THE SLOPE OF THE SMILE, AND THE COMOVEMENT
OF VOLATILITY AND RETURNS
Anthony Neuberger
Warwick Business School University of Warwick Coventry United Kingdom
August 2009
The slope of the implied volatility smile is equal in expectation to the covariation between volatility and future returns. The relationship is essentially model free. More precisely, the slope, suitably defined, is the price of a swap contract where the floating leg is the covariation between returns on the underlying asset and changes in its implied volatility. The term structure of the slope also carries information about the correlation between returns and future volatility. The results are analogous to the finding that the model free implied variance is equal to the expected realized variance under the pricing measure.
I am grateful to the Bank of England for the data, and to Peter Carr, John Crosby, Ciara Rogerson, Mark Salmon, Paul Schneider, Pierre de Vos, and to participants at the MathFinance Conference in Frankfurt and a seminar at Warwick Business School for their helpful comments. The skew observed in the implied volatility of options on equity market indices has been much discussed in the literature. The contribution of this paper is to relate the skew to expectations about the correlation between market returns and changes to implied volatility, in a way that is essentially model-free. The argument starts from an appropriate measure of the slope of the implied volatility surface with respect to the strike price of the option. This measure is shown to be the fair price of a swap contract where the floating leg is the covariation between returns on the underlying asset and changes in its volatility.
The relationship between slope and future covariation is model free in the sense that it depends only on the forward price of the underlying asset following a positive diffusion.
The term structure of the slope carries information about the correlation between returns and future volatility at different horizons. The results are analogous to the finding that the model free implied variance is equal to the expected realized variance under the pricing measure.
The investigation of the dynamics of asset prices and the behavior of risk premia is at the very core of empirical finance. The availability of a rich set of data from the options market has provided powerful help in this research. There is a voluminous literature starting with Black (1976) that uses data from option prices to investigate the dynamics of asset prices across many different markets1. The study by Bates (1996) is typical. It starts with a set of models of asset price dynamics and of risk premia. Each model
1 Some of the more recent literature includes Pan (2002), Jones (2003), Eraker (2004), Dupoyet (2006),
Bakshi, Carr and Wu (2008).
- 1 - produces a valuation formula that is fitted to a data set comprising prices both of the underlying asset and options on that asset. The models are compared in terms of goodness of fit, parameter stability and plausibility and parsimony. Conclusions are drawn.
The approach has proved productive in deepening our understanding of the behavior of option prices, and in particular option prices as expressed through the implied volatility surface. It has strengthened our intuition about the broad features of the dynamics that are needed to account for the shape of the implied volatility surface in both the strike and the maturity dimension, and the dynamics of the surface.
However, the approach has its limitations. The researcher examines the data through the lens of a model, a model that is almost certainly mis-specified and chosen as much for its tractability as its plausibility. This makes the results of tests hard to interpret with confidence. It is not clear for example whether the finding that the jump parameter in some model is statistically significant shows that there are jumps in the data, or that the model is mis-specified and features that are not captured by the model (say jumps in volatility) manifest themselves as jumps in returns.
Similar concerns arise in the use of option prices to investigate risk premia. As Jiang and
Tian (2005) argue, studies of the efficiency of option markets that compare Black-
Scholes implied volatility with realized volatility are not pure tests of market efficiency but joint tests with the Black-Scholes model. In a major methodological advance, they use the model free implied variance (MFIV) in their study. This allows them to compare the data from option prices and the data from asset returns without committing to any specific model of the dynamics of asset returns.
- 2 - The term “model free implied variance” might suggest that the reason that the MFIV is useful is that it is a measure of variance that can be extracted from option prices without the benefit of theory. That is not correct. As Breeden and Litzenberger (1978) showed forty years ago, given a complete set of option prices, the entire risk neutral distribution can be recovered in a way that is model free. There are other model-free measures of implied variance that could be used in place of the MFIV, including the (superficially more natural) variance of returns and the variance of log returns. Bakshi, Kapadia and
Madan (2003) show exactly how to recover the moments of the risk neutral distribution of log returns from European option prices. But knowledge of these moments is not particularly useful in a study of the dynamics of returns without some robust theory to link the two.
The reason that the MFIV is useful is because it is based on just such a robust theory. As demonstrated by Britten-Jones and Neuberger (2000), the MFIV is the initial wealth required to create a portfolio whose terminal value is equal to the realized variance. It is this replication of the variance swap that is substantially model free. The replication strategy does, in fact, depend on some assumptions, notably that the market is frictionless and that the asset price follows a diffusion, but they are rather weak. In short, the MFIV is useful because it is equal to the expected realized variance (under the risk neutral measure) under a very broad category of models.
In this paper, I develop an analogous measure to the MFIV, which I call the slope of the smile. If the implied volatility of options of a particular maturity are plotted as a function of strike, the MFIV can be seen as a measure of the average level of the function; the slope is a measure of the gradient. Both measures can be recovered directly from a - 3 - (complete) set of European option prices. The slope is directly related to the dynamics of the underlying; it is equal to the expected realized covariation (under the risk neutral measure) between returns and volatility shocks. It is a number that can be computed direct from option prices. It is the cost of a portfolio that replicates a covariance swap under rather weak assumptions.
