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If Black“Scholes (BS) is the correct option pricing model, then there
can only be one BS implied volatility regardless of the strike price of
the option, or whether the option is a call or a put. BS implied volatility
smile and skew are clear evidence that market option prices are not priced
according to the BS formula. This raises the important question about
the relationship between BS implied volatility and the true volatility.
The BS option price is a positive function of the volatility of the
underlying asset. If the BS model is correct, then market option price
should be the same as the BS option price and the BS implied volatility
derived from market option price will be the same as the true volatility.
If the BS price is incorrect and is lower than the market price, then BS
implied volatility overstates the true volatility. The reverse is true if the
BS price is higher than the market price. The problem is complicated
by the fact that BS implied volatility differs across strike prices. All the
theories that predict the relationship between BS price and the market
option price are all contingent on the proposed alternative option pricing
model or the proposed alternative pricing dynamic being correct. Given
that the BS implied volatility, despite all its shortcomings, has been
proven overwhelmingly to be the best forecast of volatility, it will be
useful to understand the links between BS implied volatility bias and
the true volatility. This is the objective of this chapter.
There have been a lot of efforts made to solve the BS anomalies.
The stochastic volatility (SV) option pricing model is one of the most
important extensions of Black“Scholes. The SV option pricing model
is motivated by the widespread evidence that volatility is stochastic and
that the distribution of risky asset returns has tail(s) longer than that of
a normal distribution. An SV model with correlated price and volatility
innovations can address both anomalies. The SV option pricing model
was developed roughly over a decade with contributions from Johnson
and Shanno (1987), Wiggins (1987), Hull and White (1987, 1988), Scott
(1987), Stein and Stein (1991) and Heston (1993). It was in Heston
(1993) that a closed form solution was derived using the characteristic
98 Forecasting Financial Market Volatility


function of the price distribution. Section 9.1 presents this landmark
Heston SV option pricing model while some details of the derivation are
presented in the Appendix to this chapter (Section 9.5). In Section 9.2,
we simulate a series of Heston option prices from a range of parameters.
Then we use these option prices as if they were the market option prices
to back out the corresponding BS implied volatilities. If market option
prices are priced according to the Heston formula, the simulations in
this section will give us some insight into the relationship between BS
implied volatility bias and the true volatility. In Section 9.3, we analyse
the usefulness and practicality of the Heston model by looking at the
impact of Heston model parameters on skewness and kurtosis range and
sensitivity, and some empirical tests of Heston model. Finally, Section
9.4 analyses empirical ¬ndings on the the predictive power of Heston
implied volatility as a volatility forecast.


9.1 THE HESTON STOCHASTIC VOLATILITY
OPTION PRICING MODEL
Heston (1993) speci¬es the stock price and volatility price processes as
follows:

d St = µSdt + …t Sdz s,t ,

d…t = κ [θ ’ …t ] dt + σν …t dz …,t ,

where …t is the instantaneous variance, κ is the speed of mean reversion,
θ is the long-run level of volatility and σν is the ˜volatility of volatility™.
The two Wiener processes, dz s,t and dz …,t have constant correlation ρ.
The assumption that consumption growth has a constant correlation with
spot-asset returns generates a risk premium proportional to …t . Given the
volatility risk premium, the risk-neutral volatility process can be written
as
√ *
d…t = κ [θ ’ …t ] dt ’ »…t dt + σν …t dz …,t

= κ * θ * ’ … dt + σ … dz * , ν …,t
t t

where » is the market price of (volatility) risk, and κ * = κ + » and
θ * = κθ /(κ + »). Here κ * is the risk-neutral mean reverting parameter
and θ * is the risk-neutral long-run level of volatility. The parameter σν
and ρ implicit in the risk-neutral process are the same as that in the
real volatility process. Given the price and the volatility dynamics, the
Option Pricing with Stochastic Volatility 99


Heston (1993) formula for pricing European calls is
c = S P1 ’ K e’r (T ’t) P2 ,
1∞ e’iφ ln K f i
1
Pj = + dφ, for j = 1, 2
Re
2 π0 iφ
f i = exp {C (T ’ t; φ) + D (T ’ t; φ) … + iφx} ,
where
x = ln S, „ = T ’ t,
1 ’ ged„
a
C („ ; φ) = r φi„ + 2 b j ’ ρσν φi + d „ ’ 2 ln ,
σν 1’g
b j ’ ρσν φi + d 1 ’ ed„
D („ ; φ) = ,
σν 1 ’ ged„
2

b j ’ ρσν φi + d
g= ,
b j ’ ρσν φi ’ d
2
d= ρσν φi ’ b j
’ σν 2µφi ’ φ 2 ,
2

µ1 = 1 2 , µ2 = ’1 2 , a = κθ = κ * θ * ,
b1 = κ + » ’ ρσν = κ * ’ ρσν , b2 = κ + » = κ * .


9.2 HESTON PRICE AND BLACK“SCHOLES IMPLIED
In this section, we analyse possible BS implied bias by simulating a
series of Heston option prices with parameter values similar to those in
Bakshi, Cao and Chen (1997), Nandi (1998), Das and Sundaram (1999),
Bates (2000), Lin, Strong and Xu (2001), Fiorentini, Angel and Rubio
(2002) and Andersen, Benzoni and Lund (2002). For the simulations, we
set the asset price as 100, interest rate as zero, time to maturity is 1 year,
and strike prices ranging from 50 to 150. In most simulations, and unless
otherwise stated, the current ˜instantaneous™ volatility, σt , is set equal to
the long-run level, θ, at 20%. There are ¬ve other parameters used in the
Heston formula, namely, κ, the speed of mean reversion, θ, the long-run
volatility level, », the market price of risk, σ… , volatility of volatility, and
ρ, the correlation between the price and the volatility processes. If we
set » = 0, then the volatility process becomes risk-neutral, and κ and θ
become κ * and θ * respectively.
The ¬rst set of simulations presented in Figure 9.1(a) involves repli-
cating the Black“Scholes prices as a special case. Here we set σ… = 0.
100 Forecasting Financial Market Volatility


(a) A Black“Scholes series (b) Effect of volatility of volatility (su, Kurtosis)
(S = 100, r = 0, T = 1, l = 0) (S = 100, r = 0, T = 1, l = 0, k = 0.1)

Skewness = 0, Kurtosis = 3
su = 0.1 , Kur = 3.7
0.45 su = 0.3, Kur = 9.53
0.45
0.4 0.4 su = 0.6, Kur = 29.1
0.35 0.35
su = 1.0, Kur = 75.6
0.3 0.3
BS Implied




BS Implied
0.25 0.25
0.2 0.2
0.15 0.15
0.1 0.1
0.05 0.05
0 0
50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150
Strike Price, K Strike Price, K

