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Smile dynamics III
In two articles published in 2004 and 2005 in Risk,                                        application to the market of Vix futures and options, as well as
                                                                                           options on realised variance, using the continuous version. We
Lorenzo Bergomi assessed the structural limitations of                                     next examine some properties of the vanilla smile that this model
                                                                                           produces and summarise our work in the concluding section.
existing models for equity derivatives and introduced a
new model based on the direct modelling of the joint                                       Dynamics for forward variances
                                                                                           In previous work (Bergomi, 2005), from which we borrow both
dynamics of the spot and the implied variance swap                                         vocabulary and notation, we proposed two versions of a model
volatilities. Here he presents new work on an extension                                    aimed at modelling either (a) a set of discrete forward variances or
                                                                                           (b) the full variance curve. The use of a discrete structure was
of this model, which, while remaining Markovian,                                           then motivated by the need to separately control the short for-
provides control on the smile of forward variances and                                     ward skew and the spot-volatility correlation. Our starting point
                                                                                           was the following lognormal dynamics:
can be calibrated to Vix futures and options
                                                                                                           T           T                            kn T t
                                                                                                           t           0   exp            wn e               Xn
To remedy                     some of the limitations of popular models used
                              for equity derivatives hightlighted in previous
         work (Bergomi, 2004), we proposed a stochastic volatility model

                                                                                                               2   nm
                                                                                                                           wn w m e
                                                                                                                                      kn k m T t
                                                                                                                                                       E Xn Xm

         based on a specification of the joint dynamics of the underlying                   for either:
         spot and its implied forward variance swap (VS) variances (Ber-                        continuous forward variances T: the forward instantaneous
         gomi, 2005). The aim of this model was to afford better control of:                variance for date T, observed at t; or
            the term structure of the volatility of volatility;                                discrete forward variances VtT ,T : we first set up a tenor struc-
                                                                                                                                          i   i+1

            the forward skew; and                                                          ture of dates Ti. VtT ,T is then the forward variance for the time
                                                                                                                   i   i+1

            the correlation between spot and VS volatilities                               interval [Ti, Ti+1], observed at t.
         for the pricing of options such as reverse cliquets, accumulators                     The initial values of forward variances, T are inputs of the
         and options on realised variance. In practice, as there were no                   model, just like yield curves in interest rate models. The Xn are
         traded instruments to hedge the volatility-of-volatility risk, the                correlated Ornstein-Ühlenbeck processes: dXn            knXndt + dW n,
         level of the corresponding parameter in our model was set to a                    wn are positive weights and is a global scaling factor for the
         conservative value and we concluded our article with the follow-                  volatility of forward variances.
         ing statement: “It is the hope of the author that the liquidity of                    The dynamics in equation (1) has the following properties:
         options on volatility and variance increases so that we will soon                 the T are driftless; they are lognormally distributed; and each
         be able to trade the smile of the volatility of volatility!”                          (t) is a function of the Xi. The model is Markovian and time-
            The author’s hopefulness was fulfilled, as these past few years                 homogeneous. Volatilities of forward variances are functions of
         have witnessed the growth of the market in options on realised                    T t only.
         variance and the development of the market of Vix futures and                         How many Ornstein-Ühlenbeck processes should one use? The
         options. Vix futures expire 30 days before the maturities of listed               dynamics of the variance curve generated by one-factor stochastic
         S&P 500 options. Their value at expiry is equal to the 30-day vari-               volatility models – such as the popular Heston model – is too poor
         ance swap volatility of the S&P 500 index, derived from an approx-                for practical use unless one introduces time-dependent volatilities
         imate replication of the variance swap using market prices of listed              of forward volatilities, a very undesirable device. In Bergomi (2005),
         S&P 500 options.1 Listed options on these futures with the same                   we felt that using two factors affords adequate control on the term
         maturities as those of the underlying futures trade as well.                      structure of volatilities of volatilities. We make the same choice
            In the over-the-counter market, we can soon expect to see ever-                here and use two Ornstein-Ühlenbeck processes X and Y – using
         more complex options with payouts involving both an underlying                    more processes generates no additional complexity but should be
         and its realised variance – with possibly several underlyings –                   motivated by the nature of the option one wishes to price:
         requiring adequate modelling of the smile of volatility of volatility.
            In this article, we propose a new version of our model that                                                    dX t = − k1 X t dt + dWt X
         focuses on the modelling of the dynamics of volatilities and                                                      dYt = − k 2Yt dt + dWtY
         addresses the issue of the smile of volatility of volatility, which is
         manifested, for example, in the smiles of options on Vix futures.                 where k1 > k2 , X0 = Y0 = 0. W X and W Y are correlated with each
            We first present our model in its two forms, continuous and                     other as well as with W, the Brownian motion driving the spot
         discrete, and motivate the use of either one. We then illustrate its              process, which will be specified in due course. The correlations
             For a precise definition of the Vix, see   are defined as dW XdW Y = dt, dWdW X = SXdt, dWdW Y =

