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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 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 n (1) 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 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 t 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 t=0 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, 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 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 1 For a precise deﬁnition of the Vix, see www.cboe.com/micro/vix/vixwhite.pdf are deﬁned as dW XdW Y = dt, dWdW X = SXdt, dWdW Y = 90 Risk October 2008 SY 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: t 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: T 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 2 fT T t 2 fT 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[: i 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 i 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 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 x 2 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, 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 by writing St = f(Wt, t) as the solution of equation (6) is fully 4 Calculated using monthly data provided in Gallais-Hamonno (2007) risk.net 91 CUTTING EDGE. EQUITY DERIVATIVES i 2 h t ,T 2 2 h t ,T one if FT is below a barrier l. This is also the price of the same i 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 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 ) T ; let us write it as: and where N is the cumulative density for xT , a centred Gaussian i i 2 2 random variable with variance (1 – ) E[X T ] + 2 E[Y 2 ] + 2 (1 – Ti i T 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 L 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 SdW 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- T 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 i 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 fi 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- T 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 i2 monthly variance is known, as it is given by F T . However, we do not i 28 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. i 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 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 xtT 1 Vt 1 0 f ,t dT 100 Ti Ti Ti Market 1 90 Model T T Ti,Ti+1 Because f (x , t) is a smooth function of T, V t 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- % 0 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 i to replicate any European-style option on FT , in particular its i options dies oﬀ quickly after the ﬁrst two or three listed maturi- i2 square FT . The consistency condition relating to the S&P 500 i 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 Fi 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. T 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 risk.net 93 CUTTING EDGE. EQUITY DERIVATIVES 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 80 initial term structure of VS volatilities is ﬂat ( T = 0 T). The 0 vol instantaneous volatility of volatility T of V 0,T is then given by: % 60 2 2 40 vol 2 1 e k1T 2 1 e k2T T 1 k1T k2T 20 1 (15) 1 e k1T 1 e k2T 2 0 0 6 12 18 24 2 1 k1T k2T vol T 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- 2 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 rt 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 60 60 40 % % 40 20 20 0 –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 (16) 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: risk.net 95 CUTTING EDGE. EQUITY DERIVATIVES 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 Approximate 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: 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: lorenzo.bergomi@sgcib.com 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 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