CUTTING EDGE. EQUITY DERIVATIVES 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 existing models for equity derivatives and introduced a produces and summarise our work in the concluding section. 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 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- correlation. Our starting point can be calibrated to Vix futures and options was the following lognormal dynamics:  T T  kn T t t 0 exp  wne Xn

n (1) some of the limitations of popular models used 2   kn km T t for equity derivatives hightlighted in previous   wnwm e EX n Xm To remedy 2  work (Bergomi, 2004), we proposed a model nm  based on a specification of the joint dynamics of the underlying for either: T spot and its implied forward variance swap (VS) variances (Ber- N 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

Ti,Ti+1 N the term structure of the volatility of volatility; N discrete forward variances Vt : we first set up a tenor struc- Ti,Ti+1 N the forward skew; and ture of dates Ti. Vt is then the forward variance for the time N the correlation between spot and VS volatilities interval [Ti, Ti+1], observed at t. T for the pricing of options such as reverse cliquets, accumulators The initial values of forward variances,] t=0 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 n traded instruments to hedge the volatility-of-volatility risk, the correlated Ornstein-Ühlenbeck processes: dXn ïknXndt + dW t, 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 T 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 Tït 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 – 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 one wishes to price: requiring adequate modelling of the smile of volatility of volatility. =− + X In this article, we propose a new version of our model that dXt k1Xt dt dWt focuses on the modelling of the dynamics of volatilities and =− + Y dYt k2Yt dt dWt addresses the issue of the smile of volatility of volatility, which is X Y manifested, for example, in the smiles of options on Vix futures. where k1 > k2, X0 = Y0 = 0. W and W 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 1 For a precise definition of the Vix, see www.cboe.com/micro/vix/vixwhite.pdf X Y X Y are defined as ¤dW dW ´ = Wdt, ¤dWdW ´ = WSXdt, ¤dWdW ´ =

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bergomi.indd 90 1/10/08 11:51:44 WSYdt with the following parameterisation: determined by its terminal condition at t = T, set by the market. Smiles for maturities shorter than T generated by equation (6)    1 2 1 2 (2) SY SX SX 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 models whose t k t s X t k t s Y  1  2  (3) local volatility function is set by the mapping function f. In the Xt   e dWs ,Yt   e dWs 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 1 financial observables one needs to model.2 While Vix futures x f S,t 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 N 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, T 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 T k T t k T t   1  2  (the average volatility of the Dow Jones index from 1920 to 2000 xt  1  e Xt e Yt (4) 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 eachT . One is   1/ 1   21  then able, given the values of Xt and Yt at time t, to generate the T T x is driftless by construction. Let us now introduce a function full variance curve ] t. T T T f (x, t) that maps xt on to the forward variance ] t: However, having to solve equation (6) for many values of T is impractical on one hand, and on the other hand, traded instru- T T f T xT ,t (5) t 0 t ments provide information on discrete forward variances: The condition that ]T be driftless translates into the following T ,T 1 T2 T T V 1 2   dT condition for the mapping function f (x, t): t T t T2  T1 1 T 2 T t 2 T f    f (6)  2  0 rather than instantaneous ones; Vix futures, for example, are dt 2 dx related to one-month forward variances. We now present two where X(Y) is given by: solutions to this issue that can be used practically. T N A single f per time interval. Define a tenor structure Ti, for 2 2 2 2 k 2 2 k  k k  1  2 1 2 (7) example given by the maturities of Vix futures, and assume that    1  e  e  21  e  Ti all functions fT are equal to f for T ‘ [Ti, Ti+1[: The normalisation factorF has been introduced so that X (Y = 0) T T V f ()x,t = f i ()x,t = 1. A solution to equation (6) needs to be properly rescaled so