(d) Effect of k
(c) Effect of correlation, r
(S = 100, r = 0, T = 1, l = 0, su = 0.6, q = 0.2)
(S = 100, r = 0, T = 1, l = 0)
k = 0.01, √…t = 0.7 k = 3, √…t = 0.7
r = -0.95 r = -0.5
k = 0.01, √…t = 0.15 k = 3, √…t = 0.15
r=0 r = 0.6
0.4
0.8
0.35
0.7
0.3
0.6
BS Implied




0.25
BS Implied




0.5
0.2
0.4
0.15
0.3
0.1 0.2
0.05 0.1
0.1049
0 0
50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150
Strike Price, K
Strike Price, K


(e) Effect of l on negatively correlated (f) Effect of l on positively correlated
processes (S = 100, r = 0, T = 1, r = -0.5) processes (S = 100, r = 0, T = 1, r = 0.5)
Skewness = -0.1623, Kurtosis = 3.7494
Skewness = +0.1623, Kurtosis = 3.7494

l = -2 l=0 l=2 l = -2 l=0 l=2

0.45 0.45
0.4 0.4
0.35 0.35
0.3 0.3
BS Implied




BS Implied




0.25 0.25
0.2 0.2
0.15 0.15
0.1 0.1
0.05 0.05
0 0
50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150
Strike Price, K Strike Price, K


Figure 9.1 Relationships between Heston option prices and Black“Scholes implied
volatility


Since there is no volatility risk, » = 0. This is a special case where
the Heston price and the Black“Scholes price are identical and the BS
implied volatility is the same across strike prices. In this special case,
BS implied volatility (at any strike price) is a perfect representation of
true volatility.
Option Pricing with Stochastic Volatility 101


In the second set of simulations presented in Figure 9.1(b), we al-
ter σ… , the volatility of volatility, and keep all the other parameters the
same and constant. The effect of an increase in σ… is to increase the
unconditional volatility and kurtosis of risk-neutral price distribution. It
is the risk-neutral distribution because », the market price of risk, is set
equal to zero. As σ… increases and without appropriate compensation for
volatility risk premium, ATM (at-the-money) implied volatility under-
estimates true volatility while OTM (out-of-the-money) implied volatil-
ity overestimates it. This is the same outcome as Hull and White (1987)
where the price and the volatility processes are not correlated and there
is no risk premium for volatility risk. With appropriate adjustment for
volatility (which will require a volatility risk premium input), the ATM
implied volatility will be at the right level, but Black“Scholes will con-
tinue to underprice OTM options (and OTM BS implied overestimates
true volatility) because of the BS lognormal thin tail assumptions. As-
suming that ρ = 0 or at least is constant over time, and that σ… and » are
relatively stable, a time series regression of historical ˜actual volatility™
on historical ˜implied volatility™ at a particular strike will be suf¬cient
to correct for these biases. This is basically the Ederington and Guan
(1999) approach. We will show in the next section that ρ is not likely to
be stable. When ρ is not constant, the analysis below and Figure 9.1(c)
show that ATM implied volatility is least affected by changing ρ. This
explains why ATM implied volatility is the most robust and popular
choice of volatility forecast.
In Figure 9.1(c), it is clear that changing the correlation coef¬cient
alone has no impact on ATM implied volatility. Correlation has the
greatest impact on skewness of the price distribution and determines the
shape of volatility smile or skew. Its impact on kurtosis is less marked
when compared with σ… , the volatility of volatility.
Figure 9.1(d) highlights the impact of κ, the mean reversion parameter
which we have already brie¬‚y touched on in relation to the long memory
of volatility in Chapter 5. The higher the rate of mean reversion, the more
likely the return distribution will be normal even when the volatility of

volatility, σ… , and the initial volatility, νt , are both high. When this is
the case, there is no strike price bias in BS implied (i.e. there will not be
volatility smile). When κ is low, this is when the problem starts. A low κ
corresponds with volatility persistence where BS implied volatility will
be sensitive to the current state of volatility level. At high volatility state,

high νt compensates for the low κ and the strike price bias is less severe.
Strike price effect or the volatility smile is the most acute when initial

volatility level νt is low. ATM options will be overpriced vis-` -vis a
102 Forecasting Financial Market Volatility



OTM options.1 (Note that we have set » = 0 in this set of simulations.)

When σ… = 0.6, θ = 0.2, νt = 0.15 and κ = 0.01, the ATM BS im-
plied is only 0.1049 much lower than any of the volatility parameters.
Figure 9.1(e) and 9.1(f) can be used to infer the impacts of parameter
estimates above when the volatility risk premium » is omitted. In the
literature, we often read ˜. . . volatility risk premium is negative re¬‚ect-
ing the negative correlation between the price and the volatility dyna-
mics . . . ™ (Buraschi and Jackwerth, 2001; Bakshi and Kapadia, 2003.
All series in Figure 9.1(e) have correlation ρ = ’0.5 and all series in
Figure 9.1(f) have correlation ρ = +0.5. A negative » (volatility risk
premium) produces higher Heston price and higher BS implied volatil-
ity. The impact is the same whether the correlation ρ is negative or
positive. We will see later in the next section that empirical evidence in-
dicates that Figure 9.1(f) is just as likely a scenario as Figure 9.1(e). As
κ * = κ + » and θ * = κθ /(κ + »), a negative » has the effect of reduc-
ing κ * (resulting in a smaller option price) and increasing θ * (resulting
in a bigger option price). Simulations, not reported here, show that the
price impact of θ * is much greater than that of κ * , so the outcome will
be a higher option price due to the negative ». Hence, a ˜negative risk
premium™ is to be expected whether the price and the volatility processes
are positively or negatively correlated.2 This also means that, without
accounting for the volatility risk premium, the BS option price will be
too low and the BS implied will always overstate true volatility. Both
volatility and volatility risk premium have positive impact on option
price. The omission of volatility risk premium will cause the volatil-
ity risk premium component to be ˜translated™ into higher BS implied
volatility.


9.3 MODEL ASSESSMENT
In this section, we evaluate the Heston model using simulations. In par-
ticular, we examine the skewness and kurtosis planes covered by a range
of Heston parameter values. We have no information on the volatility
risk premium. Hence, to avoid an additional dimension of complexity,
we will evaluate the risk-neutral parameters κ * and θ * instead of κ and
θ for the true volatility process.

