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Working Paper Series
Department of Economics
Alfred Lerner College of Business & Economics
University of Delaware
Working Paper No. 2005-12
The Impact of Serial Correlation on Option Prices in a NonFrictionless Environment: An Alternative Explanation for
Volatility Skew
Joel S. Sternberg
John S. Ying
The Impact of Serial Correlation on Option Prices in a Non-Frictionless Environment:
An Alternative Explanation for Volatility Skew
Joel S. Sternberg a
John S. Ying b
a
Corresponding Author
Graduate School of Management
Clark University
950 Main Street
Worcester, MA 01610
[email protected]
Ph: 508-793-7702
Fax: 240-208-7256
b
Department of Economics
313 Purnell Hall
Lerner College of Business and Economics
University of Delaware
Newark, DE 19716
[email protected]
JEL Codes: G13—Contingent Pricing; G12—Asset Pricing
Keywords: Options, Implied Volatility, Volatility Skew
The Impact of Serial Correlation on Option Prices in a Non-Frictionless Environment:
An Alternative Explanation for Volatility Skew
Abstract
A persistent anomaly in option pricing is the volatility skew. Many have attempted to
explain it with stochastic volatility and/or jump diffusion models with mixed results. We
propose a model that incorporates positive serial correlation in the stock price process
and test it on empirical data for four “momentum” stocks and their heavily traded
options. Although the notion of serial correlation seems to challenge the notion of
arbitrage enforced option valuation, in fact, the existence of transaction costs,
discontinuities and other frictions in the market allow for fairly wide arbitrage-free
bounds on option prices. Within these bounds, serial correlation seems to explain not
only the often-noted skew in stock option prices, but also the rarely noted upward bias of
implied volatilities over actual volatilities.
I. Introduction
Since the Black-Scholes (1973) model was proposed, it has become widely used professionally
and widely cited academically. This is in spite of the fact that the assumptions underlying the BlackScholes model are known to be inconsistent with empirical observation. As testimony to the robustness of
the model, however, practitioners and professionals alike have chosen to make relatively minor
modifications in the assumptions of the model rather than throwing the proverbial baby out with the bath
water. These modifications stand by the seminal Black-Scholes conception of an arbitrage based optionpricing model in which a perfect hedge can be formed between the option and the underlying asset.
The inconsistency that has received the most attention is the so-called volatility smile. The
volatility smile is the oft-noted observation that in-the-money and out-of-the-money options for many
underlying assets tend to have higher implied volatilities than at-the-money options. The most cited
explanation for this is the presence of stochastic volatility. Work by Hull and White (1987), Wiggins
(1987), Johnson and Shanno (1987) and Heston (1993) attempts to model this phenomenon. The presence
of stochastic volatility when there is no correlation with stock prices, results in leptokurtosis, or fat tails,
reflecting a greater likelihood of achieving extreme outcomes. However, the most standard versions of
these models result in lower at-the-money option prices, which may become more pronounced with longer
time to expiration.
Introducing negative correlation between volatility and stock prices can lead to the volatility skew.
It has been most noted with equity options. With the volatility skew, implied volatility declines with higher
strike prices. Some have attempted to attribute this to the impact on corporate leverage of higher stock
prices. Specifically, as stock prices rise, corporations become less leveraged with respect to their debt-toequity ratios, and therefore exhibit less volatility.
Das and Sundaram (1999) test the ability of stochastic volatility and jump diffusion models to
explain the volatility skew and the term structure of implied volatility. They find that the stochastic
volatility model cannot, under reasonable assumptions, generate sufficiently pronounced smiles for shortterms options and that it does not flatten sufficiently for longer-term options. The jump models are found to
monotonically increase (at-the-money) implied volatilities over time and flatten the implied volatility smile
much more quickly than is observed empirically.
Rubinstein (1994) suggests that the volatility skew reflects a “crash” mentality. Adding validity
to his claim is the fact that this pattern has only become the norm from a point prior to the 1987 crash, as
noted by Bates (1990), through the present period.
A third phenomenon that we observe, but is rarely noted, is that implied volatilities tend to be
higher than historical volatilities. We document this for a small sample of recently, heavily traded options.
These results are broadly consistent with Canina and Figlewski (1993), who also find that implied volatility
has no predictive power for future realized volatility. Christensen and Prabhala (1998) dispute their
findings, but still find that implied volatility is upwardly biased. Stochastic volatility, jumps, nor fear of
crashes can explain this. Regardless of the shape of the volatility curve, the plot generally lies everywhere
above measured volatility.
