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An Empirical Study on theImplied Volatility Function of S&P 500 Options
by
Shiheng Wang
An essay submitted to the Department of Economicsin partial fulfillment of the requirements for
the degree of Master of Arts
Queen’s UniversityKingston, Ontario, Canada
August 2002
copyright c©Shiheng Wang 2002
Abstract
A better understanding of the empirical dynamics of Black-Scholes implied volatil-
ity surface has long been of considerable interest to both practitioners and aca-
demics. Basing on some findings about the ad hoc Black-Scholes valuation ap-
proach suggested in Dumas, Flemming and Whaley (1998), this essay studies the
empirical performance of various volatility function forms that characterize the
regularities underlying the Black-Scholes implied volatility skew or smile.
Because the volatility function suggested in Dumas, Flemming and Whaley (1998)
is unstable over time, we propose a new class of dynamics implied volatility function
that separates a time-invariant implied volatility function from the random factors
that drive changes in the individual implied volatilities. The random factors are
incorporated through the at-the-money implied volatilities which are modelled as
a function of lagged volatility and a non-linear function of the underlying asset
return. This dynamic model is found to greatly improve the pricing performance.
Acknowledgements
I would like to thank Professor Wulin Suo for his invaluable guidance and patience
in advising me with this essay. I am also grateful to Yu Du and Jun Yuan for
useful comments and assistance in mathematics. Finally, I dedicate this essay to
my families who always support and encourage me in my endeavors.
1
Contents
1 Introduction 3
2 Black-Scholes Implied Volatility Function 8
2.1 Modelling Implied Volatility Smile . . . . . . . . . . . . . . . . . . 8
2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Estimated Parameters and In-Sample Pricing Fit . . . . . . . . . . 14
2.5 Out-of-Sample Forecasting Performance . . . . . . . . . . . . . . . . 16
3 Dynamic Black-Scholes Implied Volatility Function 19
3.1 A Brief Review of Derman’s “Sticky” Models of Implied Volatility . 20
3.2 Dynamic Black-Scholes Implied Volatility Function . . . . . . . . . 24
3.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Estimation and Forecasting Results . . . . . . . . . . . . . . . . . . 28
4 Conclusion 32
2
1 Introduction
During the last two decades the market for financial derivatives has experienced
rapid growth and many innovative product creations. For valuing these instru-
ments, the Black-Scholes (1973) model is often applied as a starting point. Black-
Scholes model assumes that the underlying asset is traded in a frictionless market
and the asset’s price follows a geometric Brownian motion with constant volatility.
To recover the volatility from option price, we invert the Black-Scholes call option
formula by replacing St with St − PV D,
C = (St − PV D)N(d1)−Ke−rτN(d2), (1)
d1 =ln[(St − PDV )/K] + (r + 0.5σ2)τ
σ√
τ, (2)
d2 = d1 − σ√
τ , (3)
where St is the spot price of underlying asset, K is the strike price, r is the risk free
interest rate, τ is time to maturity of the option, σ is the volatility of underlying as-
set price and PV D is the present value of cash dividends summed over option’s life.
For all options on the same underlying asset, the volatilities σ implied by inverting
the Black-Scholes valuation formula (hereon referred to as “Black-Scholes implied
volatility” or “BSIV” for short) would be identical regardless of strike price and
time to maturity of individual options. In practice, however, Black-Scholes im-
plied volatilities tend to differ across strike prices and time to maturity1. S&P
500 option Black-Scholes implied volatilities, for example, form a “smile” pattern
1Rubinstein(1994), Dumas, Flemming and Whaley (1998), and Bakshi, Cao and Chen(1997)examined the S&P 500 index option market. Taylor and Xu(1993) performed similar investi-gations for Philadelphia Exchange foreign currency option market. Duque and Paxson (1993)examined the stock options traded at the London International Financial Futures Exchange,Heynen(1993) investigated the European Options Exchange.
3
prior to the October 1987 market crash. Options that are deep in- and out-of-
the-money have higher implied volatilities than at-the-money options. After the
crash, a “skew” appears—the implied volatilities decrease monotonically as the
strike price rises relative to the index level, with the rate of decrease increasing for
options with shorter time to maturity. Therefore, practitioners commonly use dif-
ferent volatilities for different strike prices and maturities in order to take account
of the deviation from the Black-Scholes constant volatility assumption.
A number of new approaches have been developed to account for the nonconstant
volatility, among which a very representative one is the “deterministic volatility
function” method proposed by Derman and Kani (1994), Dupire (1994), Rubin-
stein (1994) and later empirically tested by Dumas, Flemming and Whaley (1998)
(hereon referred to as “DFW”). In this method, the volatility is assumed to be
a deterministic function of asset price and/or time. This is the “simplest method
that preserves the arbitrage argument underlying the Black-Scholes model” (DFW,
1998, p.2065). Stochastic volatility or jump diffusion models require additional as-
sumptions about investor preferences for risk or additional securities which can
be used to hedge volatility or jump risk. Thus they are difficult for practitioners
to implement in practice. In contrast, deterministic volatility function approach
assumes the volatility rate is a flexible but deterministic function of asset price
and time, and only the parameters governing the volatility processes need to be
estimated.
However, DFW (1998) found that an ad hoc Black-Scholes valuation method
(hereon referred to as “AHBS” method) even outperformed the deterministic
4
volatility function model in hedging performance. In the AHBS method, the Black-
Scholes implied volatility is constructed as a quadratic function of asset price S
and time t, in part because the Black-Scholes implied volatilities for S&P 500 op-
tions tend to have a parabolic shape. At date t, the suggested function is plugged
into Black-Scholes valuation formula to back out the parameters of the function
by minimizing differences between reported option prices and model prices. The
estimated parameters are used to calculate the volatility level at date t + 7, which
is then applied to Black-Scholes formula to predict the option price at t + 7.
