Electronic copy available at: http://ssrn.com/abstract=1524193

Valuation of VIX derivatives∗

Javier MencıaBank of Spain

<javier.mencia@bde.es>

Enrique SentanaCEMFI

<sentana@cemfi.es>

December 2009Revised: March 2011

Abstract

We conduct an extensive empirical analysis of VIX derivative valuation modelsbefore, during and after the recent financial crisis. Since the restrictive assump-tions about mean reversion and heteroskedasticity of existing models yield largedistortions during the crisis, we propose generalisations with a time varying cen-tral tendency, jumps and stochastic volatility, analyse their pricing performance,and their implications for term structures of VIX futures and options, and optionvolatility ”skews”. We find that a process without jumps for the observed VIXindex combining central tendency and stochastic volatility reliably prices VIX fu-tures and options, respectively, across bull and bear markets.

Keywords: Central Tendency, Stochastic Volatility, Jumps, Term Structure,Volatility Skews.

JEL: G13.

∗We are grateful to Dante Amengual, Torben Andersen, Max Bruche, Peter Carr, Rob Engle, JavierGil-Bazo, Ioannis Paraskevopoulos, Francisco Penaranda, Eduardo Schwartz and Neil Shephard, as wellas seminar participants at Oxford MAN Institute, Universidad de Murcia, Universitat Pompeu Fabra,the SoFiE European Conference (Geneva, June 2009), the Finance Forum (Elche, November 2010) andthe Symposium of the Spanish Economic Association (Madrid, December 2010) for helpful comments,discussions and suggestions. Of course, the usual caveat applies. Financial support from the SpanishMinistry of Science and Innovation through grant ECO 2008-00280 (Sentana) is gratefully acknowledged.The views expressed in this paper are those of the authors, and do not reflect those of the Bank of Spain.Address for correspondence: Casado del Alisal 5, E-28014, Madrid, Spain, tel: +34 91 429 05 51, fax:+34 91 429 1056.

Electronic copy available at: http://ssrn.com/abstract=1524193

1 Introduction

It is now widely accepted that the volatility of financial assets changes stochasti-

cally over time, with fairly calmed phases being followed by more turbulent periods of

uncertain length. For financial market participants, it is of the utmost importance to un-

derstand the nature of those variations because volatility is a crucial determinant of their

investment decisions. Although many model-based and model-free volatility measures

have been proposed in the academic literature (see Andersen, Bollerslev, and Diebold,

2009, for a recent survey), the Chicago Board Options Exchange (CBOE) volatility in-

dex, widely known by its ticker symbol VIX, has effectively become the standard measure

of volatility risk for investors in the US stock market. The goal of the VIX index is to

capture the volatility (i.e. standard deviation) of the S&P500 over the next month im-

plicit in stock index option prices. Formally, it is the square root of the risk neutral

expectation of the integrated variance of the S&P500 over the next 30 calendar days,

reported on an annualised basis. Despite this rather technical definition, both financial

market participants and the media pay a lot of attention to its movements. To some

extent, its popularity is due to the fact that VIX changes are negatively correlated to

changes in stock prices. The most popular explanation is that investors trade options on

the S&P500 to buy protection in periods of market turmoil, which increases the value of

the VIX. In fact, as Andersen and Bondarenko (2007) and many others show, the VIX

almost uniformly exceeds realised volatility because investors are on average willing to

pay a sizeable premium to acquire a positive exposure to future equity-index volatility.

For that reason, some commentators refer to it as the market’s fear gauge, even though

a high value does not necessarily imply negative future returns.

But apart from its role as a risk indicator, nowadays it is possible to directly invest in

volatility as an asset class by means of VIX derivatives. Specifically, on March 26, 2004,

trading in futures on the VIX began on the CBOE Futures Exchange (CFE). They are

standard futures contracts on forward 30-day implied vols that cash settle to a special

opening quotation (VRO) of the VIX on the Wednesday that is 30 days prior to the

3rd Friday of the calendar month immediately following the expiring month. Further,

on February 24, 2006, European-style options on the VIX index were also launched on

the CBOE. Like VIX futures, they are cash settled according to the difference between

1

the value of the VIX at expiration and their strike price. Importantly, they can also be

interpreted as options on VIX futures, which one can exploit to simplify their valuation

and avoid compounding errors made in valuing futures. VIX options and futures are

among the most actively traded contracts at CBOE and CFE, averaging close to 140,000

contracts combined per day, with much larger volumes on certain days. One of the main

reasons for the high interest in these products is that VIX derivative positions can be

used to hedge the risks of investments in the S&P500 index. Szado (2009) finds that such

strategies do indeed provide significant protection, especially in downturns. Moreover,

by holding VIX derivatives investors can achieve exposure to S&P500 volatility without

having to delta hedge their S&P500 option positions with the stock index. As a result,

it is often cheaper to be long in out-of-the-money VIX call options than to buy out-of-

the-money puts on the S&P500.

Although these new assets certainly offer additional investment and hedging oppor-

tunities, their correct use requires reliable valuation models that adequately capture the

features of the underlying volatility index. Somewhat surprisingly, several theoretical ap-

proaches to price VIX derivatives appeared in the academic literature long before they

could be traded. Specifically, Whaley (1993) priced volatility futures assuming that the

observed volatility index on which they are written follows a Geometric Brownian Mo-

tion (GBM). As a result, his model does not allow for mean reversion in the VIX, which,

as we shall see, is at odds with the empirical evidence. The two most prominent mean

reverting models proposed so far for volatility indices have been the square root pro-

cess (SQR) process considered by Grunbichler and Longstaff (1996), and the log-normal

Ornstein-Uhlenbeck (log-OU) process analysed by Detemple and Osakwe (2000).

Several authors have previously looked at the empirical performance of these pricing

models for VIX derivatives. In particular, Zhang and Zhu (2006) study the empirical

validity of the SQR model by first estimating its parameters from VIX historical data,

and then assessing the pricing errors of VIX futures implied by those estimates. Following

a similar estimation strategy, Dotsis, Psychoyios, and Skiadopoulos (2007) also use VIX

futures data to evaluate the gains of adding jumps to a SQR diffusion. In addition, they

estimate a GBM process. Surprisingly, this model yields reasonably good results, but

the time span of their sample is perhaps too short for the evident mean reverting features

of the VIX to play any crucial role.

2

In turn, Wang and Daigler (2011) compare the empirical fit of the SQR and GBM

models using data on options written on the VIX. They also find evidence supporting the

GBM assumption. However, one could alternatively interpret their findings as evidence

in favour of the log-OU process, which also yields the Black (1976) option formula if the

underlying instrument is a VIX futures contract, but at the same time is consistent with

mean reversion.

Despite this empirical evidence, both the SQR and the log-OU processes show some

glaring deficiencies in capturing the strong persistence of the VIX, which produces large

and lasting deviations of this index from its long run mean. In contrast, the implicit

assumption in those models is that this volatility index mean reverts at a simple, non-

negative exponential rate. Such a limitation becomes particularly apparent during bear-

ish stock markets, in which volatility measures such as the VIX typically experience large

increases and remain at high levels for long periods. Arguably, the apparent success of

those models is to a large extent due to the fact that the sample periods considered in

the existing studies only cover the relatively long and quiet bull market that ended in

the summer of 2007.

In this context, the initial objective of our paper is to study the empirical ability

of existing mean-reverting models for volatility indices to price derivatives on the VIX

over a longer sample span that includes data before, during and after the unprecedented

recent financial crisis, which provides a unique testing ground for our study. To do so,

we use an extensive database that comprises all futures and European option prices on

the VIX from February 2006, when options were introduced, until December 2010. As a

result, we can study whether the SQR and log-OU models are able to yield reliable in-

and out-of-sample prices in a variety of market circumstances.

Our findings indicate that although a log-OU process for the VIX provides a better

fit than a SQR model, especially for VIX options, its performance deteriorates during

the market turmoil of the second part of our sample. For that reason, we generalise

the log-OU process by proposing several novel but empirically relevant extensions: a

time-varying central tendency in the mean, jumps, and stochastic volatility. A central

tendency, which was first introduced by Jegadeesh and Pennacchi (1996) and Balduzzi,

Das, and Foresi (1998) in the context of term structure models, allows the “average”

volatility level to be time-varying, while stochastic volatility permits a changing disper-

3

sion for the (log) volatility index, and together with jumps, introduces non-normality in

its conditional distribution.

Following Trolle and Schwartz (2009) among others, we specify our models in state

space form and calibrate their parameters by pseudo maximum likelihood. Since our ap-

proach accommodates both real and risk-neutral measures, we can use not only derivative

prices but also historical observations on the VIX itself. In order to compute the theo-

retical derivatives prices, we often need to invert the conditional characteristic function

using Fourier methods (see e.g. Carr and Madan, 1999).

To validate the models, we analyse the discrepancies between actual and theoretical

derivatives prices. But we also go beyond pricing errors, and analyse the implications

of the aforementioned extensions for the term structures of VIX futures and options,

and the option volatility “skews”, all of which are of considerable independent interest.

Since we combine futures and options data, we can also assess which of those additional

features is more relevant for pricing futures, and which one is more important for options.

Importantly, our approach differs from Sepp (2008), who prices VIX derivatives by

using data on the S&P500 only. Although the relevant distribution of future values of the

VIX conditional on current information might arguably depend not only on the current

level of the VIX but also on the value of the S&P500, it seems odd to completely neglect

the information content of contemporaneously observed VIX values. This is particularly

relevant in view of the fact that one cannot reproduce the spot VIX index by using

the model proposed by Sepp (2008) for the purposes of obtaining the required S&P500

option values that feed in the closed-form CBOE formula for this volatility index.1 In

order to avoid compounding valuation errors, therefore, we prefer to treat the current

VIX level as our “sufficient statistic”, and to treat VIX futures likewise for the purposes

of valuing options, instead of assigning that role to the S&P500. Ideally, though, one

would like to use both, but we leave this for future research. In this sense, it is important

to emphasise that the univariate stochastic processes that we consider for the VIX are

not necessarily incompatible with models for S&P500 options capable of reproducing

this observed volatility index.

1A relevant analogy would be the use of a model for the dividends and risk premia of the 500individual stocks that constitute the index for the purposes of valuing S&P500 derivatives discardingthe information in the index itself, even though any such multivariate model would very likely fail toreproduce the value of the index.

4

Nevertheless, our analysis can still shed some light on the long-lasting debate sur-

rounding the modelling of the (instantaneous) volatility of this broad stock market index.

In particular, given the relationship between the observed VIX and the unobserved in-

tegrated volatility of the S&P500, our results imply that the sophisticated stochastic

volatility models that many researchers and practitioners use for valuing S&P options

should probably allow for a slower mean reversion than usually considered, as well as for

time-varying volatility of volatility.

The rest of the paper is organised as follows. We describe the data in Section 2, and

explain our estimation strategy in Section 3. Then, we assess the empirical performance

of existing models in Section 4 and consider our proposed extensions in Section 5. Finally,

we conclude in Section 6. Auxiliary results are gathered in an Appendix.

2 Preliminary data analysis

2.1 The CBOE Volatility Index

VIX was originally introduced in 1993 to track the Black-Scholes implied volatilities

of options on the S&P100 with near-the-money strikes (see Whaley, 1993). The CBOE

redefined the index in 2003, renamed the original index as VXO, and released a time

series of daily closing prices starting in January 1990 (see Carr and Wu, 2006). Nowadays,

VIX is computed in real time using as inputs the mid bid-ask market prices for most calls

and puts on the S&P500 index for the front month and the second month expirations

with at least 8 days left (see CBOE, 2009).2 Figure 1a displays the entire historical

evolution of the VIX. Between January 1990 and December 2010 its average closing

value was 20.4. As other volatility measures, though, it is characterised by swings from

low to high levels, with a temporal pattern that shows mean reversion over the long

run but displays strongly persistent deviations from the mean during extended periods.

The lowest closing price (9.31) corresponds to December 22, 1993. Figure 1b, which

focuses on the sample period in our derivatives database, shows that volatility was also

remarkably low between February 2006 and July 2007, with values well below 20. During

this period, the lowest value was 9.89 on January 24, 2007, in what some have called

2Currently, CBOE applies the VIX methodology to 3-month options on the S&P500 (VXV), as wellas 1-month options on the most important US stock market indices: DJIA (VXD), S&P100 (VXO),Nasdaq-100 (VXN) and Russell 2000 (RVX). They also construct analogous short term volatility indicesfor Crude Oil (OVX), Gold (GVZ) and the US $/eexchange rate (EVZ).

