volatilities from at-the-money options are virtually

unbiased, Working paper, Federal Reserve Bank of Atlanta.

Feinstein, S.P. (1989b) Forecasting stock market volatility using options on index futures,

Economic Review (Federal Reserve Bank of Atlanta), 74, 3, 12“30.

Ferreira, M.A. (1999) Forecasting interest rate volatility from the information in histori-

cal data, Working paper, Department of Finance, University of Wisconsin-Madison.

206 References

Figlewski, S. (1997) Forecasting volatility, Financial Markets, Institutions and Instru-

ments (New York University Salomon Center), 6, 1, 1“88.

Figlewski, S., and T.C. Green (1999) Market risk and model risk for a ¬nancial institution

writing options, Journal of Finance, 54, 4, 1465“1999.

Fiorentini, G., A. Leon and G. Rubio (2002) Estimation and empirical performance of

Heston™s stochastic volatility model: The case of a thinly traded market, Journal of

Empirical Finance, 9, 225“255.

Fleming, J. (1998) The quality of market voltility forecasts implied by S&P 100 index

option prices, Journal of Empirical Finance, 5, 4, 317“345.

Fleming, J., and C. Kirby (2003) A closer look at the relation between GARCH and

stochastic autoregressive volatility, Journal of Financial Econometrics, 1, 365“

419.

Fleming, J., and R.E. Whaley (1994) The value of wildcard options, Journal of Finance,

49, 1, 215“236, March.

Fleming, J., C. Kirby and B. Ostdiek (2000) The economic value of volatility timing

Journal of Finance, 56, 1.

Fleming, J., C. Kirby and B. Ostdiek (2002) The economic value of volatility timing

using realized volatility, Journal of Financial Economics, 67, 473“509.

Fleming, J., B. Ostdiek and R.E. Whaley (1995) Predicting stock market volatility: A

new measure, Journal of Futures Market, 15, 3, 265“302.

Forbes, K.J., and R. Rigobon (2002) No contagion, only interdependence: measuring

stock market co-movements, Journal of Finance, 57, 5, 2223“2262.

Fouque, J.-P., G. Papanicolaou and K.R. Sircar (2000) Derivatives in Financial Markets

with Stochastic Volatility, Cambridge University Press, Cambridge.

Franke, G., R.C. Stapleton and M.G. Subrahmanyam (1998) Who buys and who sells op-

tions: The role of options in an economy with background risk, Journal of Economic

Theory, 82, 1, 89“109.

Franses, P.H., and H. Ghijsels (1999) Additive outliers, GARCH and forecasting volatil-

ity, International Journal of Forecasting, 15, 1“9.

Franses, P.H., and D. Van Dijk (1996) Forecasting stock market volatility using (non-

linear) Garch models, Journal of Forecasting, 15, 3, 229“235.

Franses, P.H. and D. van Dijk (2000) Non-Linear Time Series Models in Empirical

Finance, Cambridge University Press, Cambridge.

French, K.R., G.W. Schwert and R.F. Stambaugh (1987) Expected stock returns and

volatility, Journal of Financial Economics, 19, 1, 3“30.

Frennberg, P., and B. Hansson (1996) An evaluation of alternative models for predicting

stock volatility, Journal of International Financial Markets, Institutions and Money,

5, 117“134.

Fridman, M., and L. Harris (1998) A maximum likelihood approach for non-Gaussian

stochastic volatility models, Journal of Business and Economic Statistics, 16, 284“

291.

Friedman, B.M., and D.I. Laibson (1989) Economic implications of extraordinary move-

ments in stock prices, Brooking Papers on Economic Activity, 2, 137“189.

Fung, H.-G., C.-J. Lie and A. Moreno (1990) The forecasting performance of the implied

standard deviation in currency options, Managerial Finance, 16, 3, 24“29.

Fung, W.K.H., and D.A. Hsieh (1991) Empirical analysis of implied volatility: Stocks,

bonds and currencies, Working paper, Department of Finance, Fuqua School of

Business.

Gallant, A.R., P.E. Rossi and G. Tauchen (1993) Nonlinear dynamic structures, Econo-

metrica, 61, 4, 871“907.

References 207

Garman, M.B., and M.J. Klass (1980) On the estimation of security price volatilities

from historical data, Journal of Business, 53, 1, 67“78.