T Ti that f (0, 0) = 1. Since f (x, t) is defined for t < Ti only, we then need to specify T T T T We also require that f (x, t) be monotonic in x. The lognormal how ] Ti evolves into ] T. We can simply assume that ] stays fro- T T Ti case (equation (2) in Bergomi, 2005) corresponds to the follow- zen: ] T = ] Ti. We then need to generate one mapping function f ing particular solution of equation (6): (x, t) per interval [Ti, Ti+1]. N An analytical ansatz for f T. Exponentials are eigenfunctions of the 2 T  2 T x d heat equation (6) and generate the lognormal solution (8). The f 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 T 2 T t els’ (Kennedy, Hunt & Pelsser, 2000), f maps a Gaussian process f x,t  de  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 dR(\) is a measure such that µ0 dR(\) = 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, 2 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 4 Calculated using monthly data provided in Gallais-Hamonno (2007) by writing St = f(Wt, t) as the solution of equation (6) is fully

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i 2 2 2 one if F is below a barrier l. This is also the price of the same ht,T ht,T Ti  T  T  T  T x  x  Ti,Ti+1 2  T 2 T  T 2 digital on V , but with barrier L = l . The price of this digital f x,t  1   T e e T e e (9) Ti in our model is N(x*(L)), where x*(L) is defined by: T 2 where h(t, T) = X(Y) dY and X(Y) is defined in equation (7). L , T ,T i * µTït T = i i +1 () L V0 f ()x L ,Ti \T, GT are functions of T. \T is a scale factor for the volatility of ]T; let us write it as: and where N is the cumulative density for xi , a centred Gaussian Ti 2 2 2 2 random variable with variance (1 – V) E[X T] + V E[Y T ] + 2V(1 – T i i T  2 V)E[X Y ]. Equating the model price to the market price deter- 1 Ti Ti T T T mines x*(L). We now have one value for f i at point x = x*(L): where we have introduced S and _ . S controls the general level of T i ()* () = L volatility of volatility in the model while _ is an adjustment fac- f x L ,Ti T ,T T V i i +1 tor, controlling the volatility of ]T. Using these new variables, the 0 dynamics of ]T at time t = 0 reads: Doing this for sufficiently many values of L completely specifies the function f i(x, t = T). Solving equation (12) generates f i for t < T . dT  2 T dxT i i T N The spot process in the discrete version. Let us start with the T For T q 0, ] has an instantaneous volatility equal to 2S_T. Ide- following dynamics for S on the interval [Ti, Ti+1[: ally, we would like the _T to be close to one: S is then the volatility T ,T of a very short VS volatility. dS = ()r − q Sdt + V i i +1 SdW Ti For the sake of calibrating LT, GT, _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 Li, Gi, _i. continuous version of the model, the level of both forward skew N 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 WSX and WSY, the correlations of W with WX and WY. dS r q Sdt S t f t xt t dW The discrete framework, however, brings us the additional capa-    0 t , t (10) bility of being able to control the short forward skew independently. X Note that correlations WSX and WSY of W with, respectively, W We introduce, as in Bergomi (2005), a ‘local volatility’ function: and WY will determine both the correlation between the spot and  S  VS volatilities as well as the skew of the vanilla smile generated by Ti ,Ti 1  i ,VT the model.  i   ST  Calibration and pricing results obtained using the analytical i T ansatz for f 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. N If we are S T ,T dS  r  q Sdt   ,V i i  SdW only interested in modelling the joint dynamics of the S&P 500  i Ti   ST and Vix futures, rather than the full dynamics of the S&P 500 i

variance curve, it is natural to define a tenor structure of equally where Xi() is defined so that: spaced dates Ti corresponding to the monthly expiries of Vix futures N the VS volatility for maturity Ti+1 observed at Ti: Ti,Ti+1 and to directly model discrete forward variances Vt . T ,T Mirroring the dynamics for instantaneous variances, let us V i i +1 i Ti define processes xt and write: T ,T T ,T i i V i i 1 V i i 1 f x t is matched; and t  0 t , (11) N the desired level of forward skew and its dependence on the k T t k T t i   1i   2 i  level of volatility are obtained. For example, it is easy to generate xt   1   e Xt  e Yt a short forward skew whose level is independent of or propor-

