97 (0.00)
Finally, there is some evidence that stock markets share periods of high volatility. Thissuggests that multivariate models will be important.8.1.2 Pricing contingent assetsConsider an asset C, with expiring date t + τ , which is a function of a generic underlying security S. Assume now that S can be described by the following geometricdiffusion process:dS = µSdt + σ Sdzso that d log S = (µ − σ 2 /2) dt + σ dz. Economists term such an asset C as “contingent”or “derivative”. A primary example of a derivative is an option, which gives the ownerthe ability but not the obligation to trade the underlying security at a given price K, calledthe strike price, in the future. European call options are the most known: the owner canbuy the underlying asset at the strike price K only when the call expires, i.e. at datet + τ . Its value at the expiration date will be:Ct + τ = max(St+τ − K, 0)Its purchase value at time t is as yet unknown, but can be determined in different ways.One of these consists of calculating the discounted expected value of the option at timet + τ.exp(−rτ )ESt+τ |St [max(St+τ − K, 0)]where r is the risk-free interest rate. However, this completely ignores the fact that this isa risky asset and traders expect higher returns than on riskless assets. This is the reasonwhy the discounted expected value is not considered by the market as a correct methodto evaluate an asset. To avoid this inconvenience it is opportune to introduce a utilityfunction into the pricing of options, letting the dealers choose the risk–expected gaincombination they prefer.It turns out that the added complexity of a utility function can be avoided by assumingcontinuous and costless trading. This statement can be shown by creating a particularportfolio, which by construction is made up of owning θ of the underlying shares andby borrowing a single contingent asset C. If the investor properly selects θ at eachtime, the stochastic component of the process disappears and ensures the portfolio hasa riskless dynamic, making its return a deterministic function of time (see Black and
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Scholes (1973)). As time passes, the portfolio will have to be continually adjusted tomaintain risklessness, hence the need for continuous costless trading.The return of this portfolio must be equal to the riskless interest rate r because the portfolio itself is risk-free, otherwise traders will have an arbitrage opportunity. This conditionis necessary to obtain the stochastic differential equation followed by the contingent asset:∂C1 ∂ 2C 2 2∂C+σ S +rS = rC∂t2 ∂S∂S∂S
with end condition C = max(S − K, 0)
This equation is quite easy to solve and does not depend on the mean parameter µ nor onthe risk preferences of the traders. Whatever the risk preferences may be, the evaluation ofthe option does not change. When solving the equation, risk-neutral preferences are usedto simplify calculations. With instantaneous variance σ 2 , the following Black–Scholesvaluation formula is obtained:√CtBS (σ 2 ) = St (d) − Ke−rτ (d − σ τ )
d=
where
log(St /K) + (r + σ 2 /2)τ√σ τ
Note that σ 2 is the only unknown parameter: St and r are observed, while τ and K areusually given by institutional norms. The price depends strictly on σ 2 which is moreimportant than the drift, as is often the case in ﬁnance, so that the price of the option canbe considered an indicator of the volatility of the underlying asset.Empirically, the Black–Scholes formula can be used in two ways: either by estimatingσ 2 (the historical volatility) and then calculating the option price or by using real pricesto determine a value for σ 2 (called implied volatility).This type of analysis has a considerable shortcoming: the basic assumption that stockreturns follow a geometric diffusion process is a poor one, as indicated in Figure 8.1, andcan affect the valuation formula, reducing the precision of the option pricing. This realisation has prompted theoretical work into option pricing theory under various changingvolatility regimes. The leading paper in this ﬁeld is Hull and White (1987), to which wewill return later.8.1.3 Risk evaluationVaR (Value at Risk) is the maximum amount that is expected to be lost over some targetperiod, i.e. the maximum likely loss. It is a statistical risk measure and represents apercentile of the probability distribution of the variable of interest.Generally speaking, VaR can be analytically deﬁned as follows. Let xt be a randomvariable of interest measure and Fx (xt ) its cumulative distribution function:a = Prob(xt ≤ x t ) = Fx (x t ) =
 xt−∞
f (xt ) dxt
VaR is the percentile deﬁned by the relation:VaRx (1 − a) = xt∗ = Fx−1(a)t