The general link between the slope of the smile and the covariation between volatility shocks and returns has been well known for a long time. For example Heston (1993) explicitly uses the correlation between volatility and returns to fit the shape of the smile and recognizes that the correlation is positively related to the slope of the smile or, equivalently, to the skew of the risk neutral distribution. While the link between the shape of the smile and the dynamics of the price process is easy to explore in the context of a particular model, this paper demonstrates how to do it generically. A specific definition of slope is linked to a specific definition of the covariation.
The theory naturally extends to a theory of the term structure of the slope. The forward slope for some future specified period, say for some period next year, is computed using the prices today of options that expire at the beginning and end of the period. The forward slope is then shown to reflect the covariation between now and the beginning of the period of returns on the underlying and changes to the forward implied variance for that period. Thus the shape of the term structure gives insight into the way in which shocks to returns are correlated with shocks to volatility of returns over different horizons.
The evidence from implied variance suggests that using the right definition matters. In their study of the predictive power of variance premia, Bollerslev, Tauchen and Zhou
- 4 - (2008) emphasize that “the results depend critically on the use of “model-free”, as opposed to standard Black-Scholes, implied variances, and realized variances constructed from high-frequency intraday, as opposed to daily data.”
The slope provides a powerful tool for understanding the “smirk” in equity index contracts, and the related question of the skewness of equity market returns, which has been extensively studied by Campbell and Hentschel (1992), Rubinstein (1994), Bakshi et al. (1997), Dumas et al. (1998), Das and Sundaram (1999), Bates (2000), and Foresi and Wu (2005). More recently researchers have turned their attention to studying the skew in other markets such as foreign exchange (Dupoyet 2006, Carr and Wu 2007,
Bakshi et al 2008), and interest rates (Deuskar et al 2008).
The aim of this paper is primarily methodological. In the first section the relationship between and the covariation of returns and volatility is established. The relationship does rest on some broad assumptions (price continuity, complete set of European option prices, non-stochastic interest rates), and the second section examines how the conclusions are affected when these assumptions are relaxed. The third section presents some descriptive statistics illustrating the behavior of the slope and its term structure in the US equity index market over recent years. The final section concludes.
1. Theory
Definition of Slope
The economy is set in a continuous time framework, characterized by a filtered probability space, with a fixed horizon T. The risk-free interest rate is normalized to zero.
- 5 - There is no arbitrage, so there exists a probability measure Q under which the gains process associated with any admissible trading strategy is a martingale (Harrison and
Krebs, 1979). References to expectations are under the Q measure unless stated otherwise.
There is a risky asset whose price at time t is Xt. Contingent claims on the terminal value
2 of X, XT, are also traded. In particular there is a Log Contract Lt and an Entropy Contract
Et with terminal pay-offs:
LX log ; TT (1) EXXTTT log .
I assume that the two derivative contracts trade continuously, their prices are finite (so in
particular Xt > 0 almost surely), and the vector of security prices XLEt,, t t follows a diffusion process. The effect of relaxing these assumptions is considered in the following section.
In a Black-Scholes world, where X has constant volatility , the price of these two contracts would be:
L log X 1 2 T t , t t 2 (2) 1 2 Et X tlog X t 2 X t T t .
2 The term “entropy” is used because the functional form recalls the definition of entropy in information theory (Shannon 1948). Carr and Lee (2007) use the term similarly.
- 6 - Accordingly, define the (Black-Scholes) implied variance of two contracts as:
l e Et vt 2 log X t L t , and v t 2 log X t , X t (3) 1l 1 e soLt Log X t 2 v t ,and E t X t Log X t 2 v t .
In a Black-Scholes world these two variances would be identical but in our much more general economy, they differ. Define the slope as:
e l st v t v t . (4)
Proposition 1: the two implied variances and the slope over a period are approximately proportional to the variance and third moment of the return over that period:
l Q 2 3 e Q 2 3 Q 1 3 4 vExOxttt tttt ;; vExOx tttt sE 3 xOx t (5)
wherext X T X t X t .
Proof: see Appendix.
The two implied variances both approximate the variance of the risk neutral distribution, while the slope reflects its skewness, though without scaling for the cube of the standard deviation.
Different authors have used different empirical measures of the slope of the smile. Bates
(2000) uses the difference in implied (Black-Scholes) volatilities of two calls with log strikes symmetrically placed around the forward price; Carr and Wu (2007) look at two
- 7 - calls whose Black-Scholes deltas are symmetrically placed around 0.5. The difference between the measures is effectively one of scaling3, but it is scaling that affects the comparison of skew across assets with different volatilities, and across maturities. Our measure of slope, which is motivated by theory, and is independent of any model, also differs in its scaling.
The following proposition brings together some of the key properties of slope; part (b) was proved by Carr and Lee (2007: see their corollary 2.7).