Ti,Ti+1 V0 are inputs given by the VS market. The mapping functions tional to the level of short forward at-the-money volatility. Any f i satisfy the following equation: type of dependence is easily accommodated. For details on the i 2 i procedure for determining X (), see Bergomi (2005). f  T  t 2 f i  i  We now have a model that calibrates exactly to the Vix market  2  0 (12) dt 2 dx and affords full control of the short forward skew. where X(Y) 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 X0 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 i minal values of the f (x, t = Ti) so that market prices of Vix futures maturity T1, then set correlations WSX and WSY 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-

Ti,Ti+1 Ti,Ti+1 i N Compute the level of forward variance V0 from the Vix ward variances V instead of working with Vix futures F future and options of maturity Ti. The corresponding expression directly, writing: is given in equation (13) further below. i = i i i Ft F0 f ()xt ,t N From the Vix smile for maturity Ti, one easily generates the i i price of a digital option of maturity Ti on Vix future F that pays since the F are driftless as well? Consider for example the option to

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bergomi.indd 92 1/10/08 11:51:46 enter at T into a VS contract of maturity T . The variance of matu- i i+n 1 Vix futures on March 18, 2008 rity Ti+n is the average of n monthly variances. At time t = Ti, the first monthly variance is known, as it is given by F i2. However, we do not 28 Ti have access to the other variances, since what we are modelling are Market Model the F j , which are expectations of square roots of forward variances. 27 Ti While forward variances are additive, forward volatilities are not. 26 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- 23 cuss the pricing of spot-starting and forward-starting options on realised variance. 22 N 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 T ,T 1 Ti 1 T T T V i i 1   f x ,t dT 100 t T 0 t i Market Ti1  Ti 90 Model T T Ti,Ti+1 Because f (xt, t) is a smooth function of T, Vt is efficiently eval- uated by Gaussian quadrature. 80 T The initial variance curve ] 0 is a basic underlier in our model, 70 along with S. It would be natural to generate ]T from the VS vola- 0 % tilities derived from the S&P 500 vanilla smile. For each Vix 60

future expiring at Ti, we then need to calibrate Li, Gi, _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- N 30 ances. The settlement value of the Vix future of maturity Ti is the 20% 25% 30% 35% VS volatility for maturity Ti+1:

T ,T An example of calibration. i = i i +1 N We show results of a calibration FT VT i i done on March 18, 2008, for Vix futures and options of maturi- i Using the Vix future Ft, 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 Fi , in particular its options dies off quickly after the first two or three listed maturi- Ti square Fi2. The consistency condition relating to the S&P 500 ties. Also, strikes that could be considered liquid, given the mar- Ti forward variance VTi,Ti+1 and the Vix market is thus: ket conditions in March 2008, ranged from 20–35%. Calibration

i of triplets (Li, Gi, _i) is done by least-squares minimisation. Figure T ,T i2 F i  i V i i 1  F  2 P t, F dK  2 C t, F dK (13) 1 shows Vix futures and figure 2 Vix implied volatilities. t t 0 K t F i K t In figure 2, the curves are stacked with i where PK(CK) is the undiscounted price of a put (call) option on F 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

Ti,Ti+1 met. Frequently, values of Vt derived through replication lie 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 S, V, k1, k2 and W. the Vix futures are too high, typically 0.5 of a volatility point. Table B shows the values of L, G and _ generated by calibration.

This is not always easily arbitrageable, as: one incurs bid/offer Parameters S, V, k1, k2 and W control the correlation structure costs in trading variance swaps and Vix options; as is well known and volatilities of forward volatilities. Multiplying S 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 S 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 G vanishes. options decays quickly with their maturity. ]T is then a shifted lognormal. In what follows, for the purpose of calibrating Vix futures and N 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

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A. Values for S, V, k1, k2 and W used in calibrating Vix smiles C. The three sets of parameters used to generate curves S 130.0% in figure 3

V 28% Set 1 Set 2 Set 3 S 130% 137% 125% k1 8.0 V 28% 29% 32% k2 0.35

W 0% k1 8.0 12.0 4.5

k2 0.35 0.30 0.60 B. Values of L, G and _ calibrated from Vix smiles W 0% 90% –70%