Since the deterministic volatility function approach has not been found to be more
successful than the AHBS method, it may be useful to re-examine the AHBS
method in details. The main reason for this investigation is that AHBS method is
a “variation of what is applied in practice as a means of predicting option prices”
(DFW, 1998, p.2086). To account for the skew pattern in Black-Scholes implied
volatilities, many market practitioners simply smooth the implied volatility rela-
tion across strike prices and time to maturity, and then value option using the
smoothed relation. The AHBS method operationalizes such practice. Same as
the deterministic volatility function, it’s simple in parameter estimation compared
to the stochastic volatility models. Moreover, it is meaningful to investigate the
AHBS methods in more detail if the following points are considered:
First, in their empirical test, DFW (1998) found that the volatility functions ap-
plied to AHBS methods are unstable over time. That is, the volatility function
implied by today’s option price is not the same one embedded in option price to-
morrow. To apply the AHBS method, we need to re-estimate the volatility function
5
every day or every week so as to remedy the observed functional instability. Before
we follow such practice, we need to first find out what the best volatility function
form to perfectly characterize the regularities underlying the Black-Scholes implied
volatility skew or smile. Although DFW (1998) proposed some volatility functions
in their research, those function forms are restrictive and arbitrary to a certain
extent, since DFW’s objective was to test the stability of the suggested functions
rather than to find a function form which best captures the daily volatility pattern.
Second, DFW (1998) only utilized the current underlying asset price S and time t
as variables in the volatility functions when implementing AHBS method. Actu-
ally, in existing literature, Black-Scholes implied volatilities are found to depend-
ing on a wide variety of contract specific variables, such as strike price K(Bates,
1995), proportional moneyness S/K − 1 (Clewlow, 19992) and maturity adjusted
proportional moneyness 1√τ
ln(K/F ) (where F is the forward price of S) (Naten-
berg, 1994).
Lastly, DFW’s (1998) conclusion of functional instability was based on a method-
ology which examined whether the levels of Black-Scholes implied volatilities for
options with a certain time to maturity (i.e. 90 days) will predict the levels of
volatilities of options with a different time to maturity (i.e. 83 days). If the rel-
ative shape of Black-Scholes implied volatility surfaces depends on the time to
maturity, the volatility function may remain stable over time when maturity of
all options are controlled to be the same. Then we don’t need to continuously
update the function every day or every week . The stable function could thus be
2Their study applied principal component analysis to first differences of Black-Scholes impliedvolatilities for fixed maturity buckets, across both strike and moneyness metrics.
6
implemented to AHBS valuation method for long-term forecasting or hedging.
The objectives of this essay are twofold. First, by incorporating different contract
specific variables in the volatility functions that are applied to AHBS valuation
method, we aim at finding out which competitive function could best characterize
the regularities in the skew pattern of the Black-Scholes implied volatility. Second,
by controlling all options to the same maturity, we investigate whether a stable
volatility function exists. This hypothesis is tested by comparing the out-of-sample
predicting performance of the assumed stable volatility function with the known
unstable volatility function, both of which are applied to the AHBS valuation
method. If the former model could outperform the latter one, this may prove from
one angle that stable volatility function exists under certain restrictions, or at least
the stability of volatility functions could be improved under certain assumptions.
This essay is organized as follows. Section 2 proposes competitive volatility func-
tions for the Black-Scholes implied volatility skew. Tests on both in-sample fit-
ting and out-of-sample forecasting accuracy are carried out by employing AHBS
valuation methods. Section 3 investigates a richer structure of Black-Scholes im-
plied volatility time series for options with given time to maturity, and presents a
volatility functional form which is assumed to be stable over time. Comparisons
of out-of-sample pricing accuracy with unstable volatility function are performed.
Section 4 concludes the paper.
7
2 Black-Scholes Implied Volatility Function
2.1 Modelling Implied Volatility Smile
As introduced in Section 1, in our study we call the volatility recovered from option
price by inverting the Black-Scholes (1973) formula “Black-Scholes implied volatil-
ity”. The Black-Scholes implied volatility smile is often modelled by a quadratic
regression of the form
σ = α0 + α1X + α2X2 (4)
where X is a contract specific state variable.
In principal, with a suitable choice of X , this function is able to capture a smile
as well as a skew pattern. The first objective of our study is to find out what
function form that applied to AHBS valuation approach could best fit reported
option prices. In our study, we call function (4) “Black-Scholes implied volatility
function” (hereon referred to as “BSIV function”). In existing literature, the fol-
lowing functions have been suggested to describe the volatility skew:
Model 1: σ(K, τ) = α0 + α1K100
+ α2(K100
)2
Model 2 : σ(K, τ) = α0 + α1Kt
St+ α2(
Kt
St)2
Model 3: σ(K, τ) = α0 + α1K100
+ α2(K100
)2 + α3τ + α4(K100
)τ
Model 4: σ(K, τ) = α0 + α1ln(K/F )√
τ+ α2[
ln(K/F )√τ
]2
where σ(K, τ) is Black-Scholes implied volatility for the option with strike price
K and time to maturity τ ; Kt is the strike price K discounted to the value at time
t; and F is the forward price of St.
8
Model 1 was utilized by Shimko (1993) and Bates (1995) to describe the relation
between Black-Scholes implied volatilities and strike prices. It was also proved to
be the best model in DFW’s study with respect to the out-of-sample forecasting
performance (We make a revision by replacing K with K/100 to avoid the esti-
mated coefficient being too small). Similar to mode 1, model 2 tries to capture
the connection between implied volatilities and moneyness. Rosenberg (2000) used
proportional moneyness KSt−1 as the variable in the volatility function. In contrast,
we replace K by Kt to ensure a consistent definition of moneyness Kt/St at differ-
ent time points. Model 3, proposed by DFW (1998), was the best model in their
study in respect of in-sample fitting, but performed poor in out-of-sample fore-
casting. DFW (1998) argued that the time variable may be an important cause of
overfitting at the estimation stage. Model 4, put forward first by Natenberg (1994)
and later adopted by Tompkin (1999), is expected to reduce the smile’s dependence
on time to maturity and thus overcome the deficiency inherited in model 3. As
the shape of the volatility smile depends on the option maturity and in most cases
the smile becomes less pronounced as the option maturity increases. Define the
volatility smile as the relationship between implied volatility and ln(K/F )√τ
usually
makes the smiles much less dependent on time to maturity.3 This is consistent
with Taylor and Xu (1993), who demonstrated that a more complex valuation
model (such as jump diffusion) can generate time-dependence in the skew even if
volatility is constant over time.