We believe that there may be a ready explanation for this phenomenon as well. A wide swath of
finance literature has documented the serial correlation or predictability of asset returns. In fact, positive
serial correlation has been noted by Lo and MacKinlay (1988) and Poterba and Summers (1988) for stock
indices, and Cutler, et al. (1990) for stocks, bonds, gold, and foreign exchange. Jegadeesh and Titman (1993)
term this serial correlation, momentum, and observe that portfolios of outperforming stocks tend to
continue to outperform in subsequent three-month to one-year periods. Along with Grundy and Martin
(2001), they attribute momentum to firm specific factors. Lo and MacKinlay (1990) point out that
momentum in indexes could be caused by any of three factors: 1) higher unconditional expected returns; 2)
positive return autocorrelations; or 3) negative, lagged, cross-covariances.
Barberis, et al. (1998), Daniel, et al. (1988) and Hong and Stein (1999) develop behavioral models
of momentum that involve underreaction due to positive autocovariances. Lewellen (2002), however,
attributes the phenomenon to overreaction characterized by negative, lagged, cross-covariances between
stocks, but Chen and Hong (2002) rebut his findings. Regardless of the current state of the debate, it has
become common practice to adjust for momentum in calculating abnormal returns on individual stocks
since first proposed by Barber, et al. (1999), a seemingly tacit acceptance of some degree of positive serial
correlation.
Consistent with option theory, Lo and Wang (1995) show that autocorrelation, or other processes
implying return predictability, should not affect option prices. However, they note that a sampling of the
unconditional volatility will yield measures of standard deviation that, if utilized unadjusted in the Black-
Scholes model, will result in incorrect option prices. They propose a correction to the measured,
unconditional volatility so that it yields, presumably, arbitrage-free option prices.
To wit, measured, unconditional variances must be scaled upward for negative serially correlated
returns, and downward for positive autocorrelation, lest the former result in prices that are too low and the
latter in prices that are too high. Yet, ironically, the implied volatilities of shorter-term options, where
positive autocorrelation manifests, tend to be consistently higher than historical, measured, realized
volatilities. To reconcile theoretical option pricing with observed option pricing under a Black-Scholes
type framework, it would appear that a scaling up of measured, realized volatilities might be in order.
Is it merely a coincidence that autocorrelations tend to switch from positive to negative over
longer intervals in a manner consistent with the flattening of the volatility smile/smirk? But, per Lo and
Wang (1995), these negative autocorrelations are specifically not an explanation for lower implied
volatilities.
We believe that, at times, the simple explanation might just be the best. First hand testimony from
the trading room of the largest provider of portfolio insurance (a.k.a. dynamic asset allocation) on
October 19, 1987, suggests that engaging in portfolio rebalancing to maintain a perfect hedge is
impossible.1 The attempt to do so for individual stocks, as opposed to an index, would be, if anything, even
more formidable. There is no reason to believe that individual option market makers or traders are any
better at perfecting the impossible job of constructing perfect hedges than the largest provider of portfolio
insurance was back in 1987. After all, market makers make markets on a full range of options for a host of
stocks, so they cannot be expected to execute rebalancing trades with perfect, robotic precision.
Professional arbitrageurs, on the other hand, have to incur greater transaction costs.
It is not our intention to throw out over 30 years of options thought. Rather, much like Leland
(1985), Boyle and Vorst (1992), and Constantinides and Zariphopoulou (1999, 2001), we suggest that
options should be seen as having arbitrage-free bounds, and that options may trade more or less freely
within those bounds. We believe that serial correlation can lead options to trade near the top or bottom
range of those bounds without attracting would-be arbitrageurs, who are fully aware of the transaction costs
and discontinuities that they encounter in attempting to form a perfect hedge.
If it is acknowledged that serial correlation or momentum may exist and be exploitable through
static momentum- or technically-based trading rules (consider that 90% of managed, futures funds trade on
trends and momentum), divergences from theoretical option prices should not cause great surprise.
Exploiting option mispricing for profit requires a dynamic rebalancing strategy, a much more formidable
obstacle to success. Furthermore, options are a unique way to capture momentum, since calls and puts
effectively increase exposure to the underlying asset as it moves in the anticipated direction.