90       Risk October 2008
   dt with the following parameterisation:                                                                                      determined by its terminal condition at t = T, set by the market.
                                                                               2                       2                        Smiles for maturities shorter than T generated by equation (6)
                         SY               SX                   1                           1           SX                 (2)
                                                                                                                                will generally not agree with market smiles. For examples of
  Xt and Yt and their increments are easily simulated using their                                                               Markov-functional models in the equity context, see Carr &
integral representation:                                                                                                        Madan (1999).
                                                                                                                                  Markov-functional models are local volatility models whose
                         t                                                         t
                Xt           e    k1 t s
                                              dW sX ,Yt                                e   k2 t s
                                                                                                        dW sY             (3)   local volatility function is set by the mapping function f. In the
                         0                                                     0
                                                                                                                                case where x is a Brownian motion, it is given by:
   Choosing to model discrete or continuous forward variances is                                                                                                                      ln f
a matter of practical convenience, motivated by the nature of the                                                                                                       t, S
                                                                                                                                                                                       x       x f      1
                                                                                                                                                                                                            S ,t
financial observables one needs to model.2 While Vix futures
naturally call for a discrete framework, the continuous form is                                                                 They are not suitable for pricing options with high sensitivity to
more suited to options involving implied or realised variances of                                                               forward volatility or volatility of volatility, but are an economical
arbitrary maturities. In what follows we develop both types of                                                                  solution to the issue of single-maturity smile modelling. This is
model, starting with the continuous one.                                                                                        the case for Vix options, whose maturities are also the expiry
   A Markovian model for continuous forward variances. We                                                                       dates of the underlying futures.
take as our starting point equation (1) – with two factors – but                                                                   The case for local volatility can in fact be argued more convinc-
relax lognormality while keeping the model Markovian. Let us                                                                    ingly for volatility than for equities themselves: from 1920 to 2000,
define xT as:
                                                                                                                                the Dow Jones index rose by a factor of around 100, while volatili-
                                                                                                                                ty’s order of magnitude has changed little in the past two centuries
            xtT                   1            e   k1 T t
                                                                       Xt                  e       k2 T t
                                                                                                            Yt            (4)   (the average volatility of the Dow Jones index from 1920 to 2000
                                                                                                                                was 16%, while the volatility of an index built using stocks trading
where:                                                                                                                          on the Paris bourse from 1801 to 1900 was around 14%4).
                                                                                                                                   Once f T (x, t = T) has been chosen, solving equation (6) gener-
                                                       2               2                                                        ates f for times t < T. This has to be done once for each T. One is
                              1/          1                                            2           1
                                                                                                                                then able, given the values of Xt and Yt at time t, to generate the
xT is driftless by construction. Let us now introduce a function                                                                full variance curve T.  t
f T (x, t) that maps xT on to the forward variance T:
                      t                            t
                                                                                                                                   However, having to solve equation (6) for many values of T is
                                          T        T                                                                            impractical on one hand, and on the other hand, traded instru-
                                          t        0       f T xtT ,t                                                     (5)
                                                                                                                                ments provide information on discrete forward variances:
  The condition that be driftless translates into the following                                                                                                                       1            T2
condition for the mapping function f T (x, t):                                                                                                                    VtT1 ,T2                              T
                                                                                                                                                                                                        t dT
                                                                                                                                                                                 T2       T1       T1
                              fT                   T           t           2
                                                                                                   0                      (6)   rather than instantaneous ones; Vix futures, for example, are
                              dt                   2                       dx 2                                                 related to one-month forward variances. We now present two
where ( ) is given by:                                                                                                          solutions to this issue that can be used practically.
                                                                                                                                   A single f T per time interval. Define a tenor structure Ti, for
  2         2                 2       2 k1         2           2 k2                                              k1 k 2         example given by the maturities of Vix futures, and assume that
                     1            e                    e                               2           1        e             (7)
                                                                                                                                all functions f T are equal to f T for T [Ti, Ti+1[:

The normalisation factor has been introduced so that ( = 0)
                                                                                                                                                                       f T ( x,t ) = f Ti ( x,t )
= 1. A solution to equation (6) needs to be properly rescaled so
that f T (0, 0) = 1.                                                                                                            Since f T (x, t) is defined for t < Ti only, we then need to specify