Although every function has been proved to capture certain characteristics of
3For more detailed discussion of this function, see Natenberg, Option Pricing and Volatility:Advanced Trading Strategies and Techniques, second edition, Chicago (1994). This approachmakes the assumption that the volatility of an option depends on the number of standard devi-ation lnK is away from the mean of lnF .
9
volatility skew in different sample periods or for different derivative products, few
studies made an investigation into which model best characterizes Black-Scholes
volatility skew, based on the same derivative product and under the same sample
period. It’s therefore meaningful for us to carry out such a test on the S& P 500
index options.
2.2 Methodology
Since the BSIV functions implemented in AHBS valuation method were found to
be unstable in DFW’s (1998) empirical study, we re-estimate it on a daily ba-
sis. The logic of our test is straightforward. First, we use today’s option prices
to estimate the parameters of each competitive function. Then, we step one day
forward in time—using the estimated parameters to forecast the Black-Scholes im-
plied volatilities and option prices on the second day. We repeat such procedures
every day for three months in 1996. This approach is similar to those employed
by DFW (1998) who re-estimated the volatility functions week to week.
We estimate each competitive model by minimizing the sum of squared dollar errors
between the reported option price and model price. Let Cmak(K, τ) be reported
price for the option with strike price K and time to maturity τ and Cest(α, K, τ)
the option’s model price determined by plugging the BSIV function into Black-
Scholes valuation formula. The objective loss function with respect to parameters
α of each competitve function is in the form of
Loss(α) = minn∑
i=1
(Cmak − Cest)2 (5)
Backing out the structural parameters by using the loss function is a common
10
approach in the existing literature. For example, Bates (1996), Bakshi, Cao and
Chen (1997), and Longstaff (1995) all used this approach. We employ the mini-
mized loss as the measure of in-sample fitting.
To assess the out-of-sample predicting accuracy, we use two measures on each day:
(1) the root mean squared error (RMSE) is the square root of the average squared
deviation of the reported option price from the model price, i.e.
RMSE =
√√√√ 1
n
n∑i=1
(Cmak − Cest)2 (6)
(2) the mean relative error (MRE) is the average ratio of the absolute pricing error
to market price, i.e.
MRE =1
n
n∑i=1
(|Cmak − Cest|/Cmak) (7)
2.3 Data
The daily record on the last bid-ask quote of S&P 500 call options from March
1, 1996 to May 31, 1996 is obtained from the Chicago Board Option Exchange
(CBOE). The daily record of closing quote of S&P 500 index level is obtained
from the Center for Research in Security Prices (CRSP) database.4
To proxy for the risk-free rate, the rate on T-bills of comparable maturity is used.
The data on the daily T-bill bid and ask discount with maturities up to three
months are hand-collected from the Wall Street Journal. The average of the bid
4Option prices and closing index levels are nonsynchronous, either because their closing quoteare simply recorded at different times or because closing prices of at least some of five hundredstocks that constitute the index are stale. Assuming such random errors in index prices to be0.25%, Jorion (1995) estimates that the error in measured implied volatility is about 1.2%.
11
and ask T-bill discounts is used and converted to an annualized interest rate. Since
T-bills mature every Thursdays while index options expire on the third Friday of
each month, we utilize two T-bill rates straddling an option’s maturity date to
obtain the interest rate corresponding to the option’s maturity. This is done for
each contract on each day in our sample.
The daily cash dividends for the S&P 500 index are collected from the S&P 500
Information Bulletin. The actual cash dividend paid during the option’s life are
used to proxy for expected dividends and are summed over the option’s life.
PV D =n∑
i=1
e−ritiDi, (8)
Where Di is the ith cash dividend payment, ti is the time to ex-dividend from
current date t, and ri is the ti-period risk free interest rate.
The forward price of S&P 500 index is therefore
F = (St − PV D)erτ (9)
Three exclusive filters are applied to the option data. First, we eliminate options
with fewer than 7 or more than 90 days to expiration. The options with maturity
fewer than 7 days usually have relatively small time premium, hence the estima-
tion of volatility is extremely sensitive to nonsynchronous option prices and other
possible measurement errors. Options with more than 90 days to maturity are
not actively traded. Second, we eliminate options whose absolute “moneyness”,
|K/S| − 1, is greater than 10 percent. Like extremely short-term options, deep in-
and out-of-the money options have small time premiums and hence contain little
information about the volatility function. Nor are they actively traded. Finally,
quotes not satisfying the arbitrage restriction
12
Ct ≥ max(0, St −Ke−rτ − PDV ) (10)
are taken out of the sample.
We divide the option data into several categories based on moneyness and time to
maturity. Define S−K as the time-t intrinsic value of a call, where S is dividend-
excluded. A call option is then said to be at-the-money (ATM) if 0.97≤K/S≤1.03,
out-of-the-money (OTM) if K/S>0.97, and in-the-money (ITM) if K/S<1.03. A
finer partition results in six moneyness categories. By the time to maturity, an
option contract can be classified as (i) short-term (7-45 days) and (ii) long-term
(45-90 days). The proposed moneyness and maturity classification produce 12 cat-
egories for which the empirical results are reported.
Table 1 describes certain properties of the call options used in our study. The
average closing bid-ask midpoint option price, the average Black-Scholes implied
volatility and number of observations are reported for each moneyness-maturity
category.
There are 1924 call options included in our study, with OTM and ATM options
respectively taking up 47 percent and 45 percent of the whole sample. The average
option price ranges from $0.68 for the short-term deep OTM option, to $67.13 for
the long-term deep ITM option.
Clearly, regardless of time to maturity, the Black-Schole implied volatility exhibits
a strong skew pattern (as the call option goes from deep ITM to ATM and then to
13
deep OTM, with the deepest ITM implied volatilities taking the highest values).
Furthermore, the volatility skew is stronger for short-term options, indicating that
short-term options are more severely mispriced by the Black-Scholes model. These
findings of clear moneyness-related and maturity-related bias associated with the
Black-Scholes, are consistent with those in the existing literature (Bates, 1996;
Bakshi, Cao and Chen, 1997).
