1hT +1  T ∼ĥT +1/TĥT +1/Tσ̂T +1/T = E exp+ expQT +1/T= expY2282With regard to the switching regime model, since σt = σ0 (1 − st ) + σ1 st , we can forecastthe volatility as follows:σ̂T +1/T = E(σT +1 |Y T ) = σ0 P (sT +1 = 0|Y T ) + σ1 P (sT +1 = 1|Y T )whereP (sT +1 = 0|Y T ) = P (hT +1 = α|Y T )
P (sT +1 = 1|Y T ) = P (hT +1 = α + β|Y T )
are the prediction probabilities43 obtained with the last iteration of the Hamilton ﬁlter.The forecasted volatility is evidenced in the output and it is saved as the last value ofthe estimated volatility.44Let us now consider some practical utilisations of the estimated volatilities.8.7.2 Option pricingAs seen in Section 8.1.2, the option price in the Black and Scholes framework can beexpressed as a conditional expectation given the current price of the underlying asset:CtBS = exp(−rτ )ESt+τ |St [max(St+τ − K, 0)]42For the ARSV model, Var1 indicates the simulated returns and Var2 the simulated volatilities. For the SRSV,Var1 indicates the simulated returns, Var2 the simulated volatilities, Var3 the smoothed probabilities of the highvolatility regime and Var4 the regime shifts.43It is possible to obtain the prediction probabilities by multiplying the transition matrix P by the ﬁlteredprobabilities (which are saved in the estimation output ﬁle as Var4), i.e.

P (st = 0|Y t )P (st+1 = 0|Y t )=PttP (st+1 = 1|Y )P (st = 1|Y )
44In the estimation output ﬁle, in the last row the non-zero value for the estimated volatility is the forecastedvolatility, while the non-zero value for the ﬁltered probability is the prediction probability of the high volatilityregime (see footnote 43). All the other variables are set equal to zero.
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where the dynamic of the asset is described by a geometric diffusion process and theexpectation is taken with respect to the risk-neutral probability measure.Since the Black and Scholes formula can be expressed as a function only of the volatility, great effort has been made in modelling its behaviour. While Black and Scholesassume that it is constant over the life of the option, a series of models proposed in thelate 1980s supposes that it varies through time in a deterministic or stochastic way, in anattempt to capture the empirical features of the option prices. In fact, an analysis of thevolatility implied in the market option prices (the so-called implied volatility) highlightsthat the volatility is neither constant through time nor independent of the strike price (theso-called “smile” and “sneer” effect) (see Rubinstein (1985)).A very simple approach consists of using the volatility estimated with stochastic volatility models as input to the Black and Scholes formula. In that case, it is sufﬁcient toconsider the forecasted volatility σ̂t/t−1 as the volatility parameter in the formula to obtainthe option price:Ct = CtBS (σ̂t/t−1 )In the following we review some stochastic volatility models for the option pricing, whichconsider the volatility as an exogenous stochastic process.The path-breaking work on stochastic volatility models applied to option pricing is thepaper by Hull and White (1987). The authors assume that both the underlying security Sand the variance σ 2 follow a geometric diffusion process:dS = µSdt + σ Sdzdσ 2 = φσ 2 dt + ξ σ 2 dωwhere the correlation ρ between the two Brownian motions dz, dω is a constant with modulus less than one. Hull and White take ρ ≡ 0. Scott (1987) considers the case in whichthe volatility follows an Ornstein–Uhlenbeck process and also imposes the restrictionρ ≡ 0. Finally, Heston (1993) proposes the familiar mean-reverting square root processfor the volatility:dS = µSdt + σ Sdzdσ 2 = γ (φ − σ 2 )dt + ξ σ dωwhere φ is the long-run average variance and he takes the assumption ρ = 0.Introducing stochastic volatility into the deﬁnition of the stochastic differential equationof the underlying asset creates several complications. A dynamic portfolio with only oneoption and one underlying asset is not sufﬁcient to create a riskless investment strategy.The problem arises since the stochastic differential equation for the option contains twosources of uncertainty. Unfortunately, it is impossible to eliminate volatility market riskpremium and correlation parameters from the partial differential equation using only oneoption and one underlying asset. Moreover, these parameters are difﬁcult to estimate45and extensive use of numerical techniques is required to solve the two-dimensional partialdifferential equation.45An exception occurs when the volatility is a deterministic function of the asset price or time. In this case itis possible to easily ﬁnd a solution to the partial differential equation.