1
When BS overprice options, the BS implied volatility will understate volatility because BS implied is
inverted from market price, which is lower than the BS price.
This is really a misnomer: while the » parameter is negative, it actually results in a higher option price. So
2

strictly speaking the volatility risk premium is positive!
Option Pricing with Stochastic Volatility 103


9.3.1 Zero correlation
We learn from the simulations in Section 9.2 and from Figure 9.1 that,
according to the Heston model, skewness in stock returns distribution
and BS implied volatility asymmetry are determined completely by the
correlation parameter, ρ. When the correlation parameter is equal to
zero, we get zero skewness and both the returns distribution and BS
implied volatility will be symmetrical. Figure 9.2 presents the kurtosis
values produced by different combinations of κ, θ and σv . One important
pattern emerged that highlights the importance of the mean reversion
parameter, κ. When κ is low we have high volatility persistence, and
vice versa for high value of κ.
At high value of κ, kurtosis is close to 3, regardless of the value of θ and
σv . This is, unfortunately, the less likely scenario for a ¬nancial market
time series that typically has high volatility persistence and low value
of κ. At low value of κ, the kurtosis is the highest at low level of σv , the
parameter for volatility of volatility. At high level of σv , kurtosis drops
to 3 very consistently, regardless of the value of the other parameters.
At low level of σv , the long-term level of volatility, θ, comes into effect.
The higher the value of θ, the lower the kurtosis value, even though it is
still much greater than 3.
When skewness is zero and kurtosis is low (i.e. relatively ¬‚at BS im-
plied volatility), it will be dif¬cult to differentiate whether it is due to
a high κ, a high σv or both. This also re¬‚ects the underlying prop-
erty that a high κ, a high σv or both make the stochastic volatility
structure less important and the BS model will be adequate in this
case.


9.3.2 Nonzero correlation
In Figures 9.3 and 9.4, we illustrate skewness and kurtosis, respectively,
for the case when the correlation coef¬cient, ρ, is greater than 0. The
case for ρ < 0 will not be discussed here as it is the re¬‚ective image
of ρ > 0 (e.g. instead of positive skewness, we get negative skewness
etc.).
Figure 9.3 shows that skewness is ¬rst ˜triggered™ by a nonzero cor-
relation coef¬cient, after which κ and σv combine to drive skewness.
High skewness occurs when σv is high and κ is low (i.e. high volatility
persistence). At relatively low skewness level, there is a huge range of
high κ, low σv or both that produce similar values of skewness. A low θ
104 Forecasting Financial Market Volatility


(a) k = 1, r = 0, Skewness = 0
80

70

60

50

40
Kurtosis 30

20

10

0
0.05 0.1 0.5
0.15 0.4
0.2 0.3
0.25 0.2
0.3 0.1
q su


(b) k = 0.1, r = 0, Skewness = 0
80.00

70.00

60.00

50.00

40.00
Kurtosis
30.00

20.00

10.00

0.00
0.1 0.1
0.2
0.2 0.3
0.3 0.4
0.5
0.4
su q
0.5

Figure 9.2 Impact of Heston parameters on kurtosis for symmetrical distribution with
zero correlation and zero skewness

and high ρ produces high skewness, but at low level of σv , skewness is
much less sensitive to these two parameters.
Figure 9.4 gives a similar pattern for kurtosis. Except when σv is very
high and κ is low, the plane for kurtosis is very ¬‚at and not sensitive to
θ or ρ.
Option Pricing with Stochastic Volatility 105


k = 1, su = 0.5 k = 0.1, su = 0.5
3.5 3.5
3 3
2.5 2.5
2 2
Skewness Skewness
1.5 1.5
1 1
0.5 0.5
0 0
0.9




0.9
0.7
0.05




0.7
0.05
0.5




0.5
0.3
0.2




0.3
0.2
0.1




0.1
q r q r


k = 0.1, su = 0.1
k = 1, su = 0.1
3.5
3.5
3
3
2.5
2.5
2
2
Skewness
Skewness
1.5
1.5
1
1
0.5
0.5
0
0
0.9
0.9




0.7
0.7




0.05
0.05




0.5
0.5




0.3
0.3




0.2
0.2

0.1




0.1




q r q r


Figure 9.3 Impact of Heston parameters on skewness


9.4 VOLATILITY FORECAST USING THE
HESTON MODEL
The thick tail and nonsymmetrical distribution found empirically could
be a result of volatility being stochastic. The simulation results in the
previous section suggest that σv , the volatility of volatility, is the main
driving force for kurtosis and skewness (if correlation is not equal to
zero). At high κ, volatility mean reversion will cancel out much of
the σv impact on kurtosis and some of that on skewness. Correlation
between the price and the volatility processes, ρ, determines the sign of
the skewness. But beyond that its impact on the magnitude of skewness
is much less compared with σv and κ. Correlation has negligible impact
on kurtosis. The long-run volatility level, θ, has very little impact on
skewness and kurtosis, except when σv is very high and κ is very low. So
a stochastic volatility pricing model is useful and will outperform Black“
Scholes only when volatility is truly stochastic (i.e. high σv ) and volatility
is persistent (i.e. low κ). The dif¬culty with the Heston model is that, once
106 Forecasting Financial Market Volatility


k = 1, su = 0.5 k = 0.1, su = 0.5
90 90
80 80
70 70
60 60
50 50
Kurtosis Kurtosis
40 40
30 30
20 20
10 10
0 0
0.9




0.9
0.7
0.05




0.7
0.05
0.5




0.5
0.3
0.2




0.3
0.2
0.1




0.1
q r q r


k = 0.1, su = 0.1
k = 1, su = 0.1
90
90
80
80
70
70
60
60
50
50
Kurtosis
Kurtosis
40
40
30
30
20
20
10
10
0
0
0.9
0.9




0.7
0.7




0.05
0.05




0.5
0.5




0.3
0.3




0.2
0.2

0.1




0.1




q r q r


Figure 9.4 Impact of Heston parameters on kurtosis



we move away from the high σv and low κ region, a large combination
of parameter values can produce similar skewness and kurtosis. This
contributes to model parameter instability and convergence dif¬culty
during estimation.
Through simulation results we can predict the degree of Black“
Scholes pricing bias as a result of stochastic volatility. In the case where
volatility is stochastic and ρ = 0, Black“Scholes overprices near-the-
money (NTM) or at-the-money (ATM) options and the degree of over-
pricing increases with maturity. On the other hand, Black“Scholes un-
derprices both in- and out-of-the-money options. In term of implied
volatility, ATM implied volatility will be lower than actual volatility
while implied volatility of far-from-the-money options (i.e. either very
high or very low strikes) will be higher than actual volatility. The pattern
of pricing bias will be much harder to predict if ρ is not zero, when there
is a premium for bearing volatility risk, and if either or both values vary
through time.
Option Pricing with Stochastic Volatility 107


Some of the early work on option implied volatility focuses on ¬nd-
ing an optimal weighting scheme to aggregate implied volatility of op-
tions across strikes. (See Bates (1996) for a comprehensive survey of
these weighting schemes.) Since the plot of implied volatility against
strikes can take many shapes, it is not likely that a single weighting
scheme will remove all pricing errors consistently. For this reason and
together with the liquidity argument, ATM option implied volatility is
often used for volatility forecast but not implied volatilities at other
strikes.