Table 1: Sample Properties of Closing Call Option QuoteMarch 1, 1996 to May 31, 1996
Moneyness Days to Maturity Number ofK/S 7-45 days 46-90 days samples
ITM [0.90, 0.94) $49.07 $67.13(0.296) (0.205) [50]
[0.94, 0.97) $32.82 $42.55(0.211) (0.187) [115]
ATM [0.97, 1.00) $17.53 $24.64(0.166) (0.158) [312]
[1.00, 1.03) $7.70 $16.89(0.152) (0.155) [546]
OTM [1.03, 1.06) $2.14 $8.10(0.146) (0.145) [465]
[1.06, 1.10] $0.68 $3.15(0.145) (0.137) [436]
2.4 Estimated Parameters and In-Sample Pricing Fit
The estimation procedures described in the Section 2.2 are separately done for
each competitive function on each day. Table 2 reports the average and standard
error of each estimated parameter as well as the average loss for each competitive
function. These reported statistics are informative and several observations are in
14
order:
First, all models basically grasp the skew pattern of Black-Scholes implied volatil-
ity, with all α1<0 (which represents tilt of the skew) and α2> 0 (which represents
curvature of the skew).
Table 2: Mean and Standard Deviation of Estimated Parameters
Coefficient Competitive ModelsEstimate model 1 model 2 model 3 model 4
α0 7.62 7.63 7.82 0.16(std err) (6.89) (8.74) (8.23) (0.01)
α1 -2.19 -13.30 -2.26 -0.78(std err) (2.58) (16.94) (2.47) (2.33)
α2 0.16 6.32 0.17 1.09(std err) (0.19) (8.28) (0.19) (1.92)
α3 -1.51(std err) (1.67)
α4 0.06(std err) (0.32)Loss(α) 35.27 28.92 29.03 27.15
Second, it’s obvious there is considerable variation in coefficient estimates, which is
suggested by the standard error of coefficient estimates. It demonstrates that even
within a very short time interval (overnight), infusion of new information could
produce great change in Black-Scholes implied volatility surface which leads to the
instability of volatility function. This result is consistent with that of DFW (1998).
Lastly, based on the minimized loss, model 2, model 3 and model 4 greatly im-
proves the in-sample fitting compared to model 1. The average minimized loss of
15
model 1 reaches 35.27, while the average minimized loss of the other three models
are all around 28. Model 4 slightly dominates both model 2 and model 3. Model
1’s poor performance is in expectation, since it contains the least information, with
only the strike price taken into consideration. By expressing the underlying asset’s
effect in the form of moneyness Kt/St, model 2 obviously infuses more information
than model 1. More importantly, model 2 actually describes the inverse relation
between index return and volatility changes. Although most empirical studies use
index return to measure volatility, DFW (1998) and Derman (1999) found that the
inverse relation is also apparent between the index return and Black-Scholes im-
plied volatility. The addition of time to maturity variable τ to model 3 appears to
be important, as its in-sample fitting is also greatly improved compared to model
1. But most of the incremental power comes from the cross-product term K100
τ ,
as we find that little difference in explanatory power occurs if we estimate model
3 without the variable τ . And it does no better than model 2, which does not
consider time to maturity. Model 4 incorporates all the advantage of model 2 and
model 3. It includes all the variables to which the implied volatilities are sensitive.
Furthermore, its superiority to other models may proves from a different angle
that the volatility of an option depends on the number of standard deviations lnK
is away from the mean of lnF . Compared to model 3, it also enjoys the advantage
of incorporating all related factors in a parsimonious function form.
2.5 Out-of-Sample Forecasting Performance
We have shown that the in-sample fit of daily option prices is increasingly better
as we move from model 1 to 3, and then to 2 and finally to 4. But in-sample fitting
actually reflects a model’s static performance and a good in-sample fit does not
16
necessarily guarantee an good out-of-sample fit. For example, one may argue that
model 3 outperforms model 1 owing to extra parameters. The presence of more
parameters may actually cause overfitting and therefore penalize the model if the
extra parameters do not improve its structural fitting. For this purpose, we rely on
the current day’s estimated parameters to compute next day’s model based option
prices. Next, compute RMSE and MRE every day to obtain the average RMSE
and MRE. Table 3 reports each model’s average RMSE and MRE with respect to
the same classification of moneyness and maturity specified in Table 1.
Two features are noticeable across all models in the forecasting results: First, for a
given maturity, the absolute pricing error measurement RMSE typically increases
from OTM to ATM and then to ITM options. But by percentage pricing error
measurement MRE, OTM options are mispriced to the greatest extent. One pos-
sible reason for this feature is that the loss function (5) is constructed in favor of
more expensive options. The value of call options is a decreasing function of strike
prices, and ITM options generally have the highest prices. To fit the options in
an overall level, loss function (5) obviously puts first priority on fitting the most
expensive option. So, even though OTM options are priced with the least absolute
error, they are actually mispriced with the largest relative error.
Second, for a given moneyness category, RMSE typically increases from short to
long term options. But MRE shows that the short-term options are mispriced to
a larger degree. The reason for this result is also inherited in the characteristics of
the loss function, which is a more favorable treatment of long-term options, whose
values are generally higher than short-term ones.
17
Table 3: Average Forecasting Error of CompetitiveBlack-Scholes Implied Volatility Functions
Days to Moneyness Competitive ModelsMaturity K/S model 1 model 2 model 3 model 4
RMSE 7-45 [0.90, 0.94) 2.87 2.57 2.93 2.43[0.94, 0.97) 2.37 1.95 2.41 1.87[0.97, 1.00) 1.79 1.55 1.84 1.62[1.00, 1.03) 1.34 1.09 1.39 1.11[1.03, 1.06) 0.66 0.47 0.71 0.40[1.06, 1.10] 0.22 0.16 0.21 0.13
46-90 [0.90,0.94) 3.38 2.97 3.42 2.87[0.94, 0.97) 2.79 2.42 2.84 2.34[0.97, 1.00) 2.15 1.82 2.17 1.87[1.00, 1.03) 1.59 1.24 1.63 1.29[1.03, 1.06) 1.17 0.89 1.12 0.74[1.06, 1.10] 0.79 0.64 0.75 0.55
MRE 7-45 [0.90, 0.94) 0.057 0.053 0.061 0.047[0.94, 0.97) 0.074 0.059 0.070 0.055[0.97, 1.00) 0.102 0.086 0.105 0.092[1.00, 1.03) 0.173 0.140 0.174 0.141[1.03, 1.06) 0.307 0.220 0.331 0.187[1.06, 1.10] 0.335 0.235 0.335 0.201
46-90 [0.90, 0.94) 0.049 0.042 0.048 0.040[0.94, 0.97) 0.064 0.054 0.063 0.052[0.97, 1.00) 0.081 0.056 0.066 0.055[1.00, 1.03) 0.094 0.075 0.094 0.076[1.03, 1.06) 0.142 0.105 0.138 0.091[1.06, 1.10] 0.251 0.203 0.239 0.174
Next we move to detailed analysis on the out-of-sample pricing accuracy of com-
petitive functions. First, model 1, which performs poorest in in-sample fitting,
outperforms model 3 for almost all option categories except for the the short-term
deep OTM options with monyness 1.06≤K/S≤ 1.10 and long-term OTM options
with moneyness 1.03≤K/S≤ 1.10. The deterioration of model 3’s performance
18
proves what DFW (1998) found: time to maturity may only serves to overfit the
data at the estimation stage.