Gemmill, G. (1986) The forecasting performance of stock options on the London Traded

Options Markets, Journal of Business Finance and Accounting, 13, 4, 535“546.

Geske, R. (1979) The valuation of compound options, Journal of Financial Economics,

7, 63“81.

Ghysels, E., A. Harvey and E. Renault (1996) Stochastic volatility, pp. 119“191, in:

Maddala, G.S., and C.R. Rao (eds), Handbook of Statistics: Statistical Methods in

Finance, Vol. 14, Elsevier Science, Amsterdam.

Glosten, L.R., R. Jagannathan and D.E. Runkle (1993) On the relation between the

expected value and the volatility of the nominal excess return on stocks, Journal of

Finance, 48, 1779“1801.

Gourieroux, C. (1997) ARCH Models and Financial Applications, Springer, New York.

Granger, C.W.R. (1999) Empirical Modeling in Economics. Speci¬cation and Evalua-

tion. Cambridge University Press, Cambridge.

Granger, C.W.J. (2001) Long memory processes “ an economist™s viewpoint, Working

paper, University of California, San Diego.

Granger, C.W.J., and Z. Ding (1995) Some properties of absolute return: An alternative

measure of risk, Annales dEconomie et de Statistique, 40, 67“91.

Granger, C.W.J., Z. Ding and S. Spear (2000) Stylized facts on the temporal and dis-

tributional properties of absolute returns: An update, Working paper, University of

California, San Diego.

Granger, C.W.J., and N. Hyung (2004) Occasional structural breaks and long memory

with an application to the S&P500 absolute stock returns, Journal of Empirical

Finance, 11, 3, 399“421.

Granger, C.W.J., and R. Joyeux (1980) An introduction to long memory time series and

fractional differencing, Journal of Time Series Analysis, 1, 15“39.

Gray, S.F. (1996) Modeling the conditional distribution of interest rates as a regime-

switching process, Journal of Financial Economics, 42, 1, 27“62.

Guo, D. (1996a) The predictive power of implied stochastic variance from currency

options, Journal of Futures Markets, 16, 8, 915“942.

Guo, D. (1996b) The information content of implied stochastic volatility from currency

options, Canadian Journal of Economics, 29, S, 559“561.

Hamao, Y., R.W. Masulis and V. Ng (1989) Correlations in price changes and volatility

across international stock markets, Review of Financial Studies, 3, 281“307.

Hamid, S. (1998) Ef¬cient consolidation of implied volatilities and a test of intertemporal

averaging, Derivatives Quarterly, 4, 3, 35“49.

Hamilton, J.D. (1989) A new approach to the economic analysis of nonstationary time

series and the business cycle, Econometrica, 57, 357“384.

Hamilton, J.D., and G. Lin (1996) Stock market volatility and the business cycle, Journal

of Applied Econometrics, 11, 5, 573“593.

Hamilton, J.D., and R. Susmel (1994) Autoregressive conditional heteroskedasticity and

changes in regime, Journal of Econometrics, 64, 1“2, 307“333.

Hansen, L.P., and R.J. Hodrick (1980) Forward exchange rates as optimal predictors

of future spot rates: An econometric analysis, Journal of Political Economy, 88,

829“853.

Hansen, P.R., and A. Lunde (2004a) A forecast comparison of volatility models: Does

anything beat a GARCH(1,1), Journal of Applied Econometrics, Forthcoming.

Hansen, P.R., and A. Lunde (2004b) Consistent ranking of volatility models, Journal of

Econometrics, Forthcoming.

208 References

Harvey, A.C. (1998) Long memory in stochastic volatility, Chapter 12, pp. 307“320,

in: Knight, J., and S. Satchell (eds), Forecasting Volatility in the Financial Markets,

Butterworth, Oxford.

Harvey, A.C., E. Ruiz and N. Shephard (1994) Multivariate stochastic variance models,

Review of Economic Studies, 61, 247“264.

Harvey, C.R., and R.E. Whaley (1992) Market volatility prediction and the ef¬-

ciency of the S&P100 Index option market, Journal of Financial Economics, 31, 1,

43“74.

Heath, D., R. Jarrow, and A. Morton (1992) Bond pricing and the term structure of

interest rates: A new methodology for contingent claim valuation, Econometrica,

60, 77“105.