9.5 APPENDIX: THE MARKET PRICE OF
VOLATILITY RISK
9.5.1 Ito™s lemma for two stochastic variables
Given two stochastic processes,3
d S1 = µ1 (S1 , S2 , t) dt + σ1 (S1 , S2 , t) d X 1 ,
d S2 = µ2 (S1 , S2 , t) dt + σ2 (S1 , S2 , t) d X 2 ,
E {d X 1 d X 2 } = ρdt,
where X 1 and X 2 are two related Brownian motions.
From Ito™s lemma, the derivative function V (S1 , S2 , t) will have the
following process:
12 12
d V = Vt + σ1 VS1 S1 + ρσ1 σ2 VS1 S2 + σ2 VS2 S2 dt
2 2
+VS1 d S1 + VS2 d S2 ,
where
‚V ‚2V ‚ ‚V
Vt = , VS1 S1 = VS1 S2 = .
and
‚t ‚ S1 ‚ S2
‚ S1
2



9.5.2 The case of stochastic volatility
Here, we assume S1 is the underlying asset and S2 is the stochastic
volatility σ1 as follows:
d S1 = µ1 (S1 , σ, t) dt + σ1 (S1 , σ, t) d X 1 , (9.1)
dσ1 = p (S1 , σ, t) dt + q (S1 , σ, t) d X 2 ,
E {d X 1 d X 2 } = ρdt,
3
I am grateful to Konstantinos Vonatsos for helping me with materials presented in this section.
108 Forecasting Financial Market Volatility


and from Ito™s lemma, we get:
12 1
d V = Vt + σ1 VS1 S1 + ρqσ1 VS1 σ + q 2 VS2 σ dt (9.2)
2 2
+VS1 d S1 + Vσ dσ.
In the following stochastic volatility derivation, µ1 = µSt is the mean
drift of the stock price process. The volatility of S1 , σ1 = f (σ ) S1 is
stochastic and is level-dependent. The mean drift of the volatility pro-
cess is more complex as volatility cannot become negative and should
be stationary in the long run. Hence an OU (Ornstein“Uhlenbeck)
process is usually recommended with p (S1 , σ, t) = ± (m ’ σ ) and
q (S1 , σ, t) = β:
dσ1 = ± (m ’ σ1 ) dt + βd X 2 .
Here, p = ± (m ’ σ ) is the mean drift of the volatility process, m is
the long-term mean level of the volatility, and ± is the speed at which
volatility reverts to m, and β is the volatility of volatility.


9.5.3 Constructing the risk-free strategy
To value an option V (S1 , σ, t) we must form a risk-free portfolio using
the underlying asset to hedge the movement in S1 and use another option
V (S1 , σ, t) to hedge the movement in σ . Let the risk-free portfolio be:
=V’ ’ S1 .
1V

Applying Ito™s lemma from (9.2) on the risk-free portfolio ,
12 1
= Vt + σ1 VS1 S1 + q 2 Vσ σ + ρqσ1 VS1 σ dt
d
2 2
12 1
’ 1 V t + σ1 V S1 S1 + q 2 V σ σ + ρqσ1 V S1 σ dt
2 2
+ VS1 ’ ’
1 V S1 d S1
+ Vσ ’ 1V σ dσ.
To eliminate dσ , set
Vσ ’ = 0,
1V σ

= ,
1

Option Pricing with Stochastic Volatility 109


and, to eliminate d S1 , set

VS1 ’ V S1 ’ = 0,


= VS1 ’ V S1 .

This results in:
12 1
= Vt + σ1 VS1 S1 + q 2 Vσ σ + ρqσ1 VS1 σ dt
d
2 2
Vσ 1 1
’ V t + σ1 2 V S1 S1 + q 2 V σ σ + ρqσ1 V S1 σ dt
2 2

= r dt
Vσ Vσ
=r V ’ V ’ VS1 ’ V S1 S1 dt.
Vσ Vσ
Dividing both sides by Vσ , we get:
1 12 1
Vt + σ1 VS1 S1 + q 2 Vσ σ + ρqσ1 VS1 σ
Vσ 2 2
1 12 1
’ V t + σ1 V S1 S1 + q 2 V σ σ + ρqσ1 V S1 σ
2 2

V V VS S1 V S S1
=r ’r ’r 1 +r 1 .
Vσ Vσ
Vσ Vσ
Now we separate the two options by moving V to one side and V to the
other:
1 12 1
Vt + σ1 VS1 S1 + q 2 Vσ σ + ρqσ1 VS1 σ ’ r V + r VS1 S1
Vσ 2 2
1 12 1
= V t + σ1 V S1 S1 + q 2 V σ σ + ρqσ1 V S1 σ ’ r V + r V S1 S1 .
2 2

Each side of the equation is a function of S1 , σ and t, and is independent
of the other option. So we may write:
1 12 1
Vt + σ1 VS1 S1 + q 2 Vσ σ + ρqσ1 VS1 σ ’ r V + r VS1 S1
Vσ 2 2
= f (S1 , σ, t) ,
12 1
Vt + σ1 VS1 S1 + q 2 Vσ σ + ρqσ1 VS1 σ ’ r V + r VS1 S1
2 2
= f (S1 , σ, t) Vσ , (9.3)
110 Forecasting Financial Market Volatility


and similarly for the RHS. In order to solve the PDE, we need to under-
stand the function f (S1 , σ, t) which depends on whether or not d S1 and
dσ are correlated.

9.5.4 Correlated processes
If two Brownian motions d X 1 and d X 2 are correlated with correlation
coef¬cient ρ, then we may write:
d X 2 = ρd X 1 + 1 ’ ρ 2d t ,
where d t is the part of d X 2 that is not related to d X 1 .
Now consider the hedged portfolio,
=V’ S1 , (9.4)
where only the risk that is due to the underlying asset and its correlated
volatility risk are hedged. Volatility risk orthogonal to d S1 is not hedged.
From Ito™s lemma, we get:
= dV ’
d d S1
12 1
= Vt + σ1 VS1 S1 + ρqσ1 VS1 σ + q 2 Vσ σ dt
2 2
+VS1 d S1 + Vσ dσ ’ d S1 . (9.5)
Now write:
d S1 = µ1 dt + σ1 d X 1 , and
dσ = pdt + qd X 2
= pdt + q ρd X 1 + 1 ’ ρ 2d .
t

Substitute this result into (9.5) and get:
12 1
= Vt + σ1 VS1 S1 + ρqσ1 VS1 σ + q 2 Vσ σ dt
d
2 2
+VS1 {µ1 dt + σ1 d X 1 } + Vσ pdt + q ρd X 1 + 1 ’ ρ 2d t

’ {µ1 dt + σ1 d X 1 }
12 12
= Vt + σ1 VS1 S1 + ρqσ1 VS1 σ + q Vσ σ dt
2 2
+ VS1 µ1 + Vσ p ’ µ1 dt + VS1 σ1 + Vσ qρ ’ σ1 d X 1
+Vσ q 1 ’ ρ 2 d t . (9.6)
Option Pricing with Stochastic Volatility 111


So to get rid of d X 1 , the hedge ratio should be:

VS1 σ1 + Vσ qρ ’ σ1 = 0,
Vσ qρ
= VS1 + .
σ1
With this hedge ratio, only the uncorrelated volatility risk,
Vσ q 1 ’ ρ 2 d t , is left in the portfolio. If ρ = 1, the portfolio would
be risk-free.
Now substitute value of into (9.6). We get:
12 1
= Vt + σ1 VS1 S1 + ρqσ1 VS1 σ + q 2 Vσ σ dt
d
2 2
Vσ qρµ1
+ VS1 µ1 + Vσ p ’ VS1 µ1 ’ dt + Vσ q 1 ’ ρ 2 d t
σ1
12 1
= Vt + σ1 VS1 S1 + ρqσ1 VS1 σ + q 2 Vσ σ dt
2 2
Vσ qρµ1
+ Vσ p ’ dt + Vσ q 1 ’ ρ 2 d t . (9.7)
σ1


9.5.5 The market price of risk
Next we made the assumption that the partially hedged portfolio in
(9.4) will earn a risk-free return plus a premium for unhedged volatility
risk, , such that

= r dt +
d
= r (V ’ S1 ) dt +
Vσ qρ
= r V ’ VS1 S1 ’ S1 dt + . (9.8)
σ1
Substituting d from (9.7), we get:
12 1
Vt + σ1 VS1 S1 + ρqσ1 VS1 σ + q 2 Vσ σ dt
2 2
Vσ qρµ1
+ Vσ p ’ dt + Vσ q 1 ’ ρ 2 d t
σ1
Vσ qρ
= r V dt ’ r VS1 S1 dt ’ r S1 dt + ,
σ1
112 Forecasting Financial Market Volatility


12 1
= Vt + σ1 VS1 S1 + ρqσ1 VS1 σ + q 2 Vσ σ ’ r V + r VS1 S1 dt
2 2
Vσ qρ
+ Vσ p + (r S1 ’ µ1 ) dt + Vσ q 1 ’ ρ 2 d t .
σ1
Now replace the {} term with f (S1 , σ, t) in (9.3) we get:
Vσ qρ
= f Vσ + Vσ p + (r S1 ’ µ1 ) dt + Vσ q 1 ’ ρ 2 d t
σ1
f + p + qρ (r S1 ’ µ1 )
σ1
= Vσ q 1 ’ ρ 2 dt + d t .
q 1 ’ ρ2
Now de¬ne ˜market price of risk™ γ :

f + p+ (r S1 ’ µ1 )
σ1
γ= , (9.9)
q 1’ρ 2

where γ is the ˜returns™ associated with each unit of risk that is due to d t
(i.e. the unhedged volatility risk), hence, the denominator q 1 ’ ρ 2 .
From (9.9), we can get an expression for f :

f = ’p + (µ1 ’ r S1 ) + γ q 1 ’ ρ 2 .
σ1
Substituting p = ± (m ’ σ ) , q = β, µ1 = µS1 , and σ1 = f (σ ) S1 :
βρ
f = ’± (m ’ σ ) + (µS1 ’ r S1 ) + γ β 1 ’ ρ 2 .
f (σ ) S1
βρ (µ ’ r )
= ’± (m ’ σ ) + + γ β 1 ’ ρ 2.
f (σ )
We can now price the option with stochastic volatility in (9.3) using the
expression for f above and get:
122 1
0 = Vt + f St VS1 S1 + β 2 Vσ σ + ρβ f St VS1 σ ’ r V + r VS1 S1
2 2
ρ (µ ’ r )
+ ± (m ’ σ ) ’ β + γ 1 ’ ρ 2 Vσ (9.10)
f (σ )
Now write
ρ (µ ’ r )
(S1 , σ, t) = + γ 1 ’ ρ 2.
f (σ )
Option Pricing with Stochastic Volatility 113


We write (9.10) as
122
Vt + f S1 VS1 S1 + r VS1 S1 ’ V +ρβ f S1 VS1 σ
2
correlation
Black“Scholes
1
+ β 2 Vσ σ + ± (m ’ σ ) Vσ ’ β Vσ = 0 (9.11)
2
premium
L ou

or, on rearrangement,4
122 1
Vt + f S1 VS1 S1 + r VS1 S1 + ρβ f S1 VS1 σ + β 2 Vσ σ + ± (m ’ σ ) Vσ
2 2
correlation
L ou
Black“Scholes
= + β Vσ
rV
risk-free return as in BS premium for volatility risk

This analysis show that when volatility is stochastic in the form in
(9.1), the option price will be higher. The additional risk premium is
related to the correlation between volatility and the stock price processes
and the mean-reverting dynamic of the volatility process.

4
This result is shown in Fouque, Papanicolaou and Sircar (2000).
10
Option Forecasting Power

Option implied volatility has always been perceived as a market™s
expectation of future volatility and hence it is a market-based volatil-
ity forecast. It makes use of a richer and more up-to-date information
set, and arguably it should be superior to time series volatility forecast.
On the other hand, we showed in the previous two chapters that op-
tion model-based forecast requires a number of assumptions to hold for
the option theory to produce a useful volatility estimate. Moreover, op-
tion implied also suffers from many market-driven pricing irregularities.
Nevertheless, the volatility forecasting contests show overwhelmingly
that option implied volatility has superior forecasting capability, out-
performing many historical price volatility models and matching the
performance of forecasts generated from time series models that use a
large amount of high-frequency data.



10.1 USING OPTION IMPLIED STANDARD
DEVIATION TO FORECAST VOLATILITY
Once an implied volatility estimate is obtained, it is usually scaled by

n to get an n-day-ahead volatility forecast. In some cases, a regression
model may be used to adjust for historical bias (e.g. Ederington and
Guan, 2000b), or the implied volatility may be parameterized within a
GARCH/ARFIMA model with or without its own persistence adjust-
ment (e.g. Day and Lewis, 1992; Blair, Poon and Taylor, 2001; Hwang
and Satchell, 1998).
Implied volatility, especially that of stock options, can be quite un-
stable across time. Beckers (1981) ¬nds taking a 5-day average improves
the forecasting power of stock option implied. Hamid (1998) ¬nds such
an intertemporal averaging is also useful for stock index option dur-
ing very turbulent periods. On a slightly different note, Xu and Taylor
(1995) ¬nd implied estimated from a sophisticated volatility term struc-
ture model produces similar forecasting performance as implied from
the shortest maturity option.
116 Forecasting Financial Market Volatility


In contrast to time series volatility forecasting models, the use of im-
plied volatility as a volatility forecast involves some extra complexities.
A test on the forecasting power of option implied standard deviation
(ISD) is a joint test of option market ef¬ciency and a correct option pric-
ing model. Since trading frictions differ across assets, some options are
easier to replicate and hedge than the others. It is therefore reasonable
to expect different levels of ef¬ciency and different forecasting power
for options written on different assets.
While each historical price constitutes an observation in the sample
used in calculating volatility forecast, each option price constitutes a
volatility forecast over the option maturity, and there can be many option
prices at any one time. The problem of volatility smile and volatility skew
means that options of different strike prices produce different Black“
Scholes implied volatility estimates.
The issue of a correct option pricing model is more fundamental
in ¬nance. Option pricing has a long history and various extensions
have been made since Black“Scholes to cope with dividend payments,
early exercise and stochastic volatility. However, none of the option
pricing models (except Heston (1993)) that appeared in the volatility
forecasting literature allows for a premium for bearing volatility risk.
In the presence of a volatility risk premium, we expect the option price
to be higher which means implied volatility derived using an option
pricing model that assumes zero volatility risk premium (such as the
Black“Scholes model) will also be higher, and hence automatically be
more biased as a volatility forecast. Section 10.3 examines the issue of
biasedness of ISD forecasts and evaluates the extent to which implied
biasedness is due to the omission of volatility risk premium.