Second, the inclusion of the index level in the volatility function (the index level
is expressed in forward price in model 4) make both model 2 and model 4 greatly
outperform model 1 and 3. This may suggest that the the stability of volatility
functions may be greatly improved by utilizing moneyness Kt/St or maturity ad-
justed moneyness ln(K/F )√τ
as state variables. Further comparison between model 2
with model 4 reveals that regardless of time to maturity, model 4 dominates model
3 in both ITM and OTM categories, but does worse for pricing ATM options.
Results in Section 2.4 and 2.5 suggest two possible avenues for us: employing time
to maturity as one of variables to construct more elaborate volatility functions
may be unnecessary; and, the stability of BSIV functions is improved most by a
quadratic regression of maturity adjusted monyness.
3 Dynamic Black-Scholes Implied Volatility Func-
tion
In last section we identify a Black-Scholes implied volatility function which greatly
improves both the in-sample fitting and out-of-sample pricing accuracy when ap-
plied to the AHBS valuation approach. However, to implement such a function,
we have to re-estimate it on a daily basis to remedy the observed functional insta-
bility. One may ask whether we could filter the random factors that produce the
instability and whether a stable BSIV function exists after we filter the random
19
factors. If the answer is yes, we could avoid continuously updating the function
everyday. What we need to do is to model the random factors and the time-
invariant BSIV function, respectively. We could call this kind of model “dynamic
BSIV function”. This is a potentially promising approach, because, as noted in
DFW (1998), an important reason for the failure of the deterministic volatility
function is that they were unable to replicate the stylized fact that “new market
information induces a shift in the level of overall market volatility from week to
week” (p.2081). This is also the main deficiency inherited in the BSIV function we
discussed in Section 2. A dynamic BSIV function that overcomes this deficiency
may be able to extract some useful information embedded in the cross-section of
option prices, while preserving the key dynamics that drive changes in individual
option implied volatilities.
Next we give a brief review of Derman’s “sticky” volatility models which sheds
light on the structural form of dynamic BSIV functions we will discuss later.
3.1 A Brief Review of Derman’s “Sticky” Models of Im-plied Volatility
Figure 1 shows the 45 days to maturity Black-Scholes implied volatilities of call
options with strikes 660, 670, 680, 690, 700 on S&P 500 for the period from March
1, 1996 to 31 December, 1996.5 Two features are noticeable: first, the volatility
skew is always negative. At any time, implied volatilities increase monotonically
as the strike levels decrease. Second, all time series of fixed strike option implied
5The 45 days to maturity implied volatility for a particular strike is obtained by interpolationfrom the closing bid-ask midpoint prices of options with maturity straddling the 45 days tomaturity.
20
volatilities have a similar trending, they move approximately linearly away from
the at-the-money implied volatility series. The greater the strike price, the smaller
the deviation.
Observation of data similar to these, but on S&P 500 index options 3-month volatil-
ities of different sample periods, has motivated Derman (1999) to investigate the
systematic connections between changes in index level and the volatility recovered
from option prices. He argued that the market view of the future volatility struc-
ture can be determined from current option price via the implied volatility tree
model, as illustrated in Derman (1996), much as forward rates can be extracted
from the market’s current bond yield. Your personal view of future volatilities can
be used to produce your own future volatility tree. The chosen tree determines all
future options values. Derman formulated three different types of market regimes
in which different volatility trees exist:
(a) Range-bounded regime, where future index moves are likely to be constrained
within a certain range and there is no significant change in the future volatility;
(b) Trending regime, where the level of the market is undergoing some significant
change, without a significant change in future volatility;
(c) Jumpy regime, where the probability of jumps in the index level is particularly
high, so future volatility increases.
Different linear parameterization of the volatility skew for pricing and hedging
options applies in each regime. These are known as “sticky” volatility models, be-
cause each parameterization implied a different type of “stickiness” for the volatil-
21
ity in a binomial tree. Denote by σ(atm, τ) the volatility of the τ -maturity ATM
option, σ0 and S0 the initial implied volatility and price used to calibrate the tree:
(a) In a range-bounded market, Derman proposed that skews are parameterized
by “sticky strike” model:
σ(K, τ) = σ0 − b(τ)(K − S0) (11)
So, for the given current skew which is determined by current option price and
index level S0, the future volatility of an option with a particular strike σ(K, τ)
remain unchanged as index moves. This model attribute to each option of a def-
inite strike a different volatility tree. As the index moves, all that happens is the
root of each tree is moved to the new index level. The same tree is still used to
price the option with fixed strike. Since σ(atm, τ) = σ0− b(τ)(S−S0), this model
implies that σ(atm, τ) decreases as the index increases. The rules are illustrated
graphically in Figure 2.
(b) For a stable trending market, skews are parameterized by the “stick moneyness”
model:
σ(K, τ) = σ0 − b(τ)(K − S) (12)
So, fixed strike volatility σ(K, τ) increases with the index level S, but the volatility
stays constant with respect to the moneyness of the option. The option that is
10% out of the money after the index moves should have the same implied volatil-
ity as the 10% OTM option before the index moves. That is, it is the moneyness
of the option that determines the future volatility in the tree. You can visualize
this in Figure 3: as the index moves, the moneyness of the option changes and we
consequently move to a tree of different volatility structure, the one corresponding
22
to the current option moneyness. Since σ(atm, τ) = σ0, this model implies that
σ(atm, τ) is independent of the index.