II. Stock Process Model
We derive a stock process in which we explicitly define the variance and embed serial correlation
into the stock price process. Through Monte Carlo simulation techniques, we are able to observe the
manner in which the drift process affects measured variance. Since simulations require the evolution of
asset prices over small time intervals to accurately reflect distributional assumptions, we will focus on
positive serial correlation or momentum, which is observed over shorter intervals. (Negative
autocorrelations have been observed over some small intervals, but much, if not all of the effect, is due to
“bid-ask bounce.”)
The standard stock price process under Black-Scholes is:
dS/S = µdt + σZ√dt,
(1)
where µ is the drift, σ is standard deviation and Z is a random draw from a standardized normal
distribution with moments (0,1).
We add a simple component for autocorrelation.
dS/S = µdt + ρ[(dS-1/S-1) - µdt] + σZ√dt,
(2)
where (dS/S)-1 is the prior period return and ρ is the first-order autocorrelation coefficient.
Sternberg and Ying (1993) have shown that the correlation component can be embedded into the
drift process. Theoretically, it will disappear in the risk-neutralized version of equation (2). However, de
facto, Lo and Wang (1995) have shown that serial correlation will affect the unconditional variance for an
Ornstein-Uhlenbeck process. The same holds for our simplified correlation process. It follows from
equation (2) that
dS-1/S-1 = µdt + ρ[(dS-2/S-2) - µdt] + σZ-1√dt,
(2a)
so the continuous time, infinite horizon expression for equation (2) is:
dS/S = µdt + ∫0∞(ρnσZ-n√dt)
(3)
where ρn is the nth order autocorrelation, while the discrete time sampling analog for the
continuous time expression is:
dS/S = µdt + Σ0∞(ρnσZ-n√dt)
(4)
or
dS/S = [µdt + Σ1∞(ρnσZ-n√dt)] + σZ√dt,
(4a)
where [µdt + Σ1∞(ρnσZ-n√dt)] is the conditional, periodic drift.
In the limiting case where ρ is zero, equation (4) collapses back to equation (1). First-order
negative autocorrelation can be similarly induced by making ρ negative, but would occur over longer
measurement periods that are difficult to simulate.
As can readily be observed above and confirmed by Lo and Wang (1995), this simple
autocorrelation process increases the effective, unconditional variance when there is positive
autocorrelation and decreases it when there is negative autocorrelation. Since Z, Z-1 and Z-n are by
definition uncorrelated, Z’s are normally distributed (0,1) and the nth order correlation is given by the
function ρn, the unconditional variance of discrete time returns is given by:
Σ0∞ (ρ2nσ2dt) = (1/(1-ρ2))σ2dt
(5)
and the annualized standard deviation by:
σ√(1/(1-ρ2))
(5a)
For this process to yield Black-Scholes prices, a scalar correction of √(1-ρ2) must be applied to
the unconditional volatility to convert it to the relevant conditional volatility for option pricing purposes.
Alternatively, the volatility can be maintained by evolving the stock price by:
dS/S = µdt + ρ[(dS-1/S-1) - µdt] + (1-ρ2)1/2σZ√dt = dS/S = µdt + ρσZ-1√dt + (1-ρ2)1/2σZ√dt,
with discrete time, infinite horizon equivalent,
dS/S = µdt + Σ0∞[ρnσZ-n(1-ρ2)1/2√dt],
yielding an unconditional variance of σ2dt.
However, this alternative stock process, while maintaining the unconditional variance, does not
comport with the fact that serial correlation imposed on an otherwise stationary process should result in a
change in measured variance.
The question is how realistic it is to actually apply the scalar correction to the measured variance
arising from equation (4). The ability to replicate options at Black-Scholes prices is predicated on
continuous, costless rebalancing.
We contend that the Monte Carlo prices, uncorrected for serial correlation by the Lo and Wang
(1995) adjustment, may well represent market prices so long as they fall within reasonable bounds. Indeed,
of note is the fact that implied volatilities for many assets exceeds even their uncorrected, unconditional,
measured standard deviation over shorter horizons when serial correlation is likely to positive. Therefore,
it appears that the market might, correctly or incorrectly, be using unconditional volatilities in the pricing of
options.