Proposition 2: slope has the following properties:
a) it is increasing in the price of high strike options and decreasing in the price of
low strike options;
b) if the prices of options display put-call symmetry in the sense of Carr, Ellis and
Gupta (1998) (so implied volatilities are symmetric around the forward price),
then the slope is zero;
c) if implied volatility is linear in the log strike, then to first order slope is
proportional to:
I 2 d I 3 T t (6) t dlog k
3 The ratio between the two measures of slope is proportional to logk 1 T t for close to the money options.
- 8 - I where t is the implied volatility at-the-money.
Proof: A European claim G with pay-off g(XT) can be replicated using a portfolio of puts and calls, together with cash and forward contracts:
x Ggx 1 gx' F x gk '' P kdk gk '' C kdk , (7) 0 x with arbitrary x, where P(k)(C(k)) is a put (call) with strike k and maturity T, 1 is cash, and F(x) is a forward contract with strike x (Bakshi and Madan 2000).
With x = Xt, and equating the prices of equivalent portfolios at time t:
Xt 1 1 L log X P k dk C k dk , t t0 2 tX 2 t kt k (8) Xt 1 1 EXt tlog X t Pkdk t Ckdk t , 0 kXt k
where Ct(K) and Pt(K) are the prices of calls and puts at time t. Comparing (8) with (3) gives:
2 Xt k X X k svv e l t Pkdk t Ckdk (9) t t t 0 2 tX 2 t t Xt k k
Part (a) follows immediately.
Carr, Ellis and Gupta (1998) define put-call symmetry to mean that prices of out-of-the- money puts and calls are related:
CXPXt t t t for all 0. (10)
- 9 - I Put-call symmetry is equivalent to requiring that t k , the Black-Scholes implied volatility of a call or put option with strike k and maturity T at time t, is symmetrical about the forward price:
II t XX t t t . (11)
Putting (10) into (9) gives st = 0.
Part (c) is proved in the Appendix. ■
The Slope and the Covariance Swap
The Log and Entropy Contracts provide a link between option prices and the dynamics of the asset price process. The implied variance of the Log Contract is not only related to the variance of the total return of the asset over the life of the option (Proposition 1); more significantly it is related to the realized volatility of the asset over the period (Dupire,
1994 and Neuberger, 1994). Using a similar argument, I show that that the slope is related to the realized covariation between returns and volatility. To introduce the argument and notation, I first reproduce the standard result that the implied variance of the Log Contract is also the fair price of a variance swap.
Proposition 3: the implied variance of the Log Contract is the fair price of a variance swap that pays the realized variance of the asset from now till the horizon date.
Equivalently, the implied variance is equal to the expectation of the realized variance
(under the measure Q).
- 10 - Proof: since prices follow a diffusion, apply Ito’s Lemma to equation (3). Dropping time subscripts for clarity:
2 dX1 dX 1 l dL dv , XX2 2 2 (12) dX1 dX 1 1 dX dE X1 ln X 1 ve dv e dv e . 2 XXX2 2 2
4 Consider the strategy of going short two log contracts and long 2/Xt of the underlying .
The wealth process is:
dX dW 2 dL 2 X 2 (13) dX l dv . X
Start with zero wealth at time t, and use P for pay-off and V for variance. The strategy replicates the pay-off to a variance swap PV:
2 TT V dX l P dW vt , (14) t t X where the agent receives the realized variance (or squared volatility times time) and pays fixed. This gives:
2 T dX vl E Q , (15) t t t X
4 The portfolio can be regarded as a delta hedged log contract, scaled to give unit gamma.
- 11 - since the gains process is a martingale under Q. ■
The following Proposition shows how the log and entropy contracts can be used to replicate a covariance swap whose pay-off PC is the difference between a fixed leg, and a floating leg which is the covariation between the returns to the asset and innovations to implied variance.
Proposition 4: the slope is the fair price of a covariance swap that pays the realized covariation of the asset return with innovations to implied variance from now till the horizon date. Equivalently, the slope is equal to the expectation of the realized covariation.
5 Proof: Consider the following strategy : go long 2 log contracts, and 2/Xt entropy
1 e contracts, and short 2 2 lnXt 2 v t / X t of the underlying. The change in wealth is:
2 4 2ln X ve dW 2 dL dE dX XX (16) dX dve dv l dv e . X
Terminal wealth is given by:
5 This portfolio can be described as a gamma and delta hedged Entropy Contract, where the gamma hedging is done using the Log Contract.
- 12 - TTdX dX PC W v l v e u dv e u dv e s ; T t tt u t u t XXu u (17) T dX sos EQu dv e . t tt u X u
■
Proposition 4 interprets the slope of the implied volatility surface in terms of the dynamics of the underlying process. The slope is equal to the expectation of the covariation between returns and shocks to implied variance.
In replicating a variance swap, the holding of the log contract is held constant over the life of the swap, and all trading is in the underlying. By contrast, in the covariance swap, the holding of the entropy contract is continuously rebalanced to keep the log contract gamma hedged. If the entropy contract is not traded but simply delta hedged, and there is no log contract, the portfolio replicates a so-called gamma swap with pay-off
2 T dX u X u (see Lee 2008). t X u
The implied variance at time u is the square of the implied volatility times (T-u). If the correlation between returns and volatility is constant over time, and if volatility shocks are persistent, then the impact of a return shock on variance increases with (T-u), and the integrand in (17) is decreasing in magnitude as u approaches T. It is therefore useful to think in terms of forward slope, where the tenor does not change with time.