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In the Hull and White (1987) formula, the option price is determined assuming thatthe volatility market risk premium is zero and there is zero correlation between the twoBrownian motions describing the underlying asset and the volatility, i.e. the volatilityis uncorrelated with the asset price. With these assumptions and using a risk-neutralvaluation procedure, they show that the price of an option with stochastic volatility is theBlack and Scholes price integrated over the distribution of the mean volatility:CtHW = CtBS (σ 2 )g(σ 2 |σt2 )dσ 2whereσ2 =
1τ
 t+τσ 2 (u)dut
and g(σ |σt2 ) is the conditional probability density of the mean variance σ 2 over theperiod τ .In the more general case of non-zero correlation, the framework becomes more complex,allowing only numerical solutions.It can be observed that continuous time stochastic volatility provides an attractiveand intuitive explanation for observed volatility patterns and for observed biases inimplied volatility. Precisely, smiles, skews, upward and downward term structuresof implied volatility arise naturally from a stochastic volatility model. However the factthat stochastic volatility models ﬁt empirical patterns does not mean that those models arecorrect and the biases in market prices may be the result of other factors, not considered,such as liquidity problems.A work related to that of Hull and White is Naik (1993). While in the Hull and Whitespeciﬁcation the volatility follows a continuous diffusion process, Naik analyses the casewhere the instantaneous variance of an asset is subject to random discontinuous shifts.In particular, the volatility is described with a right-continuous Markov chain process: itremains in the same state for a random amount of time and then shifts to another statewith transition probabilities determined by a matrix. In the case of a two-state volatilityprocess the transition matrix is simply:p001 − p11P=1 − p00p11Assuming that the underlying process is continuous, the risk of a shift in volatility isdiversiﬁable, and therefore not priced, and that the two processes are uncorrelated, theoption valuation equation can be expressed in a closed form as follows: τCtN (σ1 ) =CtBS (σ 2 (x))g(x|σ1 ) dx0
where σ1 indicates the high volatility level, σ 2 (x) = [σ12 x + σ02 (τ − x)]/τ , 0 ≤ x ≤ τ andg(x|σ1 ) denotes the unconditional density of the time spent by the volatility process in thehigh volatility state, given the current high volatility state. In the same way it is possibleto determine the option price conditional on the current low volatility state (CtN (σ0 ) withσ0 the low volatility level).
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As in the Hull and White model, the option price is the expectation of the Black andScholes formula computed for the average future volatility, given the current state. Sincetwo regimes are considered, the ﬁnal option price can be obtained by a weighted averageof the two conditional values CtN (σ1 ) and CtN (σ0 ).This analysis can be extended by considering multiple states and correlation betweenchanges of the underlying and shifts in volatility, but unfortunately in these cases theoption price can be obtained only via numerical methods. In this kind of procedure adiscrete time Markov chain is used as an approximation of the volatility process.We brieﬂy present two examples of this approach, due to Billio and Pelizzon (1997)and Bollen (1998) (see also Bollen et al. (2000)). Both these works are based on thehypothesis that the returns of the underlying asset follow a switching regime model. Thedistribution is characterised by a mixture of distributions with different variance, wherethe weights depend on a hidden Markov chain process which represents possible differentvolatility regimes of the market. To obtain the numerical solution, the basic idea is toapproximate the distribution through a multinomial approach, considering a binomial treefor each of the two distributions characterised by different variance.Following Cox et al. (1979), the future value of the underlying process can beexpressed as:uSt pSt+t =dSt 1 − pOne possible speciﬁcation of the parameters that guarantees asymptotic normality andconvergence√ to the desired mean and variance of the continuously compounded returns isu = exp( σ 2 t + r 2 t 2 ), d = u−1 and p = (ert − d)/(u − d). In this case the processis fully characterised by the parameter representing the variance σ 2 , since the step sizet and the risk-free interest rate r are given. Once the variance is estimated it is possibleto construct a tree to represent the possible future paths of the variable and hence thedistribution of returns (at maturity).If a regime switching model is considered, the distribution of the returns is simply amixture of the distributions characterising each state. Therefore a discrete process can beused to approximate the continuous time process, and hence the distribution of returns,in each state. In this case, two binomial distributions are used as an approximation of themixture of the two distributions:u S |s = 1 p11 1 t tu0 St |st = 0 p00St+t =d0 St |st = 0 1 − p00d1 St |st = 1 1 − p11where st denotes the regime.In this case it is necessary to invoke the risk-neutralhypothesis to determine
the values of the parameters that are u1 = exp( σ12 t + r 2 t 2 ), d1 = u−11 , p11 =(ert − d1 )/(u1 − d1 ) for the high volatility regime and u0 = exp( σ02 t + r 2 t 2 ), d0 =
rt− d0 )/(u0 − d0 ) for the low volatility regime. This model is usuallyu−10 and p00 = (ecalled a quadrinomial tree (or lattice). The inner two branches correspond to the lowvolatility regime while the outer ones correspond to the high volatility regime. Each setof probabilities (pii , 1 − pii ) must be interpreted as the branch probability conditional onthe regime i, with i = 0, 1.