  We also require that f T (x, t) be monotonic in x. The lognormal                                                              how T evolves into T . We can simply assume that T stays fro-
                                                                                                                                        T   i             T
case (equation (2) in Bergomi, 2005) corresponds to the follow-                                                                 zen: T = T . We then need to generate one mapping function f T
                                                                                                                                        T     T     i

ing particular solution of equation (6):                                                                                        (x, t) per interval [Ti, Ti+1].
                                                                                                                                  An analytical ansatz for f T. Exponentials are eigenfunctions of the
                                                                   2       T
                                                                                                   d                            heat equation (6) and generate the lognormal solution (8). The
                              f T x,t              e               2       T       t                                      (8)
                                                                                                                                familiar representation of space-time harmonic functions suggests
   Mapping a Gaussian process on to a financial instrument is a                                                                  that we use as an ansatz for f T a linear combination of exponentials:
popular technique in fixed income. In ‘Markov-functional mod-                                                                                                                                                2   T       2
                                                                                                                                                                                                    x                       d
els’ (Kennedy, Hunt & Pelsser, 2000), f maps a Gaussian process                                                                                         f T x,t                  d             e        2       T   t

on to a Libor or swap rate setting at t = T. f T (x, t = T) is chosen so                                                                                                     0

that the market caplet or swaption smile is calibrated. Different                                                                where d ( ) is a measure such that 0 d ( ) = 1. We expect that
Libor rates of the same yield curve can be driven by either differ-                                                              mixing several exponentials with positive weights will generate a
ent processes or by the same process, although this may generate                                                                positively sloping volatility, which is what Vix smiles exhibit. Let
peculiar behaviour.3                                                                                                            us use a combination of two exponentials and write:
   Using the ansatz St = f(Wt, t), where Wt is a Brownian motion,
                                                                                                                                  In fixed income, one similarly chooses to model either continuous forward rates or discrete Libor rates
for equity underlyings is not as popular. Indeed, in the equity                                                                 3
                                                                                                                                  For example, it was shown (Kennedy, Hunt & Pelsser, 2000) that in a Markov functional model forward
context one requires that the model match market smiles for sev-                                                                rates cannot be all lognormal. This, however, does not mean that a Markov functional model is incapable of
                                                                                                                                generating flat smiles for Libor rates. It actually will, but while Libor smiles are flat, the associated local
eral maturities, for the same underlying. This cannot be achieved                                                               volatilities are not flat, except for the particular Libor rate whose numeraire is chosen as base numeraire
by writing St = f(Wt, t) as the solution of equation (6) is fully                                                               4
                                                                                                                                  Calculated using monthly data provided in Gallais-Hamonno (2007)


                                                                                 h t ,T
                                                                                                                                2    2
                                                                                                                                       h t ,T
                                                                                                                                                       one if FT is below a barrier l. This is also the price of the same
                                                                               T                                                T    T
                    f   T
                              x,t          1                 e    Tx
                                                                       e                                         Tx
                                                                                                                                                 (9)   digital on V TT ,T , but with barrier L = l2 . The price of this digital
                                                                                                                                                                                  i   i+1

                                                                                                       Te             e
                                                                                  2                         T                         2
                                                     T                                                                                                                        i

                                                                                                                                                       in our model is N(x*(L)), where x*(L) is defined by:
        where h(t, T) = T t ( )2 d and ( ) is defined in equation (7). T,
            , T are functions of T. T is a scale factor for the volatility of
                                                                                                                                                                                                  L = V0 i
                                                                                                                                                                                                             T ,Ti +1
                                                                                                                                                                                                                        f i x * ( L ) ,Ti       )
           ; let us write it as:                                                                                                                       and where N is the cumulative density for xT , a centred Gaussian
                                                                                                                                                                                             2     2
                                                                                                                                                       random variable with variance (1 – ) E[X T ] + 2 E[Y 2 ] + 2 (1 –
                                                                                                                                                                                                              Ti                                i
                                                     T           2                                                                                      )E[XT Y T ]. Equating the model price to the market price deter-
                                                                       1          T               T    T
                                                                                                                                                              i       i

                                                                                                                                                       mines x*(L). We now have one value for f i at point x = x*(L):
        where we have introduced and T. controls the general level of
        volatility of volatility in the model while T is an adjustment fac-                                                                                                                             (
                                                                                                                                                                                                    f i x * ( L ) ,Ti =       )    V0 i
                                                                                                                                                                                                                                         T ,Ti +1
        tor, controlling the volatility of T. Using these new variables, the
        dynamics of T at time t = 0 reads:                                                                                                             Doing this for sufficiently many values of L completely specifies the
                                                              T                           T        T                                                   function f i(x, t = Ti). Solving equation (12) generates f i for t < Ti.
                                                       d               2                      dx
                                                                                 T                                                                        The spot process in the discrete version. Let us start with the
        For T      0, T has an instantaneous volatility equal to 2 T. Ide-                                                                             following dynamics for S on the interval [Ti, Ti+1[:
        ally, we would like the T to be close to one: is then the volatility
        of a very short VS volatility.                                                                                                                                                      dS = ( r − q ) Sdt + VTi i
                                                                                                                                                                                                                                     T ,Ti +1
           For the sake of calibrating T, T, T to Vix futures and Vix
        options expiring at time Ti, we will take them to be piece-wise                                                                                   This dynamics for S is lognormal over [Ti, Ti+1[. Just like in the
        constant over the interval [Ti, Ti+1[, equal to i, i, i.                                                                                       continuous version of the model, the level of both forward skew
           The spot process in the continuous version. The dynamics                                                                                    and vanilla skew as well as the correlation between S and VS vola-
        for S is given by:                                                                                                                             tilities will be controlled by SX and SY, the correlations of W
                                                                                                                                                       with W X and W Y.
                                  dS           r     q Sdt                 S          t
                                                                                      0           f t xtt ,t dWt                                (10)      The discrete framework, however, brings us the additional capa-
                                                                                                                                                       bility of being able to control the short forward skew independently.
          Note that correlations SX and SY of W with, respectively, W X                                                                                We introduce, as in Bergomi (2005), a ‘local volatility’ function:
        and W Y will determine both the correlation between the spot and
        VS volatilities as well as the skew of the vanilla smile generated by                                                                                                                                      S   T ,T
                                                                                                                                                                                                              i      ,V i i          1