10.2 AT-THE-MONEY OR WEIGHTED IMPLIED?
Since options of different strikes have been known to produce differ-
ent implied volatilities, a decision has to be made as to which of these
implied volatilities should be used, or which weighting scheme should
be adopted, that will produce a forecast that is most superior. The most
common strategy is to choose the implied derived from an ATM op-
tion based on the argument that an ATM option is the most liquid and
hence ATM implied is least prone to measurement errors. The analysis
in Chapter 9 shows that, omitting volatility risk premium, ATM implied
is also least likely to be biased.
If ATM implied is not available, then an NTM (nearest-to-the-money)
option is used instead. Sometimes, to reduce measurement errors and
Option Forecasting Power 117


the effect of bid“ask bounce, an average is taken from a group of NTM
implied volatilities. Weighting schemes that also give greater weight
to ATM implied are vega (i.e. the partial derivative of option price
w.r.t. volatility) weighted or trading volume weighted, weighted least
squares (WLS) and some multiplicative versions of these three. The
WLS method, ¬rst appeared in Whaley (1982), aims to minimize the
sum of squared errors between the market and the theoretical prices of a
group of options. Since the ATM option has the highest trading volume
and the ATM option price is the most sensitive to volatility input, all
three weighting schemes (and the combinations thereof) have the ef-
fect of placing the greatest weight on ATM implied. Other less popular
weighting schemes include equally weighted, and weight based on the
elasticity of option price to volatility.
The forecasting power of individual and composite implied volatilities
has been tested in Ederington and Guan (2000b), Fung, Lie and Moreno
(1990), Gemmill (1986), Kroner, Kneafsey and Claessens (1995), Scott
and Tucker (1989) and Vasilellis and Meade (1996). The general con-
sensus is that among the weighted implied volatilities, those that favour
the ATM option such as the WLS and the vega weighted implied are
better. The worst performing ones are equally weighted and elastic-
ity weighted implied using options across all strikes. Different ¬ndings
emerged as to whether an individual implied volatility forecasts better
than a composite implied. Beckers (1981) Feinstein (1989b), Fung, Lie
and Moreno (1990) and Gemmill (1986) ¬nd evidence to support indi-
vidual implied although they all prefer a different implied (viz. ATM,
Just-OTM, OTM and ITM respectively for the four studies). Kroner,
Kneafsey and Claessens ¬nd composite implied volatility forecasts bet-
ter than ATM implied. On the other hand, Scott and Tucker (1989) con-
clude that when emphasis is placed on ATM implied, which weighting
scheme one chooses does not really matter.
A series of studies by Ederington and Guan have reported some inter-
esting ¬ndings. Ederington and Guan (1999) report that the information
content of implied volatility of S&P500 futures options exhibits a frown
shape across strikes with options that are NTM and have moderately
high strike (i.e. OTM calls and ITM puts) possess the largest informa-
tion content with R 2 equal to 17% for calls and 36% for puts.

10.3 IMPLIED BIASEDNESS
Usually, forecast unbiasedness is not an overriding issue in any forecast-
ing exercise. Forecast bias can be estimated and corrected if the degree
118 Forecasting Financial Market Volatility


of bias remains stable through time. Testing for biasedness is usually
carried out using the regression equation (2.3), where X i = X t is the
implied forecast of period t volatility. For a forecast to be unbiased, one
would require ± = 0 and β = 1. Implied forecast is upwardly biased if
± > 0 and β = 1, or ± = 0 and β > 1. In the case where ± > 0 and
β < 1, which is the most common scenario, implied underforecasts low
volatility and overforecasts high volatility.
It has been argued that implied bias will persist only if it is dif¬cult
to perform arbitrage trades that are needed to remove the mispricing.
This is more likely in the case of stock index options and less likely
for futures options. Stocks and stock options are traded in different
markets. Since trading of a basket of stocks is cumbersome, arbitrage
trades in relation to a mispriced stock index option may have to be
done indirectly via index futures. On the other hand, futures and futures
options are traded alongside each other. Trading in these two contracts
are highly liquid. Despite these differences in trading friction, implied
biasedness is reported in both the S&P100 OEX market (Canina and
Figlewski, 1993; Christensen and Prabhala, 1998; Fleming, Ostdiek and
Whaley, 1995; Fleming, 1998) and the S&P500 futures options market
(Feinstein, 1989b; Ederington and Guan, 1999, 2002).
Biasedness is equally widespread among implied volatilities of cur-
rency options (see Guo, 1996b; Jorion, 1995; Li, 2002; Scott and Tucker,
1989; Wei and Frankel, 1991). The only exception is Jorion (1996) who
cannot reject the null hypothesis that the one-day-ahead forecasts from
implied are unbiased. The ¬ve studies listed earlier use implied to fore-
cast exchange rate volatility over a much longer horizon ranging from
one to nine months.
Unbiasedness of implied forecast was not rejected in the Swedish mar-
ket (Frennberg and Hansson, 1996). Unbiasedness of implied forecast
was rejected for UK stock options (Gemmill, 1986), US stock options
(Lamoureux and Lastrapes, 1993), options and futures options across a
range of assets in Australia (Edey and Elliot, 1992) and for 35 futures
options contracts traded over nine markets ranging from interest rate
to livestock futures (Szakmary, Ors, Kim and Davidson, 2002). On the
other hand, Amin and Ng (1997) ¬nd the hypothesis that ± = 0 and
β = 1 cannot be rejected for the Eurodollar futures options market.
Where unbiasedness was rejected, the bias in all but two cases was
due to ± > 0 and β < 1. These two exceptions are Fleming (1998) who
reports ± = 0 and β < 1 for S&P100 OEX options, and Day and Lewis
(1993) who ¬nd ± > 0 and β = 1 for distant-term oil futures options
contracts.
Option Forecasting Power 119


Christensen and Prabhala (1998) argue that implied is biased because
of error-in-variable caused by measurement errors. Using last period
implied and last period historical volatility as instrumental variables to
correct for these measurement errors, Christensen and Prabhala (1998)
¬nd unbiasedness cannot be rejected for implied volatility of the S&P100
OEX option. Ederington and Guan (1999, 2002) ¬nd bias in S&P500
futures options implied also disappeared when similar instrument vari-
ables were used.