Hentschel, L. (2001) Errors in implied volatility estimation, working paper, University

of Rochester.

Heston, S.L (1993) A closed solution for options with stochastic volatility, with ap-

plication to bond and currency options, Review of Financial Studies, 6, 2, 327“

343.

Heynen, R.C. (1995) Essays on Derivatives Pricing Theory, Thesis Publishers, Amster-

dam.

Heynen, R.C., and H.M. Kat (1994) Volatility prediction: A comparison of stochastic

volatility, GARCH(1,1) and EGARCH(1,1) models, Journal of Derivatives, 2, 50“

65.

Hol, E., and S.J. Koopman (2002) Forecasting the variability of stock index returns with

stochastic volatility models and implied volatility, Working paper, Free University,

Amsterdam.

Hong, Y. (2001) A test for volatility spillover with application to exchange rates, Journal

of Econometrics, 103, 183“224.

Hosking, J.R.M. (1981) Fractional differencing, Biometrika, 68, 165“176.

Hsieh, D.A. (1989) Modeling heteroscedasticity in daily foreign exchange rates, Journal

of Business and Economic Statistics, 7, 3, 307“317.

Huber, P.J. (1981) Robust Statistics, John Wiley and Sons Canada Ltd, Ontario.

Hull, J. (2002) Options, Futures and Other Derivative Securities, 5th edn, Prentice Hall,

Englewood Cliffs, NJ.

Hull, J., and A. White (1987) The pricing of options on assets with stochastic volatilities,

Journal of Finance, 42, 2, 281“300.

Hull, J., and A. White (1988) An analysis of the bias in option pricing caused by a

stochastic volatility, Advances in Futures and Options Research, 3, 27“61.

Hwang, S., and S. Satchell (1998) Implied volatility forecasting: A comparison of dif-

ferent procedures including fractionally integrated models with applications to UK

equity options, Chapter 7, pp. 193“225, in: Knight, J., and S. Satchell (eds), Fore-

casting Volatility in the Financial Markets, Butterworth, Oxford.

Jacquier, E., N.G. Polson and P.E. Rossi (1994) Bayesian analysis of stochastic

volatility models: reply, Journal of Business and Economic Statistics, 12, 4, 413“

417.

Jarrow, R. (ed) (1998) Volatility: New Estimation Techniques for Pricing Derivatives,

Risk Books, London.

Johnson, H., and D. Shanno (1987) Option pricing when the variance is changing,

Journal of Financial and Quantitative Analysis, 22, 143“151.

Jones, C.S. (2001) The dynamics of stochastic volatility: Evidence from underlying

and options markets, Working paper, Simon School of Business, University of

Rochester.

References 209

Jones, C., O. Lamont and R. Lumsdaine (1998) Macroeconomic news and bond market

volatility, Journal of Financial Economics, 47, 315“337.

Jorion, P. (1995) Predicting volatility in the foreign exchange market, Journal of Finance,

50, 2, 507“528.

Jorion, P. (1996) Risk and turnover in the foreign exchange market, in: Frankel, J.A.,

G. Galli and A. Giovannini (eds), The Microstructure of Foreign Exchange Markets,

The University of Chicago Press, Chicago.

Jorion, P. (2001) Value at Risk: The New Benchmark for Managing Financial Risk, 2nd

edn, McGraw-Hill, New York.

Karatzas, I., and S.E. Shreve (1988) Brownian Motion and Stochastic Calculus, Springer

Verlag, New York.

Karolyi, G.A. (1993) A Bayesian approach to modeling stock return volatility and option

valuation, Journal of Financial and Quantitative Analysis, 28, 4, 579“595.

Karolyi, G.A. (1995) A multivariate GARCH model of international transimissions of

stock returns and volatility: The case of the United States and Canada, Journal of

Business and Economic Statistics, 13, 1, 11“25.

Karpoff, J.M. (1987) The relation between price changes and trading volume: A survey,

Journal of Financial and Quantitative Analysis, 22, 1, 109“126.

Kearns, P., and A.R. Pagan (1993) Australian stock market volatility, 1875“1987, Eco-

nomic Record, 69, 163“178.

Kim, S., N. Shephard and S. Chib (1998) Stochastic volatility: likelihood inference and

comparison with ARCH models, Review of Economic Studies, 65, 361“393.