        the model.                                                                                                                                                                                                STi Ti
          Calibration and pricing results obtained using the analytical
        ansatz for f T in the continuous version of our model will be discussed                                                                        and write the dynamics of S over [Ti, Ti+1[ as:
        below. Prior to this, let us present the discrete form of our model.
           A Markovian model for discrete forward variances. If we are                                                                                                                                                             S   T ,T
        only interested in modelling the joint dynamics of the S&P 500                                                                                                            dS          r         q Sdt             i          ,V i i             SdW
                                                                                                                                                                                                                                  STi Ti
        and Vix futures, rather than the full dynamics of the S&P 500
        variance curve, it is natural to define a tenor structure of equally                                                                            where i() is defined so that:
        spaced dates Ti corresponding to the monthly expiries of Vix futures                                                                             the VS volatility for maturity Ti+1 observed at Ti:
        and to directly model discrete forward variances VtT ,T .                                                          i   i+1

          Mirroring the dynamics for instantaneous variances, let us                                                                                                                                               VTi i
                                                                                                                                                                                                                        T ,Ti +1
        define processes xti and write:
                            VtTi ,Ti   1
                                           V0 i
                                               T ,Ti     1
                                                             f i xti ,t                                                                                is matched; and
                                                                                                                                                (11)      the desired level of forward skew and its dependence on the
                                    xti                1              e
                                                                           k1 Ti t
                                                                                          Xt               e
                                                                                                                k 2 Ti t
                                                                                                                                Yt                     level of volatility are obtained. For example, it is easy to generate
                                                                                                                                                       a short forward skew whose level is independent of or propor-
        V0 ,T are inputs given by the VS market. The mapping functions
          i   i+1
                                                                                                                                                       tional to the level of short forward at-the-money volatility. Any
        f satisfy the following equation:                                                                                                              type of dependence is easily accommodated. For details on the
                                                                  2                                                                                    procedure for determining i(), see Bergomi (2005).
                                                fi                    Ti         t            2
                                                                                                            0                                   (12)      We now have a model that calibrates exactly to the Vix market
                                               dt                      2                  dx 2                                                         and affords full control of the short forward skew.
        where ( ) is defined in (7).                                                                                                                       To calibrate the vanilla smile of the S&P 500 in addition to the
           One of the benefits of using the discrete framework is that cali-                                                                            Vix market, choose the local volatility 0 for the first interval [T0,
        bration to the smiles of Vix options is exact. To generate the ter-                                                                            T1[ so as to exactly match the smile of the S&P 500 index for
        minal values of the f i(x, t = Ti) so that market prices of Vix futures                                                                        maturity T1, then set correlations SX and SY to calibrate at best
        and options are matched, use the following simple procedure                                                                                    the S&P 500 smile for longer-dated maturities.
        based on Kennedy, Hunt & Pelsser (2000).                                                                                                          A natural question in the discrete context is: why model for-
            Compute the level of forward variance V0 ,T from the Vix                                             i   i+1
                                                                                                                                                       ward variances V T ,T instead of working with Vix futures Fi
                                                                                                                                                                                              i   i+1

        future and options of maturity Ti. The corresponding expression                                                                                directly, writing:
        is given in equation (13) further below.
           From the Vix smile for maturity Ti, one easily generates the
                                                                                                                                                                                                            Fti = F0i f i xti ,t  ( )
        price of a digital option of maturity Ti on Vix future Fi that pays                                                                                               i
                                                                                                                                                       since the F are driftless as well? Consider for example the option to

92      Risk October 2008
enter at Ti into a VS contract of maturity Ti+n. The variance of matu-
rity Ti+n is the average of n monthly variances. At time t = Ti, the first                                                   1 Vix futures on March 18, 2008
monthly variance is known, as it is given by F T . However, we do not                       i
have access to the other variances, since what we are modelling are                                                                                                                                            Market
      j                                                                                                                          27                                                                            Model
the FT , which are expectations of square roots of forward variances.