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where (1 − a) is the VaR conﬁdence level, for instance 95% or 99%, and Fx−1(a) is thetinverse of the cumulative distribution function.Given its generality, the VaR method can be applied for different types of riskmeasurement, such as market risk, credit risk, operational risk and commodity risk (seeAlexander (1996)). Moreover, for its versatility, VaR allows us to obtain an intuitiverisk measure, to deﬁne homogeneous risk measures that permit a comparison amongdifferent ﬁnancial instruments, to determine limiting positions and to construct riskadjusted proﬁtability measures.Let us concentrate on its application to market risk. Market risk means the possibilitythat an unexpected variation of market factors (interest rates, exchange rates, stock prices,etc.) causes an increase or a reduction in the value of a position or in the value of a ﬁnancialportfolio. VaR, in this context, is the maximum expected loss of a marketable ﬁnancialinstruments portfolio which could be experienced, for a speciﬁed time horizon period anda speciﬁed conﬁdence level.We now consider a general portfolio model which allows us to set all the hypothesesdiscriminating a risk measurement model like VaR in a systematic manner, by payingparticular attention to the role of the volatility.Let xτ be a random variable which represents the value of a portfolio in a future periodτ . It is deﬁned by the following relation:xτ =
N
wi,t Pi,τ
i=1
where the random variables Pi,τ represent the future value of the N assets in the portfolioand wi,t are their weights at time t. If we suppose that the N assets will be subjected toK risk market factors χj,τ , the future value of the portfolio can be expressed as a functionof the K stochastic risk factors by the following pricing formula:xτ =
N
wi,t Pi,τ (χ1,τ , . . . , χK,τ )
i=1
The hypothesis characterising the model therefore concerns: the endogenous variablechoice; the pricing formula; the risk factors deﬁnition and their distributions; the riskfactors volatility; the risk factors mapping; the conﬁdence level and the choice of thetime horizon.In the literature, the following approaches are suggested to estimate the VaR (Best,1998): parametric methods; historical simulation; Monte Carlo simulation; stress testing.Concerning the parametric methods and the Monte Carlo simulation, it is crucial to properly describe the volatility dynamics of the risk factors to obtain correct estimates of theVaR (see, for example, Lehar et al. (2002)).
8.2 MODELS OF CHANGING VOLATILITYFollowing Cox (1981) and Shephard (1996) models of changing volatility can be usefully partitioned into observation-driven and parameter-driven models. They both can begenerally expressed using the following parametric framework:yt |zt ∼ N (µt , σt2 )
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where µt is often set equal to zero (as we do not intend to focus on that feature ofthe model).In the ﬁrst class, i.e. in observation-driven models, zt is a function of lagged values ofyt . The autoregressive conditional heteroskedasticity (ARCH) models introduced by Engle(1982) are the most representative example of observation-driven models. They describethe variance as a linear function of the squares of past observations:22σt2 = α0 + α1 yt−1+ · · · + αp yt−p
and so the model is deﬁned by the conditional density (one-step-ahead forecast density):yt |Y t−1 ∼ N (0, σt2 )where Y t−1 is the set of observations up to time t − 1. This allows today’s variance todepend on the variability of recent observations and then one type of shock alone drivesboth the series itself and its volatility.The use of models described by their one-step-ahead forecast offers remarkable advantages that are worth being highlighted. First, the likelihood expression can be simplyobtained by combining these densities, making the estimation and testing easy to handle,at least in principle. Second, conditional densities imply the use of conditional momentswhich are used widely to specify ﬁnance theory, although this is conditional on economicagents’, if not econometricians’, information. Finally, the observation-driven models parallel the autoregressive and moving average ones which are commonly used for modelsof changing means.In the second class, i.e. in parameter-driven (or parameter dynamic latent variable orstate space) models, zt is a function of an unobserved or latent component. The lognormal stochastic volatility model created by Taylor (1986) is the simplest and bestknown example:yt |ht ∼ N (0, exp(ht ))
ht = α + βht−1 + ηt ,
ηt ∼ NID(0, ση2 )
(8.1)
where ht represents the log-volatility, which is unobserved but can be estimated using theobservations. With respect to the previous class, these models are driven by two types ofshock, one of which inﬂuences the volatility (i.e. conditional variance equations). Thesemodels parallel the Gaussian state space models of means dealt with by Kalman (1960).In spite of this, a shortcoming of parameter-driven volatility models is that they generally lack analytic one-step-ahead forecast densities y t |Y t−1 , unlike the models of themean which ﬁt into the Gaussian state space form. Hence either an approximation or anumerically intensive method is required to deal with these models.Although SV models are harder to handle statistically than the correspondingobservation-driven models, there are still some good reasons for investigating them. Wewill see that their properties are easier to ﬁnd, understand, manipulate and generalise tothe multivariate case. They also have simpler analogous continuous time representations,which is important given that much modern ﬁnance employs diffusions. An example ofthis is the work by Hull and White (1987) which uses a log-normal SV model, replacingthe discrete time AR(1) for ht with an Ornstein–Uhlenbeck process.