- 13 - Making the maturity date T explicit, vt T is the total variance over the period [t, T].
The instantaneous forward volatility and forward slope for maturity T can then be defined as:
i2 d i tT v t T i l,; e dT (18) d sf T s T e2 T l 2 T tdT t t t
Proposition 5: the forward slope for maturity T is the fair price of a forward covariance swap. Equivalently, the forward slope is equal to the expectation of the realized covariation between returns and changes in forward implied (squared) volatility.
Proof: differentiate equation (17) with respect to T (this corresponds to going long 1/dt covariance replication strategies with maturity T+dt, and short the same number with maturity T):
dW T dX PFC T u d e2 T s f T ; t u t dT X u (19) T dX sosf T E Qu d e2 T . t tt u X u
■
Higher Order Contracts
The Log and Entropy contracts were apparently conjured from nowhere. In fact they are
n special cases of power contracts. Define an n-power claim as one that pays X T and
- 14 - n n denote its time t price by Yt . Define the n-power variance as the value of vt that solves the equation:
1 n n n 2 n n1 vt Yt X t e , (20) so v is the Black-Scholes implied variance of the power contract.
Proposition 6: the difference between the n-power and 0-power variance is the fair price of a swap that has a pay-off which reflects both the realized covariation of the asset return with innovations to implied n-power variance, and the variance of the n-power variance.
2 n0 TT ndX n vvnE QQ dv u 1 nnE 1 dv . (21) t t tt u 8 t t u X u
Proof: the argument is the same as in propositions 3 and 4. ■
The cases where n = 0 and 1 are degenerate, and correspond to the Log and Entropy contracts respectively6. Proposition 4 can then be seen as a special case of proposition 6, with n = 1. There is a potentially rich vein of investigation of these higher order contracts.
In particular the difference between the 0, 1 and 2 power contracts contains information about the volatility of volatility, but that goes beyond the scope of the current paper.
n n 6 XXX1 To see this, note that: logXXX lim and log lim . n0 n n 1 n
- 15 - One other observation is pertinent. If the underlying is an exchange rate, it is normal to treat either currency as a numeraire. An n-power contract in $/€, using $ as the numeraire,
n pays $X T . From the perspective of the euro investor, who sees a rate at time t of Zt = 1/Xt
n 1n €/$, the contract pays $ZT or € ZT . The two agents see the same price for the contract, so the n-power variance using the dollar as numeraire is the same as the (1-n)-power variance using the euro as the numeraire. Taking the case where n = 0, this gives:
Proposition 7: in Foreign Exchange markets, the slope is equal to the difference between the MFIV in the foreign currency, and that in the home currency.
The Term Structure of the Slope and its Interpretation
To provide some intuitive interpretation of the slope and of its term structure, assume for the moment that a return shock is associated with a parallel shift in the term structure of volatility, and the relationship is linear. More specifically, assume that:
e2 dXt dX t Et d t T t dt . (22) XXt t
Proposition 5, and in particular equation (19), then imply that:
2 T f Q dX u st T E t t X u (23) l vt T .
- 16 - If is indeed constant, equation (23) implies that forward slope is proportional to variance, and hence to T-t. That means in turn that slope, which is the integral of forward slope, increases with (T-t)2.
But the assumption that is independent of T-t is extreme; in a Heston (1993) stochastic volatility model for example, it holds only if volatility shocks are very long-lived, with negligible mean reversion. If, on the contrary, volatility shocks die out quickly, the in
(22) is zero for T-t > some then the forward slope is constant for T > t+ and slope then increases linearly rather than quadratically with maturity. The term structure of the slope carries useful information about the temporal structure of the relationship between returns and volatility changes.
The slope is the difference between two variances. It does not admit of a simple intuitive interpretation. However equation (23) suggests that the slope can be linked to the beta of implied volatility on returns where the beta is the expected change in annualized volatility per unit return on the asset. Assume that the term structure is reasonably flat so that we
e 2 can approximate vt T by t T t .
Proposition 8: defined the normalized slope for horizon T as:
s T snormalized T t , (24) t 2 3 T t t
- 17 - where t is the implied volatility at time t. If the term structure of volatility is flat, so that
e 2 vt T can be approximated by t T t , the normalized slope can be interpreted as the beta of implied volatility of the asset on its returns.
Proof: see Appendix.
Comparing propositions 2 and 8, it can be seen that the normalized slope is similar to the gradient of implied Black-Scholes volatility with log strike. Also observe from
Proposition 1 that the slope is approximately one third of the third moment of returns, while the denominator of (24) is approximately the square root of the tenor multiplied by the cube of the standard deviation of returns. The normalized slope is therefore approximately equal to the skewness of the probability density function of returns divided by the tenor to the power of three over two.