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279
Although the quadrinomial tree represents both distributions accurately, its branches donot recombine efﬁciently, exhibiting an exponential growth of the computational time asthe number of steps increases.Bollen (1998) proposes a method to overcome this problem and develops a pentanomialtree. The deﬁnition of the parameters is modiﬁed to yield both the possibility to recombine the tree and the convergence of the discrete process to the mixture of distributions.This is obtained by approximating each regime density by a trinomial distribution insteadof the binomial one. The modiﬁed tree has ﬁve evenly spacedbranches because thestep sizes of the two regimes are in the ratio 1:2 (u1 = exp( σ12 t + r 2 t 2 ), u0 =exp( 12 σ12 t + r 2 t 2 ) and the middle branch is shared by the two regimes:u1 St |st = 1u0 St |st = 0St+t = Std0 St |st = 0d1 St |st = 1
p1,up0,upmp0,dp1,d
where pst ,u , pst ,d are the probabilities to go up and down conditional on the regime stand pm is the probability to remain at the same level price St .Once the tree is generated, the option value is calculated operating backward from theterminal values, i.e. the payoffs, to the valuation time. In the regime switching approaches,for simplicity two conditional option values are calculated at each step, where the conditioning information is the regime at the valuation time t:Ct (st = 0) = [p00 Ct+1 (st+1 = 0) + (1 − p00 )Ct+1 (st+1 = 1)]e−rtCt (st = 1) = [(1 − p11 )Ct+1 (st+1 = 0) + p11 Ct+1 (st+1 = 1)]e−rtAt the valuation time the value of the option is obtained as a weighted average of thetwo conditional option values, where the weights depend on the knowledge of the currentregime. If the regime is unknown, the unconditional probabilities:1 − p112 − p00 − p11
1 − p002 − p00 − p11
are used.Let us take an example of the pentanomial approach by considering the parametersestimated in Section 8.7.1.1.We deal with a European call option on the FTSE100 index quoted at LIFFE on22 August 2000 (when the FTSE100 index quoted 6584.82), with maturity June 2001and strike price 5900. The risk-free interest rate r is approximated with three-monthLibor46 and the time to maturity in terms of trading days is 213. In this example t = 1.46
London Interbank Offered Rate.