10.4 VOLATILITY RISK PREMIUM
It has been suggested that implied biasedness could not have been caused
by model misspeci¬cation or measurement errors because this has rel-
atively small effects for ATM options, used in most of the studies that
report implied biasedness. In addition, the clientele effect cannot explain
the bias either because it only affects OTM options. The volatility risk
premium analysed in Chapter 9 is now often cited as an explanation.
Poteshman (2000) ¬nds half of the bias in S&P500 futures options
implied was removed when actual volatility was estimated with a more
ef¬cient volatility estimator based on intraday 5-minute returns. The
other half of the bias was almost completely removed when a more
sophisticated and less restrictive option pricing model, i.e. the Heston
(1993) model, was used. Further research on option volatility risk pre-
mium is currently under way in Benzoni (2001) and Chernov (2001).
Chernov (2001) ¬nds, similarly to Poteshman (2000), that when implied
volatility is discounted by a volatility risk premium and when the errors-
in-variables problems in historical and realized volatility are removed,
the unbiasedness of the S&P100 index option implied volatility cannot
be rejected over the sample period from 1986 to 2000. The volatility risk
premium debate continues if we are able to predict the magnitude and
the variations of the volatility premium and if implied from an option
pricing model that permits a nonzero market price of risk will outperform
time series models when all forecasts (including forecasts of volatility
risk premium) are made in an ex ante manner.
Ederington and Guan (2000b) ¬nd that using regression coef¬cients
produced from in-sample regression of forecast against realized volatil-
ity is very effective in correcting implied forecasting bias. They also
¬nd that after such a bias correction, there is little to be gained from
averaging implied across strikes. This means that ATM implied together
with a bias correction scheme could be the simplest, and yet the best,
way forward.
11
Volatility Forecasting Records

11.1 WHICH VOLATILITY FORECASTING MODEL?
Our JEL survey has concentrated on two questions: is volatility fore-
castable? If it is, which method will provide the best forecasts? To con-
sider these questions, a number of basic methodological viewpoints need
to be discussed, mostly about the evaluation of forecasts. What exactly is
being forecast? Does the time interval (the observation interval) matter?
Are the results similar for different speculative markets? How does one
measure predictive performance?
Volatility forecasts are classi¬ed in this section as belonging in one
of the following four categories:
r HISVOL: for historical volatility, which include random walk, histor-
ical averages of squared returns, or absolute returns. Also included
in this category are time series models based on historical volatility
using moving averages, exponential weights, autoregressive models,
or even fractionally integrated autoregressive absolute returns, for ex-
ample. Note that HISVOL models can be highly sophisticated. The
multivariate VAR realized volatility model in Andersen, Bollerslev,
Diebold and Labys (2001) is classi¬ed here as a ˜HISVOL™ model. All
models in this group model volatility directly, omitting the goodness
of ¬t of the returns distribution or any other variables such as option
prices.
r GARCH: any member of the ARCH, GARCH, EGARCH and so forth
family is included.
r SV: for stochastic volatility model forecasts.
r ISD: for option implied standard deviation, based on the Black“
Scholes model and various generalizations.

The survey of papers includes 93 studies, but 25 of them did not
involve comparisons between methods from at least two of these groups,
and so were not helpful for comparison purposes.
Table 11.1 involves just pairwise comparisons. Of the 66 studies that
were relevant, some compared just one pair of forecasting techniques,
122 Forecasting Financial Market Volatility


Table 11.1 Pair-wise comparisons of forecasting performance
of various volatility models

Number of studies Studies percentage

HISVOL > GARCH 22 56%
GARCH > HISVOL 17 44%

HISVOL > ISD 8 24%
ISD > HISVOL 26 76%

GARCH > ISD 1 6%
ISD > GARCH 17 94%

SV > HISVOL 3
SV > GARCH 3
GARCH > SV 1
ISD > SV 1

Note: “A > B” means model A™s forecasting performance is better than
that of model B™s


other compared several. For those involving both HISVOL and GARCH
models, 22 found HISVOL better at forecasting than GARCH (56% of
the total), and 17 found GARCH superior to HISVOL (44%).
The combination of forecasts has a mixed picture. Two studies ¬nd it
to be helpful but another does not.
The overall ranking suggests that ISD provides the best forecasting
with HISVOL and GARCH roughly equal, although possibly HISVOL
does somewhat better in the comparisons. The success of the implied
volatility should not be surprising as these forecasts use a larger, and
more relevant, information set than the alternative methods as they use
option prices. They are also less practical, not being available for all
assets.
Among the 93 papers, 17 studies compared alternative version of
GARCH. It is clear that GARCH dominates ARCH. In general, mod-
els that incorporate volatility asymmetry such as EGARCH and GJR-
GARCH, perform better than GARCH. But certain specialized speci¬-
cations, such as fractionally integrated GARCH (FIGARCH) and regime
switching GARCH (RSGARCH) do better in some studies. However,
it seems clear that one form of study that is included is conducted just
to support a viewpoint that a particular method is useful. It might not
have been submitted for publication if the required result had not been
reached. This is one of the obvious weaknesses of a comparison such as
this: the papers being reported have been prepared for different reasons
Volatility Forecasting Records 123


and use different data sets, many kinds of assets, various intervals and
a variety of evaluation techniques. Rarely discussed is if one method
is signi¬cantly better than another. Thus, although a suggestion can be
made that a particular method of forecasting volatility is the best, no
statement is available about the cost“bene¬t from using it rather than
something simpler or how far ahead the bene¬ts will occur.
Financial market volatility is clearly forecastable. The debate is on
how far ahead one can accurately forecast and to what extent volatility
changes can be predicted. This conclusion does not violate market ef¬-
ciency since accurate volatility forecast is not in con¬‚ict with underlying
asset and option prices being correct. The option implied volatility, being
a market-based volatility forecast, has been shown to contain most in-
formation about future volatility. The supremacy among historical time
series models depends on the type of asset being modelled. But, as a
rule of thumb, historical volatility methods work equally well compared
with more sophisticated ARCH class and SV models. Better reward
could be gained by making sure that actual volatility is measured accu-
rately. These are broad-brush conclusions, omitting the ¬ne details that
we outline in this book. Because of the complex issues involved and the
importance of volatility measure, volatility forecasting will continue to
remain a specialist subject and to be studied vigorously.