III. Methodology
Valuation will be done by Monte Carlo simulation, supplemented by Boyle’s (1977) control
variate method. Under the assumption of risk neutrality, the incremental movement in the stock price is
given by,
dS = rSdt + σSZ√dt,
while for the simple autocorrelation process in discrete time,
dS = rSdt + ρ[(dS-1/S-1) - rdt]S + σSZ√dt,
where r is the logarithmic risk-free rate of interest.2
We introduce values of ρ = .05 and σ = .4 and begin the simulation with a stock price of 100 at
time 0 under the assumption that Σ1∞(ρnσZ-n√dt) = 0. An autocorrelation coefficient of zero, of course,
reverts to the standard Black-Scholes stock process.
The stock is grown for 100 periods, where dt = .0025, corresponding to three months or 1/4 years.
Each scenario is run 10,000 times to obtain 10,000 possible stock outcomes. Strike prices ranging from
50% (of the stock price) in-the-money to 50% out-of-the-money3, are subtracted from each of the stock
price outcomes where the call price at expiration is calculated as the Max[S-X, 0]. Each of the call prices is
then discounted to obtain present values for them. The call prices are then averaged for each strike price to
obtain their simulated values. The process is repeated using a variable discount rate, where the periodic
discount rate is reset each period to rdt + Σ1∞(ρnσZ-n√dt), consistent with the periodic, conditional drift
from equation (4a). The results are substantially similar and are therefore not reported in the paper.
The same procedure is applied to the standard stock process when ρ = 0 to obtain a Black-Scholes
value from the simulation. The actual Black-Scholes values are then calculated by the Black-Scholes
closed-form equation, and the difference between the two methods is ascertained. Under Boyle’s (1977)
control variate method, this correction is then applied to each of the other option values obtained via Monte
Carlo simulation. The control variate method vastly increases the efficiency of the simulation, reducing the
number of simulations that must be run.
Lo and Wang (1995) suggest that the autocorrelation should not result in different option prices.
More specifically, they point out that allowing autocorrelation to affect option prices violates the theoretical
arbitrage-free condition for option pricing. For example, if positive autocorrelation did result in higher call
prices, arbitrageurs would sell those options and replicate them dynamically. However, this not only fails
to consider the vega risk and jump risk in replicating options, but transaction costs as well.
While by no means minimizing vega and jump risk, we focus on the transaction costs involved in
performing dynamic replication. For comparison purposes, we will apply bounds established by Boyle and
Vorst (1992) in discrete time using a binomial lattice and proportional transaction costs.
Their formula for boundaries on implied variance is σ2[1 +/- (2k√n)/(σ√T)], where k is the
transaction cost as a percent of the stock price and n is the number of times the hedge is rebalanced. If we
assume that the hedge is rebalanced twice a day for the three-month option, √n is approximately 12. If k is
.25%4, implied option volatilities for stock with a standard deviation of 46% could lie anywhere within a
range of 39% and 52% without offering an arbitrage opportunity. Only our 40-50% in-the-money options
fail to fit within the boundary, but they are consistent with observed, implied volatilities. Furthermore,
deep-in-the-money strike prices are likely to be crossed only if the stock experiences a discontinuous
decline, an event that cannot be dynamically hedged.
Boyle and Vorst (1992) point out that risk adverse option traders might be willing to sacrifice a
degree of hedging accuracy for reduced transaction costs.5 However, the transaction costs required in their
binomial model do not, in fact, assure a perfect hedge outside of the theoretical binomial framework (no
matter how often the rebalancing occurs), so an even wider band may actually be appropriate.
To compare our simulation results to an empirical data sample, we analyze data for options with
90 days to expiration on Amazon.com, Cisco Systems, eBay and Yahoo! for the period 2001-2004. We
chose these options for two reasons. First, they represent some of the most active options trading at that
time. Secondly, none of them pay dividends, which enables the use of the Black-Scholes model for option
pricing.
Data was obtained from IVolatility.com.6 We gathered implied volatility data on options ranging
from 50% out-of-the-money to 50% in-the-money with 90 days to expiration. In addition, we calculated the
volatilities that were realized by the stocks over those same periods. That information was combined to
form a single plot of implied volatilities relative to moneyness. Similarly, the realized volatilities, which
are independent of moneyness, were combined to form a basis of comparison.
First-order autocorrelations were calculated on monthly, logarithmic returns from the first January
these stocks began trading publicly. For Cisco, Yahoo!, Amazon.com and eBay, those dates are January
1991, 1997, 1998 and 1999, respectively. The use of monthly return data minimizes the impact of bid/ask
bounce, which can account for a large portion of daily, and even weekly, return volatility.