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8.3 STOCHASTIC VOLATILITY MODELSFor these models the volatility depends on some unobserved components or a latent structure. One interpretation for the latent ht is to represent the random and uneven ﬂow of newinformation, which is very difﬁcult to model directly, into ﬁnancial markets (Clark, 1973).The most popular of these parameter-driven stochastic volatility models, from Taylor(1986), putsyt = εt exp(ht /2)(8.2)ht = α + βht−1 + ηtwhere εt and ηt are two independent Gaussian white noises, with variances 1 and ση2 ,respectively. Due to the Gaussianity of ηt , this model is called a log-normal SV model.Another possible interpretation for ht is to characterise the regime in which ﬁnancialmarkets are operating and then it could be described by a discrete valued variable. Themost popular approach to modelling changes in regime is the class of Markov switchingmodels introduced by Hamilton (1989) in the econometrics literature. In that case thesimplest model is:3yt = εt exp(ht /2)(8.3)ht = α + βstwhere st is a two-state ﬁrst-order Markov chain which can take values 0, 1 and is independent of εt . The value of the time series st , for all t, depends only on the last valuest−1 , i.e. for i, j = 0, 1:P (st = j |st−1 = i, st−2 = i, . . .) = P (st = j |st−1 = i) = pijThe probabilities (pij )i,j =0,1 are called transition probabilities of moving from one stateto the other. Obviously, we get that:p00 + p01 = p10 + p11 = 1and these transition probabilities are collected in the transition matrix P:P=
p001 − p00
1 − p11p11

which fully describes the Markov chain.A two-state Markov chain can easily be represented by a simple AR(1) process asfollows:st = (1 − p00 ) + (−1 + p00 + p11 )st−1 + vt(8.4)where vt = st − E(st |st−1 , st−2 , . . .). Although vt can take only a ﬁnite set of values, onaverage vt is zero. The innovation vt is thus a martingale difference sequence. Given the3The representation yt = σst εt , with σ0 = exp(α/2) and σ1 = exp((α + β)/2), is clearly equivalent. To identifythe regime 1 as the high volatility regime, we set β > 0.
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autoregressive representation of the Markov chain, it is possible to rewrite the volatilityequation of model (8.3) in the following way:ht = α + βst= α + β[(1 − p00 ) + (−1 + p00 + p11 )st−1 + vt ]= α(2 − p00 − p11 ) + β(1 − p00 ) + (−1 + p00 + p11 )ht−1 + βvtThe SV model with discrete volatility has therefore the same structure as model (8.2) butwith a noise that can take only a ﬁnite set of values:
yt = εt exp(ht /2)ht = a + bht−1 + ωt
(8.5)
We describe the basic properties of both types of model in the following sections.8.3.1 SV models with continuous volatilityWe consider εt and ηt independent, Gaussian white noises. The properties of model (8.2)are discussed in Taylor (1986, 1994) (see also Shephard (1996)). Broadly speaking, giventhe product process nature of the model, these properties are easy to derive, but estimationis substantially harder than for the corresponding ARCH models.As ηt is Gaussian, ht is a standard Gaussian autoregression. It will be stationary(covariance4 and strictly5 ) if |β| < 1 with:µh = E(ht ) =
α1−β
σh2 = Var(ht ) =
ση21 − β2
As εt is always stationary, yt will be stationary if and only if ht is stationary, yt being theproduct of two stationary processes. Using the properties of the log-normal distribution,all the moments exist if ht is stationary and in particular the kurtosis is:E(yt4 )= 3 exp(σh2 ) ≥ 3(E(yt2 ))2which shows that the SV model has fatter tails than the corresponding normal distributionand all the odd moments are zero.4A stochastic process yt is covariance stationary if the degree of covariance amongst its observations dependsonly on the time gap between them, i.e. Cov(yt , yt+r ) = γ (r) for all t.5For some processes there will exist no moments, even in cases where the corresponding unconditional distributions are perfectly well-behaved. The strict stationarity of yt is then deﬁned as follows:F (yt+r , yt+r+1 , . . . , yt+r+p ) = F (yt , yt+1 , . . . , yt+p ) for all p and r.