While forward variances are additive, forward volatilities are not.

Calibration of Vix futures and their smiles – options on                                                                         25

realised variance
Here, we demonstrate the capabilities of the continuous version                                                                  24
of our model. We test its calibration to the Vix market, then dis-
cuss the pricing of spot-starting and forward-starting options on
realised variance.                                                                                                               22
   Calibration of Vix futures and their smiles. We model the                                                                                April            May             June             July           August
full set of instantaneous forward variances. A discrete forward
variance over [Ti, Ti+1], such as the 30-day variance that underlies
Vix futures, is given by:
                                                                                                                            2 Vix smiles on March 18, 2008
                          Ti ,Ti                    1              Ti        T         T
                     Vt            1
                                                                             0     f             ,t dT                        100
                                             Ti         Ti     Ti                                                                             Market
                                                                                                                                90            Model
           T     T                                                                          Ti,Ti+1
Because f (x , t) is a smooth function of T, V
                                                  is efficiently eval-                       t
uated by Gaussian quadrature.                                                                                                   80
   The initial variance curve T is a basic underlier in our model,
                                 0                                                                                              70
along with S. It would be natural to generate T from the VS vola-

tilities derived from the S&P 500 vanilla smile. For each Vix                                                                   60
future expiring at Ti, we then need to calibrate i, i, i so that the                                                            50
Vix future itself and the implied volatilities of its options are
matched. Under which conditions is this possible?                                                                               40
   Consistency of Vix futures/options with S&P 500 VS vari-                                                                     30
ances. The settlement value of the Vix future of maturity Ti is the                                                               20%                         25%                          30%                         35%
VS volatility for maturity Ti+1:

                                                   FTii = VTi i
                                                                        T ,Ti +1                                            An example of calibration. We show results of a calibration
                                                                                                                        done on March 18, 2008, for Vix futures and options of maturi-
Using the Vix future Fti, along with its calls and puts, one is able                                                    ties April to August. Note that the liquidity of Vix futures and
to replicate any European-style option on FT , in particular its                                  i
                                                                                                                        options dies off quickly after the first two or three listed maturi-
square FT . The consistency condition relating to the S&P 500
                                                                                                                        ties. Also, strikes that could be considered liquid, given the mar-
forward variance V T ,T and the Vix market is thus:
                                   i   i+1
                                                                                                                        ket conditions in March 2008, ranged from 20–35%. Calibration
                                                                                                                        of triplets ( i, i, i) is done by least-squares minimisation. Figure
      VtTi ,Ti   1
                      Fti 2            2           PK t, Fti dK                        2         C K t, Fti dK   (13)   1 shows Vix futures and figure 2 Vix implied volatilities.
                                             0                                             Fi
                                                                                                                            In figure 2, the implied volatility curves are stacked with
where PK(CK) is the undiscounted price of a put (call) option on Fi                                                     respect to their maturities. The higher volatilities correspond to
of maturity Ti. Practically, strikes used in the replication do not                                                     the April maturity, the lowest to the August maturity. While cali-
range from zero to infinity: we discard extreme strikes, which                                                           bration is not perfect, it is very acceptable.
contribute little as Vix smiles are not very steep.                                                                         Had we used the discrete version of our model, calibration
  In our experience, the consistency condition above is not always                                                      would be perfect by construction. Market and model curves
met. Frequently, values of VtT ,T derived through replication lie
                                                        i    i+1
                                                                                                                        would overlap exactly.
above the values we get from the S&P 500 VS market, meaning                                                                 Table A shows the values we use for parameters , , k1, k2 and .
the Vix futures are too high, typically 0.5 of a volatility point.                                                      Table B shows the values of , and generated by calibration.
This is not always easily arbitrageable, as: one incurs bid/offer                                                            Parameters , , k1, k2 and control the correlation structure
costs in trading variance swaps and Vix options; as is well known                                                       and volatilities of forward volatilities. Multiplying and dividing
to practitioners, the settlement values of Vix futures, which                                                             i
                                                                                                                            by the same factor leaves the dynamics unchanged. We have
involve market prices of traded S&P 500 options and are very                                                            chosen so that the i lie around one. As table B shows, does
sensitive to quotes of out-of-the-money puts often do not match                                                         not vary much across maturities. This is desirable on the grounds
one-month VS volatilities5; and the liquidity of Vix futures and                                                        of time-homogeneity.6 Note that for some maturities vanishes.
options decays quickly with their maturity.                                                                                 is then a shifted lognormal.
  In what follows, for the purpose of calibrating Vix futures and                                                           Options on realised variance. First, we consider the influence
options, we use the variance curve T generated by the replication
                                      0                                                                                 5
                                                                                                                          Even after taking into account the difference in the definitions of VS volatilities and Vix volatilities.
above rather than that given by the S&P 500 smile. Doing other-                                                         Standard VS contracts define realised annualised variance as the sum of daily squared returns multiplied
                                                                                                                        by factor 252/N where N is the number of observed returns, instead of dividing the sum of squared returns
wise would be akin to trying to calibrate option prices using an                                                        by the maturity of the VS contract
incorrect value for the underlying.                                                                                     6
                                                                                                                          This is generally the case and is not specific to Vix smiles of March 18, 2008


