Smile dynamics -- a theory of the implied leverage effect
aa r X i v : . [ q -f i n . P R ] S e p Smile dynamics – a theory of the implied leverage effect
Stefano Ciliberti, Jean-Philippe Bouchaud, Marc Potters
Science & Finance, Capital Fund Management,6 Bd Haussmann, 75009 Paris, France (Dated: December 2, 2008)
Abstract
We study in details the skew of stock option smiles, which is induced by the so-called leverageeffect on the underlying – i.e. the correlation between past returns and future square returns.This naturally explains the anomalous dependence of the skew as a function of maturity of theoption. The market cap dependence of the leverage effect is analyzed using a one-factor model.We show how this leverage correlation gives rise to a non-trivial smile dynamics, which turns outto be intermediate between the “sticky strike” and the “sticky delta” rules. Finally, we compareour result with stock option data, and find that option markets overestimate the leverage effect bya large factor, in particular for long dated options. . INTRODUCTION It is by now well known that the Black-Scholes model is a gross oversimplification of realprice changes. Non Gaussian effects have to be factored in order to explain (or possiblyto predict) the shape of the so-called volatility smile, i.e. the dependence of the impliedvolatility of options on the strike price and maturity. Market prices of options, once convertedinto an effective Black-Scholes volatility, indeed lead to volatility surfaces – the impliedvolatility depends both on the strike price K and the maturity T . But matters are evenmore complicated since the whole surface is itself time dependent: the implied volatilityassociated to a given strike and maturity changes from one day to the next (see e.g. [1]).An adequate model for the dynamics of the volatility surface is crucial for volatility riskmanagement and for market making, for instance. Market practice on this issue followssimple rules of thumb [2], such as the “sticky strike” rule where the volatility of a givenstrike and maturity remains constant as time evolves. Another rule is “sticky delta” (or“sticky moneyness”) where the volatility associated to a given moneyness stays constant,which means, to a first approximation, that the volatility smile is anchored to the underlyingasset price.From a theoretical point of view, different routes have been suggested over the years tohandle smile dynamics. One is to rely on local volatility models, popularized by Dupire [3]and Derman and Kani [4], where the underlying is assumed to follow a (geometric) Brownianmotion with a local value of the volatility that depends deterministically on the price leveland time. As shown by Dupire, it is always possible to choose this dependence such thatthe smile surface is fitted exactly. This has been considered by many to be a very desirablefeature, and such an approach has had considerable success among quants. Unfortunately,this idea suffers from lethal drawbacks. For one thing, this approach predicts that when theprice of the underlying asset decreases, the smile shifts to higher prices and vice versa (seethe detailed discussion of this point in [5]). This is completely opposite to what is observedin reality, where asset prices and market smiles tend to move in the same direction. From amore fundamental point of view, local volatility models cannot possibly represent a plausibledynamics for the underlying. The whole approach is, in our mind, a victory of the “fit only”approach to derivative pricing and the demise of theory , in the sense of a true, first principleunderstanding of option prices in terms of realistic models of asset prices. Contrarily to2hat many quants believe, even a perfect fit of the smile is not necessarily a good model ofthe smile.A more promising path starts from realistic models for the true dynamics of the under-lying. Many models have been proposed in order to account for the non-Gaussian nature ofprice changes: jumps and L´evy processes [6], GARCH and stochastic volatility models [7, 8],multiscale multifractal models [9, 10], mixed jumps/stochastic vol models, etc. One partic-ularly elaborate model of this kind is the so-called SABR model, where the log-volatilityfollows a random walk correlated with the price itself in order to capture the leverage effect,i.e. the rise of volatility when prices fall (and vice-versa, see below). Interestingly, all thesemodels predict a certain term structure