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The dynamic properties of yt are easy to ﬁnd. First, as εt is iid, yt is a martingaledifference6 and is a white noise7 if |β| < 1. As ht is a Gaussian AR(1):22Cov(yt2 , yt−r) = E(yt2 yt−r) − (E(yt2 ))2
= E(exp(ht + ht−r )) − (E(exp(ht )))2= exp(2µh + σh2 )(exp(σh2 β r ) − 1)and so:ρyt2 (r) =
2Cov(yt2 , yt−r)exp(σh2 β r ) − 1 ∼ exp(σh2 ) − 1 r=β=Var(yt2 )3 exp(σh2 ) − 13 exp(σh2 ) − 1
Hence, the memory of the yt is deﬁned by the memory of the latent ht , in this case anAR(1). Moreover, note that if β < 0, ρyt2 (r) can be negative, unlike the ARCH models.This is the autocorrelation function of an ARMA(1,1) process, thus the SV model behavesin a manner similar to the GARCH(1,1) model. Finally, note that there is no need fornon-negativity constraints nor for bounded kurtosis constraints on the coefﬁcients. Thisis a great advantage with respect to GARCH models.Insights on the dynamic properties of the SV model can also be obtained by squaringand taking logs, getting:log(yt2 ) = ht + log(εt2 )(8.6)ht = α + βht−1 + ηta linear process, which adds the iid log(εt2 ) to the AR(1) ht . As a result log(yt2 ) ∼ARMA(1,1). If εt is Gaussian, then log(εt2 ) has a mean of −1.27 and variance 4.93, butits distribution is far from being normal, as it is heavily skewed with a long left-hand tail,caused by taking the logs of very small numbers, an operation which generates outliers.The autocorrelation function for log(yt2 ) is:ρlog(yt2 ) (r) =
βr1 + 4.93/σh2
8.3.2 SV models with discrete volatilityWe consider a two-state Markov chain st independent of εt , which is a Gaussian whitenoise.Assuming stationarity,8 the unconditional probabilities to be in the regime 0(P (st = 0)= π0 ) or 1 (P (st = 1) = π1 ) are deﬁned as follows:π0 = p00 π0 + (1 − p11 )π1π1 = (1 − p00 )π0 + p11 π16yt being a martingale difference stipulates that E|yt | < ∞ and that E(yt |yt−1 , yt−2 , . . .) = 0. All martingaledifferences have zero means and are uncorrelated over time. If the unconditional variance of the martingaledifference is constant over time, then the series is also a white noise.7This means E(yt ) = µ, Var(yt ) = σ 2 and Cov(yt , yt+r ) = 0 for all r = 0. Often µ will be taken to be zero.These unconditional moment conditions are sometimes strengthened to include yt being independent, ratherthan uncorrelated, over time. This will be called strong white noise, a special case of which is independent andidentically distributed (iid).8An ergodic Markov chain is a covariance stationary process. For some basic properties of Markov chains, seeHamilton (1994, chap. 22).