         A. Values for , , k1, k2 and used in calibrating Vix smiles                              C. The three sets of parameters used to generate curves
                                                                         130.0%                   in figure 3
                                                                          28%                                                         Set 1                      Set 2                     Set 3

         k1                                                               8.0                                                     130%                           137%                      125%

         k2                                                               0.35                                                        28%                        29%                       32%

                                                                          0%                      k1                                   8.0                       12.0                       4.5

                                                                                                  k2                                  0.35                       0.30                      0.60

                                                                                                                                       0%                        90%                       –70%
         B. Values of , and calibrated from Vix smiles
                                    i                           i                   i

         Apr 16, 2008              87%                         21%                106%            D. Prices of at-the-money call options on spot-starting
         May 21, 2008              36%                         11%                94%
                                                                                                  realised variance, forward realised variance and at-the-
         Jun 18, 2008              35%                         0%                 96%
                                                                                                  money swaptions
         Jul 16, 2008              30%                         0%                 99%                                                                    Set 1               Set 2             Set 3

         Aug 20, 2008              24%                         0%                 100%            Spot-starting realised – six months                    2.03%               2.03%             2.03%

                                                                                                  Spot-starting realised – one year                      2.22%               2.21%             2.24%

                                                                                                  Six months realised in six months                      3.04%               2.89%             3.25%
          3 Term structure of the instantaneous volatility
                                                                                                  Six months in six months swaption                      2.28%               2.10%             2.57%
          of volatility generated by three different sets of
          parameters , , k1, k2,                                                                 component of the option’s value if VS volatilities were frozen. Here
              120                                                                                our focus is on the contribution of volatility of volatility. Let us
                                                                                  Set 1          assume no smile for forward variances ( = 0, = 1), use values for
              100                                                                 Set 2
                                                                                  Set 3            , , k1, k2 and in table A and consider the instantaneous volatil-
                                                                                                 ity of a VS volatility as a function of its maturity, assuming the
                                                                                                 initial term structure of VS volatilities is flat ( T = 0 T). The
                                                                                                 instantaneous volatility of volatility T of V 0,T is then given by:

                                                                                                                                                     2                                     2
                  40                                                                                       vol                2   1 e k1T                                2   1 e k2T
                                                                                                           T                                                 1
                                                                                                                                    k1T                                        k2T
                                                                                                                                                                                       1           (15)
                                                                                                                                                    1 e k1T              1 e k2T
                       0       6                     12                  18               24                                 2         1
                                                                                                                                                      k1T                  k2T
                                                                                                       is plotted in figure 3 for the set of parameters in table A along
        of the correlation structure of forward variances, then use infor-                       with two other curves obtained using other sets of parameters.
        mation embedded in the skew of Vix options to generate a skew                            The unit for T is one month. Table C shows the corresponding
        for realised variance.                                                                   values of , , k1, k2 and . We have chosen three values for ,
           The correlation structure of forward variances. We turn                               0%, 90% and –70%, and have set the remaining parameters so
        here to the pricing of options on realised variance, which are liq-                      that the term structures of instantaneous volatility of VS volatil-
        uid on indexes. The standard definition of their payout is:                               ity are almost identical.
                                      1                                                             Table D shows prices for at-the-money options on realised vari-
                                                 rT        K                                     ance for maturities of six months and one year, for an initial flat
                                   2 ˆ 0,T
                                                                                                 term structure of VS volatilities equal to 20%. In addition, table
        where ^ 0, T is the VS volatility for maturity T and rT is the realised                  D lists prices of: an option on forward realised variance (the real-
        volatility. Mirroring a common approximation for Asian options,                          ised variance is sampled over an interval of six months starting in
        a popular approach among practitioners starts with the definition                         six months); and a VS swaption (the option to enter in six months
        of an effective underlying St:                                                            a VS contract of six months’ maturity struck at today’s level of
                                            2                                                    forward VS volatility).
                                        t   rt   T        t ˆ t2,T   t
                              St                                                          (14)      While prices of options on realised variance starting today are
                                                     T                                           practically identical, prices for forward-starting options are differ-
        where rt is the annualised realised volatility up to t and ^ t,T t is                    ent. This highlights the fact that while a term structure of volatility
        the VS volatility for maturity T seen at time t. The variation of                        of volatility such as those shown in figure 3 is all one needs to price
          t,T t
                is hedged by trading variance swaps of maturity T. This also                     spot-starting options on realised variance, it does not uniquely
        hedges the variation of 2 . The dynamics of St then depends on
                                                                                                 determine volatilities of forward volatilities. These depend on the
        that of ^ t,T t . One only needs to model the volatility of ^ t,T t.                     correlation structure of forward volatilities. As an illustration, we
            As explained in Bergomi (2005), since variance is measured                           plot in figure 4 the instantaneous correlation at t = 0 of Vt with:
        using discretely sampled returns, one also needs to include the con-
                                                                                                                                              Vti    , i 1
        tribution from the kurtosis of daily returns. This contribution
        mostly matters for shorter maturities and would stand as the sole                        for    = one month and i = 0 to 11, for the three sets of parameters.