for the cumulants (skewness, kurtosis) of the returndistribution over different time scales, which in turn allows one to calculate the parabolicshape of the smile for near-the-money options of different maturities T [5, 11, 12, 13]. Forexample, models where returns are independent, identically distributed – IID – variables(like L´evy processes), the skewness is predicted to decay like T − / and the kurtosis as T − .As we show below, the leverage effect leads to a much richer term structure of the skewness,whereas long-ranged volatility clustering leads to a non trivial term structure of the kurtosis[13].The aim of this paper is to explore the dynamics of the smile around the money, withinthe lowest order approximation that only retains the skewness effect. We show that suchan approximation leads to an explicit prediction for dynamics of the smile, in particular ofthe implied leverage effect, i.e., the correlation between returns and at the money impliedvolatilities. Our result only depends on the historical leverage correlation, and predicts avolatility shift intermediate between “sticky strike” (for short maturities) and “sticky delta”(for long maturities), in a way that we detail below. We then compare our result with marketoption data on stocks and indices, and find that the market on average overestimates theleverage effect by a rather large factor. II. VOLATILITY SMILE: CUMULANT EXPANSION AND HISTORICALLEVERAGE
Let us first recall the cumulant expansion of the volatility smile, worked out in slightlydifferent form in several papers (see [11, 12, 13] and also [5, 14, 15]). Converted into a Black-3choles volatility Σ, the price of a near-the-money option of maturity T can be generallyexpressed as: Σ( K, T ) = σ " ζ ( T )6 M + κ ( T )24 ( M −
1) + O ( M ) (1)where K is the strike, S the price of the underlying, σ the true volatility of the stock and M = ( Ke − rT /S − /σ √ T is the moneyness . ζ ( T ) and κ ( T ) are respectively the skewnessand the kurtosis of the forward looking, un-drifted probability of price changes over lag T [13]. In the above formula we neglect various terms that are usually small (for example, aterm in ζ M that is small compared to the kurtosis contribution). In the following we willin fact discard the quadratic contribution and study the assymmetry of the volatility smilefor options in the immediate vicinity of the money, where the smile is entirely described bythe volatility and the skewness. Although both these quantities should be interpreted asforward looking, it proves very useful to see what a purely historical approach has to say.The unconditional historical skewness can be written in full generality as [13]:[18] ζ ( T ) = ζ √ T + 3 √ T T X t =1 (cid:18) − tT (cid:19) g L ( t ) , (2)where ζ is the skewness of daily returns and g L ( t ) is the leverage correlation function ofdaily returns g L ( t ) = h r i r i + t i c /σ , which was studied in, e.g. [16, 17]. For IID returns, g L ( t ) ≡ T − / . Negative price-volatility correlations produceanomalous skewness, that can even grow with maturity before decaying to zero. An exampleof this is shown in Fig. 1 for a collection of international indices. A good fit of g L ( t ) can beobtained with a pure exponential: g L ( t ) = − A exp( − t/t L ), as also shown in Fig. 1 for theOEX. This functional form for the leverage correlation leads to the following explicit shapefor the skewness: ζ ( T ) = ζ √ T − AT / (cid:16) T t L − t L (1 − exp( − T /t L )) (cid:17) . (3)It is easy to check that the leverage induced term first increases as T / , reaches a maximumfor T ≈ t L and decays back to zero as T − / for large maturities.It is also interesting to measure the historical skewness of individual stocks, which ismuch less pronounced than the index leverage [16] – see Figs. 2-a,b,c for small, mid andlarge caps, and Fig. 2-d for a direct comparison between different market caps. Here againa purely exponential fit is acceptable, with however parameters A and t L that depend on4 sk e w ne ss T [ work days ] OEXtheoretical fit to OEXNASDAQMIDNIKKEIFTSE
FIG. 1: Historical skewness of some of the major indexes as a function of the horizon T , computedfrom Eq. 2. The data used refer to the period 1990-2006. The SPX skewness (not shown) is nearlyindistinguishable from that of the OEX. The actual shape of this curve is however found to varyquite significantly with the time period. We also show the fit for the OEX with Eq. 3, leading to A = 0 .