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Taking the estimated volatilities of Section 8.7.1.1, the parameters of the pentanomialmodel can be computed as follows:
u1 = e
√ 2
σ1 +r 2
1√ 2 2σ1 +r
u0 = e 2
1√ 2 2d0 = e− 2 σ1 +r√ 2 2d1 = e− σ1 +r
√ 2 2er − e− σ1 +rp1,u = √ 2 2√ 2 2e σ1 +r − e− σ1 +r1√ 2 21√ 2 2−σ1 +r−σ1 +rr− pm 1 − e 2e −e 2p0,u =1√ 2 21√ 2 2e 2 σ1 +r − e− 2 σ1 +r222 σ0 + r pm = 1 − 4  σ12 + r 2p0,d = 1 − p0,u − pmp1,d = 1 − p1,u
Once the tree is generated, the payoffs are calculated with the usual procedure (seeAppendix A for details). The conditional option values Ct (st = 0), Ct (st = 1) are obtainedusing the estimated transition probabilities, while the ﬁnal value of the option is a weightedaverage where the weights are the unconditional probabilities:1 − p112 − p00 − p11
1 − p002 − p00 − p11
The pentanomial option value is therefore 1055.14, while the market value was 1047.5.8.7.3 Value at riskVaR is a very intuitive measure to evaluate market risk because it indicates the maximumpotential loss at a given level of conﬁdence (a) for a portfolio of ﬁnancial assets over aspeciﬁed time horizon (h).In #practice, the value of a portfolio is expressed as a function of K risk factors,xτ = Ni=1 wi,t Pi,τ (χ1,τ , . . . , χK,τ ). The factors inﬂuencing the portfolio value are usually identiﬁed with some market variables such as interest rates, exchange rates or stockindexes. If their distribution is known in a closed form, we need to estimate the distribution of the future value of the portfolio conditional on the available information and theVaR is then the solution to:a=
 VaR(h,a)−∞
f (xt+h ) dx
Different parametric models can be used to forecast the portfolio return distribution. Thesimple way to calculate VaR involves assuming that the risk factor returns follow a multivariate normal distribution conditional on the available information. If the portfolio returnis linearly dependent on them, its probability distribution is also normal and the VaR is
Stochastic Volatility Models
281
simply the quantile of this analytic distribution. If the linear assumption is inappropriate, the portfolio return can be approximated as a quadratic function of the risk factorreturns.An alternative way to handle the non-linearity is to use Monte Carlo simulation. Theidea is to simulate repeatedly the random processes governing the risk factors. Eachsimulation gives us a possible value for the portfolio at the end of our time horizon. Ifenough of these simulations are considered, it is possible to infer the VaR, as the relevantquantile of the simulated distribution.Since market risk factors usually have fatter tails than the normal distribution, it is alsopossible to use historical simulation rather than a parametric approach. The idea behindthis technique is to use the historical distribution of returns to the assets in the portfolioto simulate the portfolio’s VaR, on the hypothetical assumption that we held this portfolioover the period of time covered by our historical data set. Thus, the historical simulationinvolves collecting historic asset returns over some observation period and using theweights of the current portfolio to simulate the hypothetical returns we would have had ifwe had held our current portfolio over the observation period. It is then assumed that thishistorical distribution of returns is also a good proxy for the portfolio return distributionit will face over the next holding period and VaR is calculated as the relevant quantile ofthis distribution.The advantage of the parametric approach is that the factors variance–covariance matrixcan be updated using a general model of changing or stochastic volatility. The main disadvantage is that the factor returns are usually assumed to be conditionally normal, losingthe possibility to take into account non-linear correlations among them. Historical simulation has the advantage of reﬂecting the historical multivariate probability distributionof the risk factor returns, avoiding ad hoc assumptions. However the method suffers aserious drawback. Its main disadvantage is that it does not incorporate volatility updating. Moreover, extreme quantiles are difﬁcult to estimate, as extrapolation beyond pastobservations is impossible. Finally, quantile estimates tend to be very volatile whenevera large observation enters the sample and the database is not sufﬁciently large.The advantage of the parametric approach to update the volatility suggests the simplestutilisation of the SV models for the VaR computation. Having chosen the asset or portfoliodistribution (usually the normal one), it is possible to use the forecasted volatility tocharacterise the future return distribution. Thus, σ̂T +1/T can be used to calculate the VaRover the next period.A different approach using the SV model is to devolatilise the observed returns seriesand to revolatilise it with an appropriate forecasted value, obtained with a particular modelof changing volatility. This approach is considered in several recent works (Barone-Adesiet al., 1998; Hull and White, 1998) and is a way of combining different methods andpartially overcoming the drawbacks of each.To make the historical simulation consistent with empirical ﬁndings, the log-normalSV model and the regime switching model may be considered to describe the volatilitybehaviour. Past returns are standardised by the estimated volatility to obtain standardisedresiduals. Statistical tests can conﬁrm that these standardised residuals behave approximately as an iid series which exhibits heavy tails. Historical simulation can then be used.Finally, to adjust them to the current market conditions, the randomly selected standardisedresiduals are multiplied by the forecasted volatility obtained with the SV model.