5

“the calm before the storm”. Indeed, although the Dow Jones Industrial Average closed

above 14,000 for the first time in history in the summer of 2007, some warning signals

were observed around this period. In particular, Bear Stearns disclosed that its two

subprime hedge funds had lost all of their value on July 18, 2007, and one day later

Fed Chairman Ben Bernanke warned the US Senate Banking Committee that subprime

losses could top $100 billion. Over the following year, VIX increased to values between

20 and 35. Finally, in the autumn of 2008 it reached unprecedented levels. In particular,

the largest historical closing price (80.86) took place on November 20, 2008, although on

October 24 the VIX reached an intraday value of 89.53. After this peak, VIX followed

a decreasing trend over the following months until the beginning of April, 2010, when

the Greek debt crisis started worsening. These markedly different regimes offer a very

interesting testing ground to analyse the performance of valuation models for volatility

derivatives. In particular, we can assess if the models calibrated with pre-crisis data

perform well under the extreme conditions of the 2008-2009 financial crisis. Similarly,

we can assess if models estimated with data that covers the 2008-2009 financial crisis

perform well during the 2010 sovereign debt crisis.

In order to characterise the time series dynamics of the VIX, we have estimated several

ARMA models using the whole VIX historical observations from 1990 to 2010. Figure 2a

compares the sample autocorrelations of the log VIX with those implied by the estimated

models. There is clear evidence of high persistence, with a first-order autocorrelation

above 0.98 and a slow rate of decay for higher orders. Consequently, an AR(1) model

seems unable to capture the shape of the sample correlogram. An alternative illustration

of the failure of this model is provided by the presence of positive partial autocorrelations

of orders higher than one, as Figure 2b confirms. Therefore, it is necessary to introduce

a moving average component to take into account this feature. An ARMA(1,1) model,

though, only offers a slight improvement. In contrast, an ARMA(2,1) model turns out

to yield autocorrelations and partial autocorrelations that are much closer to the sample

values. As we shall see below, our preferred continuous time models have ARMA(2,1)

representations in discrete time. Importantly, this result is not due to the behaviour of

the VIX during the financial crisis, since an ARMA(2,1) is also necessary to capture the

autocorrelation profile when we only consider data from 1990 until the summer of 2007.

We have also analysed the presence of time varying features in the volatility of the log

6

VIX. In particular, we have tested for Garch effects in the residuals of the ARMA(2,1)

estimation.3 Interestingly, we can easily reject the absence of Garch-type effects at

all conventional levels. In this sense, we shall see that our preferred continuous time

specification allows for time varying volatility.

2.2 VIX derivatives

Our sample contains daily closing bid-ask mid prices of futures and European put and

call options on the VIX, which we downloaded from Bloomberg. We consider the entire

history of these series since options were introduced in February 2006 until December

2010. In terms of maturity, we have data on all the contracts with expirations between

March 2006 and May (July) 2011 for options (futures). CFE may list futures for up to 9

near-term serial months, as well as 5 months on the February quarterly cycle associated

to the March quarterly cycle for options on the S&P500. In turn, CBOE initially lists

some in-, at- and out-of-the-money strike prices, and then adds new strikes as the VIX

index moves up or down. Generally, the options expiration dates are up to 3 near-term

months plus up to 3 additional months on the February quarterly cycle.

Following Dumas, Fleming, and Whaley (1998), we exclude derivatives with fewer

than six days to expiration due to their illiquidity. In addition, we only consider those

prices for which open interests and volumes are available. All in all, we have 8,665

and 87,870 prices of futures and options, respectively. Of those option prices, 58,099

correspond to calls and 29,771 to puts. We proxy for the riskless interest rate by using

the daily Eurodollar rates at 1-week, 1, 3 and 6 months and 1 year, which we interpolate

to match the maturities of the futures and option contracts that we observe.

Figure 3a shows the number of futures prices per day in our database. We have

between 4 and 6 daily prices during 2006; afterwards, 8 prices per day become available

on average. Figure 3b indicates that the number of option prices per day is also smaller

at the beginning of the sample, but it tends to stabilise in 2007 at around 20 for puts

and 40 for calls. The number of call prices per day increases again after mid 2009.

Figure 4a shows the evolution of the VIX term structure implicit in VIX futures.

This figure clearly confirms that there was a substantial level shift in the last quarter

3Following Demos and Sentana (1998), we carry out a one-sided LM test of conditional homoskedas-ticity against Garch as TR2 from the regression of the squared ARMA(2,1) residuals on a constantand the first lag of the Riskmetrics volatility estimate (see RiskMetrics Group, 1996).

7

of 2008. The negative slope during that period, though, indicates that the market did

not expect the VIX to remain at such high values forever. Figure 4b compares futures

prices with the VIX. As can be seen, most future prices were above the volatility index

until mid 2007, which suggests that market participants perceived that the VIX was too

low during those years. For instance, on October 12, 2006, the VIX was 11.09, while the

price of VIX futures expiring on February 14, 2007, was 15.72. The spot price reflected

the expected volatility for the period of October 12 to November 11, while the futures

price reflected the expected volatility for the period of February 14 to March 16, 2007.

In contrast, most prices were below the VIX in the autumn of 2008, which confirms that

volatility was then perceived to be above its long run mean.

Figure 4c focuses on the VIX term structure at four particularly informative days.

Futures prices were extremely low on July 16, 2007, even though the first warning signals

about the impending crisis were starting to appear. Prices had already risen significantly

by August 15, 2008, one month before the collapse of Lehman Brothers. Nevertheless,

they were much higher on November 20, 2008, which is the day in which the VIX reached

its maximum historical closing value at 80.86. The increase is particularly remarkable

at the short end of the curve. Since then, though, VIX futures prices significantly came

down until the beginning of April 2010, right before the European sovereign debt crisis.

3 Pricing and estimation strategy

We assume that there is a risk free asset with instantaneous rate r. Let V (t) be

the VIX value at time t. We define F (t, T ) as the actual strike price of a futures

contract on V (t) that matures at T > t. Similarly, we will denote the prices at t of

call and put options maturing at T with strike price K by c(t, T,K) and p(t, T,K),

respectively. Importantly, since the VIX index is a risk neutral volatility forecast, not

a directly traded asset, there is no cost of carry relationship between the price of the

futures and the VIX (see Grunbichler and Longstaff, 1996, for more details). There is

no convenience yield either, as in the case of futures on commodities. Therefore, absent

any other market information, VIX derivatives must be priced according to some model

for the risk neutral evolution of the VIX. This situation is similar, but not identical, to

term structure models.

Let M index the asset pricing models that we consider. Then, the theoretical futures

8

price implied by model M will be:

FM(t, T, V (t),φ) = EQM [V (T )|I(t),φ], (1)

where Q indicates that the expectation is evaluated at the risk neutral measure, φ is

the vector of free parameters of model M , and I(t) denotes the information available at

time t, which includes V (t) and its past values.

We can analogously express the theoretical value of a European call option with strike

K and maturity at T under this model as

cM [t, T,K, F (t, T ),φ] = exp(−rτ)EQM [max(V (T )−K, 0)|I(t),φ], (2)

where τ = T−t. Nevertheless, we can exploit the fact that V (T ) = F (T, T ) to price calls

using the futures contract that expires on the same date as the underlying instrument,

instead of the actual volatility index. In this way, we make sure that the pricing errors

of options are not caused by distortions in our futures valuation formulas. Similarly,

European put prices p(t, T,K) under M can be easily obtained from the put-call-forward

parity relationship

pM [t, T,K, F (t, T ),φ] = cM [t, T,K, F (t, T ),φ]− exp(−rτ)[F (t, T )−K].

We estimate the parameters of all the models that we consider by pseudo maximum

likelihood using futures and options prices as well as data from the VIX index. To do

so, we assume the existence of pricing errors such that

F (t, T ) = FM(t, T, V (t),φ) + εft,T ,

c(t, T,K) = cM [t, T,K, F (t, T ),φ] + εct,T (K),

p(t, T,K) = pM [t, T,K, F (t, T ),φ] + εpt,T (K),

where εft,T ∼ N(0, σ2f ), εct,T (K) ∼ N(0, σ2

o) and εpt,T (K) ∼ N(0, σ2o) are iid over time

and across strikes and maturities. A non-trivial advantage of this method over tradi-

tional calibration procedures is that we do not have to treat the model parameters as

deterministic functions of time. In addition, we can also obtain standard errors for our

parameter estimators, which can in turn be used to conduct formal hypothesis tests.

Furthermore, we can exploit the relationship between actual and risk-neutral measures

to estimate prices of risk (see the Appendix for details).

9

For interpretation purposes, it is convenient to write the log likelihood at a particular

time t as

l[ot, ft, V (t)|I(t− 1)] = lo(ot|ft, V (t), I(t− 1)) + lf (ft|V (t), I(t− 1)) + lv[V (t)|I(t− 1)],

where ot and ft are, respectively, the set of option and futures prices traded at t,

lo(ot|ft, V (t), I(t−1)) is the log density of the option prices conditional on futures prices

and VIX, lf (ft|V (t), I(t−1)) is the log likelihood of the futures prices conditional on the

volatility index, and lv[V (t)|I(t− 1)] is the log density of the volatility index.

We consider three estimation samples that correspond to the distinct volatility phases

described in Section 2: (i) until 15-Aug-2008, (ii) until 31-March-2010 and (iii) full

sample. Finally, we evaluate the in- and out-of-sample empirical fit of the models over

those periods. Thus, we can assess model performance following major volatility increases

in global stock markets.

4 Existing one-factor models

4.1 Model specification

We first compare the two mean-reverting volatility models that have been used so

far in the literature: the square root and the log Ornstein-Uhlenbeck processes. As we

mentioned before, Grunbichler and Longstaff (1996) proposed the square root process

(SQR) to model a standard deviation index. This model, which was used by Cox,

Ingersoll, and Ross (1985) for interest rates and Heston (1993) for the instantaneous

variance of stock prices, satisfies the diffusion

dV (t) = κP [θP − V (t)]dt+ σ√V (t)dW P (t),

where W P (t) is a Brownian motion under the real measure. Following Dai and Singleton

(2000), we specify a price of risk that is proportional to the instantaneous volatility.

Specifically, we assume that Λ(t) = ς√V (t), so that dWQ(t) = dW P (t) + ς

√V (t)dt.

Then, it is straightforward to show that V (t) satisfies the square root diffusion

dV (t) = κ[θ − V (t)]dt+ σ√V (t)dWQ(t),

under the equivalent risk neutral measure, where κ = κP + σς and θ = κP θP/κ.

10

As is well known, the risk-neutral distribution of 2cV (T ) given V (t) is a non-central

chi-square with ν = 4κθ/σ2 degrees of freedom and non-centrality parameter ψ =

2cV (t) exp(−κτ), where

c =2κ

σ2(1− exp(−κτ)).

As a result, the price of the futures contract (1) can be expressed in this case as

FSQR(t, T, V (t), κ, θ, σ) = θ + exp(−κτ)[V (t)− θ]. (3)

We can interpret θ as the long-run mean of V (t), since the conditional expected value

of the volatility index converges to θ as τ goes to infinity. In addition, κ is usually

interpreted as a mean-reversion parameter because the higher it is, the more quickly the

process reverts to its long run mean.4

The call price formula (2) for this model becomes

cSQR(t, T,K, κ, θ, σ) = V (t) exp(−(κ+ r)τ)[1− FNC2(2cK; ν + 4, ψ)]

+θ[1− exp(−κτ)] exp(−rτ)[1− FNC2(2cK; ν + 2, ψ)]

−K exp(−rτ)[1− FNC2(2cK; ν, ψ)], (4)

where FNC2(·; ν, ψ) is the cumulative distribution function (cdf) of a non-central chi-

square distribution with ν degrees of freedom and non-centrality parameter ψ. Hence,

it is straightforward to express the call price as a function of F (t, T ) by exploiting the

relationship between futures prices and V (t) in (3).

More recently, Detemple and Osakwe (2000) considered the log-normal Ornstein-

Uhlenbeck (log-OU) diffusion:

d log V (t) = κ[θP − log V (t)]dt+ σdW P (t).

If once again we follow Dai and Singleton (2000) in specifying the price of risk as Λ(t) = ς,

then V (t) will satisfy the log-OU diffusion

d log V (t) = κ[θ − log V (t)]dt+ σdWQ(t).

under the risk neutral measure, where θ = θP − σς/κ

As is well known, this model implies that log V (t) would follow a conditionally ho-

moskedastic Gaussian AR(1) process if sampled at equally spaced discrete intervals.

4As remarked by Hansen and Scheinkman (2009), the density of this distribution will be positive atV (T ) = 0 if the Feller condition 2κθ ≥ σ2 is violated.

11

More generally, the conditional risk-neutral distribution of log V (T ) given V (t) would be

Gaussian with mean

µ(t, τ) = θ + exp(−κτ)[log V (t)− θ]

and variance

ϕ2(τ) =σ2

2κ[1− exp(−2κτ)]. (5)

As in the SQR process, θ and κ can be interpreted as the long-run mean and mean-

reversion parameters, respectively, but now it is the log of V (t) that mean reverts to θ.