IV. Results
Monthly autocorrelations are computed for Cisco, Yahoo!, Amazon.com and eBay over several
time periods by regressing returns on lagged, monthly returns. A summary of the results is presented in
Table 1. Since individual stock returns are related to market return, and the market index itself has been
found to be autocorrelated, a second regression is run in which returns are regressed on monthly, lagged
returns and the contemporaneous monthly market return. A summary of these results is presented in Table
2. The few negative correlations are highlighted.
It is evident from both Tables 1 and 2 that the autocorrelations are overwhelmingly positive.
Furthermore, the correlations become more positive over longer sampling periods. However, since both the
magnitude of the correlations and the sample size is relatively small, few are significant at the 5% level.
Nevertheless, the preponderance of positive correlations over several time periods lends support to positive
serial correlation as a basis for momentum trading.
Dennis and Mayhew (2002) find that stocks with higher betas tend to have steeper smiles. All of
the stocks in our sample are generally recognized as high beta stocks and, anecdotally, were popular with
“momentum” traders. Perhaps serial correlation and momentum are both related to a stock’s beta.
Our simulation uses a 46.13% standard deviation, a 2% risk-free rate (ln[1.02]) and first-order,
positive serial correlation of 5%. The standard deviation was chosen to conform to the actual, 46.25%,
unconditional volatility observed for the four stocks over the sample period, an assumed autocorrelation of
5% and equations (5) and (5a). The simulated option prices are presented in Table 3. These values
generate a volatility skew that is remarkably similar to what we observe empirically.
Figure 1 presents a comparison of the implied volatilities observed for the four stock options and
those derived from the simulation. Both demonstrate the volatility skew that has been noted by many
others. The implied volatilities decline markedly moving from the 50% in-the-money options toward
higher strike prices, leveling out at substantially at higher strike prices.7 The primary difference between
the two plots is that, empirically, the implied volatilities turn up slightly at the highest strike prices.
However, the increase is marginal, and may well be explained by the super-imposition of a
complementary return-generating model that accommodates discrete jumps.
Perhaps most notable is a comparison of these implied volatilities to actual volatilities realized
from 90 days before expiration to the expiration date, which are also noted in Figure 1. These measured
volatilities are everywhere below the implied volatilities. Therefore, in addition to skew that has been so
often noted, there also appears to be a distinct implied volatility bias. In fact, any cursory look at the web
site IVolatility.com confirms that implied volatilities on 90-day options are consistently above the trailing
90-day historical volatilities. Our graph simply goes one step further by comparing implied volatilities to
the more meaningful realized volatility over the forward-looking period, since implied volatilities in
theory proxy for expected, future, volatility.
It is important to note that, to the extent there is positive serial correlation, the realized,
unconditional variances, per Lo and Wang (1995), actually overstate the likely conditional volatility that is
presumably relevant for option pricing. However, the correction we derive in equations (5) and (5a)
magnifies the bias when applied to the unconditional volatility by lowering the conditional volatility even
further below the implied volatilities observed.
While models of stochastic volatility have a problem explaining the more distant expirations of the
implied volatility term structure, a serial correlation-based model offers a potential solution to the
conundrum. Since autocorrelation tend to go from positive to negative over longer periods of time, we
would expect to see a flattening of the skew over longer-term options.
Unfortunately, it is difficult to perform simulations embodying negative autocorrelation. In order
for the simulation to achieve a degree of accuracy, time steps must be sufficiently small. Yet, developing a
process that can switch from positive to negative autocorrelation over time is difficult. Long-term negative
serial correlation cannot be easily induced through a simple process without also inducing short-term
negative serial correlation.
V. Summary
Efforts to explain the volatility skew observed in stock options have fallen short of success. We
consider the fact that the non-frictionless nature of stock and options markets allow for broad arbitrage-free
bounds on option prices, and that option prices may settle anywhere within those bounds. In recognition of
its role as the focus of many professional trading strategies, momentum is imposed on the standard stock
process through autocorrelation. A Monte Carlo simulation is performed to discern the impact of this
process on option prices. We find that autocorrelation fairly accurately reproduces the volatility skew
under the most parsimonious assumptions. Furthermore, since autocorrelation tends to shift from positive
to negative when moving from monthly to yearly return intervals, it may also explain the mitigation of
skew observed in longer-term options.