16 and t L = 31 days. the stock, mostly through market capitalisation M . For the period 2001 – 2006, we findthat t L ∼
12 days across all M s, whereas A increases by a factor 2 to 3 between M = 5 10 and M = 5 10 $. A possible intuitive explanation for this increase of A is that the influenceof the market mode is stronger on large cap stocks than on small cap stocks, for whichthe idiosyncratic contribution is larger. In this case, it would indeed be expected that theleverage of large cap stocks is more akin to an index leverage effect.In order to better understand this phenomenon, we postulate a simple one-factor modelfor the returns of a given stock: r t = β Φ t + ε t , where Φ is the market return (which weapproximate by the S&P500), and ε t is the idiosyncratic term, uncorrelated with Φ t . Thetotal volatility σ is decomposed into σ = β σ + σ ε ; typically the second term is two tothree times larger than the first one. The total skewness of a stock can be also be decomposedinto three different contributions: market-induced vol on the market mode, market-inducedvol on the idiosyncratic part and idiosyncratic-induced vol on itself. More precisely, one can5 sk e w ne ss T [ work days ] stk -> stkres -> residx -> residx -> idx -1.6-1.4-1.2-1-0.8-0.6-0.4-0.2 0 0 50 100 150 200 250 sk e w ne ss T [ work days ] stk -> stkres -> residx -> residx -> idx-1.6-1.4-1.2-1-0.8-0.6-0.4-0.2 0 0 50 100 150 200 250 sk e w ne ss T [ work days ] stk -> stkres -> residx -> residx -> idx -1.6-1.4-1.2-1-0.8-0.6-0.4-0.2 0 0 50 100 150 200 250 sk e w ne ss T [ work days ] small cap mid caplarge capSP 500
FIG. 2: Historical skewness for individual stocks (symbols with error bars) and decompositionusing a one-factor model, Eq. (4). Top left, top right, and bottom left panels refer to (resp)small, mid, and large cap US stocks, in the period 2002-2008. In each panel, we show the threecontributions appearing in Eq. 4, together with the total stock skewness. The notable feature isthat the index/idiosyncratic contribution ζ Φ → ε ( T ) increases with market cap. The bottom rightcompares the skewness of different market capitalisations and the index skewness. write:[19] ζ ( T ) = ζ ε → ε ( T ) + βσ Φ σ ! ζ Φ → ε ( T ) + βσ Φ σ ! ζ Φ → Φ ( T ) , (4)with obvious notations. Note that the market-induced vol contributions are weighted bythe ratio of the market factor vol to the total volatility, which decreases for smaller capstocks. The three ζ ’s are all of the same order of magnitude, with ζ Φ → Φ ( T ) a factor 2-3larger than the other two. Interestingly the other two contributions cross as a function of T : the idiosyncratic-induced vol on itself is dominant at small T , whereas the market- andtheir maturity dependence is shown in Fig. 2 for small, mid and large cap stocks, togetherwith the average total leverage effect. We have checked that the weighted contribution ofthe three terms add up the total effect, as should be. In Fig. 3 we also show the ratio βσ Φ /σ as a function of the market cap M . Although this ratio indeed increases with M , another6 be t a * m a r k e t v o l / s t o ck v o l log ( market cap [M$] ) FIG. 3: Ratio βσ Φ /σ as a function of market cap. When raised to the power three as in Eq. 4, themarket leverage effect contributes to roughly a fourth of the total skewness ζ ( T ). effect that explains the increase of the leverage effect with market cap is the growth of theindex/idiosyncratic leverage effect ζ Φ → ε ( T ) (see Fig. 2). III. THE IMPLIED LEVERAGE EFFECT
We now turn to the dynamics of the smile, in particular of the implied leverage effectthat measures how the at-the-money (ATM) implied volatility is correlated with the stockreturn. For a given day t , the implied vol reads, to first order in moneyness:Σ t ( M , T ) ≈ σ t ( T ) " ζ ( T )6 M (5)where σ t ( T ) = E [ 1 T Z t + Tt r t ′ dt ′ ] (6)is the expected average squared volatility between now and maturity, and is equal, withinthis approximation, to the ATM vol:Σ t ( M = 0 , T ) = σ t ( T ) (7)Now, between t and t + 1, the price evolves as S t → S t + r t S t . How is the smile expectedto react? There are two simple rules of thumb commonly used in the market place:7 Sticky Strike (ss):