In this context, it is straightforward to show that the futures price is

FLog−OU(t, T, V (t), κ, θ, σ) = exp(µ(t, τ) + 0.5ϕ2(τ)),

while the call price can be expressed as

cLog−OU(t, T,K, κ, θ, σ) = exp(−rτ)F (t, T )Φ

[log(F (t, T )/K) + 1

2ϕ2(τ)

ϕ(τ)

]−K exp(−rτ)Φ

[log(F (t, T )/K)− 1

2ϕ2(τ)

ϕ(τ)

](6)

if we take the futures contract as the underlying instrument, where Φ(·) denotes the

standard normal cdf. This is the well known Black (1976) formula, although in this case

the implied volatility ϕ(τ) in (5) is not constant across maturities. In this sense, the

pricing formula proposed by Whaley (1993) based on a geometric Brownian motion can

also be expressed as (6) if ϕ(τ) is taken as a constant irrespective of τ .

4.2 Empirical performance

We have estimated the parameters of these two models over the three sample periods

described at the end of Section 3. Table 1 reports the in- and out-of-sample root mean

square pricing errors (RMSE). The SQR model clearly yields larger price distortions

for futures than the log-OU, especially during the financial crisis, with almost doubled

RMSE’s. The log-OU model also displays smaller RMSE’s for options, both in and out

of sample. This is confirmed in Figure 5, which compares the daily RMSE’s of the two

models. As can be seen, the SQR yields smaller futures price errors until the Summer

of 2007, but from then on the log-OU is clearly superior. For options, the log-OU is

generally superior over the whole sample. Nevertheless, the performance of both models

deteriorates with the financial crisis, especially during its most critical episodes.

12

Table 2 reports the parameter estimates that we obtain. Although the values are not

directly comparable because volatility is expressed in levels in the SQR model and in logs

in the log-OU model, in both cases the mean reversion parameter κ is quite sensitive to

the sample period used for estimation purposes. In contrast, the volatility parameter σ

is more stable for the log-OU process, probably because the log transformation is more

appropriate to capture the distortions produced by the large movements of the VIX that

took place at the end of our sample. Interestingly, the price of risk turns out to be

insignificant in the log-OU case, and is only significant for the SQR in the two larger

samples.

An alternative, more illustrative way to assess the validity of the log-OU model is by

computing the implied volatilities with the Black (1976) formula (6). As we mentioned

before, the implied volatility of the log-OU process is constant for different degrees of

moneyness, but not across different maturities because (5) depends on τ . Hence, if

this model were correct then we should obtain constant implied volatilities for a given

maturity regardless of moneyness. Figure 6 shows that this is not at all the case. In fact,

we generally observe that implied vols have a positive moneyness slope. In addition,

implied volatilities are much higher in November 2008 than in the other three dates for a

similar range of maturities. These empirical results suggest that the volatility of the VIX

might not be constant over time, as the log-OU models assume. In this sense, Figure

6 clearly shows that the log-OU process is not only unable to generate the observed

volatility skews, but it also fails to capture the average level of implied volatilities.

5 Extensions

5.1 Model specification

The results of the previous section indicate that a log-OU process offers a better

empirical fit than a SQR process. Unfortunately, its performance tends to deteriorate

during the recent financial crisis. For that reason, in this section we explore several

extensions of the log-OU model. Specifically, we introduce three empirically relevant

features: a time varying mean, jumps and stochastic volatility. We consider these ex-

tensions first in isolation, and then in combination. As a general rule, we model the

price of risk of the continuous part of the diffusions as in the one-factor models, while we

assume that jump risk is not priced. For the sake of brevity, though, we only describe

13

the risk-neutral measure in this section. Further details about prices of risk and real

measures can be found in the Appendix. All in all, we compare the following cases:

• Central tendency (CT):

d log V (t) = κ[θ(t)− log V (t)]dt+ σdWQv (t),

with

dθ(t) = κ[θ − θ(t)]dt+ σdWQθ (t), (7)

where WQv (t) and WQ

θ (t) are independent Brownian motions. As we mentioned

before, an important deficiency of previously existing models is that they assume

that a volatility index mean reverts at a simple, non-negative exponential rate,

which is in fact 0 in the GBM model proposed by Whaley (1993). However, the

long, persistent swings in the VIX in Figure 1a suggest that we need to allow for

more complex dynamics. In this sense, we show in the Appendix that the exact

discretisation of log V (t) in the above model is a Gaussian ARMA(2,1) process,

which is consistent with the evidence reported in Section 2 (see Figure 2). As

discussed in Jegadeesh and Pennacchi (1996) and Balduzzi, Das, and Foresi (1998),

(7) allows the volatility index to revert towards a central tendency, which in turn,

fluctuates stochastically over time around a long run mean θ. As a consequence,

the conditional mean of log V (T ) is a function of the distance between log V (t)

and the central tendency θ(t), as well as the distance between θ(t) and θ.5 More

explicitly,

EQ[log V (T )|I(t)] = θ + δ(τ)[θ(t)− θ]

+ exp(−κτ)[log V (t)− θ(t)], (8)

where

δ(τ) =κ

κ− κexp(−κτ)− κ

κ− κexp(−κτ).

Importantly, this model can also be expressed as the “superposition” (i.e. sum) of

two log-OU processes. Therefore, the nested structure in (7) is not restrictive.6

5Unlike in the log-OU model, the convergence of (8) to the long run mean is not necessarily amonotonic function of τ .

6The “component” Garch model proposed by Ding and Granger (1996) to capture “long memory”features of volatility (see also Engle and Lee, 1999) can be regarded as a discrete time analogue to (7).

14

• Jumps (J):

d log V (t) = κ[θ − log V (t)]dt+ σdWQ(t) + dZ(t)− λ

κδdt, (9)

where Z(t) is a pure jump process independent of WQ(t), with intensity λ, and

whose jump amplitudes are exponentially distributed with mean 1/δ, or Exp(δ)

for short. Note that the last term in (9) simply introduces a constant shift in the

distribution of log V (t) which ensures that θ remains the long-run mean of log V (t).

Jumps in models for instantaneous volatility have been previously considered by

Duffie, Pan, and Singleton (2000) and Eraker, Johannes, and Polson (2003), among

others, while Todorov and Tauchen (2010) also consider jumps in modelling the

VIX. Unlike pure diffusions, this model allows for sudden movements in volatility

indices, which nevertheless have lasting effects due to the fact that the mean-

reversion parameter κ is bounded.

• Stochastic volatility (SV):

d log V (t) = κ[θ − log V (t)]dt+√ω(t)dWQ(t)

where ω(t) follows an OU-Γ process, which belongs to the class of Levy OU pro-

cesses considered by Barndorff-Nielsen and Shephard (2001). Specifically,

dω(t) = −λω(t)dt+ dZ(t) (10)

where Z(t) is a pure jump process with intensity λ and Exp(δ) jump amplitude,

while WQ(t) is an independent Brownian motion. We use this extension to assess to

what extent the price distortions in the previous models are due to the assumption

of constant volatility over time. Importantly, the model that we adopt is consistent

with the presence of mean reversion in ω(t), since

EQ [ω(T ))|ω(t)] = δ−1 + exp(−λτ)[ω(t)− δ−1

]. (11)

Hence, δ−1 can be interpreted as the long run mean of the instantaneous volatility

of the log VIX, while λ will be the corresponding mean reversion parameter. An-

other non-trivial advantage of this model over other alternatives such as a square

root process for ω(t) is that it allows the valuation of derivatives by inverting the

conditional characteristic function (see the Appendix for details).

15

• Central tendency and jumps (CTJ):

d log V (t) = κ[θ(t)− log V (t)]dt+ σdWQv (t) + dZ(t)− λ

κδdt (12)

where θ(t) follows the diffusion (7) and Z(t) is a pure jump process with intensity

λ and Exp(δ) jump amplitude, while WQv (t) and WQ

θ (t) are independent Brownian

motions. We again introduce a constant shift in (12) to ensure that θ is the long

run mean of both θ(t) and log V (t).

• Central tendency and stochastic volatility (CTSV):

d log V (t) = κ[θ(t)− log V (t)]dt+√ω(t)dWQ

V (t)

where θ(t) follows the diffusion (7) while ω(t) is defined in (10). As in previous

cases, the jump variable Z(t) and the Brownian motions are mutually independent.

Except for the CT model, it is not generally possible to price derivative contracts for

these extensions in closed form. However, it is possible to obtain the required prices by

Fourier inversion of the conditional characteristic function. In particular, we use formula

(5) in Carr and Madan (1999) to invert the relevant characteristic functions, which we

derive in the Appendix.

Some of the previous extensions introduce as additional factors a time varying ten-

dency (CT, CTJ), a time varying volatility (SV) or both (CTSV), which we need to

filter out for estimation purposes. In this regard, we use the standard Kalman filter to

“estimate” θ(t). And although the jump variable Z(t) in (9) can also be interpreted as

an additional factor, its impact cannot be separately identified from the impact of the

diffusion shocks dWQ(t) because both variables share the same mean reversion coefficient

κ. Finally, following Trolle and Schwartz (2009) and others, we employ the extended

Kalman filter to deal with ω(t) (see again the Appendix for further details).

5.2 Empirical performance

Table 3 reports the in- and out-of-sample RMSE’s of the extensions introduced in

the previous section. As expected, the CT model is able to yield much smaller RMSE’s

for futures than the standard log-OU model, both in- and out-of-sample. Therefore,

the inclusion of a time-varying mean seems to be crucial for adequately capturing the

16

behaviour of futures prices. At the same time, given that we show in the Appendix that

option prices do not depend on θ(t) once we condition on the current futures price, it is

not surprising that the difference in parameter estimates hardly affects the RMSE’s of

options.

In contrast, the introduction of jumps in the baseline log-OU process only introduces

a slight improvement for options. As for futures prices, jumps yield almost no change,

which is again to be expected because we can also show that jumps generate identical

futures prices as a log-OU model up to first order.7 Similarly, adding jumps to the

CT model does not introduce any remarkable improvements on futures, and only minor

gains for options. In this sense, it is important to emphasise that Table 3 does not

assess the importance of jumps on the historical dynamics of the VIX, only their pricing

implications.

But when we consider stochastic volatility, we observe the opposite effect: the im-

provement over the standard log-OU model is greater for options than for futures. Hence,

stochastic volatility turns out to be more important to value options, while central ten-

dency is crucial to price futures. Intuitively, the reason is that the futures formula (1)

only involves the conditional mean of V (T ), whereas (2) is more sensitive to higher order

moments, such as the volatility of volatility. Not surprisingly, we obtain the best results

when we combine central tendency and stochastic volatility. In particular, the pricing

performance of the CTSV model during the crisis (August 08 - March 10) is remarkably

good compared to the other models, even if we only focus on out-of-sample statistics. In

this sense, we would like to emphasise the good out-of-sample performance of the CTSV

model when we use the parameters estimated with data prior to August 08, which does

not include the more extreme period of the crisis. Hence, the CTSV is not only able to

describe the behaviour of the VIX during the financial crisis, but it also seems to be a

good model for the different phases of the VIX in bull and bear markets.

Figure 7 compares the daily RMSE’s of the CTSV model with the analogous series

for the standard log-OU model. Apart from the superior fit, we can clearly see that

the performance of the CTSV model is relatively stable over the entire sample. As an

alternative comparison, we also compute the empirical cumulative distribution functions

7Formally, when we consider a Taylor expansion of the futures price formula of model J around λ/δfor λ/δ > 0 but small, we only observe deviations from the standard log-OU expression for the secondand higher terms.

17

of the daily square pricing errors that the different models generate. The smaller the

errors, the more shifted to the left those distributions will be. In this respect, Figure 8

shows that the CTSV model dominates all the other models in the first order stochastic

sense. The ordering of the remaining models, though, depends on the type of derivative

asset considered. In the case of futures, the models with central tendency (CT, CTJ and

CTSV) provide an almost identical fit. In turn, they are followed by the SV, standard

log-OU, J, and SQR models. But for options, the second best model is SV, which is

followed by the CTJ, J, CT, standard log-OU and SQR models. Thus, we observe

again that a central tendency is relatively more important for futures, while stochastic

volatility offers greater gains on options. At the same time, a central tendency does not

harm the option pricing performance of the CTSV model, while stochastic volatility does

not cause any distortions to futures prices. As for jumps, Figure 8 confirms that they do

indeed help in pricing options but they hardly provide any improvement for futures. In

any case, compared to stochastic volatility, jumps seem to yield a minor improvement

even for options. Lastly, the SQR model undoubtedly offers the worst results, especially

for options.