94      Risk October 2008
  4 Instantaneous correlation at t = 0 of forward one-                       5 Implied smile for realised variance generated by
  month variances with spot-starting one-month variance                      calibration to Vix futures and options
    100                                                                        100
                                                             Set 1
       80                                                    Set 2               80
                                                             Set 3


    –20                                                                             0
            0   1   2    3    4    5    6    7     8    9   10    11                 60%    80%            100%          120%              140%         160%           180%

   In set 3, correlations of forward VS volatilities are much lower        tility. At first order in the skewness ST of lnST the at-the-money-
than in sets 1 and 2. For a given term structure of volatility of          forward skew is given by Backus, Foresi & Wu (1997):
spot-starting volatility, the volatility of forward volatilities will be                                     dˆ                   ST
higher in set 3, explaining why prices of options on forward real-
                                                                                                           d ln K             6 T
ised variance in table D increase when going from set 2 ( = 90%)                                                     F

through set 1 ( = 0%) to set 3 (               ).                          Let us here consider the case of a flat initial term structure of
   The impact on pricing would be even larger for more complex             variances and a lognormal dynamics:
structures such as options on the spread of forward variances,
                                                                                                                              2   T             2
which would presumably require a larger number of driving fac-                                                           x                          d
                                                                                                  f T x,t            e        2   T    t

tors. Note that one-factor stochastic volatility models impose 100%
instantanenous correlation between all forward volatilities.               In line with our parameterisation, we write = 2 , where is the
   Table D confirms the intuition that an option on realised for-           volatility of a very short VS volatility. At first order in the volatil-
ward variance should be more expensive than a variance swaption            ity of volatility we get for the at-the-money-forward skew:
with the same forward date, maturity and strike, the difference in
price being contributed by the randomness with which the under-              dˆ
lying realises until maturity the variance that was observed at the        d ln K   F
forward date.
                                                                                                                     k1T                                         k2T
   The smile of realised variance. Just as it is natural to incorpo-                             k1T        1 e                                 k2T      1 e
rate information about the smiles of the basket’s underlyings in a                      1   SX                                             SY
                                                                                                           k12 T 2                                       2
                                                                                                                                                        k2 T 2
basket option’s price, it is also natural to use information embed-
ded in Vix smiles to price options on realised variance on the                This expression is recovered from equation (8) in Bergomi
S&P 500, which can be thought of as options on a basket of for-            (2005) by taking the limit         0, N = T/ in the definition of
ward variances, with part of their value coming from the kurtosis           (x, n).
of daily returns.                                                             In the limit T       , the spot/volatility correlation function
   Figure 5 shows the implied smile of realised variance for the S&P       tends to zero, so ST decays like 1/ T and the skew decays as 1/T,
500 using the values in table B, which were calibrated on Vix mar-         which is what we expect for independent returns.
ket smiles on March 18, for an option of six months’ maturity,               For T 0, the skew does not vanish and tends to a finite value
using an initial flat VS term structure. This is the implied volatility     given in our model by:
of the underlying S in equation (14), assuming it has zero drift. The
x-axis represents the moneyness of the volatility: K/ ^0,T. Prices for                                           1           SX             SY
the option on realised variance have been produced by simulating                                       2
the spot process according to equation (10).                               This is common to all stochastic volatility models: the density they
   The smile in figure 5 should be taken with a pinch of salt. We           generate for lnS becomes Gaussian for T 0. However, because ST
know for example that equity market skews for indexes cannot be            vanishes like T, the skew, which is proportional to ST/ T, tends to
recovered by pricing them in a multi-asset local volatility frame-         a finite value. This expression for the skew when T 0 agrees with
work with constant correlations between the underlying stocks –            the general result for stochastic volatility models:
an issue known as correlation smile.                                                                dˆ                1   dS dV
    The vanilla smile. The dynamics of the variance curve not
                                                                                                  d ln K     S       4dt S V V
only affects the pricing of options on realised or implied variance
but also determines the smile of vanilla options generated by the          where V is the instantaneous variance.
model. Let us derive an approximate expression for the at-the-                We now check the accuracy of the approximate expression for
money-forward skew at order one in the volatility of volatility            the skew in equation (16) against its actual value. Figure 6 shows:
using the same procedure as in Bergomi (2005), based on a calcu-           the actual skew measured as the difference of the implied volatili-
lation of the skewness of lnST at order one in the volatility of vola-     ties of the 95% and 105% strikes; and:


                                                                                  relations SX and SY, once values for parameters , , k1, k2 and
          6 The 95–105% at-the-money-forward skew as a                              have been chosen. SX and SY can be used to control the over-
          function of maturity                                                    all level of the vanilla skew. However, as is made manifest in
                5                                                                 expressions (16) and (15), the decay of the skew and the term
                                                              Actual              structure of volatilities of volatilities both depend on k1 and k2
                4                                                                 as k1, k2 control both the volatility/volatility and spot return/
                                                                                  volatility correlation functions. It will thus be difficult to have
                3                                                                 separate handles on the term structure of the vanilla skew and
                                                                                  the term structure of volatility of volatility, at least in a two-fac-

                2                                                                 tor framework.
                                                                                  In this article, we propose a model driven by easy-to-simulate
                    0        0.5             1.0             1.5        2.0       Ornstein-Ühlenbeck processes that, in addition to providing con-
                                                                                  trol of the term structure of volatilities of volatilities and the cor-
                                                                                  relation structure of forward volatilities, allows us to control the
                                                                                  smile of forward volatilities, while remaining Markovian. We
          7 ST as a function of T                                                 illustrate the impact of the correlation structure of forward vola-
                                                                                  tilities on the pricing of options involving implied or realised for-
                                                                                  ward variances.
           –0.2                                                                      We demonstrate how the model can be calibrated to Vix
                                                                                  futures and options – exactly in the discrete version of the model
           –0.4                                                                   – and use Vix market information to price options on the real-
                                                                                  ised variance of the S&P 500 index. As liquidity of Vix futures
                                                                                  and options expands to longer-dated maturities, and similar
           –0.8                                                                   volatility derivatives on other indexes such as the Vstoxx gain
                                                                                  popularity, we can expect an active market for payouts combin-
           –1.0                                                                   ing forward variances and the underlying index to take shape,
                                                                                  requiring adequate modelling of the joint dynamics of the spot
           –1.2                                                                   and its forward variances.
                    0        0.5             1.0             1.5            2.0      Finally, as the diversity of traded instruments grows, it is tempt-
                                                                                  ing to try to calibrate market prices of all available instruments,
                                                                                  forcing on to the model’s parameters as much time-dependence as
                                             dˆ                                   is needed to achieve this goal. We would like to caution against
                                     0.1                                          the excessive psychological benefit that perfect calibration may
                                           d ln K    F                            generate and the elation of push-button pricing. In our opinion,
        for maturities up to two years. We have used the parameters in set        the issue of specifying the general dynamics of the model so that
        1 (see table C) and the following spot/volatility correlations: SX =      the carry levels for all gammas (be they spot, volatility or cross
                , SY                        . Even though it slightly overesti-   gammas) and forward skew are predictable and comfortable is
        mates the at-the-money-forward skew, approximation (16) is of             much more relevant.
        acceptable quality. Raising the volatility of volatility or the level
        of spot/volatility correlations will cause it to deteriorate.             Lorenzo Bergomi is head of quantitative research in the equity derivatives
           It is instructive to look at the dependence of the skewness ST of      department at Société Générale. He warmly thanks members of his team for
        lnST on T. Figure 7 shows:                                                useful discussions and suggestions. Email:
                                    6 T
                                            d ln K       F
        as a function of T with:
                                                                                    Backus D, S Foresi and L Wu, 1997          Carr P and D Madan, 1999
                                         dˆ                                         Accounting for biases in Black-Scholes     Determining volatility surfaces
                                                                                    Available at http://faculty.baruch.cuny.   and option values from an implied
                                       d ln K   F                                   edu/lwu/papers/bias.pdf                    volatility smile
        given by equation (16). We can see that, while skews in figure 6 are                                                    Quantitative Analysis in Financial
                                                                                    Bergomi L, 2004                            Markets II, pages 163–191, World
        not particularly large, ST, a dimensionless number, is of order one.        Smile dynamics                             Scientific Publishing
        Notice how parameters in set 1 cause the skewness for the longer            Risk September, pages 117–123
        maturities in the figure to be flat, thus making the long-term skew                                                      Gallais-Hamonno G, 2007
                                                                                    Bergomi L, 2005                            Le marché financier français au
        decay like 1/ T, which is typical of equity market smiles.                  Smile dynamics II                          XIXème siècle
          As mentioned above, as T grows ST will eventually decay like              Risk October, pages 67–73                  Publications de la Sorbonne, Paris
        1/ T for T       1/k2 . The characteristic time of the second factor
        should then be chosen to be long enough so that for maturities of                                                      Kennedy J, P Hunt and A Pelsser, 2000
                                                                                                                               Markov-functional interest rate models
        interest this limiting behaviour for ST is not reached.                                                                Finance & Stochastics 4, pages 391–408
          Equation (16) highlights the dependence of the skew on cor-

96      Risk October 2008

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