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Applied Quantitative Methods for Trading and InvestmentTable 8.2
VaR at different conﬁdence levels for the FTSE100 index return
Conﬁdence level
Log-normal SV model
Regime switching model
0.10.050.01
2.54423.92985.3417
2.75033.58744.6502
For example, Table 8.2 shows the results obtained with the FTSE100 index returnby considering 1 000 000 historical simulations47 (see Appendix B for details). Clearly,this approach allows a wide range of stochastic and changing volatility models, such asARCH–GARCH models, to be considered. Moreover, it must be pointed out that insteadof using historical simulation, an appropriate standard distribution can also be consideredto model the transformed returns and then several probability distributions can be assumedfor the unconditional returns (McNeil and Frey, 2000; Eberlein et al., 2001).Another example of a parametric approach to VaR calculation considers the hypothesisthat a regime switching model governs the asset returns (Billio and Pelizzon, 2000):µ + σ0 εt st = 0yt =µ + σ1 εt st = 1where εt ∼ N (0,1) is independent of st .To calculate the VaR it is necessary to determine the value of the conditional distributionfor which the cumulative density is a, i.e. VaR(h,a)P (st+h |It )fst+h (yt+h |It ) dya=st+h =0,1
−∞
where fst+h (yt+h |It ) is the probability density of yt+h when the regime is st+h and conditional on the available information set It (usually containing past returns), P (st+h |It ) isthe prediction probability obtained by the Hamilton ﬁlter.Given the parameters estimated in Section 8.7.1.1 for the switching regime model andthe prediction probabilities at time t + h (obtained by the product of the transition matrix,for h − 1 steps, and the conditional probabilities P (st+1 |It ) given by the Hamilton ﬁlter),the VaR is the relevant quantile of the mixture distribution.The model can be generalised to the case of N risky assets providing an explicit linkbetween the return on the asset and the return on the market index, thus by explicitlymodelling the correlation between different assets. The Multivariate Switching RegimeBeta Model (Billio and Pelizzon, 2000) is a sort of market model or better a single factormodel in the Arbitrage Pricing Theory framework where the return of a single stock i ischaracterised by the regime switching of the market index and the regime switching ofthe speciﬁc risk of the asset. It can be written as:εt ∼ IIN (0,1)ymt = µm (st ) + σm (st )εty=µ(s)+β(s,s)y+σ(s)εε1t ∼ IIN (0,1)1t11t1t1tmt11t1ty2t = µ2 (s2t ) + β2 (st , s2t )ymt + σ2 (s2t )ε2tε2t ∼ IIN (0,1)...yNt = µN (sNt ) + βN (st , sNt )ymt + σN (sNt )εNt εNt ∼ IIN (0,1)47
Each operation takes about 12 seconds.