Table 4 reports the parameter estimates that we obtain in the different extensions

of the log-OU model introduced in Section 5. In the models with central tendency we

observe fast mean reversion of log V (t) to θ(t), which in turn mean-reverts rather more

slowly to its long rung mean θ (i.e. κ � κ). Interestingly, while the estimates of these

parameters are very stable, jump intensities vary substantially for the VIX depending

on the sample period considered for estimation. Specifically, we obtain smaller values

of λ when we include the crisis period. For the full sample, we estimate around 5

jumps per year in the J specification, and 12 jumps per year in the CTJ model. On the

other hand, the estimates of the mean reversion parameter λ in the stochastic volatility

models tend to be larger when central tendency is not simultaneously included. Since

this parameter is also responsible for jump intensity in the OU-Γ model, this result has

two interesting implications. First, a smaller value of λ tends to reduce jump activity

in Z(t). Specifically, the expected number of jumps per year decreases from 12 in the

SV model to less than 2 in the CTSV extension. Second, the deviations of ω(t) from

its long run mean are more persistent in the CTSV case because mean reversion is

slower the smaller λ is, as (11) indicates. These features are also important from a time

18

series perspective. As mentioned before, central tendency is consistent with ARMA(2,1)

dynamics in discrete time, while stochastic volatility introduces Garch-type persistent

variances for the log VIX. Lastly, the prices of risk turn out to be insignificant for these

extensions, which implies that the real and risk neutral measures are not statistically

different.

Finally, Table 5 shows the contribution of futures, options and VIX to the log-

likelihood. Once again, notice that the introduction of central tendency leads to a

substantial increase in the contribution of futures to the log-likelihood but causes much

smaller increases in the other components. A similar effect is observed for options when

we introduce stochastic volatility. In addition, we can also observe that jumps are re-

sponsible for significant improvements in the goodness of fit of the model. Thus, jumps

seem to be a relevant feature to describe the process followed by the VIX, even though

they only yield second order gains for pricing VIX derivatives.

5.3 Evolution of the factors

Figure 9a shows that the filtered values of the central tendency factor θ(t) are rather

insensitive to the particular specification that we use. Similarly, Figure 9b indicates that

our preferred CTSV model also generates almost indistinguishable filtered values for

θ(t) across the different estimation subsamples that we consider. Not surprisingly, these

figures also confirm that the log of the VIX oscillates around θ(t), which in turn changes

over time rather more slowly. In fact, the main reason for central tendency models to

work so well during the recent financial crisis is because they allow for large temporal

deviations of θ(t) from its long term value θ, thereby reconciling the large increases of

the VIX observed during that period with mean reversion over the long term.

To assess the extent to which the filtered values of θ(t) are realistic, Figure 9c com-

pares the “actual” futures prices for a constant 30-day maturity, which we obtain by

interpolation of the adjacent contracts, with the daily estimates of 30-day futures prices

generated by the log-OU, CT and CTSV models. By focusing on a constant maturity,

we can not only compare the absolute magnitude of the pricing errors, as in Figure 7a,

but also their sign and persistence. As can be seen, the pricing errors of the log-OU

process are not only substantially larger than those of the two CT extensions, but they

also display much stronger persistence. For instance, the log-OU model systematically

19

overprices futures from the beginning of the sample until the summer of 2007, but it

has the opposite bias from October 2008 until the end of the sample. The slower mean

reverting properties of the VIX are probably responsible for these persistent biases in

the log-OU model. We can also observe that the pricing errors of the CT and CTSV

models are almost identical. This feature shows once again that central tendency is the

most relevant extension for pricing futures.

Figure 10a compares the filtered instantaneous volatilities of the SV and CTSV mod-

els. Interestingly, both series display an almost identical pattern, with substantial per-

sistent oscillations over time. As it was the case for the central tendency θ(t) in Figure

9b, Figure 10b confirms that the filtered values of ω(t) obtained from the CTSV model

are also quite insensitive to the sample period used to estimate the model parameters.

Finally, we compute the 30-day ahead standard deviations of log V (t) implied by

the CTSV model to assess the extent to which the filtered values of ω(t) make sense.

In Figure 10c we compare those standard deviations with the Black (1976) implied

volatilities of at-the-money options that are exactly 30-day from expiration, which we

again obtain by interpolation. The high correlation between the two series shows that

the filtered values of stochastic volatility are indeed related to the changing perceptions

of the market about the standard deviation of the VIX.

5.4 Term structures of derivatives and implied volatility skews

So far, we have focused on the overall empirical performance of the different models.

In principle, though, our results could change for different time horizons or different

degrees of moneyness. For that reason, Table 6 shows the RMSE’s of futures contracts

for different ranges of maturity. We observe that the models without central tendency

tend to yield larger distortions for longer maturities. In contrast, the models with central

tendency are relatively worse at pricing the shortest maturity. Nevertheless, even the

worst RMSE of the CTSV model is still much smaller than the best RMSE of either the

SQR or the log-OU models without central tendency.

To gain some additional insight, in Figure 11 we look at the term structure of fu-

tures prices for the four days considered in Figure 4c. This figure confirms that central

tendency is crucial for the purposes of reproducing the changes in the level and slope of

the actual term structure of VIX futures prices. We can also observe that the J model

20

and the standard log-OU process yields prices which are almost identical, while the CTJ

model can be barely distinguished from the CT extension.

Tables 7 and 8 provide the RMSE’s of calls and puts for different ranges of maturity

and moneyness. In this case, we generally observe the highest price distortions for at-the-

money call and put options. The SQR, standard log-OU, J, CT and CTJ models seem

to do a better job for moneyness smaller than −0.3. However, once again we can confirm

that stochastic volatility models provide the best fit uniformly across all moneyness and

maturity ranges.

In Figure 12 we consider the term structure of call prices for a given strike for the

same four days as in Figures 4c and 11. All models manage to capture reasonably well

the shape of the term structure of option prices, which was upward sloping on 16-July-

2007, 15-August-2008 and 31-March-2010, and downward sloping on 20-November-2008.

However, only those models with stochastic volatility are able to reproduce the level of

call options market prices. The remaining models either underprice (20-November-2008)

or overprice (remaining dates) calls for all maturities.

In Figures 13 and 14 we assess whether jumps or stochastic volatility are better

at capturing the actual implied volatility skews depicted in Figure 6. For the sake of

completeness, we have also plotted the implied volatilities generated by the standard

log-OU model and its CT extension. These figures show that while the introduction

of jumps can reproduce the positive slope of the actual implied vols, only stochastic

volatility seems to correctly capture both their level and slope. Finally, Figures 15 and

16 confirm that the CTSV captures rather well the term structure of implied volatilities

skews across different strikes and maturities.

6 Conclusions

We carry out an extensive empirical analysis of VIX derivatives valuation models.

We consider daily prices of futures and European options from February 2006 until

December 2010. Therefore we not only cover an unusually tranquil period, but also

the early turbulences that took place between August 2007 and August 2008, the worst

months of the recent financial crisis (Autumn 2008), as well as the months in 2010 in

which the European sovereign debt crisis aggravated. These markedly different periods

provide a very useful testing ground to assess the empirical performance of the different

21

pricing models. We estimate the models by pseudo maximum likelihood using not only

futures and options data, but also historical data on the VIX itself, which allows us

to look at the relationship between real and risk neutral measures. We initially focus

on the two existing mean-reversion models: the square root (SQR) and the log-normal

Ornstein-Uhlenbeck processes (log-OU). Although SQR is more popular in the empirical

literature, we find that the log-OU model yields a better fit for future prices, especially

during the crisis, as well as a better fit for options. However, both models yield large price

distortions during the crisis. In addition, they do not seem to capture either the level

or the slope of the term structure of futures and option prices, or indeed the volatility

skew. Part of the problem is that these models implicitly assume that volatility mean

reverts at a simple exponential rate, which cannot accommodate the long and persistent

swings of the VIX observed in our sample. In this sense, we show that the simple AR(1)

structure that they imply in discrete time is not consistent with the empirical evidence

of ARMA dynamics for the VIX. In addition, these models are also inconsistent with

the strong presence of Garch-type heteroskedasticity that we find in the VIX.

We investigate the potential sources of mispricing by considering several empirically

relevant generalisations of the standard log-OU model. In particular, we introduce a

time varying central tendency, jumps and stochastic volatility. Our parameter estimates

indicate that the VIX rapidly mean-reverts to a central tendency, which in turn reverts

more slowly to a long run constant mean. This flexible structure can reconcile the large

variations of the VIX over our sample period with mean-reversion to a long run constant

value. Except for the central tendency model, though, it is not generally possible to

price derivatives in closed form for the extensions that we consider. For that reason, we

obtain the required prices by Fourier inversion of the conditional characteristic function.

Interestingly, our results indicate that a time varying central tendency is crucial to

price futures. At the same time, it is not detrimental for the valuation of options. We

also find evidence of time varying volatility in the VIX. But stochastic volatility plays

a much more important role for options while leaving futures prices almost unaffected.

In contrast, jumps do not change futures prices (up to first order), and only provide

a minor improvement for options. Importantly, our results remain valid when we focus

exclusively on the out-of-sample performance with parameters estimated using data prior

to the Autumn of 2008. In view of these findings, we conclude that a generalised log-OU

22

model that combines a time varying central tendency with stochastic volatility is needed

to obtain a good pricing performance during bull and bear markets, as well as to capture

the term structures of VIX futures and options, and the implied volatilities of options.

Another interesting finding is that we do not obtain statistically significant values for

the prices of risk.

Given the relationship between the observed VIX index and the unobserved integrated

volatility of the S&P500, our analysis also has important implications for models of this

broad stock index commonly used by participants in markets for stock index options.

Specifically, our results imply that stochastic volatility models for the S&P500 should

allow for slow mean reversion, and a time-varying volatility of volatility. Amengual

(2009) considers such a model for volatility swaps.

We could extend our empirical exercise to other recently introduced volatility deriva-

tives such as binary options or American options on VIX futures, or even the futures

and options on the CBOE Gold ETF Volatility Index. The use of more sophisticated

filtering procedures for the volatility of VIX might also be worth exploring. It would also

be interesting to investigate the incremental information content of the contemporaneous

observations of the S&P500 over and above the spot VIX for pricing VIX futures, and

above and beyond VIX futures for pricing VIX options. This question is also relevant

for the purposes of integrating the valuation of VIX derivatives with the valuation of

the underlying options on the S&P500 that are used to compute this volatility index,

as suggested by Lin and Chang (2009). We plan to address these points in subsequent

research.

23

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26

A Extensions of log-OU processes

A.1 General case

The extensions that we consider belong to the class of affine jump-diffusion state

processes analysed by Duffie, Pan, and Singleton (2000). In particular, consider an

N -dimensional vector Y(t) that satisfies the diffusion

dY(t) = K(Θ−Y(t))dt+√

S(t)dW(t) + dZ(t),

where W(t) is an N -dimensional vector of independent standard Brownian motions, K

is an N×N matrix, Θ is a vector of dimension N , S(t) is a diagonal matrix of dimension

N whose ith diagonal element is ci0 + c′i1Y(t), and finally, Z(t) is a multivariate pure

jump process with intensity λ whose jump amplitudes have joint density fJ(·).

Duffie, Pan, and Singleton (2000) show that the conditional characteristic function

of Y(T ) can be expressed as

φY (t, T,u) = E[exp(iu′Y(T ))|I(t)]

= exp(ϕ0(τ) +ϕ′Y (τ)Y(t)),

where ϕ0(τ) and ϕY (τ) satisfy the following system of differential equations:

ϕY (τ) = −K′ϕY (τ) +1

2ςY (τ),

ϕ0(τ) = Θ′KϕY (τ) +1

2ϕ′Y (τ)diag(c0)ϕY (τ) + λ[J(ϕY (τ))− 1],

where

J(u) =

∫exp(u′x)fJ(x)dx,

and ςY (τ) is an N -dimensional vector whose kth element is

ςY,k(τ) = ϕ′Y (τ)diag(ck1)ϕY (τ).

If we place log V (t) as the first element in of Y(t), then the conditional characteristic

function of log V (t) will be

φ(t, T, u) = φY (t, T,u0),

where u0 = (u, 0, · · · , 0)′. Following Carr and Madan (1999), the price of a call option

with strike K can then be expressed as

c(t, T,K) =exp(−α log(K))

π

∫ ∞0

exp(−iu log(K))ψ(u)du, (A1)

27

where

ψ(u) =exp(−rτ)φ(t, T, u− (1 + α)i)

α2 + α− u2 + i(1 + 2α)u

and α is a smoothing parameter. We evaluate (A1) by numerical integration. In our

experience, α = 1.1 yields good results.