(c) In jumpy markets, skews are parameterized by the “sticky tree” model:
σ(K, τ) = σ0 − b(τ)(K + S) + 2b(τ)S0 (13)
So fixed strike future volatility σ(K, τ) decreases as the index increases. In this
model, the future volatilities are no longer constant regardless of strike or money-
ness. However, current volatility skew determines one unique tree for the evolution
of future volatilities (compared to multiple trees in (a) and (b)) that can be used
to price all options. Since σ(atm, τ) = σ0 − 2b(τ)(S − S0), the σ(atm, τ) also
decreases as index increases, twice as fast as the fixed strike volatilities. Figure 4
illustrate how the unique tree is built.
In fact, Derman’s models yield the same relationship between fixed strike volatility
deviation from at-the-money volatility and the current index price, namely,
σ(K, τ)− σ(atm, τ) = −b(τ)(K − S) (14)
with different specification for the behavior of at-the-money volatility in relation
to the index in each regime, namely,
(a)Range-bounded: σ(atm, τ) = σ0 − b(τ)(S − S0)
(b) Stable trending: σ(atm, τ) = σ0
(c) Jumpy market: σ(atm, τ) = σ0 − 2b(τ)(S − S0)
So all three models imply the same, positive correlation between the index level S
and the deviation σ(K, τ)− σ(atm, τ).
23
Derman’s models provide insight on how to remedy the instability of BSIV func-
tion in two respects:
First, both DFW (1998) and our study in section 2 examine whether the levels of
Black-Scholes implied volatilities of today’s options with a certain time to maturity
(τ) could predict the levels of volatilities of tomorrow’s options with different time
to maturity (τ − 1). The conclusion of instability is made on the empirical results
for answering this question. However, the stability of BSIV function or the rela-
tive shape of volatility surface may depend on the maturity. In other words, stable
function may exist if the maturity of all options are controlled to be the same. All
sticky volatility models are constructed on the restriction of same time to maturity.
More importantly, this model set up a time-invariant volatility function—the pos-
itive correlation between the index level S and the deviation σ(K, τ)− σ(atm, τ),
at the same time identifying and explicitly modelling the random factor σ(atm, τ)
which drive changes in the individual implied volatility.
3.2 Dynamic Black-Scholes Implied Volatility Function
Further investigation on our data raises some doubt on the validity of a linear
parameterization given by equation (14). For most of the trading days from March
1, 1996 to 31 December 31, 1996, the data reveals that the volatility skew, with
respect to the strike price, actually implies a similar skew of the σ(K, τ)−σ(atm, τ).
The skew becomes smoother with the increase in time to maturity. Figure 5-1
and 5-2 illustrate typical patterns in S&P 500 implied volatilities skew and the
skew of σ(K, τ)−σ(atm, τ). If Derman’s linear parameterization is valid, then the
24
σ(K, τ)−σ(atm, τ) should be a negative straight line with slope −b(τ) rather than
a curve in figure 5-2. In this connection, we should take the tilt and curvature
into consideration when modelling this skew pattern. Considering we proved in
Section 2 that maturity adjusted moneyness ln(K/F )√τ
could best characterize the
Black-Scholes volatility skew, we refine equation (14) as:
σ(K, τ)− σ(atm, τ) = α0 + α1ln(K/F )√
τ+ α2[
ln(K/F )√τ
]2 (15)
As ATM option implied volatility has different sensitivity to changes in index level
under different market regime, to implement Derman’s sticky models, we have
to first identify which volatility regime is likely to prevail in the near future. In
fact, equation(14) is a generalization of all market regimes, regardless of how ATM
option implied volatility trends in individual market regimes. Then it’s valid to
generalize the process of ATM option implied volatility in one model.
Two properties of Black-Scholes implied volatility provide insight on how the pro-
cess of at-the-money option implied volatility could be described. First, an inverse
time series relation exists between the stock returns and volatility changes (This
feature is also obvious in Figure 1). A more interesting feature of this negative
relation is the leverage or asymmetric effect of bad news and good news on the
volatility. An unexpected drop in the underlying asset price (bad news) increases
volatility by a larger amount than an unexpected rise in price (good news) de-
creases the volatility. This asymmetric effect was first discovered by Black (1976)
and confirmed by findings of French, Schwert and Stambaugh (1987) and Schw-
ert (1990). Second, volatility tends to be mean-reverting. If current volatility is
25
historically low, then there is an expectation that it will increase. If the current
volatility is historically high, there is an expectation that it will decrease. Consid-
ering these properties, we could specify the ATM option implied volatility in the
process of
σ(atm, t) = ω + β1σ(atm, t− 1) + β2 max(0,−Rt) (16)
where Rt = St/St−1 − 1.
The first lag of at-the-money volatility is to incorporate the volatility mean re-
version, thus β1 is expected to be less than 1. The most recent asset return is
to incorporate the asymmetric effect of index return on implied volatility and β2
is expected to be bigger than zero. When index level rises, implied volatility is
expected to drop, then β1< 1 and max(0,−Rt) = 0 is consistent with such inverse
relation. When index level drops, implied volatility is expected to rise, this is
consistent with max(0,−Rt)>0.
To this end, the dynamic BSIV function is defined so that each option’s Black-
Scholes implied volatility depends on the level of the ATM volatility. This is
accomplished by identifying a time-invariant function (15). The ATM implied
volatility is treated as the random factor for the implied volatility function dy-
namics and modelled by equation (16).
3.3 Methodology
Following our basic assumption of given time to maturity, we restrict our analysis
to BSIV function with 30, 45 and 60 calendar days to maturity, which is the most
26
typical and relevant period length in our study samples.
Obviously, options with the desired time to maturity are not always available, so
we use the linear interpolation of the two latest options whose time to maturity
straddle the given days to maturity:
σ(K, τ0 = 30, 45, 60) =τ0 − τ1
τ2 − τ1
σ(K, τ2) +τ0 − τ2
τ1 − τ2
σ(K, τ1) (17)
Where τ1 is the latest available expiration date before or at τ0 and τ2 is the earliest
available expiration date after τ0.
The ATM Black-Scholes implied volatility, with given time to maturity on every
day, is also obtained by linear interpolation of two options with strikes straddle
the index level.