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with π0 + π1 = 1, or in a vector form:
π = Pπ1π =1

where 1 = (1, 1) . Thus, they are:π0 =
1 − p112 − p00 − p11
π1 =
1 − p002 − p00 − p11
From the deﬁnition of b in equation (8.5), we can note that when p00 + p11 > 1 the htprocess is likely to persist in its current state and it would be positively serially correlated.Its unconditional moments are:E(ht ) = α + βE(st )= α + βπ1Var(ht ) = β 2 π1 (1 − π1 )Under stationarity,9 as for the SV model with continuous volatility, all the moments exist,all the odd moments are zero and the kurtosis is:E(yt4 )(π0 + exp(2β)π1 )=3≥32 2(π0 + exp(β)π1 )2(E(yt ))Moreover, as εt is iid, yt is a martingale difference and its dynamic properties are describedby the covariances of squares:22Cov(yt2 , yt−r) = E(yt2 yt−r) − (E(yt2 ))2
= E(exp(ht + ht−r )) − (exp(α)π0 + exp(α + β)π1 )2= exp(2α)P (st = 0, st−r = 0) + exp(2α + β)P (st = 0, st−r = 1)+ exp(2α + β)P (st = 1, st−r = 0) + exp(2α + 2β)P (st = 1, st−r = 1)− (exp(α)π0 + exp(α + β)π1 )2where the vector of unconditional joint probabilities P (st , st−r ) can be computedas follows:P (st , st−r ) = P (st |st−r )P (st−r )= Pr π9
For this see Francq and Zakoı̈an (2001) and Francq et al. (2001).
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 (1 − p ) + λr (1 − p )11002 − p00 − p11Pr =  (1 − p ) + λr (1 − p )00
00
2 − p00 − p11
(1 − p11 ) + λr (1 − p11 ) 2 − p00 − p11(1 − p ) + λr (1 − p ) 00
11
2 − p00 − p11
and λ = −1 + p00 + p11 .Finally, it is useful to note that ht is itself a Markov chain which can take the valuesα and α + β with the same transition matrix P.
8.4 ESTIMATIONThe difﬁculties in estimating SV models lie in the latent nature of the volatility. Inferencemay be difﬁcult, because the distribution of yt |Y t−1 is speciﬁed implicitly rather thanexplicitly and the likelihood function appears as a multivariate integral the size of whichis equal to the number of observations multiplied by the size of the latent variables, whichis 1 for the described models.Like most non-Gaussian parameter-driven models, there are many different ways ofperforming estimation: some involve estimating or approximating the likelihood, othersuse the method of moments procedures (see Ghysels et al. (1996) and Shephard (1996)).Let us ﬁrst of all clearly state the problem of computing the likelihood function forthe general class of parametric dynamic latent variable or non-linear and/or non-Gaussianstate space models.8.4.1 A general ﬁlter for non-Gaussian parameter-driven modelsBoth SV models (with continuous and discrete volatility) ﬁt in the following framework:yt = φt (ht , εt ; θ )measurement equation(8.7)ht = ϕt (ht−1 , ηt ; θ ) transition equationwhere εt and ηt are independent white noises, with marginal distributions which maydepend on θ , the vector of parameters. Let H t and Y t denote (h1 , h2 , . . . , ht ) and(y1 , y2 , . . . , yt ) , respectively.There are serious difﬁculties in computing the likelihood function; in fact, with T thenumber of observations, we have:T
f (Y T , H T ; θ) =
f (yt |Y t−1 , H t ; θ )f (ht |Y t−1 , H t−1 ; θ)t=1
and the likelihood function is:
T
T (θ ) ≡ f (Y ; θ ) =
T
f (yt |Y t−1 ,H t ; θ )f (ht |Y t−1 ,H t−1 ; θ )
T
t=1
dht
(8.8)
t=1
which is an integral whose size is equal to the number of observations multiplied by thedimension of the unobserved variable ht , and thus it is practically unfeasible.