2. Implementation and Robustness
The key assumptions driving the results are:
prices follow a diffusion process;
the market is open continuously;
log and entropy contracts are traded;
the interest rate is zero.
In this section the implications of relaxing these assumptions are investigated.
- 18 - Jumps
The analysis assumed that the portfolio is rebalanced continuously, and the asset price follows a diffusion. By examining the outcome when the portfolio is rebalanced at defined intervals, we can investigate the sensitivity of the results to both assumptions.
Suppose that the portfolio used in Propositions 4 and 5 is only rebalanced N times over the period, at intervals of length t = T/N. The notation is modified slightly. The covariance swap matures at time T; the forward swap is constructed by going long the
T+t swap and short the T swap. Differences are forward looking, so X t means
e XXt t t . t T is the implied forward volatility of the entropy contract, so
e2 e e tT v t T t v t T /. t xt is used to denote the return on X, so xt X t X t .
Using exactly the same strategies as in propositions 4 and 5, we get:
Proposition 9: by trading the underlying, log and entropy contracts, rebalancing at discrete intervals, the covariance swap PC(T) and the forward covariance swap PFC(T) with the following pay-offs can be synthesized:
NN1 1 C e P T xn t v n t T s0 T f x n t ; n0 n 0 N FC e2 f e 2 PT xn t n t TsTfx 0 T T Txt T ; (25) n0 wheref x 4 2 x log 1 x 4 x 1 x3 O x 4 . 3
- 19 - The terms in square brackets are the discrete counterparts of the pay-offs in propositions
4 and 5; discrete rebalancing induces a disturbance of f xt into the covariance swap.
The magnitude of the disturbance relative to the covariance term is an empirical question.
The presence of the disturbance term shows why the assumption of a diffusion is necessary. If jumps occur, then as the differencing interval goes to zero, proposition 9 does not converge to its diffusion counterpart.
Proposition 10: if, under measure Q, X follows a jump diffusion process with jump rate
t, and jump return distributed with cumulative density Jt, then the forward slope is:
T dX sf T E Qt d e2 T f x dJ x . (26) 0 0 0 t T T X t
The effect of jumps on the forward slope is proportional to the jump frequency and the third moment of the jump return distribution. So, in the presence of jumps, slope reflects not only the correlation between volatility and returns, but also the skewness of the jump component of returns.
Synthesizing the Log and Entropy Contracts
We have assumed that the agent can trade log and entropy contracts. Equation (8) shows that while they can be synthesized from conventional call options, a continuum of strikes ranging from zero to infinity is required. If there are no log or entropy contracts traded, but only European options with a restricted range of strikes, replication of the contracts is imperfect, and so is the replication of the covariance swap. Proposition 11 shows how the covariance swap can be approximated using a limited palette of conventional options. - 20 - Suppose that there are European calls and puts traded with maturity T and strikes k in some finite set K. Suppose that K has N+1 members ordered from k0 to kN. Define the function I(k) as:
1 k k if kK with k 2 k k , k 2 k k ; dI k 2 n1 n 1 1 0 1 N 1 N N 1 (27) 0 otherwise.
Define the approximated implied variances as:
Xt P k C k vl* 2t dI k t dI k ; t 0 2X 2 kt k
Xt P k C k ve* 2t dI k t dI k ; (28) t 0 X t kXt kX t ***e l st v t v t .
The relationship between the slope and the realized covariation continues to hold using approximated quantities:
Proposition 11: There exists a trading strategy with terminal pay-off:
T dX PC*** t dv e s so: 0 t 0 X t (29) T dX s** EQt dv e . 0 0 0 t X t
Proof: see Appendix. ■
Proposition 11 is simple, but it is only useful insofar as the approximated variance is close to the true variance. Whether this the case depends on how rich the range of strikes is. Figure 1 shows the approximated variance when there are strikes every 5 points from - 21 - 70 to 130. The term structure of implied volatility is flat, so the true variances are independent of the asset price. The approximated variance is plotted as a function of the spot price for two maturities – 1 year and 1 month. The true implied variances are 0.05 and 0.0042 respectively, corresponding to an implied volatility of 22% annually.
When the spot is close to 100, and with a maturity of 1 year, the approximated variance is close to the true variance. This reflects the fact that the options traded span a fairly wide range of moneyness (deltas of 0.14 to 0.96). The distance between the strikes leads to minor variations. But moving away from 100, the approximated variance ceases to be close to the true variance. At 70, the approximated variance is about 0.033; the approximation is poor because of the limited range of vanilla options used to compute it – none of the options has a strike below the spot.
With only one month to maturity, the approximation is good over a much wider range, reflecting the relatively wide range of deltas traded. But the gaps between the strikes becomes more important (at a tenth of a year, a 5 point gap corresponds to about 0.7 standard deviations) and the implied variances oscillate around the true variance by around 10%.
The strategy described in the Proposition is not necessarily the best way of approximating the covariance swap. Indeed, it cannot be optimal to stay with a fixed portfolio of options if the spot price moves substantially and if new strikes are introduced. But it is a strategy that is easy to analyze, and provides an upper bound on the amount of damage that this form of market incompleteness can cause for the covariance swap.