The implied volatility of an option is only a function of thestrike, but does not depend on the price of the underlying (at least locally). Formally,Σ t +1 ( K | S t +1 ) = Σ t ( K | S t ). From the above general formula, the change of volatilityshould be proportional to: ∂ Σ t ∂S t = ∂σ t ( T ) ∂S t " ζ ( T )6 M + σ t ( T ) ζ ( T )6 ∂ M ∂S t (8)Setting this derivative to zero and focusing to the ATM vol ( M = 0), one deduces: ∂ Σ t (0 , T ) | ss ≈ ζ ( T )6 S √ T ∂S (9) • Sticky Delta (s ∆ ): In this case, the smile is assumed to move with the underlying,so that the implied volatility of a given moneyness does not change. In particular, theATM vol does not change: ∂ Σ t (0 , T ) | s ∆ = 0 (10)A purely historical theory of the implied volatility leads to a prediction that is in betweenthe above rules of thumb. One starts from the above implied vol expansion Eq. (5) andconsider the impact of the change in the price both on the expected future realized vol andon the moneyness. More precisely, the ATM vol evolves as: ∂ ln Σ t (0 , T ) = ∂ ln σ t ( T ) − ζ ( T )6 σ t √ T ∂ ln S. (11)From the definition of the leverage correlation function, and neglecting higher order (kur-tosis) correlations, the expected relative change of the future realized volatility is givenby: δσ t ( T ) = " T Z T dug L ( u ) r t (12)Collecting all contributions, one finds that the change of ATM vol for a given stock return r and a given maturity T reads (we drop the t dependence): δ Σ(0 , T )Σ(0 , T ) = 12 σ ( T ) T " T Z T du u g L ( u ) + ζ r ≡ γ ( T ) r, (13)where we have defined the implied leverage coefficient γ ( T ). We assume, as above, anexponential behaviour for the leverage correlation function, g L ( u ) = − A exp( − t/t L ) and8eglect the ζ contribution (see below). We then find that the theoretical implied leverageis given by: γ ( T ) | th = − α − (1 + ˜ T ) e − ˜ T ˜ T ; ˜ T ≡ T /t L (14)with α ≡ A/ σ (0). This is the central result of this study. It should be compared to onesobtained from the sticky strike/sticky delta prescriptions: γ ( T ) | ss ≈ − α T − (1 − e − ˜ T )˜ T ; γ ( T ) | s ∆ = 0 . (15)The asymptotic behaviour of these quantities can be compared, one finds γ ( T → | th = γ ( T → | ss = − A/ σ (0) (in agreement with the general result of Durrlemann [15]) and | γ ( T → ∞ ) | th | ∝ T − ≪ | γ ( T → ∞ ) | ss | ∝ T − . In other words, the sticky strike alwaysoverestimates the implied leverage, but becomes exact for short maturities compared to theleverage correlation time: T ≪ t L . The sticky delta procedure, on the other hand, alwaysunderestimates the true implied leverage, but becomes a better approximation than thesticky strike for large maturities. IV. COMPARISON WITH EMPIRICAL DATA: INDICES AND INDIVIDUALSTOCKS
These predictions are compared with empirical data on the implied leverage effect on theOEX index, large cap, mid cap and small cap stocks, in Figs. 4-7. On these plots we showthe implied leverage coefficient γ ( T ) as a function of maturity. The three curves correspondto (a) the theoretical prediction γ ( T ) | th computed using the historically determined leveragecorrelation g L ( t ), (b) the sticky strike procedure γ ( T ) | ss using the same historical param-eters and (c) the data obtained using from the daily change of ATM implied volatilities, γ ( T ) | imp . We also show the γ = 0 line corresponding to sticky delta. The implied data isobtained by regressing the relative daily change of ATM implied vols on the correspondingstock or index return, for each maturity. The result is then averaged over all stocks withina given tranche of market capitalisation. Similarly, the coefficient α needed to compute thetheoretical prediction is obtained by fitting the leverage correlation for each stock individu-ally, normalizing it by the realized volatility over the same period, and then averaging theratio A/σ t across different stocks. It turns out that A itself is to a good approximationproportional to σ t anyway [16]. 