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It is however possible to derive a general algorithm which allows the formal computation of the likelihood function by decomposing the calculation of integral (8.8) into asequence of integrals of lower dimension.Let f (ht−1 |Y t−1 ) be the input of the iteration,10 t = 1, 2, . . . , T . First of all, we candecompose the joint conditional density of ht , ht−1 into the product of the transitiondensity by the input density:step 1
f (ht , ht−1 |Y t−1 ) = f (ht |ht−1 )f (ht−1 |Y t−1 )
By marginalisation we obtain the prediction density of ht :t−1t−1step 2f (ht |Y ) = f (ht , ht−1 |Y ) dht−1 = f (ht |ht−1 )f (ht−1 |Y t−1 ) dht−1Let us now consider the joint density of yt , ht . It can be decomposed into the product ofthe measurement density and the prediction density:step 3
f (yt , ht |Y t−1 ) = f (yt |ht )f (ht |Y t−1 )
and, again, by marginalisation we obtain the one-step-ahead forecast density of yt :t−1t−1step 4f (yt |Y ) = f (yt , ht |Y ) dht = f (yt |ht )f (ht |Y t−1 ) dhtwhich is particularly useful, since by the combination of these densities it is possibleto obtain the likelihood function. Finally, by conditioning we obtain the ﬁltering density (output):step 5
f (ht |Y t ) =
f (yt |ht )f (ht |Y t−1 )f (yt , ht |Y t−1 )=f (yt |Y t−1 )f (yt |ht )f (ht |Y t−1 ) dht
which ends the iteration.The previous algorithm allows us to obtain several important elements. Step 2 givesthe estimation of ht given all the information available until t − 1 (prediction density).Step 5 provides the estimation of ht given all the information currently available (ﬁlteringdensity). Finally step 4, by providing the one-step-ahead forecast density, allows us tocompute the likelihood function.Unfortunately, only in very special cases is it possible to obtain analytic recursivealgorithms11 from this general ﬁltering algorithm: the Kalman ﬁlter in the Gaussian andlinear case and the Hamilton ﬁlter in the Markovian and discrete case.In the Gaussian and linear cases, the initial input f (h1 |Y 0 ) and the measurement andtransition densities are assumed to be Gaussian and at each step of the algorithm Gaussianity is preserved, then also all the outputs are Gaussian. The normal distribution is10For the ﬁrst iteration (t = 1) it is possible to consider the unconditional distribution of ht , f (h1 ). For thesake of simplicity we omit the dependence on the parameter θ .11See also Shephard (1994) for another particular case in which ht is set to be a random walk and exp(ηt ) ahighly contrived scaled beta distribution. This delivers a one-step-ahead prediction distribution which has somesimilarities to the ARCH models.
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completely described by its ﬁrst two moments and then the algorithm can be rewritten byrelating means and variances of the different densities involved. This is the Kalman ﬁlter.For the switching regime models introduced by Hamilton (1989), which represent theMarkovian and discrete case, the integrals which appear at steps 2 and 4 become a simplesum over the possible regimes, and then the whole algorithm is analytically tractable.In all the other cases, it is necessary to consider approximated solutions or simulationbased methods. Examples of approximations are the extended Kalman ﬁlter (Andersonand Moore, 1979; Harvey, 1989; Fridman and Harris, 1998), the Gaussian sumﬁlter (Sorenson and Alspach, 1971), the numerical integration (Kitagawa, 1987), theMonte Carlo integration (Tanizaki and Mariano, 1994, 1998), or the particle ﬁlter (Gordonet al., 1993); Kitagawa, 1996; Pitt and Shephard, 1999a). The simulation-based solutionsare certainly more time-consuming and demanding in terms of computing, but they aredeﬁnitely more general. We will see these methods in greater detail later.However, for the two presented models (8.2) and (8.3), the general ﬁlter introduced hereis useful for estimation. In fact, for the linearised version (8.6), the Kalman ﬁlter allowsa quasi-maximum likelihood estimation of the parameters and the discrete version (8.3)is a particular case of switching regime models for which the Hamilton ﬁlter gives thelikelihood function.8.4.1.1 The Kalman ﬁlter for quasi-maximum likelihood (QML) estimationof continuous SV modelsWe can consider the log-transformation (8.6) of the continuous SV model. As log(εt2 ) ∼iid, we obtain a linear state space model.Let LY τ = (log(y12 ), log(y22 ), . . . , log(yτ2 )) , ĥt/τ = E(ht |LY τ ) = E(ht |Y τ ) and Qt/τ =ττMSE(ht |LY ) = MSE(ht |Y ). The Kalman ﬁlter (see, for example,12 Harvey (1989))computes these quantities recursively for t = 1, . . . , T :ĥt/t−1 = α + β ĥt−1/t−1Qt/t−1 = β 2 Qt−1/t−1 + ση2et/t−1 = log(yt2 ) − ĥt/t−1Ft/t−1 = Qt/t−1 + π 2 /2ĥt/t = ĥt/t−1 + Kt et/t−1Qt/t = (1 − Kt )2 Qt/t−1−1where Kt = Qt/t−1 Ft/t−1is the Kalman gain. However, as log