- 22 - Zero Interest Rates
The assumption of zero interest rates is innocuous with a single maturity (as in propositions 1 to 4). All trading can be done in the time T forward market. If implied variances are computed using the forward prices of the log and entropy contracts, and if
Xt is interpreted as the forward price to time T, the results go through exactly. However the assumption is not innocuous in the case of a forward covariance swap.
Proposition 12: in the presence of stochastic interest rates, the forward slope for maturity T is the fair price of a forward covariance swap plus a correction term that depends on two factors: the covariation between interest rate shocks and volatility shocks, and the degree to which the covariation between volatility and returns depends on the level of interest rates.
Proof: see Appendix. ■
3. Behavior of the Slope
The contribution of this paper is theoretical. Since it is model free, it is not testable.
However, the paper would be incomplete without giving at least some description of the behavior of the slope. This section documents the behavior of the slope in the equity index option market where the slope is most persistent and pronounced. More specifically, it is based on the implied constant maturity probability density functions
(pdf) computed by the Bank of England from the daily closing prices for options on the
S&P 500 index over the period 1 January 1992 to 16 January 2009 inclusive. The methodology used is described in Clews, Panigirtzoglou and Proudman (2000). The
- 23 - methodology involves fitting a pdf to quoted option prices by plotting implied volatility of options as a function of option delta and fitting using cubic splines, and then interpolating between option expiry dates to produce constant maturity pdf’s.
When the pdf cannot be estimated reliably, because of inconsistencies in option prices, the data are omitted. With much of the liquidity in the options market being in the near term contracts, the fullest data set is for 3 month maturity where we have pdf’s for 94.4% of trading days (97.7% from 2000). By contrast we have the one month pdf data only for
70.1% of days (72.7% post 2000). So we focus primarily on the three month horizon.
We first examine the relationship between the slope, the normalized slope and other related measures that have been used in the literature to measure the asymmetry of the distribution. We focus specifically on three:
1. the difference in Black-Scholes implied volatilities of a put and a call option that
are similarly out-of-the money. Specifically, we look at options where the strike
is set at F exp 0.05 , where F is the forward price;
2. the “risk reversal”, where again the difference between the implied volatilities of
an out of the money put and call is computed, but in this case the options are
chosen to have an equal and opposite delta. The definition of delta used here is
based on the percentiles of the risk neutral density, so the two options have
strikes set at the 25th and 75th percentile of the distribution;
3. the skewness of the probability density function.
- 24 - The five different measures are computed daily over the period 1 January 2000 to 16
January 2009 using the three month density. Table I shows that the normalized slope is
quite highly correlated with the skewness of the pdf (0.92) and the gradient of the smile
when measured with fixed strike (0.86). It is poorly correlated with the risk reversal (-
0.02), but the risk reversal is quite highly correlated with the raw slope (0.89). In
regressing the slope of the implied volatility surface on other variables, the precise
choice of measure therefore makes a material difference even when options with a single
maturity are used.
The time series behavior of implied variance and slope is shown in Figure 2. Implied variance has varied widely over the period. The log and entropy variances have tracked each other closely. This is significant because the covariance swap is based on the covariation of changes in the entropy variance with returns; with the correlation in changes between the two being over 0.999, this suggests that it would make little difference if the swap were based on changes in the log variance instead.
The slope has been negative throughout the period. The correlation between slope and variance is apparent from the time series plot. Figure 3 plots both entropy variance and slope against log variance. The slope – the difference between the log and entropy variance - is clearly strongly related to the log variance, but the relationship is non-linear.
The fitted line is proportional to the log variance raised to the power of 3/2, and thus represents constant normalized slope.
The normalized slope, shown as a function of time in Figure 4, is clearly stationary (the augmented Dickey-Fuller test statistic is -5.4). It has a coefficient of variation of 0.18.
- 25 - The normalized slope averages -0.805, implying that a 1% fall in the market is accompanied by an increase of 0.8% in implied volatility. The normalized slope has never approached zero. Interestingly, the peak in implied volatility in 2008-9 has not been accompanied by a rise in the normalized slope.
The pdf’s are computed using an elaborate algorithm. To give some sense of their reliability it is useful to compare the implied variance of the log contract at 1 month maturity with that computed from the VIX index. The VIX index is designed to measure the implied volatility of a log contract, and so is closely related to our vl, and it is also based on the prices of options that are to expire within the next two months, but the methodology used by the Chicago Board Options Exchange (CBOE) for interpolation and extrapolation from quoted option prices is somewhat different from the Bank of
England’s methodology (see CBOE 2003 for a description). Figure 5, which plots the percentage difference between them, shows that the two measures track each other reasonably closely. More significantly, the daily change in implied variance computed in the two different ways are highly correlated (0.941 over the period as a whole, with the correlation improving from 0.876 in the 1990’s to 0.955 from 2000).
The negative instantaneous correlation between the VIX and the S&P 500 index is well attested in the literature (starting with Fleming, Ostdiek and Whaley 1995). Figure 6 shows the normalized 3 month slope, as in Figure 4, and superimposes a rolling three month estimate of the beta of the VIX on the S&P. The magnitudes are broadly similar.