S&P 500 closing call option data from March 1, 1996 to December 31, 1996 is also
obtained from CBOE and the same data processing procedures as Section 2.3 are
performed. We employ the data on March 1, 1996 through December 23, 1996
to estimate the dynamic BSIV function, and then use the estimated parameter to
forecast the volatilities of options with the same maturity on the last five trad-
ing days during the year. Finally, by interpolating the forecasted volatilities with
given maturity, we get the volatilities with specific maturity, which are applied to
Black-Scholes formula to predict option prices.
Using a standard Ordinary Least Square(OLS) estimation approach to estimate
equation (15), we face the problems of heteroskedasticity and serial correlation. To
address the problem of heteroskedasticity, we use the Newey-West (1987) weight-
27
ing covariance matrix. However, the Newey-West estimator fails to correct for
the serial correlation and the model was re-specified along the lines suggested
by Hendry and Mizon (1978) and Mizon (1995) by including additional variables
[(σ(K, τ)− σ(atm, τ)]t−1 in the model.6 In summary, we employ the Newey-West
approach to estimate the re-specified equation (15):
[σ(K, τ)−σ(atm, τ)]t = γ0+γ1ln(K/F )√
τ+γ2[
ln(K/F )√τ
]2+γ3[σ(K, τ)−σ(atm, τ)]t−1
(18)
Compared to the loss functions adopted in estimating the unstable BSIV functions
in Section 2, we use a different approach in estimating the dynamic BSIV function.
Therefore we could not make a comparison on the in-sample goodness of fit. For
out-of-sample pricing accuracy, we compare it with the best BSIV function (model
4) we identified in Section 2. The same re-estimation and forecasting procedures
as Section 2 are carried out for model 4.
3.4 Estimation and Forecasting Results
Table 4 reports estimates for the ATM implied volatility process used in the dynam-
ics BSIV function. This model effectively characterizes S&P 500 ATM volatility
dynamics with the adjusted R2 around 80% for all maturities. As expected, lagged
implied volatilities are helpful in predicting future implied volatility , as all β1 are
less than 1. In addition, the leverage effect is apparent. A negative return results
in an increase in ATM implied volatility, which is reflected in all β2 significantly
greater than zero.
6They argue that the existence of serial correlation may indicate model misspecification, andthus, the model should be re-specified instead of adopting the faulty alternative of correctingfor serial correlation. They demonstrate that this simple re-specification of equation (15) fromstatic one to a dynamic equation (18) will yield more consistent estimates.
28
Table 4: Estimation of At-the-money Implied Volatility Functions
Coefficient Days to Maturity30 days 45 days 60 days
ω 0.0364 0.0372 0.0416(t-value) (5.33) (5.32) (6.26)
β1 0.7481 0.7454 0.7552(t-value) 16.92 16.85 (17.79)
β2 1.0096 1.3294 1.4296(t-value) (5.92) (7.54) (8.05)
Adjusted R2 0.7781 0.8133 0.8029
Table 5: Estimation of Dynamics Black-ScholesImplied Volatility Function
Coefficient Equation (15) Equation (18)30 days 45 days 60 days 30 days 45 days 60 days
γ0 -0.0002 -0.0023 -0.0019 -0.0002 -0.0022 -0.0017(t-value) (-4.93) (-7.29) (-5.65) (-4.67) (-6.94) (-5.09)
γ1 -0.2044 -0.2152 -0.2248 -0.1892 -0.2068 -0.2066(t-value) ( 69.37) (88.31) (70.75) (45.89) (54.29) (45.10)
γ2 0.5339 0.4280 0.3669 0.4968 0.4122 0.3410(t-value) (31.26) (25.59) (14.91) (27.01) (23.42) (13.68)
γ3 0.0878 0.0432 0.0907(t-value) (5.25) (2.86) (5.45)
Durbin-Waston 1.62 1.82 1.62 2.01 2.07 2.03Adjusted R2 0.7290 0.7807 0.7321 0.7328 0.7815 0.7359
To test the validity of including a lagged dependent variable in equation (15), we
apply Newey-West method to both equation (15) and equation (18), with the re-
sults reported in Table 5. The inclusion of [σ(K, τ)− σ(atm, τ)]t−1 does not lend
itself to significant economic interpretation, demonstrated by minor changes in the
adjusted R2. Moreover, there are no material changes in all parameter estimates
and the serial correlation has been well accounted, as the Durbin-Waston statistics
29
are all around 2 in equation (18). So the re-specified function proved to be effective.
For model 4, we re-estimate the function everyday from December 23, 1996 to De-
cember 30, 1996 and then use today’s parameter estimates to forecast tomorrow’s
volatilities, which are applied to forecast option prices. RMSE and MRE are then
calculated everyday and averaged over the last five days. For dynamic BSIV func-
tion, we use the estimation result in Table 5 to forecast all the call option closing
prices on the last five days at one time. Table 6 compares the average pricing error
of the dynamic BSIV function and the model 4.
Although the results of forecasting accuracy for ITM options are mixed between the
two models, with each model outperforming the other in some maturity categories,
the dynamic BSIV function is clearly superior to model 4 in pricing ATM and OTM
options, with the pricing error averagely decreasing 4 percent for OTM options.
Actually, our estimation method is a more favorable treatment for model 4, which
incorporates the new information everyday by the re-estimation. Therefore we
would expect it to price the options more accurately. One possible reason for the
better performance of the dynamic BSIV function is that the effect of random
factors on both the σ(K, τ) and σ(atm, τ) are partially cancelled out by σ(K, τ)−
σ(atm, τ), the skew of σ(K, τ)− σ(atm, τ) thus may remain relatively stable over
time, and isn’t affected by the arrival of new information.