9t is clear from these plots that on average, implied volatilities overreact to changesof prices compared to the prediction calibrated on the historical leverage effect, exceptmaybe for small cap stocks where the level of γ is in the right range at short maturities.The overestimation tends to grow with maturity, since the theoretical prediction is that γ ( T → ∞ ) | th ∼ T − whereas the implied value γ ( T ) | imp appears to saturate at large T . Infact, the γ ( T ) | imp curve appears to be well fitted by a sticky-strike prediction γ ( T ) | ss , butwith an effective value of the parameter A substantially larger than its historical value. Thiswould be compatible with the fact that market makers use a simple sticky strike procedure,but with a smile that is significantly more skewed than justified by historical data. Weonly have partial evidence that the implied skew is indeed too large, but cannot check thisdirectly with the data at our disposal at present. However, we believe that this is a veryplausible explanation to our findings. We have furthermore checked that our conclusions arestable over different time periods.In the above theory for γ , we have explicitely neglected the one-day skewness ζ . Couldthis term explain the above discrepancy? We have checked that this term gives a smallcontribution to γ – at most 10% for short maturities. In fact, the daily skewness of individualstocks is even slightly positive, which should lead to a further (small) reduction of the impliedskew and of the implied leverage effect.Finally, we have studied more systematically the dependence of γ ( T ) | imp on market cap-italization M . The results are shown in Fig. 8. As expected (and already clear from Figs.4-6), γ ( T ) | imp increases (in absolute value) with M ; a good fit of the dependence is: γ ( T ) | imp ≈ a ( T ) + b ( T ) ln M (16)where a and b are maturity dependent coefficients. V. CONCLUSION
In this paper, we have provided a first principle theory for the implied volatility skewand at-the-money implied leverage effect in terms of the historical leverage correlation, i.e.the lagged correlation between past returns and future squared returns. We have comparedthe theoretical prediction for the implied leverage to two well known rules of thumb tomanage the smile dynamics: sticky strike or sticky delta. The sticky strike is exact for small10aturities, but is an upper bound to the theoretical result otherwise. The sticky delta isa (trivial) lower bound, which however becomes more accurate than sticky strike at largematurities.We have then compared these theoretical results to data coming from option markets. Wefind that the implied volatility strongly over-reacts, on average, to change of prices, especiallyfor long dated options. A plausible interpretation is that market makers tend to use a stickystrike rule, with an exaggerated skewness of the volatility smile. It would be interesting totest this hypothesis directly, with full option smile data and not only at-the-money vols likein this study.We have also provided an empirical study of the market cap dependence of these effects.We find that both the historical and implied leverage effect is stronger for larger cap stocks,with a roughly logarithmic dependence on market cap. Although this is partly explainedin terms of the ratio of the idiosyncratic vol to the total vol (smaller for larger cap stocks),we find that the index/idiosyncratic leverage effect is stronger for these large caps. Thismay relate to the risk aversion interpretation of the leverage effect put forth in [16]: sincelarge cap stocks are followed by more market participants, feedback effects could be strongerthere. [1] Cont R., da Fonseca J.,
Dynamics of implied volatility surfaces , Quantitative Finance, ,45, (2002); Cont R., Durrelmann V., da Fonseca J., Stochastic Models of Implied VolatilitySurfaces , Economic Notes, , (2002)[2] For a nice review on these rules of thumb, see: T. Daglish, J. Hull, W. Suo, Volatility surfaces:theory, rules of thumb and empirical evidence , Quantitative Finance, , 507, (2007).[3] Dupire B., Pricing with a Smile , Risk Magazine, , 18-20 (1994).[4] Derman, E., Kani I, Riding on a Smile
Risk Magazine, , 32-39 (1994); Derman E., Kani I.,Zou J. Z., The Local Volatility Surface: Unlocking the Information in Index Options Prices
Financial Analysts Journal, (July-Aug 1996), pp. 25-36.[5] Hagan P., Kumar D., Lesniewski A., and Woodward D.,
Managing smile risk , Wilmott mag-azine, 84-108, September 2002.[6] Cont R., Tankov P. : Financial Modelling with Jump Processes, Chapman & Hall / CRC γ T [ w days ] theoretical valuesticky strikesticky deltaoption market value