This is interesting because, as shown in Proposition 8, the normalized slope is the expected beta under the risk neutral measure while the distribution of the realized beta is clearly under the objective measure. The question of whether there are risk premia in the - 26 - pricing of covariance swaps is important, but goes beyond the scope of the present paper.
All we have shown is that the negative correlation observed in practice between changes in the VIX and market returns does explain at least in part the pronounced slope of the implied volatility smile.
Turning to the term structure of the slope, Figure 7 plots the normalized slope at 1 and 6 month horizons against the slope at 3 months. They are highly correlated – the one month slope has a correlation of 0.96 with the 3-month, and 0.75 with the 6-month slopes. So if the beta of volatility on the underlying is high at one maturity it will tend to be high at other maturities. A second observation is that the slope is steeper the shorter the maturity suggesting that forward implied volatility is less sensitive to spot returns than is spot implied volatility. A third, and somewhat more tentative observation, is this diminution effect is more pronounced when the slope is more negative, suggesting possibly that a steeper slope is associated with faster mean reversion in volatility.
4. CONCLUSIONS
The principal purpose of the paper is to show how the slope of the implied volatility smile, when measured in a suitable way, gives important information about the covariation between returns on the underlying and changes to implied volatility. The relationship is model free, and depends only on the continuity of the price path. In the presence of jumps, the slope also reflects the skewness of the jump distribution. The estimation of the slope and the implementation of the corresponding trading strategy depends on the existence of contingent claims (the Log and Entropy Contracts) that can be synthesized from standard put and call options. While exact synthesis requires the
- 27 - existence of a options with all possible strikes, we have shown how the analysis is affected if a limited number of strikes is available.
There are two obvious areas in which the slope could usefully help empirical work on asset pricing. First, the time series and cross-sectional properties of the slope in any particular asset market can be used to understand the dynamics of the underlying asset.
Second, the profitability of covariance swaps provides a clean, model free way of estimating the premium associated with taking covariance risk.
I have not dwelt on the merits of the covariance swap as an instrument for traders.
Covariance swaps would be useful in hedging portfolios against correlated movements in the asset price and its volatility – the so-called vanna or the cross derivative of portfolio value with the price of the underlying and the implied volatility (Wystup 2008). While the analogy with variance swaps is clear, it is important to stress some of the differences that might militate against an active market in covariance swaps. First, covariance is a more complex concept than variance, and it is correspondingly harder to develop an intuition for its behavior. Second, the pay-out to a covariance swap depends on the periodic measurement not only of the asset price (as in a variance swap) but also of its implied volatility. In some markets there is a reliable measure of implied volatility that might be sufficient robust to form the basis of payments in a swap contract; for example in the US stock market, the VIX index could be used even though it does not correspond exactly to the definition of implied variance envisaged in the covariance swap. But in many other markets the measurement issue is likely to be problematic.
- 28 - The analysis of the hedging of the covariance swap also provides a rigorous basis for analyzing dynamic option strategies where traders trade the risk reversal (long an out of the money call, short an out of the money put, or the reverse), rolling their options as the spot price moves to keep the risk exposure of the position constant.
- 29 - Appendix
Proof of Proposition 1
From (3):
l Q e Q X T vt 2 E t log X t log X T ,and v t 2 E t log X T log X t . (A-1) X t
Writing xt for the return (XT-Xt)/Xt:
l Q e Q vt 2 E t log1 x t ,and v t 2 E t 1 x t log1 x t x t log X t . (A-2)
Using a Taylor series expansion, and recalling that the expectation of xt under Q is zero:
vl E Q x2 2 x 3...,and v e E Q x 2 1 x 3 .... (A-3) t t t3 t t t t 3 t
The result then follows.
Proof of Proposition 2(c)
Suppose the implied volatility smile is approximately flat with:
I k 0 log k X , (A-4) for some small which is the natural measure of slope if implied volatility is plotted against log strike, as in Bates (2000). Then our measure of slope is:
- 30 - X 2 k XBS I X k BS I st 2 Pkkdk ;; 2 Ckkdk X 0 kX k
k X 3 2 2 T t Z d log k X dk 0 T t , where 0 kX (A-5) CBS, P BS Black-Scholes price of call/put with strike k and volatility ; logk X Z. standard normal density, and d 1 T t. T t 2
The approximation is a first-order Taylor expansion of the Black-Scholes price with respect to the implied volatility.
Proof of Proposition 8
If the volatility term structure is flat, we can write:
l 2 vt T T t t (A-6)
Then from (23):
f I 2 st T T t t ; (A-7) 1 2 I 2 sost T 2 T t t .
We need the beta of volatility, not of volatility squared, on returns:
dXt dX t Et d t XXt t 1 e2 dXt dX t Et d t T (A-8) 2 t XXt t s T t 2 2 3 t T t t
- 31 - Proof of Proposition 11
Consider the strategy of holding the following portfolio of traded puts and calls, forward contracts and cash at all times t:
2 Xt k X X k tP k dI k t C k dI k 0 2X 2 X kt k t (A-9) e* v t dX tF X u dve** s 1. t 0 u 0 XXt u
The initial value of the portfolio is zero, and its terminal value is:
T dX u dve** s . (A-10) 0 u 0 X u
To prove the proposition, it is sufficient to show that the strategy is also self-financing.