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1
One co-author was an officer of the company working at the portfolio insurance trading desk at the time of the
Crash.
2
For sufficiently small sampling intervals approaching continuous time, dS = (r - σ2/2)Sdt + σSZ√dt, provides
more accurate estimation for option pricing purposes. The application of the control variate method, however,
obviates this concern.
3
Hull and White’s (1987) model for stochastic volatility with no correlation between the stock and volatility
processes results in lower option values from 10% in-the-money to 10% out-of-the-money, but greater values
outside of that range.
4
The transaction costs are somewhat mitigated by the ability of the option market maker in the normal course of
business to net out delta exposure through offsetting option positions.
5
For this reason others, like Constantinides and Zariphopoulou (1999,2001) and Constantinides and Perrakis (2002)
create bounds specifically derived from a broad class of utility functions.
6
Ivolatility.com interpolates from longer-term options to estimate implied volatilities for deep in-the-money and far
out-of-the-money options. Actual 90-day options are used for all other calculations.
7
It is interesting to speculate how this pattern would manifest itself in currency options, since a far out-of-the-
money call on a upward-trending Euro is effectively the same as a far out-of-the-money put on a downward-trending
dollar. This could lead to a full-blown smile.
Table 1: Regression of Monthly Returns on Lagged Returns
CSCO
Number
of Years
Period
Start
4 Jan-91
Jan-92
Jan-93
Jan-94
Jan-95
Jan-96
Jan-97
Jan-98
Jan-99
Jan-00
Jan-01
End
Dec-94
Dec-95
Dec-96
Dec-97
Dec-98
Dec-99
Dec-00
Dec-01
Dec-02
Dec-03
Dec-04
Mean
5
Jan-91
Jan-92
Jan-93
Jan-94
Jan-95
Jan-96
Jan-97
Jan-98
Jan-99
Jan-00
Dec-95
Dec-96
Dec-97
Dec-98
Dec-99
Dec-00
Dec-01
Dec-02
Dec-03
Dec-04
Jan-91
Jan-92
Jan-93
Jan-94
Jan-95
Jan-96
Jan-97
Jan-98
Jan-99
Dec-96
Dec-97
Dec-98
Dec-99
Dec-00
Dec-01
Dec-02
Dec-03
Dec-04
Jan-91
Jan-92
Jan-93
Jan-94
Jan-95
Jan-96
Jan-97
Jan-98
Dec-97
Dec-98
Dec-99
Dec-00
Dec-01
Dec-02
Dec-03
Dec-04
Mean
6
Mean
7
Mean
YHOO
Rho
1.69%
15.14%
18.82%
16.20%
-4.41%
-8.29%
12.31%
21.71%
1.98%
-0.05%
-3.81%
6.48%
2.51%
8.00%
12.60%
12.95%
-7.19%
5.67%
20.07%
5.42%
3.20%
-0.61%
6.26%
-0.86%
6.28%
10.29%
8.47%
6.39%
16.33%
6.25%
5.40%
2.70%
6.81%
0.11%
4.78%
6.82%
14.68%
17.27%
3.92%
6.03%
5.19%
7.35%
Period
Start
End
Jan-97
Jan-98
Jan-99
Jan-00
Jan-01
Dec-00
Dec-01
Dec-02
Dec-03
Dec-04
Jan-97
Jan-98
Jan-99
Jan-00
AMZN
Rho
4.84%
10.32%
10.27%
24.06%
17.24%
13.34%
Period
Start
Dec-01
Dec-02
Dec-03
Dec-04
Rho
End
Rho
16.62%
10.91%
18.21%
10.37%