Given that the estimation algorithm requires the evaluation of the objective function

at many different parameter values, we linearise option prices with respect to ω(t) in

the models with stochastic volatility to speed up the calculations. Our procedure is

analogous to the treatment of other stochastic volatility models, which are sometimes

linearised to employ the Kalman filter (see e.g. Trolle and Schwartz, 2009). Specifically,

we linearise call prices for day t around the volatility of the previous day as follows:

cSV (t, T,K, ω(t)) ≈ cSV

(t, T,K, ω

(t− 1

360

))+∂cSV (t, T,K, x)

∂x

∣∣∣∣x=ω(t− 1

360)

[ω (t)− ω

(t− 1

360

)].

Due to the high persistence of ω(t), its previous day value turns out to be a very good

predictor, which reduces the approximation error of the above expansion. In fact, the

linearisation error, expressed in terms of the RMSE’s of options, is very small: around

0.57% when we use data up to July 2007, and below 0.25% in the remaining samples. In

any case, we calculate the exact pricing errors once we have obtained the final parameter

estimates.

A.2 Central tendency

We specify the prices of risk that link the Wiener processes under P and Q as

dWQv (t) = dW P

v (t) + Λv(t)dt,

dWQθ (t) = dW P

θ (t) + Λθ(t)dt,

where Λv(t) = ς and Λθ(t) = ς. Hence, the diffusions under the real measure can be

expressed as

d log V (t) = κ[θP (t)− log V (t)]dt+ σdW Pv (t),

dθP (t) = κ[θP − θP (t)]dt+ σdWQθ (t),

where

θP = θ +σς

κ+σς

κ,

28

and

θP (t) = θ(t) +σς

κ.

Following Leon and Sentana (1997), it can be shown that the conditional distribution

of log V (T ) given information up to time t is Gaussian with mean µCT (t, τ) given in (8)

and variance

ϕ2CT (τ) =

σ2

2κ[1− exp(−2κτ)]

+σ2

(κ

κ− κ

)2[

1−exp(−2κτ)2κ

+ 1−exp(−2κτ)2κ

−21−exp(−(κ+κ)τ)κ+κ

].

By exploiting log-normality, we can write futures prices as

FCT (t, T, V (t), κ, θ, σ) = exp(µCT (t, τ) + 0.5ϕ2CT (τ)),

while call prices follow the usual Black (1976) formula with volatility ϕCT (τ).

In terms of time series dynamics, it can be shown that θ(t) and log V (t) jointly follow

a Gaussian VAR(1) if sampled at equally spaced intervals. Specifically,(θ(T )

log V (T )

)= gτ + Fτ

(θ(t)

log V (t)

)+ ετ ,

where

gτ =

[1− exp(−κτ)

1− exp(−κτ)− κκ−κ (exp(−κτ)− exp(−κτ))

]θ,

Fτ =

[exp(−κτ) 0

κκ−κ [exp(−κτ)− exp(−κτ)] exp(−κτ)

].

and ετ ∼ iid N(0,Στ ), where Στ is a symmetric 2× 2 matrix with elements

Στ (1, 1) =σ2

2κ[1− exp(−2κτ)]

Στ (1, 2) =κσ2

κ− κ

[1− exp(−2κτ)

2κ− 1− exp(−(κ+ κ)τ)

κ+ κ

]and Στ (2, 2) = ϕ2

CT (τ). From here, it is straightforward to obtain the marginal process

followed by log V (t), which corresponds to the following ARMA(2,1) model:

log V (t) = h0(τ) + h1(τ) log V (t− τ) + h2(τ) log V (t− 2τ) + u(t) + g(τ)u(t− τ)

29

where u(t), u(t− τ), · · · ∼ iid N(0, p2(τ)) and

h0(τ) = θ (1− h1(τ)− h2(τ)) ,

h1(τ) = Fτ (1, 1) + Fτ (2, 2),

h2(τ) = −Fτ (1, 1)Fτ (2, 2),

g(τ)p2(τ) =(

Fτ (2, 1) −Fτ (1, 1))Στ

(01

),

(1 + g2(τ))p2(τ) =(

Fτ (2, 1) −Fτ (1, 1))

Στ

(Fτ (2, 1)−Fτ (1, 1)

)+ Στ (2, 2).

A.3 Jumps

Once again, we assume that dWQv (t) = dW P

v (t) + ςdt holds and that jump risk is not

priced. Then, it can be shown that log V (t) satisfies the diffusion

d log V (t) = κ[θP − log V (t)]dt+ σdW Pv (t)− λ

κδdt

under the real measure, where θP = θ + σς/κ.

The conditional characteristic function reduces to

φJ(t, T, u) = exp [ϕJ,0(τ) + ϕJ,V (τ) log V (t)] ,

where

ϕJ,0(τ) = iu

(θ − λ

κδ

)[1− exp(−κτ)]− σ2u2

4κ[1− exp(−2κτ)]

+λ

κlog

[δ − iu exp(−κτ)

δ − iu

]and

ϕJ,V (τ) = iu exp(−κτ).

A.4 Stochastic volatility

We consider a price of risk such that dWQv (t) = dW P

v (t) + ς√ω(t)dt, but we again

assume that jump risk related to Z(t) is not priced. Then, the process under the real

measure can be expressed as

d log V (t) = κ[θ +

ςγ

κω(t)− log V (t)

]dt+ γ

√ω(t)dW P (t),

dω(t) = −λω(t)dt+ dZP (t).

Since in this case the processes under the real and risk neutral measures are different,

we will describe them separately.

30

A.4.1 Risk neutral measure

The conditional characteristic function simplifies to

φSV (t, T, u) = exp [ϕSV,0(τ) + ϕSV,V (τ) log V (t) + ϕSV,ω(τ)ω(t)] ,

where

ϕSV,V (τ) = iu exp(−κτ),

ϕSV,ω(τ) =u2

2(2κ− λ)[exp(−2κτ)− exp(−λτ)] ,

and

ϕSV,0(τ) = iθu[1− exp(−κτ)] + λ [κ(τ, u)− τ ] ,

with

κ(τ, u) =

∫ τ

0

δ

δ − u2

2(2κ−λ)[exp(−2κx)− exp(−λx)]

dx. (A2)

A.4.2 Real measure

We can write the characteristic function under the real measure as

φPSV (t, T, u) = EP [exp(iu log V (T ) + ivω(T ))|V (t), ω(t)]

= exp[ϕPSV,0(τ) + ϕPSV,V (τ) log V (t) + ϕPSV,ω(τ)ω(t)

],

where

ϕPSV,0(τ) = iuθ[1− exp(−κτ)] + λ[κ(τ)− τ ],

ϕPSV,V (τ) = iu exp(−κτ),

ϕPSV,ω(τ) = iv exp(−λτ)− iu ς

κ− λ[exp(−κτ)− exp(−λτ)

]+

u2

2(2κ− λ)

[exp(−2κτ)− exp(−λτ)

]and

κ(τ) =

∫ τ

0

δ

δ − iv exp(−λx) + iuς[exp(−κx)−exp(−λx)]

κ−λ − u2[exp(−2κx)−exp(−λx)]2(2κ−λ)

dx.

31

Based on the characteristic function, it is possible to show that

E [log V (T )|V (t), ω(t)] = θ[1− exp(−κτ)]

− λς

δ(κ− λ)

[1− exp(−κτ)

κ− 1− exp(−λτ)

λ

]+ exp(−κτ) log V (t)

− ς

κ− λ[exp(−κτ)− exp(−λτ)

]ω(t)

E [ω(T )|V (t), ω(t)] =1

δ[1− exp(−λτ)] + exp(−λτ)ω(t),

V [log V (T )|V (t), ω(t)] =λ

(2κ− λ)δ

[1− exp(−λτ)

λ− 1− exp(−2κτ)

2κ

]+

2ς2λ

δ2(κ− λ)2

[1− exp(−2κτ)

2κ+

1− exp(−2λτ)

2λ− 2

1− exp(−(κ+ λ)τ)

κ+ λ

]+

1

2κ− λ[exp(−λτ)− exp(−2κτ)

]ω(t),

V P [ω(T )|V (t), ω(t)] =1− exp(−2λτ)

δ2,

and

covP [log V (T ), ω(T )|V (t), ω(t)] = − 2ςλ

δ2(κ− λ)

[1− exp(−(κ+ λ)τ)

κ+ λ− 1− exp(−2λτ)

2λ

]

A.5 Central tendency and jumps

For this case we use the same prices of risk as in the CT model and assume that jump

risk is not priced. The, we obtain

d log V (t) = κ[θP (t)− log V (t)]dt+ σdW Pv (t) + dZ(t)− λ

κδdt,

dθP (t) = κ[θP − θP (t)]dt+ σdWQθ (t),

under the real measure, where

θP = θ +σς

κ+σς

κ,

and

θP (t) = θ(t) +σς

κ.

The conditional characteristic function becomes

φCTJ(t, T, u) = E [exp(iu log V (T ) + ivθ(T )|V (t), θ(t)]

= exp [ϕCTJ,0(τ) + ϕCTJ,θ(τ)θ(t) + ϕCTJ,V (τ) log V (t)] ,

32

where

ϕCTJ,0(τ) = ivθ [1− exp(−κτ)]

+iu

(θ − λ

κδ

)κκ

κ− κ

[1− exp(−κτ)

κ− 1− exp(−κτ)

κ

]− σ

2v2

4κ[1− exp(−2κτ)]− σ2u2

4κ[1− exp(−2κτ)]

− σ2u2

2

(κ

κ− κ

)2 [1− exp(−2κτ)

2κ+

1− exp(−2κτ)

2κ− 2

1− exp(−(κ+ κ)τ)

κ+ κ

]−uvσ2 κ

κ− κ

[1− exp(−2κτ)

2κ− 1− exp(−(κ+ κ)τ)

κ+ κ

]+λ

κlog

[δ − iu exp(−κτ)

δ − iu

]−iu λ

κδ

κ

κ− κ[exp(−κτ)− exp(−κτ)] ,

ϕCTJ,θ(τ) = iv exp(−κτ) + iuκ

κ− κ[exp(−κτ)− exp(−κτ)] ,

and

ϕCTJ,V (τ) = iu exp(−κτ).

Using the characteristic function, we can show that

E [log V (T )|V (t), θ(t)] = θ

[1− exp(−κτ)− κ

κ− κ(exp(−κτ)− exp(−κτ))

]+

κ

κ− κ[exp(−κτ)− exp(−κτ)] θ(t) + exp(−κτ) log V (t),

V [log V (T )|V (t), θ(t)] =

(σ2

2κ+

λ

κδ2

)[1− exp(−2κτ)]

+σ2

(κ

κ− κ

)2 [1− exp(−2κτ)

2κ+

1− exp(−2κτ)

2κ− 2

1− exp(−(κ+ κ)τ)

κ+ κ

],

V [log V (T ), θ(T )|V (t), θ(t)] = σ2 κ

κ− κ

[1− exp(−2κτ)

2κ− 1− exp(−(κ+ κ)τ)

κ+ κ

].

A.6 Central tendency and stochastic volatility

The price of risk is such that dWQV (t) = dW P

V (t)+ς√ω(t)dt and dWQ

θ (t) = dW Pθ (t)+

ςdt, which yields

d log V (t) = κ[θ(t) +

ςγ

κω(t)− log V (t)

]dt+ γ

√ω(t)dW P

V (t),

dθ(t) = κ[θP − θ(t)]dt+ σdW Pθ (t),

dω(t) = −λω(t)dt+ dZP (t),

33

under the real measure, where

θP = θ +ς σ

κ.

A.6.1 Risk neutral measure

The conditional characteristic function is

φCTSV (t, T, u) = exp

[ϕCTSV,0(τ) + ϕCTSV,V (τ) log V (t)+ϕCTSV,ω(τ)ω(t) + ϕCTSV,θ(τ)θ(t)

],

where

ϕCTSV,V (τ) = iu exp(−κτ),

ϕCTSV,ω(τ) =u2

2(2κ− λ)[exp(−2κτ)− exp(−λτ)] ,

ϕCTSV,θ(τ) = iuκ

κ− κ[exp(−κτ)− exp(−κτ)] ,

and

ϕCTSV,0(τ) = iuκθκ

κ− κ

[1− exp(−κτ)

κ− 1− exp(−κτ)

κ

]−1

2σ2u2

(κ

κ− κ

)2[

1−exp(−2κτ)2κ

+ 1−exp(−2κτ)2κ

−21−exp(−(κ+κ)τ)κ+κ

]+λ [κ(τ, u)− τ ] ,

with κ(τ, u) defined in (A2).