FIG. 4: Comparison between the implied leverage coefficient γ ( T ) | imp and the various theoreticalpredictions, sticky strike γ ( T ) | ss , striky delta γ ( T ) | s ∆ = 0, and historical γ ( T ) | th , for large capUS stocks in the period 2004-2008. Note that the implied volatility over-reacts to stock moves, inparticular for large maturities. The effect can be roughly accounted for by the use of a sticky-strikerule with an implied skewness overestimated by ∼
50% at short maturities and ∼ -1.2-1-0.8-0.6-0.4-0.2 0 0 50 100 150 200 250 γ T [ w days ] theoretical valuesticky strikesticky deltaoption market value
FIG. 5: Same as Fig. 4, but for mid-cap US stocks. γ T [ w days ] theoretical valuesticky strikesticky deltaoption market value
FIG. 6: Same as Fig. 4, but for small-cap US stocks. In this case, the implied leverage appearsto have the correct amplitude for small maturities, but the term structure of γ ( T ) is anomalouslyflat here.Press, (2003).[7] Gatheral J. : The Volatility Surface: A Practitioner’s Guide, Wiley Finance (2006).[8] Henry-Labord`ere P. : Analysis, Geometry, and Modeling in Finance: Advanced Methods inOption Pricing, Chapman & Hall / CRC Press, forthcoming.[9] Muzy J.-F., Delour J., Bacry E., Modelling fluctuations of financial time series: from cascadeprocess to stochastic volatility model , Eur. Phys. J. B , 537-548 (2000); Bacry E., Delour J.and Muzy J.-F., Multifractal random walk , Phys. Rev. E , 026103 (2001).[10] Borland L., Bouchaud J.-P., Muzy J.-F., Zumbach G., The dynamics of Financial Mar-kets: Mandelbrot’s multifractal cascades, and beyond , Wilmott Magazine, March 2005. Bor-land L., Bouchaud J.-P.,
On a multi-timescale feedback model for volatility fluctuations ,arXiv/physics.0507073[11] Backus D., Foresi S., Li K. and Wu L.,
Accounting for Biases in Black-Scholes (1997). CRIFWorking Paper series. Paper 30.[12] Potters M., Cont R., Bouchaud J.-P.,
Financial Markets as Adaptive Ecosystems , Europhys.Lett. , 239 (1998).[13] Bouchaud J.-P., Potters M.: Theory of Financial Risk and Derivative Pricing, Cambridge γ T [ w days ] theoretical valuesticky strikesticky deltaoption market value
FIG. 7: Same as Fig. 4, but for the OEX index. Now the implied skewness needed to explain thedata with sticky strike is overestimated by a factor 2 to 3.University Press, (2000 & 2004)[14] Fouque J-P, Papanicolaou G., Sircar R. : Derivatives in Financial Markets with StochasticVolatility, Cambridge University Press, (2000)[15] Durrleman V,
From Implied to Spot Volatilities . PhD thesis, Department of Operations Re-search & Financial Engineering, Princeton University, 2004.[16] Bouchaud J.-P., Matacz A., Potters M.,
The leverage effect in financial markets: retardedvolatility and market panic
Physical Review Letters, , 228701 (2001)[17] Perello J., Masoliver J., Bouchaud J.-P., Multiple time scales in volatility and leverage corre-lations: a stochastic volatility model , Appl. Math. Fin. , 1 (2004).[18] In fact, this formula assumes that the three-point return cumulant, h r i r j r k i c , is zero when i = j = k . We have checked empirically that this term is small, even summed over noncoinciding times.[19] There is in principle a fourth contribution describing the effect of the idiosyncratic part of thereturn of future market volatility, but is is found to be extremely small, as expected intuitively. γ log ( market cap [M$] ) T = 1 monthT = 1 year FIG. 8: Scatter plot of the implied leverage coefficient γ ( T ) | imp as a function of the log market cap,and linear regression for T = 1 months and T = 1 year. The linear regressions are, respectively: γ ( T ) | imp = 1 . − .
60 log M and γ ( T ) | imp = 0 . − .
25 log M . γ T [ w days ]OEX index : 1996-2000computed from historical datamarket value for atm optionssticky strikesticky delta6-5-4-3-2-1 0 1 0 50 100 150 200 250 300 350 400 γ T [ w days ]OEX index : 2000-2004computed from historical datamarket value for atm optionssticky strikesticky delta6-5-4-3-2-1 0 1 0 50 100 150 200 250 300 350 400 γ T [ w days ]OEX index : 2004-2008computed from historical datamarket value for atm optionssticky strikesticky delta
200 300 400 500 600 700 800 9001996 1998 2000 2002 2004 2006 2008 2010 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 i nde x v a l ue v o l a t ili t y dateOEX indexdateOEX index