30
Table 6: Average Forecasting Error of Dynamic Black-ScholesImplied Volatility Function
Days to Moneyness Competitive ModelsMaturity K/S Dynamic Function Model 4
RMSE 7-45 [0.90, 0.94) 2.41 2.27[0.94, 0.97) 1.74 1.69[0.97, 1.00) 1.40 1.32[1.00, 1.03) 0.76 0.80[1.03, 1.06) 0.32 0.41[1.06, 1.10] 0.17 0.22
46-90 [0.90, 0.94) 2.73 2.64[0.94, 0.97) 2.13 2.20[0.07, 1.00) 1.86 1.80[1.00, 1.03) 1.33 1.36[1.03, 1.06) 0.72 0.81[1.06, 1.10] 0.40 0.52
MRE 7-45 [0.90, 0.94) 0.045 0.042[0.94, 0.97) 0.053 0.051[0.97, 1.00) 0.084 0.079[1.00, 1.03) 0.096 0.103[1.03, 1.06) 0.135 0.172[1.06, 1.10] 0.173 0.223
46-90 [0.90, 0.94) 0.039 0.038[0.94, 0.97) 0.051 0.052[0.97, 1.00) 0.063 0.061[1.00, 1.03) 0.067 0.068[1.03, 1.06) 0.105 0.119[1.06, 1.10] 0.112 0.147
31
4 Conclusion
Of considerable interest to both practitioners and academics is a better under-
standing of the empirical dynamics of implied volatility surface. Basing on some
findings about the ad hoc Black-Scholes valuation approach suggested in Dumas,
Flemming and Whaley (1998), our study investigates and proposes some volatility
function forms which are applied to this valuation approach.
Basing on the instability inherited in the volatility function, we first investigate
what function form best characterizes the Black-Scholes implied volatility skew
and thus reduce the fitting or forecasting error in option prices. We suggest four
competitive implied volatility functions and re-estimate each of them day to day.
The results of both in-sample fitting and out-of-sample pricing tests demonstrate
that a quadratic function of maturity adjusted moneyness ln(K/F )√τ
could reduce the
maturity-related and moneyness-related pricing error to the largest degree.
Restricted by the assumption of given time to maturity, we propose and implement
a dynamic Black-Scholes implied volatility function which is expected to remedy
the instability of previous suggested volatility functions. This dynamic function
separates a time-invariant implied volatility relation from the random factors that
drive changes in the individual implied volatilities. The random factors are incor-
porated through the ATM implied volatilities which are modelled as a function
of lagged volatility and a non-linear function of the underlying asset return. The
out-of-sample predicting results proves that the dynamic function outperforms the
known unstable BSIV function, especially in reducing the pricing error of OTM
options. This result provides strong evidence that stable BSIV function in AHBS
32
valuation method may exist or at least the stability of the volatility function could
be greatly improved under certain restrictions.
It is clear that the dynamics of option prices have not been fully explained by the
suggested Black-Scholes implied volatility function, which offer promising areas for
future research.
33
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36
Figure 1: Fixed Strike Black-Scholes Implied Volatility Time Series from March 1, 1996 to December 31, 1996 The figure graphs the time series of S&P 500 index level, the at-the-money Black-Scholes implied volatility and the Black-Scholes implied volatility for options with strike 660, 670, 680, 690 and 700. The period extends from March 1, 1996 to December 31, 1996. All the options are of 45 days to maturity. The 45-day Black-Scholes implied volatility for a particular strike is obtained by interpolation from the two latest options whose time to maturity straddles the 45 days to maturity.
0.05
0.1
0.15
0.2
0.25
0.3
0.35
960301
960401
960501
960531
960701
960731
960829
960930
961029
961127
961230
Implied Volatility
600
650
700
750
800
Index Level
ATM K=660K=670K=680K=690K=700Index Level
Figure 2: The Evolution of Fixed Strike Volatility in the Stick-Strike Model
Each binomial tree represents the evolution of future volatility which is determined by current index level and particular strike. The size of each binomial fork in the tree is intended to represent the magnitude of the future instantaneous volatility. Thus, the binomial trees of a same row illustrate the changes in volatility structure of the fixed strike option when index level moves. We assume the center column match the current negative volatility skew for strikes of 90, 100 and 110, when index level is 100. In this model, when index level move from 100 to 90 or from 100 to 110, all trees in the same row (or for fixed strike) have the same volatility structure, except that the root of the tree is relocated to the new index level. That is, the volatilities keep constant for a given strike. When we move from the top left to the bottom right in this figure, we could see that at-the-money implied volatility decreases as index level increases.
Index 90
90
100
110
Strike100 110
Current Trees
Figure 3: The Evolution of Fixed Strike Volatility in the Stick-Moneyness Model
Each binomial tree represents the evolution of future volatility which is determined by current index level and particular strike. The size of each binomial fork in the tree is intended to represent the magnitude of the future instantaneous volatility. Thus, the binomial trees of a same row illustrate the changes in volatility structure of the fixed strike option when index level moves. We assume the center column match the current negative volatility skew for strikes of 90, 100 and 110, when index level is 100. In this model, the tree has a same structure for fixed moneyness as the index level moves. So when we move from the top left to the bottom right in this figure, we could see that at-the-money implied volatility keep constant as index level increases.
IndexStrike
100
110
100 110Current Trees
90
90
Figure 4: The Evolution of Fixed Strike Volatility in the Stick-Implied Tree Model
Each binomial tree represents the evolution of future volatility which is determined by current index level and particular strike. The size of each binomial fork in the tree is intended to represent the magnitude of the future instantaneous volatility. We assume the center column match the current negative volatility skew for strikes of 90, 100 and 110, when index level is 100. In this model, current skew determines there is only one volatility tree for all the options regardless of moneyness or strike. As the index moves, we simply slide along the tree to node at corresponding index level.
Index 90 100 110Strike Current Tree
90
100
110
Figure 5-1: Black-Scholes Implied Volatility on May 16, 1996 The figure graphs the Black-Scholes Implied Volatilities for options with 30, 45 and 60 days to maturity on May 16, 1996. The implied volatility of option with given time to maturity is obtained by interpolation from the two latest options whose time to maturity straddle the given time to maturity.
0.1000
0.1400
0.1800
0.2200
0.2600
60061
062
062
563
063
564
064
565
065
566
066
567
067
568
068
569
069
570
0
Strike
Impl
ied
Vola
tility
30 days45 days60 days
Figure 5-2: Black-Scholes Implied Volatility Deviation on May 16, 1996 The figure graphs the deviation σ(K ,τ) - σ(atm ,τ) for options with 30, 45 and 60 days to maturity on May 16, 1996.
-0.0600
-0.0400
-0.0200
0.0000
0.0200
0.0400
0.0600
0.0800
0.1000
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610
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635
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Strike
Devi
atio
n 30 days45 days60 days