King, M.A., and S. Wadhwani (1990) Transmission of volatility between stock markets,

Review of Financial Studies, 3, 1, 5“33.

Klaassen, F. (1998) Improving GARCH volatility forecasts, Empirical Economics, 27,

363“394.

Knight, J., and S. Satchell (ed) (2002) Forecasting Volatility in the Financial Markets,

2nd edn, Butterworth, Oxford.

Koutmos, G., and G.G. Booth (1995) Asymmetric volatility transmission in international

stock markets, Journal of International Money and Finance, 14, 6, 747“762.

Kroner, K., Kneafsey K. and S. Claessens (1995) Forecasting volatility in commodity

markets, Journal of Forecasting, 14, 77“95.

Kroner, K.F., and V.K. Ng (1998) Modeling asymmetric co-movements of asset returns,

Review of Financial Studies, 11, 4, 817“844.

Lamoureux, C.G., and W.D. Lastrapes (1990) Persistence in variance, structural change

and the GARCH model, Journal of Business and Economic Statistics, 8, 2, 225“

234.

Lamoureux, C., and W. Lastrapes (1993) Forecasting stock-return variance: toward an

understanding of stochastic implied volatilities, Review of Financial Studies, 6, 2,

293“326.

Latane, H., and R.J. Rendleman (1976) Standard deviations of stock price ratios implied

in option prices, Journal of Finance, 31, 2, 369“381.

Lee, K.Y. (1991) Are the GARCH models best in out-of-sample performance? Eco-

nomics Letters, 37, 3, 305“308.

Li, K. (2002) Long-memory versus option-implied volatility prediction, Journal of

Derivatives, 9, 3, 9“25.

Liesenfeld, R., and J.-F. Richard (2003) Univariate and multivariate stochastic volatility

models: estimation and diagnostics, Journal of Empirical Finance, 10, 4, 505“531.

Lin, Y., N. Strong and X. Xu (2001) Pricing FTSE-100 index options under stochastic

volatility, Journal of Futures Markets, 21, 3, 197“211.

210 References

Lopez, J.A. (1998) Methods for evaluating value-at-risk estimates, Economic Policy

Review (Federal Reserve Bank of New York), 4, 3, 119“129.

Lopez, J.A. (2001) Evaluating the predictive accuracy of volatility models, Journal of

Forecasting, 20, 2, 87“109.

Loudon, G.F., W.H. Watt and P.K. Yadav (2000) An empirical analysis of alternative

parametric ARCH models, Journal of Applied Econometrics, 15, 117“136.

Martens, M., and S.-H. Poon (2001) Returns synchronization and daily correlation dy-

namics between international stock markets, Journal of Banking and Finance, 25,

10, 1805“1827.

Martens, M., and J. Zein (2004) Predicting ¬nancial volatility: High-frequency time-

series forecasts vis-` -vis implied volatility, Journal of Futures Markets, 24, 11, 1005“

a

1028.

Mayhew, S. (1995) Implied volatility, Financial Analyst Journal, 51, 8“20.

McCurdy, T.H., and I. Morgan (1987) Tests of the martingale hypothesis for foreign

currency futures with time varying volatility, International Journal of Forecasting,

3, 131“148.

McKenzie M.D. (1999) Power transformation and forecasting the magnitude of exchange

rate changes, International Journal of Forecasting, 15, 49“55.

McMillan, D.G., A.H. Speight and O.A.P. Gwilym (2000) Forecasting UK stock market

volatility, Journal of Applied Economics, 10, 435“448.

McNeil, A.J., and R. Frey (2000) Estimation of tailed-related risk measures for het-

eroscedastic ¬nancial time series: An extreme value approach, Journal of Empirical

Finance, 7, 271“300.

Merton, R.C. (1980) On estimating expected return on the market: An exploratory

investigation, Journal of Financial Economics, 8, 323“361.

Milhoj, A. (1987) A conditional variance model for daily observations of an exchange

rate, Journal of Business and Economic Statistics, 5, 99“103.

Nandi, S. (1998) How important is the correlation between returns and volatility in a

stochastic volatility model? Empirical evidence from pricing and hedging S&P500

index option market, Journal of Banking and Finance, 22, 5, 589“610.

Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: A new approach,

Econometrica, 59, 2, 347“370.

Nelson, D.B. (1992) Filtering and forecasting with misspeci¬ed ARCH models I: Getting

the right variance with the wrong model, Journal of Econometrics, 52, 61“90.

Nelson, D.B., and C.Q. Cao (1992) Inequality constraints in the univariate GARCH

model, Journal of Business and Economic Statistics, 10, 2, 229“235.

Nelson, D.B., and D.P. Foster (1995) Filtering and forecasting with misspeci¬ed ARCH

models II: Making the right forecast with the wrong model, Journal of Econometrics,

67, 2, 303“335.

Noh, J. R.F., Engle and A. Kane (1994) Forecasting volatility and option prices of the

S&P 500 index, Journal of Derivatives, 2, 17“30.

Ohanissian, A., J.R. Russell and R.S. Tsay (2003) True or spurious long memory in

volatility: Does it matter for pricing options? Working Paper, University of Chicago.

Pagan, A.R., and G.W. Schwert (1990) Alternative models for conditional models for

conditional stock volatility, Journal of Econometrics, 45, 1“2, 267“290.

Parkinson, M. (1980) The extreme value method for estimating the variance of the rate

of return, Journal of Business, 53, 61“65.

Pitt, M.J., and N. Shephard (1997) Likelihood analysis of non-Gaussian measurement

time series, Biometrika, 84, 653“667.

Peria, M.S.M. (2001) A regime-switching approach to the study of speculative attacks:

A focus on EMS crises. Working paper, World Bank.

References 211

Pong, S., M.B. Shackleton, S.J. Taylor and X. Xu (2004) Forecasting Sterling/Dollar

volatility: A comparison of implied volatilities and AR(FI)MA models, Journal of

Banking and Finance, 28, 2541“2563.

Poon, S.-H., and C.W.J. Granger (2003) Forecasting ¬nancial market volatility: A review,

Journal of Economic Literature, 41, 2, 478“539.

Poon, S.-H., and C.W.J. Granger (2005) Practical issues in forecasting volatility, Finan-

cial Analyst Journal, 61, 1, 45“65.

Poon, S.-H., M. Rockinger and J. Tawn (2003) Extreme value dependence in international

stock markets and ¬nancial applications, Statistica Sinica, 13, 929“953.

Poon, S.-H., M. Rockinger and J. Tawn (2004) Extreme-value dependence in ¬nancial

markets: Diagnostics, models and ¬nancial implications, Review of Financial Studies,

17, 2, 581“610.

Poon, S., and S.J. Taylor (1992) Stock returns and stock market volatilities, Journal of

Banking and Finance, 16, 37“59.

Poteshman, A.M. (2000) Forecasting future volatility from option prices, Working paper,

University of Illinois at Urbana-Champaign.

Randolph, W.L., and M. Najand (1991) A test of two models in forecasting stock index

futures price volatility, Journal of Futures Markets, 11, 2, 179“190.

Robinson, P.M. (ed.) (2003) Time Series with Long Memory. Oxford University Press,

Oxford.

Rogers, L.C.G., and S.E. Satchell (1991) Estimating variance from high, low and closing

prices, Annals of Applied Probability, 1, 504“512.

Rogers, L.C.G., S.E. Satchell and Y. Yoon (1994) Estimating the volatility of stock

prices: A comparison of methods that use high and low prices, Applied Financial

Economics, 4, 3, 241“248.

Roll, R. (1977) An analytic valuation formula for unprotected American call options on

stocks with known dividends, Journal of Financial Economics, 5, 251“258.

Rossi, P. (ed.) (1996) Modelling Stock Market Volatility: Bridging the Gap to Continuous

Time, Academic Press, London.

Schmalensee, R., and R.R. Trippi (1978) Common stock volatility expectations implied

by option premia, Journal of Finance, 33, 1, 129“147.

Scott, E., and A.L. Tucker (1989) Predicting currency return volatility, Journal of Bank-

ing and Finance, 13, 6, 839“851.

Scott, L.O. (1987) Option pricing when the variance changes randomly: Theory, es-

timation and an application, Journal of Financial and Quantitative Analysis, 22,

419“438.

Sentana, E. (1998) The relation between conditionally heteroskedastic factor models