11.2 GETTING THE RIGHT CONDITIONAL
VARIANCE AND FORECAST WITH THE
˜WRONG™ MODELS
Many of the time series volatility models, including the GARCH mod-
els, can be thought of as approximating a deeper time-varying volatility
construction, possibly involving several important economic explana-
tory variables. Since time series models involve only lagged returns it
seems likely that they will provide an adequate, possibly even a very
good, approximation to actuality for long periods but not at all times.
This means that they will forecast well on some occasions, but less well
on others, depending on ¬‚uctuations in the underlying driving variables.
Nelson (1992) proves that if the true process is a diffusion or near-
diffusion model with no jumps, then even when misspeci¬ed, appropri-
ately de¬ned sequences of ARCH terms with a large number of lagged
residuals may still serve as consistent estimators for the volatility of
the true underlying diffusion, in the sense that the difference between
the true instantaneous volatility and the ARCH estimates converges to
124 Forecasting Financial Market Volatility


zero in probability as the length of the sampling frequency diminishes.
Nelson (1992) shows that such ARCH models may misspecify both the
conditional mean and the dynamic of the conditional variance; in fact the
misspeci¬cation may be so severe that the models make no sense as data-
generating processes, they could still produce consistent one-step-ahead
conditional variance estimates and short-term forecasts.
Nelson and Foster (1995) provide further conditions for such mis-
speci¬ed ARCH models to produce consistent forecasts over the medium
and long term. They show that forecasts by these misspeci¬ed models
will converge in probability to the forecast generated by the true diffusion
or near-diffusion process, provided that all unobservable state variables
are consistently estimated and that the conditional mean and conditional
covariances of all state variables are correctly speci¬ed. An example
of a true diffusion process given by Nelson and Foster (1995) is the
stochastic volatility model described in Chapter 6.
These important theoretical results con¬rm our empirical observa-
tions that under normal circumstances, i.e. no big jumps in prices, there
may be little practical difference in choosing between volatility mod-
els, provided that the sampling frequency is small and that, whichever
model one has chosen, it must contain suf¬ciently long lagged residuals.
This might be an explanation for the success of high-frequency and long
memory volatility models (e.g. Blair, Poon and Taylor, 2001; Andersen,
Bollerslev, Diebold and Labys, 2001).

11.3 PREDICTABILITY ACROSS DIFFERENT ASSETS
Early studies that test the forecasting power of option ISD are fraught
with many estimation de¬ciencies. Despite these complexities, option
ISD has been found empirically to contain a signi¬cant amount of in-
formation about future volatility and it often beats volatility forecasts
produced by sophisticated time series models. Such a superior perfor-
mance appears to be common across assets.

11.3.1 Individual stocks
Latane and Rendleman (1976) were the ¬rst to discover the forecast-
ing capability of option ISD. They ¬nd actual volatilities of 24 stocks
calculated from in-sample period and extended partially into the future
are more closely related to implied than historical volatility. Chiras and
Manaster (1978) and Beckers (1981) ¬nd prediction from implied can
Volatility Forecasting Records 125


explain a large amount of the cross-sectional variations of individual
stock volatilities. Chiras and Manaster (1978) document an R 2 of 34“
70% for a large sample of stock options traded on CBOE whereas
Beckers (1981) reports an R 2 of 13“50% for a sample that varies from
62 to 116 US stocks over the sample period. Gemmill (1986) produces
an R 2 of 12“40% for a sample of 13 UK stocks. Schmalensee and
Trippi (1978) ¬nd implied volatility rises when stock price falls and that
implied volatilities of different stocks tend to move together. From a
time series perspective, Lamoureux and Lastrapes (1993) and Vasilellis
and Meade (1996) ¬nd implied volatility could also predict time series
variations of equity volatility better than forecasts produced from time
series models.
The forecast horizons of this group of studies that forecast equity
volatility are usually quite long, ranging from 3 months to 3 years. Stud-
ies that examine incremental information content of time series fore-
casts ¬nd volatility historical average provides signi¬cant incremental
information in both cross-sectional (Beckers, 1981; Chiras and Man-
aster, 1978; Gemmill, 1986) and time series settings (Lamoureux and
Lastrapes, 1993) and that combining GARCH and implied volatility
produces the best forecast (Vasilellis and Meade, 1996). These ¬ndings
have been interpreted as an evidence of stock option market inef¬ciency
since option implied does not subsume all information. In general, stock
option implied volatility exhibits instability and suffers most from mea-
surement errors and bid“ask spread because of the lower liquidity.

11.3.2 Stock market index
There are 22 studies that use index option ISD to forecast stock index
volatility; seven of these forecast volatility of S&P100, ten forecast
volatility of S&P500 and the remaining ¬ve forecast index volatility
of smaller stock markets. The S&P100 and S&P500 forecasting results
make an interesting contrast as almost all studies that forecast S&P500
volatility use S&P500 futures options which is more liquid and less
prone to measurement errors than the OEX stock index option written
on S&P100. We have dealt with the issue of measurement errors in the
discussion of biasness in Section 10.3.
All but one study (viz. Canina and Figlewski, 1993) conclude that
implied volatility contains useful information about future volatility.
Blair, Poon and Taylor (2001) and Poteshman (2000) record the highest
R 2 for S&P100 and S&P500 respectively. About 50% of index volatility
126 Forecasting Financial Market Volatility


is predictable up to a 4-week horizon when actual volatility is estimated
more accurately using very high-frequency intraday returns.
Similar, but less marked, forecasting performance emerged from the
smaller stock markets, which include the German, Australian, Canadian
and Swedish markets. For a small market such as the Swedish market,
Frennberg and Hansson (1996) ¬nd seasonality to be prominent and that
implied volatility forecast cannot beat simple historical models such as
the autoregressive model and random walk. Very erratic and unstable
forecasting results were reported in Brace and Hodgson (1991) for the
Australian market. Doidge and Wei (1998) ¬nd the Canadian Toronto
index is best forecast with GARCH and implied volatility combined,
whereas Bluhm and Yu (2000) ¬nd VDAX, the German version of VIX,
produces the best forecast for the German stock index volatility.
A range of forecast horizons were tested among this group of stud-
ies, though the most popular choice is 1 month. There is evidence that
the S&P implied contains more information after the 1987 crash (see
Christensen and Prabhala (1998) for S&P100 and Ederington and Guan
(2002) for S&P500). Some described this as the ˜awakening™ of the S&P
option markets.
About half of the papers in this group test if there is incremental
information contained in time series forecasts. Day and Lewis (1992),
Ederington and Guan (1999, 2004), and Martens and Zein (2004) ¬nd
ARCH class models and volatility historical average add a few percent-
age points to the R 2 , whereas Blair, Poon and Taylor (2001), Christensen
and Prabhala (1998), Fleming (1998), Fleming, Ostdiek and Whaley
(1995), Hol and Koopman (2001) and Szakmary, Ors, Kim and Davidson
(2002) all ¬nd option implied dominates time series forecasts.

11.3.3 Exchange rate
The strong forecasting power of implied volatility is again con¬rmed
in the currency markets. Sixteen papers study currency options for a
number of major currencies, the most popular of which are DM/US$
and ¥/US$. Most studies ¬nd implied volatility to contain information
about future volatility for a short horizon up to 3 months. Li (2002) and
Scott and Tucker (1989) ¬nd implied volatility forecast well for up to a
6“9-month horizon. Both studies register the highest R 2 in the region of
40“50%.
A number of studies in this group ¬nd implied volatility beats time
series forecasts including volatility historical average (see Fung, Lie and
Volatility Forecasting Records 127


Moreno, 1990; Wei and Frankel, 1991) and ARCH class models (see

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