Li Gi _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% 3 Term structure of the instantaneous volatility Six months realised in six months 3.04% 2.89% 3.25% of volatility generatedbythree different sets of Six months in six months swaption 2.28% 2.10% 2.57% parameters S, V, k , k , W 1 2 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 (L = 0, _ = 1), use values for 100 Set 2 Set 3 S, V, k1, k2 and W in table A and consider the instantaneous volatil- ity of a VS volatility as a function of its maturity, assuming the 80 T initial term structure of VS volatilities is flat (]0 = ]0T). The instantaneous volatility of volatility Xvol of ™V 0,T is then given by: % 60 T 2 2  k1T  k2T 40 vol 2 1  e 2 1  e T       1     k T  k T  20 1 2 1 (15) k T k T 2 0 1  e 1 1  e 2 0 6 12 18 24 2 1       k1T  k2T   vol XT 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 forT is one month. Table C shows the corresponding

for realised variance. values of S, V, k1, k2 and W. We have chosen three values for W, N 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. Table D shows prices for at-the-money options on realised vari- 1   2  K ance for maturities of six months and one year, for an initial flat 2 ˆ rT  0,T term structure of VS volatilities equal to 20%. In addition, table ^ where X0, T is the VS volatility for maturity T and XrT 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 ˆ 2 forward VS volatility). t rt  T  t  t,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 Xrt is the annualised realised volatility up to t and Xt,Tït 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 ^ Xt,Tït is hedged by trading variance swaps of maturity T. This also spot-starting options on realised variance, it does not uniquely 2 hedges the variation of Xrt. The dynamics of St then depends on determine volatilities of forward volatilities. These depend on the ^ ^ that of Xt,Tït . One only needs to model the volatility of Xt,Tït. correlation structure of forward volatilities. As an illustration, we Ƌ 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- V i,i1  tribution from the kurtosis of daily returns. This contribution t 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.

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bergomi.indd 94 1/10/08 11:51:48 4 Instantaneous correlation at t =0offorwardone- 5 Implied smile for realised variance generatedby month variances with spot-starting one-month variance calibration to Vix futures and options 100 100 Set 1 80 Set 2 80 Set 3 60 60 %

40 % 40 20 20 0

–20 0 01234567891011 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 S higher in set 3, explaining why prices of options on forward real-  T d ln K 6 T ised variance in table D increase when going from set 2 (W = 90%) F through set 1 (W = 0%) to set 3 (W ï). 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 x  d which would presumably require a larger number of driving fac- T 2 T t f x,t  e  tors. Note that one-factor stochastic volatility models impose 100%  instantanenous correlation between all forward volatilities. In line with our parameterisation, we write \ = 2S, where S 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 S 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 (16) lying realises until maturity the variance that was observed at the d ln K F forward date. k T k T The smile of realised variance. k T e 1 k T e 2 N Just as it is natural to incorpo- 1  1   2  1    1  SX  SY  rate information about the smiles of the basket’s underlyings in a  k 2T 2 k 2T 2  basket option’s price, it is also natural to use information embed-  1 2 ded in Vix smiles to price options on realised variance on the N 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 )q 0, N = T/) in the definition of ward variances, with part of their value coming from the kurtosis _(x, n). of daily returns. N In the limit T qh, 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, N For T q 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 of the volatility: K/ . Prices for  ™ X0,T 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 q 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 q 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 N The vanilla smile. The dynamics of the variance curve not  d ln K 4dt SVV only affects the pricing of options on realised or implied variance S 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:

risk.net 95

bergomi.indd 95 1/10/08 11:51:49 CUTTING EDGE. EQUITY DERIVATIVES

relations W and W , once values for parameters S, V, k , k and 6 The 95–105% at-the-money-forward skewasa SX SY 1 2 W have been chosen. WSX and WSY 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 k and k Approximate 1 2 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.

1 Conclusion In this article, we propose a model driven by easy-to-simulate 0 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- 0.0 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 –0.6 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: WSX = the carry levels for all gammas (be they spot, volatility or cross ï, WSY ï H ï . 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. N 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: [email protected] dˆ 6 T d ln K F References 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 d ln K Available at http://faculty.baruch.cuny. and option values from an implied 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 decay like 1/™T, which is typical of equity market smiles. Bergomi L, 2005 Le marché financier français au 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-

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