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283
where ymt is the market return, the market regime variable st and the single stock regimevariables sj t , j = 1, . . . , N are independent Markov chains, εt and εj t , j = 1, . . . , N , areindependently distributed.Using this approach it is possible to take into account the correlation between differentassets. In fact, the variance–covariance matrix between the two assets i and j is:(st , sit , sj t ) =
βi2 (st , sit )σm2 (st ) + σi2 (sit )βi (st , sit )βj (st , sj t )σm2 (st )
βi (st , sit )βj (st , sj t )σm2 (st )βj2 (st , sj t )σm2 (st ) + σj2 (sj t )

then the correlation between different assets depends on the extent to which each asset islinked, through the factor loading β, to the market index.To calculate VaR for a portfolio based on N assets it is sufﬁcient to use the approachpresented above. In particular, considering two assets and assuming that the number ofregimes is two for all three Markov chains we have:a=



P (st+h , si,t+h , sj,t+h |It )
st+h =0,1 si,t+h =0,1 sj,t+h =0,1
 VaR(h,a)−∞
fst+h ,si,t+h ,sj,t+h (y|It ) dy
where fst+h ,si,t+h ,sj,t+h (y|It ) is the probability density of the portfolio return y when theregimes are st+h , si,t+h , sj,t+h and conditional on the available information set It . This distribution has mean w µ(st+h , si,t+h , sj,t+h ) and variance w (st+h , si,t+h , sj,t+h )w where wis the vector of the percentage of wealth invested in the two assets and µ(st+h , si,t+h , sj,t+h )is the vector of risky asset mean returns, i.e.µ(st+h , si,t+h , sj,t+h ) =
µi (si,t+h ) + βi (st+h , si,t+h )µm (st+h )µj (sj,t+h ) + βj (st+h , sj,t+h )µm (st+h )
The drawback of this approach is that it requires the estimation of a number of parametersthat grows exponentially with the number of assets. In fact, the number of possible regimesgenerated by this model is 2N+1 .One possible solution is to consider the idiosyncratic risk distributed as IIN (0,σi2 )(without a speciﬁc Markov chain dependency) and to characterise the systematic riskwith more than one source of risk. This approach is in line with the Arbitrage PricingTheory model where the risky factors are characterised by switching regime processes.Formally, we can write this model as:Fj t = αj (sj t ) + θj (sj t )εj t#yit = µi + Kj =1 βij (sj t )Fj t + σi εit
εj t ∼ IIN (0,1), j = 1, 2, . . . , Kεit ∼ IIN (0,1), i = 1, 2, . . . , N
where Fj t is the value of factor j at time t (j = 1, 2, . . . , K), βij (sj t ) is the factor loadingof the asset i on factor j , sj t , j = 1, 2, . . . , K, are independent Markov chains, and εj t ,j = 1, 2, . . . , K, and εj t , j = 1, 2, . . . , N , are independently distributed.This model is more parsimonious, in fact the introduction of an extra asset implies thatonly K + 2 parameters need to be estimated. This approach is valid when the number ofassets in the portfolio is high and the speciﬁc risk is easily eliminated by diversiﬁcation.
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8.8 CONCLUDING REMARKSWe have tried to develop an introduction to the current literature on stochastic volatilitymodels. Other than the classical log-normal model introduced by Taylor (1986), we havealso presented the discrete volatility model in which the latent stochastic structure of thevolatility is described by a Markov chain.Both models (with continuous and discrete volatility) ﬁt in the framework of a nonlinear and non-Gaussian state space model, thus the estimation and smoothing problemsare developed along the same lines. Only for the discrete case does the general algorithmintroduced allow us to compute the likelihood function and then to obtain maximum likelihood estimates. In the continuous case, approximations or simulations must be introduced.Some extensions and multivariate models are also presented, however there is still agreat deal of work to be done.Finally, the estimation program presented considers the two basic models and allows anestimation of the latent volatility. Some possible applications are suggested and discussed.
APPENDIX A: APPLICATION OF THE PENTANOMIAL MODELThe example considers a European call option on the FTSE100 index, with maturity June2001 and strike price 5900 traded at LIFFE on 22 August 2000. Three-month Libor isused as an approximation of the risk-free interest rate r. The FTSE100 index quoted6584.82 and Libor was 6.22%.The FTSE100 index being a weighted average of the prices of 100 stocks, the dividendeffect must be considered. In the example, this parameter is considered constant48 andequal to q = 3%. The time to maturity in terms of trading days is 213 and the step sizeis the single trading day (t = 1).Taking the estimated volatilities of Section 8.7.1.1 (σ̂0 = 0.0108036 and σ̂1 =0.0234427), the parameters of the pentanomial model can be computed as follows:
er−q − e−
√ 2
σ̂1 +(r−q)2
p1,u = √ 2= 0.4968√ 222e σ̂1 +(r−q) − e− σ̂1 +(r−q)1√ 21√ 2−σ̂1 +(r−q)2−σ̂1 +(r−q)2r−q√ 2221−e−e−pem2u1 = e σ̂1 +(r−q) = 1.0237p0,u =1√ 21√ 21√ 2222e 2 σ̂1 +(r−q) − e− 2 σ̂1 +(r−q)u0 = e 2 σ̂1 +(r−q) = 1.0118= 0.42771√ 22d0 = e− 2 σ̂1 +(r−q) = 0.98832√22σ̂+(r−q)− σ̂12 +(