A.6.2 Real measure

We can write the characteristic function under the real measure as

φP (t, T, u1, u2, u3) = E [exp[iu1 log V (T ) + iu2θ(T ) + iu3ω(T )]|V (t), θ(t), ω(t)]

= exp[ϕPCTSV,0(τ) + ϕPCTSV,V (τ) log V (t) + ϕPCTSV,θ(τ)θ(t) + ϕPCTSV,ω(τ)ω(t)

],

where

ϕPCTSV,0(τ) = iu2θ(1− exp(−κτ))

+iu1θκκ

κ− κ

[1− exp(−κτ)

κ− 1− exp(−κτ)

κ

]−u2

2

σ2

2

1− exp(−2κτ)

2κ

−u21

σ2

2

(κ

κ− κ

)2 [1− exp(−2κτ)

2κ+

1− exp(−2κτ)

2κ− 2

1− exp(−(κ+ κ)τ)

κ+ κ

]−u1u2σ

2 κ

κ− κ

[1− exp(−2κτ)

2κ− 1− exp(−(κ+ κ)τ)

κ+ κ

]+λ[κ(τ)− τ ],

34

ϕPCTSV,V (τ) = iu1 exp(−κτ),

ϕPCTSV,θ(τ) = iu2 exp(−κτ) + iu1κ

κ− κ[exp(−κτ)− exp(−κτ)]

ϕPCTSV,ω(τ) = iu3 exp(−λτ)− iu1ς

κ− λ[exp(−κτ)− exp(−λτ)

]+

u21

2(2κ− λ)

[exp(−2κτ)− exp(−λτ)

]and

κ(τ) =

∫ τ

0

δ

δ − iu3 exp(−λx) + iu1ς[exp(−κx)−exp(−λx)]

κ−λ − u21[exp(−2κx)−exp(−λx)]2(2κ−λ)

dx.

Based on the characteristic function, it is possible to show that

E [log V (T )|V (t), θ(t), ω(t)] = θPκκ

κ− κ

[1− exp(−κτ)

κ− 1− exp(−κτ)

κ

]− λς

δ(κ− λ)

[1− exp(−κτ)

κ− 1− exp(−λτ)

λ

]+ exp(−κτ) log V (t)

+κ

κ− κ[exp(−κτ)− exp(−κτ)] θ(t)

− ς

κ− λ[exp(−κτ)− exp(−λτ)

]ω(t)

E [θ(T )|V (t), θ(t), ω(t)] = θP [1− exp(−κτ)] + exp(−κτ)θ(t)

E [ω(T )|V (t), θ(t), ω(t)] =1

δ[1− exp(−λτ)] + exp(−λτ)ω(t),

V [log V (T )|V (t), θ(t), ω(t)] =λ

(2κ− λ)δ

[1− exp(−λτ)

λ− 1− exp(−2κτ)

2κ

]+σ2

(κ

κ− κ

)2 [1− exp(−2κτ)

2κ+

1− exp(−2κτ)

2κ− 2

1− exp(−(κ+ κ)τ)

κ+ κ

]+

2ς2λ

δ2(κ− λ)2

[1− exp(−2κτ)

2κ+

1− exp(−2λτ)

2λ− 2

1− exp(−(κ+ λ)τ)

κ+ λ

]+

1

2κ− λ[exp(−λτ)− exp(−2κτ)

]ω(t),

V [θ(T )|V (t), θ(t), ω(t)] = σ2 1− exp(−2κτ)

2κ

V [ω(T )|V (t), θ(t), ω(t)] =1− exp(−2λτ)

δ2,

cov [log V (T ), θ(T )|V (t), θ(t), ω(t)] =σ2κ

κ− κ

[1− exp(−2κτ)

2κ− 1− exp(−(κ+ κ)τ)

κ+ κ

]and

cov [log V (T ), ω(T )|V (t), θ(t), ω(t)] = − 2ςλ

δ2(κ− λ)

[1− exp(−(κ+ λ)τ)

κ+ λ− 1− exp(−2λτ)

2λ

]

35

Table 1

Root mean square pricing errors in standard one-factor models

(a) Futures

ValidationModel Estimation Feb 06-Aug 08 Aug 08-Mar 10 Apr 10-Dec 10 Full sample

SQR Full sample 3.818 11.675 10.645 8.517Until Mar 10 4.108 12.491 11.120 9.065Until Aug 08 2.602 7.480 8.514 5.850

Log-OU Full sample 3.597 5.914 5.056 4.755Until Mar 10 3.033 6.380 5.699 4.883Until Aug 08 1.958 6.659 7.123 5.032

(a) Options

ValidationModel Estimation Feb 06-Aug 08 Aug 08-Mar 10 Apr 10-Dec 10 Full sample

SQR Full sample 0.480 0.721 0.594 0.616Until Mar 10 0.451 0.726 0.607 0.614Until Aug 08 0.353 0.962 0.821 0.767

Log-OU Full sample 0.322 0.494 0.471 0.436Until Mar 10 0.313 0.497 0.487 0.440Until Aug 08 0.306 0.553 0.557 0.484

Notes: The root mean square errors that correspond to out-of-sample periods are reported in italics.

“SQR” denotes square root model while “log-OU” refers to the log-normal Ornstein-Uhlenbeck process.

36

Table 2

Parameters estimates of the standard one-factor models under the real measure

Model Estimation κ θ σ ς σf σoSQR Full Sample 2.977 24.313 5.226 0.283 0.358 0.616

(0.360) (2.936) (0.006) (0.069) (0.001) (0.000)

Until Mar 10 3.310 24.303 5.303 0.327 0.396 0.616(0.404) (2.957) (0.008) (0.076) (0.002) (0.000)

Until Aug 08 1.769 19.234 3.535 0.068 0.171 0.353(0.368) (3.983) (0.006) (0.104) (0.001) (0.000)

Log-OU Full Sample 2.936 3.099 0.958 0.018 0.189 0.436(0.005) (0.128) (0.000) (0.392) (0.001) (0.000)

Until Mar 10 2.788 3.093 0.927 0.146 0.182 0.423(0.007) (0.140) (0.001) (0.421) (0.001) (0.000)

Until Aug 08 1.795 2.923 0.793 0.135 0.100 0.306(0.010) (0.205) (0.001) (0.463) (0.001) (0.000)

Notes: “SQR” denotes square root model while “log-OU” refers to the log-normal Ornstein-Uhlenbeck

process. Standard errors, displayed in parentheses, have been obtained by using the outer-product of

the score to estimate the information matrix.

37

Table 3

Root mean square pricing errors in the extended log-OU models

(a) Futures

ValidationExtension Estimation Feb 06-Aug 08 Aug 08-Mar 10 Apr 10-Dec 10 Full sample

CT Full sample 0.372 0.847 0.792 0.646Until Mar 10 0.348 0.773 1.004 0.653Until Aug 08 0.332 0.786 1.056 0.667

J Full sample 3.748 6.173 5.080 4.928Until Mar 10 3.191 6.669 5.731 5.070Until Aug 08 2.036 7.000 7.270 5.238

SV Full sample 2.234 4.821 4.873 3.782Until Mar 10 1.853 5.134 5.344 3.921Until Aug 08 1.756 5.720 5.662 4.239

CTJ Full sample 0.382 0.900 0.747 0.665Until Mar 10 0.353 0.804 0.913 0.646Until Aug 08 0.339 0.817 0.956 0.658

CTSV Full sample 0.377 0.839 0.901 0.666Until Mar 10 0.346 0.786 1.025 0.663Until Aug 08 0.359 0.787 1.147 0.697

(b) Options

ValidationExtension Estimation Feb 06-Aug 08 Aug 08-Mar 10 Apr 10-Dec 10 Full sample

CT Full sample 0.312 0.482 0.471 0.428Until Mar 10 0.299 0.483 0.493 0.432Until Aug 08 0.286 0.517 0.554 0.462

J Full sample 0.285 0.459 0.386 0.389Until Mar 10 0.280 0.457 0.412 0.393Until Aug 08 0.272 0.506 0.467 0.429

SV Full sample 0.201 0.241 0.275 0.237Until Mar 10 0.199 0.242 0.277 0.238Until Aug 08 0.196 0.247 0.281 0.240

CTJ Full sample 0.271 0.449 0.389 0.381Until Mar 10 0.263 0.448 0.424 0.388Until Aug 08 0.250 0.476 0.468 0.410

CTSV Full sample 0.178 0.212 0.227 0.205Until Mar 10 0.177 0.212 0.231 0.205Until Aug 08 0.174 0.219 0.236 0.209

Notes: The root mean square errors that correspond to out-of-sample periods are reported in italics.

“CT” indicates time varying Central Tendency. “J” refers to jumps. “SV” denotes Stochastic Volatility.

“CTJ” combines CT and J. “CTSV” combines CT and SV.

38

Table

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0.3

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(0.0

07)

-(0

.125)

(0.3

70)

(0.1

31)

(0.0

06)

--

-(0

.548)

-(0

.001)

(0.0

00)

Unti

lA

ug

081.

946

-2.

913

20.5

68

9.1

98

0.3

38

--

-0.2

70

-0.1

04

0.2

72

(0.0

08)

-(0

.180)

(1.3

63)

(0.2

11)

(0.0

16)

--

-(1

.035)

-(0

.001)

(0.0

00)

SV

Fu

llS

amp

le1.

117

-3.

063

--

--

12.7

89

2.2

20

0.7

41

-0.1

38

0.2

39

(0.0

04)

-(0

.002)

--

--

(0.0

36)

(0.0

04)

(0.3

85)

-(0

.001)

(0.0

00)

Unti

lM

ar10

0.96

0-

2.978

--

--

13.2

78

2.4

52

0.7

73

-0.1

23

0.2

25

(0.0

04)

-(0

.002)

--

--

(0.0

38)

(0.0

05)

(0.4

26)

-(0

.001)

(0.0

00)

Unti

lA

ug

080.

696

-2.

874

--

--

14.3

21

3.0

52

0.6

78

-0.0

91

0.1

98

(0.0

06)

-(0

.003)

--

--

(0.1

20)

(0.0

11)

(0.5

52)

-(0

.000)

(0.0

00)

CT

JF

ull

Sam

ple

5.94

70.

334

2.961

12.3

15

5.9

44

0.6

72

0.4

38

--

-0.1

87

0.1

09

0.0

24

0.3

81

(0.0

15)

(0.0

04)

(0.5

94)

(0.1

57)

(0.0

17)

(0.0

05)

(0.0

01)

--

(0.7

43)

(0.4

62)

(0.0

00)

(0.0

00)

Unti

lM

ar10

7.20

50.

404

2.953

20.4

80

7.2

07

0.6

60

0.4

59

--

-0.0

10

0.1

15

0.0

21

0.3

75

(0.0

23)

(0.0

03)

(0.5

63)

(0.3

20)

(0.0

25)

(0.0

07)

(0.0

01)

--

(0.9

07)

(0.5

10)

(0.0

00)

(0.0

00)

Unti

lA

ug

087.

088

0.43

72.

834

19.3

61

6.9

96

0.5

82

0.4

82

--

-0.0

68

0.0

46

0.0

19

0.2

50

(0.0

38)

(0.0

07)

(0.8

03)

(0.5

17)

(0.0

41)

(0.0

12)

(0.0

01)

--

(1.2

28)

(0.7

44)

(0.0

00)

(0.0

00)

CT

SV

Fu

llS

amp

le7.

813

0.35

42.

827

--

-0.2

26

1.3

83

0.2

73

0.2

47

0.0

26

0.0

24

0.2

06

(0.0

10)

(0.0

03)

(0.1

77)

--

-(0

.001)

(0.0

06)

(0.0

01)

(0.4

25)

(0.2

77)

(0.0

00)

(0.0

00)

Unti

lM

ar10

7.98

90.

425

2.783

--

-0.2

67

1.2

26

0.2

69

0.3

24

0.0

04

0.0

21

0.1

98

(0.0

13)

(0.0

03)

(0.2

06)

--

-(0

.001)

(0.0

07)

(0.0

01)

(0.4

80)

(0.3

28)

(0.0

00)

(0.0

00)

Unti

lA

ug

088.

376

0.46

92.

855

--

-0.2

69

0.5

42

0.1

73

-1.4

12

0.1

78

0.0

19

0.1

74

(0.0

32)

(0.0

06)

(0.2

73)

--

-(0

.002)

(0.0

11)

(0.0

02)

(0.1

86)

(0.4

77)

(0.0

00)

(0.0

00)

Not

es:

“CT

”in

dic

ates

Cen

tral

Ten

den

cy,

alo

g-O

Up

roce

ssw

her

eth

elo

ng

run

mea

nfo

llow

san

oth

erlo

g-O

Ud

iffu

sion

.“J”

intr

od

uce

sju

mp

sin

the

stan

dard

log-

OU

mod

el,

wh

ose

size

foll

ows

anex

pon

enti

ald

istr

ibu

tion

.“S

V”

den

ote

sS

toch

ast

icV

ola

tili

ty,

alo

g-O

Um

od

elw

her

eth

evo

lati

lity

isa

Gam

ma

OU

Lev

y

pro

cess

.“C

TJ”

com

bin

esce

ntr

alte

nd

ency

and

jum

ps.