14.03%
Jan-99
Jan-00
Jan-01
Dec-02 4.03%
Dec-03 12.98% Grand
Dec-04 14.84% Mean
10.62% 11.12%
Dec-01 8.28%
Dec-02 8.31%
Dec-03 12.89%
Dec-04 24.32%
13.45%
Jan-98 Dec-02 15.62%
Jan-99 Dec-03 12.36%
Jan-00 Dec-04 13.97%
13.98%
Jan-99
Jan-00
Dec-03 4.57% Grand
Dec-04 12.51% Mean
8.54% 10.56%
Jan-97
Jan-98
Jan-99
Dec-02 6.84%
Dec-03 8.61%
Dec-04 13.18%
9.54%
Jan-98 Dec-03 14.84%
Jan-99 Dec-04 9.35%
12.09%
Jan-99
Dec-04
Jan-97
Jan-98
Dec-03
Dec-04
6.83%
8.44%
7.63%
Jan-98
Jan-99
Jan-00
Jan-01
End
eBAY
Period
Start
Jan-98 Dec-04 13.24%
13.24%
Grand
4.60% Mean
4.60% 8.26%
Grand
Mean
9.41%
Table 2: Regression of Monthly Returns on Lagged Returns and S&P 500 Returns
CSCO
Number
Period
of Years
Start
4 Jan-94
Jan-95
Jan-96
Jan-97
Jan-98
Jan-99
Jan-00
Jan-01
Mean
5 Jan-94
Jan-95
Jan-96
Jan-97
Jan-98
Jan-99
Jan-00
Mean
6 Jan-94
Jan-95
Jan-96
Jan-97
Jan-98
Jan-99
Mean
7 Jan-94
Jan-95
Jan-96
Jan-97
Jan-98
Mean
End
Rho
Dec-97 16.50%
Dec-98 -4.01%
Dec-99 -3.84%
Dec-00 11.04%
Dec-01 17.86%
Dec-02 9.66%
Dec-03 3.32%
Dec-04 -0.40%
6.27%
Dec-98 13.09%
Dec-99 -3.66%
Dec-00 6.06%
Dec-01 15.14%
Dec-02 9.84%
Dec-03 7.06%
Dec-04 2.12%
7.09%
Dec-99 10.90%
Dec-00 6.05%
Dec-01 12.27%
Dec-02 8.12%
Dec-03 7.80%
Dec-04 6.14%
8.55%
Dec-00 14.83%
Dec-01 12.47%
Dec-02 5.87%
Dec-03 6.55%
Dec-04 7.37%
9.42%
YHOO
Period
Start
End
AMZN
Rho
Jan-97
Jan-98
Jan-99
Jan-00
Jan-01
Dec-00 5.92%
Dec-01 5.06%
Dec-02 6.48%
Dec-03 12.73%
Dec-04 2.85%
6.61%
Jan-97
Jan-98
Jan-99
Jan-00
Dec-01 6.22%
Dec-02 2.40%
Dec-03 6.43%
Dec-04 13.00%
7.01%
Jan-97
Jan-98
Jan-99
Dec-02
Dec-03
Dec-04
Jan-97
Jan-98
Dec-03
Dec-04
Period
Start
End
eBAY
Rho
Jan-98 Dec-01 8.55%
Jan-99 Dec-02 6.46%
Jan-00 Dec-03 11.49%
Jan-01 Dec-04 -2.53%
5.99%
Period
Start
End
Rho
Jan-99
Jan-00
Jan-01
Dec-02 5.02%
Dec-03 17.88% Grand
Dec-04 7.21% Mean
10.04% 7.23%
Jan-98
Jan-99
Jan-00
Dec-02
Dec-03
Dec-04
9.11%
5.25%
8.55%
7.64%
Jan-99
Jan-00
Dec-03 4.64% Grand
Dec-04 15.08% Mean
9.86% 7.90%
3.34%
1.96%
6.67%
3.99%
Jan-98
Jan-99
Dec-03
Dec-04
7.46%
3.23%
5.35%
Jan-99
Dec-04
2.79%
1.99%
2.39%
Jan-98
Dec-04
6.74%
6.74%
Grand
3.76% Mean
3.76% 5.41%
Grand
Mean
6.18%
Table 3: Simulated Option Values w/46.13% Volatility, 5% Serial Correlation, 3-Month Expiration
Strike Price
% (in)out-of-the-money
Simulated Value
Black-Scholes Value
(50)
50.52
50.25
(40)
40.67
40.37
(30)
31.20
30.83
(20)
22.63
22.14
(10)
15.49
14.90
0
10.03
9.41
10
6.18
5.61
20
3.65
3.19
30
2.08
1.74
40
1.15
0.91
50
0.62
0.47
Figure 1: Comparison of Historical, Implied and Simulated Volatilities
90%
80%
70%
60%
50%
Implied Volatility
Historical Volatility
Simulated Volatility
40%
30%
20%
10%
10
%
20
%
30
%
40
%
50
%
0%
-5
0%
-4
0%
-3
0%
-2
0%
-1
0%
0%
% in/out-of-the-money