“CT

SV

”co

mb

ines

centr

al

ten

den

cyan

dst

och

ast

icvola

tility

.S

tan

dard

erro

rs,

dis

pla

yed

inp

are

nth

eses

,

hav

eb

een

obta

ined

by

usi

ng

the

oute

r-p

rod

uct

ofth

esc

ore

toes

tim

ate

the

info

rmati

on

matr

ix.

39

Table 5

Contribution to the log-likelihood of futures, options and VIX

Log-OU modelsData SQR Standard CT J SV CTJ CTSV

Futures -3398.6 2145.4 19222.4 1838.0 4919.5 19140.3 19363.2Options -82136.7 -51811.8 -50119.4 -41647.9 61.8 -39919.7 12207.8VIX -2460.4 1313.2 1410.7 1294.0 1323.7 1385.7 1417.6Total -87995.7 -48353.2 -29486.3 -38515.9 6305.0 -19393.7 32988.6

Notes: “SQR” denotes square root model and “log-OU” refers to a log-normal Ornstein-Uhlenbeck

process. “CT” indicates Central Tendency, a log-OU process where the long run mean follows an-

other log-OU diffusion. “J” introduces jumps in the standard log-OU model, whose size follows an

exponential distribution. “SV” denotes Stochastic Volatility, a log-OU model where the volatility

is a Gamma OU Levy process. “CTJ” combines central tendency and jumps. “CTSV” combines

central tendency and stochastic volatility.

40

Table 6

Root mean square pricing errors of futures prices by maturity

Log-OU modelsMaturity N SQR Standard CT J SV CTJ CTSV

Less than 1 month 1036 2.846 1.922 0.999 1.954 2.344 1.035 0.975From 1 to 3 months 2290 6.760 3.811 0.819 3.954 3.837 0.865 0.821From 3 to 6 months 3093 9.786 5.015 0.421 5.232 4.043 0.415 0.468More than 6 months 2246 9.880 6.007 0.471 6.189 3.887 0.460 0.532Total 8665 8.517 4.755 0.646 4.928 3.782 0.665 0.666

Notes: “SQR” denotes square root model and “log-OU” refers to a log-normal Ornstein-Uhlenbeck

process. “CT” indicates Central Tendency, a log-OU process where the long run mean follows an-

other log-OU diffusion. “J” introduces jumps in the standard log-OU model, whose size follows an

exponential distribution. “SV” denotes Stochastic Volatility, a log-OU model where the volatility

is a Gamma OU Levy process. “CTJ” combines central tendency and jumps. “CTSV” combines

central tendency and stochastic volatility. The column labeled N gives the number of prices per

category. Results are based on full sample estimates.

41

Table 7

Root mean square pricing errors of call prices by moneyness and maturities

(a) τ < 1 monthLog-OU models

Moneyness N SQR Standard CT J SV CTJ CTSV

< −0.3 2199 0.165 0.133 0.131 0.145 0.120 0.145 0.123[−0.3,−0.1) 2080 0.529 0.354 0.364 0.401 0.183 0.381 0.206[−0.1, 0.1) 2260 0.792 0.534 0.516 0.567 0.241 0.512 0.240[0.1, 0.3) 2254 0.694 0.542 0.462 0.436 0.288 0.408 0.241≥ 0.3 3085 0.372 0.339 0.307 0.215 0.223 0.215 0.189

(b) 1 month< τ < 3 monthsLog-OU models

Moneyness N SQR Standard CT J SV CTJ CTSV

< −0.3 3553 0.324 0.205 0.202 0.172 0.194 0.172 0.184[−0.3,−0.1) 4217 0.665 0.452 0.447 0.380 0.254 0.374 0.219[−0.1, 0.1) 4701 0.728 0.492 0.488 0.478 0.161 0.475 0.151[0.1, 0.3) 4638 0.711 0.469 0.467 0.437 0.227 0.435 0.200≥ 0.3 8089 0.505 0.377 0.382 0.261 0.239 0.262 0.170

(c) τ > 3 monthsLog-OU models

Moneyness N SQR Standard CT J SV CTJ CTSV

< −0.3 3229 0.435 0.256 0.261 0.185 0.230 0.199 0.213[−0.3,−0.1) 3719 0.626 0.451 0.426 0.362 0.276 0.353 0.188[−0.1, 0.1) 4491 0.570 0.435 0.412 0.422 0.207 0.402 0.197[0.1, 0.3) 3971 0.677 0.474 0.480 0.451 0.270 0.441 0.231≥ 0.3 5613 0.651 0.447 0.470 0.313 0.284 0.333 0.230

(d) All maturitiesLog-OU models

Moneyness N SQR Standard CT J SV CTJ CTSV

< −0.3 8981 0.341 0.211 0.212 0.171 0.193 0.176 0.183[−0.3,−0.1) 10016 0.624 0.433 0.423 0.378 0.250 0.368 0.205[−0.1, 0.1) 11452 0.685 0.480 0.466 0.477 0.197 0.456 0.190[0.1, 0.3) 10863 0.695 0.487 0.471 0.442 0.257 0.432 0.221≥ 0.3 16787 0.539 0.395 0.402 0.272 0.252 0.280 0.196

Notes: Moneyness is defined as log(K/F (t, T )), where K and F (t, T ) are the strike and futures prices,

respectively. “SQR” denotes square root model and “log-OU” refers to a log-normal Ornstein-Uhlenbeck

process. “CT” indicates Central Tendency, a log-OU process where the long run mean follows another

log-OU diffusion. “J” introduces jumps in the standard log-OU model, whose size follows an exponential

distribution. “SV” denotes Stochastic Volatility, a log-OU model where the volatility is a Gamma OU

Levy process. “CTJ” combines central tendency and jumps. “CTSV” combines central tendency and

stochastic volatility. The column labeled N gives the number of prices per category. τ denotes time to

maturity. Results are based on full sample estimates.

42

Table 8

Root mean square pricing errors of put prices by moneyness and maturities

(a) τ < 1 monthLog-OU models

Moneyness N SQR Standard CT J SV CTJ CTSV

< −0.3 457 0.262 0.213 0.193 0.256 0.139 0.256 0.151[−0.3,−0.1) 1657 0.574 0.394 0.405 0.447 0.196 0.425 0.223[−0.1, 0.1) 2239 0.800 0.540 0.529 0.569 0.246 0.520 0.251[0.1, 0.3) 1787 0.753 0.586 0.507 0.484 0.296 0.459 0.253≥ 0.3 1385 0.424 0.386 0.355 0.283 0.240 0.283 0.216

(b) 1 month< τ < 3 monthsLog-OU models

Moneyness N SQR Standard CT J SV CTJ CTSV

< −0.3 1970 0.347 0.245 0.240 0.237 0.234 0.244 0.211[−0.3,−0.1) 4274 0.662 0.466 0.460 0.399 0.253 0.393 0.213[−0.1, 0.1) 3966 0.743 0.512 0.511 0.496 0.169 0.495 0.152[0.1, 0.3) 2272 0.802 0.550 0.546 0.504 0.236 0.507 0.211≥ 0.3 1505 0.575 0.436 0.431 0.323 0.252 0.321 0.204

(c) τ > 3 monthsLog-OU models

Moneyness N SQR Standard CT J SV CTJ CTSV

< −0.3 1698 0.428 0.294 0.292 0.280 0.249 0.287 0.218[−0.3,−0.1) 2764 0.642 0.487 0.461 0.412 0.294 0.408 0.206[−0.1, 0.1) 2174 0.661 0.493 0.480 0.484 0.229 0.477 0.214[0.1, 0.3) 990 0.867 0.592 0.609 0.556 0.293 0.561 0.260≥ 0.3 633 0.637 0.457 0.469 0.374 0.268 0.376 0.319

(d) All maturitiesLog-OU models

Moneyness N SQR Standard CT J SV CTJ CTSV

< −0.3 4125 0.375 0.263 0.258 0.257 0.232 0.264 0.208[−0.3,−0.1) 8695 0.639 0.460 0.450 0.413 0.257 0.404 0.213[−0.1, 0.1) 8379 0.739 0.515 0.508 0.513 0.208 0.497 0.199[0.1, 0.3) 5049 0.798 0.571 0.546 0.508 0.270 0.502 0.237≥ 0.3 3523 0.534 0.421 0.410 0.318 0.251 0.318 0.233

Notes: Moneyness is defined as log(K/F (t, T )), where K and F (t, T ) are the strike and futures prices,

respectively. “SQR” denotes square root model and “log-OU” refers to a log-normal Ornstein-Uhlenbeck

process. “CT” indicates Central Tendency, a log-OU process where the long run mean follows another

log-OU diffusion. “J” introduces jumps in the standard log-OU model, whose size follows an exponential

distribution. “SV” denotes Stochastic Volatility, a log-OU model where the volatility is a Gamma OU

Levy process. “CTJ” combines central tendency and jumps. “CTSV” combines central tendency and

stochastic volatility. The column labeled N gives the number of prices per category. τ denotes time to

maturity. Results are based on full sample estimates.

43

Figure 1: Historical evolution of the VIX index

(a) 1990-2010

May90 Jan93 Oct95 Jul98 Apr01 Jan04 Oct06 Jul09

10

20

30

40

50

60

70

80

(b) 2006-2010

Mar06 Oct06 Apr07 Nov07 Jun08 Dec08 Jul09 Jan10 Aug10

10

20

30

40

50

60

70

80

90

16−Jul−07 15−Aug−08

20−Nov−08

31−Mar−10

Figure 2: Time series autocorrelations of the log-VIX and estimated ARMA models

(a) Autocorrelations

1 2 3 4 5 6 7 8 9 100.86

0.88

0.9

0.92

0.94

0.96

0.98

1

AR(1)ARMA(1,1)ARMA(2,1)Actual

(b) Partial autocorrelations

2 3 4 5 6 7 8 9 10−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12AR(1)ARMA(1,1)ARMA(2,1)Actual

Note: Results are based on the 1990-2010 sample (5280 daily observations).

Figure 3a: Number of futures prices per day

Mar06 Oct06 Apr07 Nov07 Jun08 Dec08 Jul09 Jan10 Aug101

2

3

4

5

6

7

8

9

10

Figure 3b: Number of option prices per day

Mar06 Oct06 Apr07 Nov07 Jun08 Dec08 Jul09 Jan10 Aug100

20

40

60

80

100

120

140

CallsPuts

Figure 4a: Term structure of VIX futures

Oct06Feb08

Jul09Nov10

24

68

20

40

60

DateMonths to expiration

Figure 4b: VIX futures prices vs. VIX index

Mar06 Oct06 Apr07 Nov07 Jun08 Dec08 Jul09 Jan10 Aug1010

20

30

40

50

60

70

80VIXFutures

Figure 4c: Term structure of VIX futures at four particular days

1 2 3 4 5 6 7 815

20

25

30

351 2 3 4 5 6 7 8

10

20

30

40

50

60

7016/07/2007 (left)15/08/2008 (left)20/11/2008 (right)31/03/2010 (left)

Figure 5: Daily root mean square errors of standard one-factor pricing models

(a) Futures

Mar06 Oct06 Apr07 Nov07 Jun08 Dec08 Jul09 Jan10 Aug100

5

10

15

20

25SQRLog−OU

(b) Options

Mar06 Oct06 Apr07 Nov07 Jun08 Dec08 Jul09 Jan10 Aug100

0.5

1

1.5

2

2.5SQRLog−OU

Note: “SQR” denotes square root model and “log-OU” refers to a log-normal Ornstein-Uhlenbeck

process. Results are based on full sample estimates.

Figure 6: Implied volatilities of call prices

(a) 16-July-2007

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.250.15

0.2

0.25

0.3

0.35

0.4

0.45

36 days

63 days91 days

Log−OU, 36 days

Log−OU, 63 days

Log−OU, 91 days

Moneyness

(b) 15-August-2008

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.250.15

0.2

0.25

0.3

0.35

0.4

0.45

32 days

67 days94 days

Log−OU, 32 days

Log−OU, 67 days

Log−OU, 94 days

Moneyness

(c) 20-November-2008

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.250.15

0.2

0.25

0.3

0.35

0.4

0.45

27 days

61 days

88 days

Log−OU, 27 days

Log−OU, 61 days

Log−OU, 88 days

Moneyness

(d) 31-March-2010

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.250.15

0.2

0.25

0.3

0.35

0.4

0.45

21 days

76 days

111 days

Log−OU, 21 days

Log−OU, 76 days

Log−OU, 111 days

Moneyness

Note: Implied volatilities have been obtained by inverting the Black (1976) call price formula (5).

Moneyness is defined as log(K/F (t, T )).

Figure 7: Daily root mean square errors of standard log-OU and central tendency log-OUwith stochastic volatility

(a) Futures

Mar06 Oct06 Apr07 Nov07 Jun08 Dec08 Jul09 Jan10 Aug10

2

4

6

8

10

12

Standard log−OUCTSV

(b) Options

Mar06 Oct06 Apr07 Nov07 Jun08 Dec08 Jul09 Jan10 Aug10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Standard log−OUCTSV

Notes: “Standard log-OU” refers to a log-normal Ornstein-Uhlenbeck process. “CTSV” generalises

the standard log-OU by introducing a time varying central tendency and stochastic volatility.

Results are based on full sample estimates.

Figure 8: Empirical cumulative distribution function of the square pricing errors

(a) Futures

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SQRStandard log−OUCTJSVCTJCTSV

(b) Options

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SQRStandard log−OUCTJSVCTJCTSV

Notes: “SQR” denotes square root model and “log-OU” refers to a log-normal Ornstein-Uhlenbeck

process. “J” introduces jumps in the standard log-OU model, whose size follows an exponential

distribution. “CT” indicates Central Tendency, a log-OU process where the long run mean follows

another log-OU diffusion. “SV” denotes Stochastic Volatility, a log-OU model where the volatility

is a Gamma OU Levy process. “CTJ” combines central tendency and jumps. “CTSV” combines

central tendency and stochastic volatility. Results are based on full sample estimates.

Figure 9a: Filtered θ(t) for different log-OU models with central tendency

Mar06 Oct06 Apr07 Nov07 Jun08 Dec08 Jul09 Jan10 Aug102

2.5

3

3.5

4

4.5log(VIX)CTCTJCTSV

Figure 9b: Filtered θ(t) for different estimation samples in the log-OU model with centraltendency and stochastic volatility

Mar06 Oct06 Apr07 Nov07 Jun08 Dec08 Jul09 Jan10 Aug102

2.5

3

3.5

4

4.5log(VIX)Until Aug08Until Mar10Full Sample

Figure 9c: Differences between model-based and actual one-month futures prices

Mar06 Oct06 Apr07 Nov07 Jun08 Dec08 Jul09 Jan10 Aug10−10

−8

−6

−4

−2

0

2

4

6

8

10

12log−OUCTCTSV

Notes: θ(t) denotes the time varying central tendency around which the VIX mean-reverts in central

tendency models. “CT” indicates Central Tendency, a log-OU process where the long run mean follows

another log-OU diffusion. “CTJ” combines central tendency and jumps with exponential sizes. “CTSV”

combines central tendency and stochastic volatility. Results in Figures 9a and 9c are based on full sample

estimates. One month actual futures prices in Figure 9c have been obtained by interpolation of the prices

at the adjacent maturities.

Figure 10a: Filtered ω(t) for different log-OU models with stochastic volatility

Mar06 Oct06 Apr07 Nov07 Jun08 Dec08 Jul09 Jan10 Aug102

2.5

3

3.5

4

4.5Mar06 Oct06 Apr07 Nov07 Jun08 Dec08 Jul09 Jan10 Aug10

0

0.5

1

1.5

2

2.5

3

3.5

4log(VIX) (left)SV (right)CTSV (right)

Figure 10b: Filtered ω(t) for different estimation samples in the log-OU model with centraltendency and stochastic volatility

Mar06 Oct06 Apr07 Nov07 Jun08 Dec08 Jul09 Jan10 Aug100

0.5

1

1.5

2

2.5

3

3.5

4

Until Aug08Until Mar10Full sample

Figure 10c: One month volatilities of the VIX

Mar06 Oct06 Apr07 Nov07 Jun08 Dec08 Jul09 Jan10 Aug100.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Implied volatility of one−month at−the−money optionsOne month ahead std. dev. of log(VIX) under CTSV

Notes: Different vertical scales are used for the VIX and the volatilities in Figure 9a. “SV” denotes a

log-OU model with stochastic volatility, while “CTSV” refers to a log-OU model with central tendency

and stochastic volatility. One-month implied vols in Figure 9c have been obtained by interpolation of the

implied vols of options with moneyness | log(K/F (t, T ))| < .1, which in turn result from inverting the

Black (1976) call price formula. Results in Figures 10a and 10c are based on full sample estimates.

Figure 11: Fit of the term structure of futures prices

(a) 16-July-2007

2 3 4 5 6 7 8 9 10 1116

17

18

19

20

21

22

23

24

25

ActualSQRStandard log−OUCTJSVCTJCTSV

Months to maturity

(b) 15-August-2008

2 3 4 5 6 716

17

18

19

20

21

22

23

24

25

Months to maturity

(c) 20-November-2008

1 2 3 4 5 6 7 825

30

35

40

45

50

55

60

65

70

Months to maturity

(d) 31-March-2010

1 2 3 4 5 6 716

17

18

19

20

21

22

23

24

25

Months to maturity

Notes: “SQR” denotes square root model and “log-OU” refers to a log-normal Ornstein-Uhlenbeck process.

“J” introduces jumps in the standard log-OU model, whose size follows an exponential distribution. “CT”

indicates Central Tendency, a log-OU process where the long run mean follows another log-OU diffusion.

“SV” denotes Stochastic Volatility, a log-OU model where the volatility is a Gamma OU Levy process.

“CTJ” combines central tendency and jumps. “CTSV” combines central tendency and stochastic volatility.

Results are based on full sample estimates.

Figure 12: Fit of the term structure of call prices for fixed strikes

(a) 16-July-2007, Strike=15

2 3 4 5 6 7 8 9 102

2.5

3

3.5

4

4.5

5

5.5

ActualSQRStandard log−OUCTJSVCTJCTSV

Months to maturity

(b) 15-August-2008, Strike=20

1.5 2 2.5 3 3.5 4 4.5 52

2.5

3

3.5

4

4.5

5

5.5

Months to maturity

(c) 20-November-2008, Strike=60

1 1.5 2 2.5 3 3.5 4 4.52

3

4

5

6

7

8

9

10

11

12

Months to maturity

(d) 31-March-2010, Strike=25

1 1.5 2 2.5 3 3.5 4 4.5 5 5.50

0.5

1

1.5

2

2.5

3

3.5

Months to maturity

Notes: “SQR” denotes square root model and “log-OU” refers to a log-normal Ornstein-Uhlenbeck process.

“J” introduces jumps in the standard log-OU model, whose size follows an exponential distribution. “CT”

indicates Central Tendency, a log-OU process where the long run mean follows another log-OU diffusion.

“SV” denotes Stochastic Volatility, a log-OU model where the volatility is a Gamma OU Levy process.

“CTJ” combines central tendency and jumps. “CTSV” combines central tendency and stochastic volatility.

Results are based on full sample estimates.

Fig

ure

13:

Fit

ofth

eim

pli

edvo

lati

liti

esof

call

pri

ces

(i)

(a)

16-J

uly

-2007.τ

=36

day

s

−0.

2−

0.1

00.

10.

20.

150.2

0.250.

3

0.350.

4

0.45

Act

ual

Sta

ndar

d lo

g−O

UC

TC

TJ

CT

SV

Mon

eyn

ess

(b)

16-J

uly

-200

7.τ

=63

day

s

−0.

2−

0.1

00.

10.

20.

150.2

0.250.

3

0.350.

4

0.45

Mon

eyn

ess

(c)

16-J

uly

-200

7.τ

=91

day

s

−0.

2−

0.1

00.

10.

20.

150.2

0.250.

3

0.350.

4

0.45

Mon

eyn

ess

(d)

15-

Au

gust

-200

8.τ

=32

day

s

−0.

2−

0.1

00.

10.

20.

150.2

0.250.

3

0.350.

4

0.45

Mon

eyn

ess

(e)

15-A

ugu

st-2

008.τ

=67

day

s

−0.

2−

0.1

00.

10.

20.

150.2

0.250.

3

0.350.

4

0.45

Mon

eyn

ess

(f)

15-A

ugu

st-2

008.τ

=94

day

s

−0.

2−

0.1

00.

10.

20.

150.2

0.250.

3

0.350.

4

0.45

Mon

eyn

ess

Not

e:Im

plied

vol

atilit

ies

hav

eb

een

obta

ined

by

inve

rtin

gth

eB

lack

(1976)

call

pri

cefo

rmu

la.

Mon

eyn

ess

isd

efin

edas

log(K/F

(t,T

)).

“L

og-

OU

”re

fers

toa

log-

nor

mal

Orn

stei

n-U

hle

nb

eck

pro

cess

.“C

TJ”

com

bin

esce

ntr

al

ten

den

cyan

dju

mp

s.“C

TS

V”

com

bin

esce

ntr

al

ten

den

cyan

d

stoch

asti

cvol

atil

ity.

Res

ult

sar

eb

ased

onfu

llsa

mp

lees

tim

ate

s.

Fig

ure

14:

Fit

ofth

eim

pli

edvo

lati

liti

esof

call

pri

ces

(ii)

(a)

20-N

ovem

ber

-2008.τ

=27

day

s

−0.

2−

0.1

00.

10.

20.

150.2

0.250.

3

0.350.

4

0.45

Act

ual

Sta

ndar

d lo

g−O

UC

TC

TJ

CT

SV

Mon

eyn

ess

(b)

20-N

ovem

ber

-200

8.τ

=61

day

s

−0.

2−

0.1

00.

10.

20.

150.2

0.250.

3

0.350.

4

0.45

Mon

eyn

ess

(c)

20-N

ovem

ber

-200

8.τ

=88

day

s

−0.

2−

0.1

00.

10.

20.

150.2

0.250.

3

0.350.

4

0.45

Mon

eyn

ess

(d)

31-M

arc

h-2

010

.τ

=21

day

s

−0.

2−

0.1

00.

10.

20.

150.2

0.250.

3

0.350.

4

0.45

Mon

eyn

ess

(e)

31-M

arch

-201

0.τ

=76

day

s

−0.

2−

0.1

00.

10.

20.

150.2

0.250.

3

0.350.

4

0.45

Mon

eyn

ess

(f)

31-M

arch

-201

0.τ

=11

1d

ays

−0.

2−

0.1

00.

10.

20.

150.2

0.250.

3

0.350.

4

0.45

Mon

eyn

ess

Not

e:Im

plied

vol

atilit

ies

hav

eb

een

obta

ined

by

inve

rtin

gth

eB

lack

(1976)

call

pri

cefo

rmu

la.

Mon

eyn

ess

isd

efin

edas

log(K/F

(t,T

)).

“L

og-

OU

”re

fers

toa

log-

nor

mal

Orn

stei

n-U

hle

nb

eck

pro

cess

.“C

TJ”

com

bin

esce

ntr

al

ten

den

cyan

dju

mp

s.“C

TS

V”

com

bin

esce

ntr

al

ten

den

cyan

d

stoch

asti

cvol

atil

ity.

Res

ult

sar

eb

ased

onfu

llsa

mp

lees

tim

ate

s.

Figure 15: Fit of the implied volatilities of call prices (i)

(a) 16-July-2007, Actual data

12 13 14 15 16 17 18 19 20 21 220

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

36 63 91125305

Strike

(b) 16-July-2007, CTSV

12 13 14 15 16 17 18 19 20 21 220

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Strike

(c) 15-August-2008, Actual data

16 18 20 22 24 26 28 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

32 67 94122156

Strike

(d) 15-August-2008, CTSV

16 18 20 22 24 26 28 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Strike

Note: Implied volatilities have been obtained by inverting the Black (1976) call price formula. The

legends indicate the number of remaining days until maturity for each implied volatility smirk.

The range of strikes is such that it covers the range of moneyness values from -.25 to .25, where

moneyness is defined as log(K/F (t, T )). “CTSV” is a log-normal Ornstein Uhlenbeck process that

combines central tendency and stochastic volatility. Results are based on full sample estimates.

Figure 16: Fit of the implied volatilities of call prices (ii)

(a) 20-November-2008, Actual data

40 45 50 55 60 65 70 75 80 850

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

27 61 88118145

Strike

(b) 20-November-2008, CTSV

40 45 50 55 60 65 70 75 80 850

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Strike

(c) 31-March-2010, Actual data

15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

21 49 76111138165

Strike

(d) 31-March-2010, CTSV

15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Strike

Note: Implied volatilities have been obtained by inverting the Black (1976) call price formula. The

legends indicate the number of remaining days until maturity for each implied volatility smirk.

The range of strikes is such that it covers the range of moneyness values from -.25 to .25, where

moneyness is defined as log(K/F (t, T )). “CTSV” is a log-normal Ornstein Uhlenbeck process that

combines central tendency and stochastic volatility. Results are based on full sample estimates.