The cost of rolling the options is:
X 2 t dt k Xt dt X t dt k Pt dt k dI k C t dt k dI k 0 2X 2 X kt dt k t dt (A-11) X 2 t k Xt X t k Pt dt k dI k C t dt k dI k 0 2X 2 t Xt k k
Rearranging the terms, this becomes:
X dX t t dt 2 2 Pt dt k dI k C t dt k dI k 0 X t dt Xt kX t dt kX t dt (A-12) X t dt k X t 2 Pt dt k C t dt k dI k X 2 t k X t
- 32 - The term in the first brackets is the approximated variance; the second can be simplified using put-call parity to give:
dX Xt dt k X k X t ve * 2t t dt dI k . (A-13) t dt X 2 t Xt k X t
If X follows a diffusion the second term is zero. The cost of rolling forward the other components of the portfolio is:
e** e v vt dt dX t dtF X t F X u dve*. (A-14) tdttdt tdtt t u XXXt dt t u
Since a forward contract with strike k has value k-Xt at time t, this is equal to:
dXte** dX t e vt dv t . (A-15) XXt t
Putting (A-13) and (A-15) together, the cost of rolling the portfolio is zero.
Proof of Proposition 12
Consider the strategy of going long synthetic covariance swaps with maturity T+t and short swaps with maturity T. Let rt denote the forward interest rate for time T is at time t, so at time t the agent can arrange to borrow 1 at time T and repay 1+ rtt at time T+t.
Let Xt be the forward price with maturity T. Consider the strategy of going short
1t 1 r0 t swaps with maturity T and long 1/t swaps with maturity T+t. Hedge
- 33 - using the forward interest rate market to ensure that all cash flows occur at time T+t.
Terminal wealth measured in time T+t cash is:
1 T t d1 rt t X t vl T t v e T t dv e T t 0 0 0 t t 1 rt t X t (A-16) T 1 l e1 rt t dX t e v0 T v 0 T dvt T 0 t 1 r0 t X t
Taking limits as t goes to zero, terminal wealth is:
TTdX dX td e2 TsT f drrr t dvT e . (A-17) 0t0 0 t t 0 t XXt t
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- 38 - Table I: the correlation between the slope and other measures of the skewness in the implied probability density.
Fixed Risk Normalized Skew X Strike Reversal T Slope
Risk Reversal 0.399
Skew X T 0.654 -0.384
Normalized Slope 0.863 -0.018 0.920
Slope 0.227 0.893 -0.371 - 0.046
The table shows the correlation over the period 1 January 2000-16 January 2009 between five different measures: the “fixed strike” and the “risk reversal” are both computed from the difference in implied volatilities between an out of the call and an out of the money put; in the former case the strikes are 5% above and below the forward price, while in the latter the strikes are set at the 25th and 75th percentile of the probability density function. The skew is the skewness coefficient of the implied probability density function and slope and normalized slope are defined in the paper.
The measures are calculated daily using the constant 3 month maturity pdf computed from the prices of options on the S&P 500.
- 39 - Figure 1: the impact of a limited range of strikes
0.05
0.04 ) * e v (
e 0.03 c n a i r a v
d 0.02 e i l p m
I 1 year 0.01 1 month
0 50 75 100 125 150 Spot price
The chart shows the implied variance as a function of the spot price for two different maturities. All options trade on an implied Black-Scholes volatility of 22%, so the true variance is 0.05 at 1 year and 0.0042 at 1 month. Option with strikes at intervals of 5 from 70 to 130 are traded.
- 40 - Figure 2: The Implied Variance of 3-month Log and Entropy Contracts, and the Slope, for S&P-500 options 1992-2009
0.1 0.01
0.08 0.008
0.06 0.006
0.04 0.004
0.02 0.002 e c e n p a o
i 0 0 l r S a V -0.02 -0.002
-0.04 -0.004 v(log) (lhs) -0.06 -0.006 v(entropy) (lhs) -0.08 s (rhs) -0.008 -0.1 -0.01 92 94 96 98 00 02 04 06 08
The top half of the graph plots the implied variance of the Log and Entropy contracts computed from 3 month options on the S&P 500 index. The two variances are barely distinguishable. Their difference, the slope, is plotted on the bottom half using the right hand scale.
- 41 - Figure 3: The Entropy Variance and Slope plotted as a function of the Log Variance for 3 month contracts on the S&P 5000 1992-2009
0.15 0
t 0.12 -0.003 c a r t n o c
y 0.09 -0.006 p o e r p t v(entropy)(lhs) o n l e S f slope (rhs) o
0.06 -0.009 e c n a i r a
V 0.03 -0.012
0 -0.015 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Variance of log contract
The variance of the Entropy Contract (left hand scale) and the slope (right hand scale) plotted as a function of the variance of the Log Contract, using daily data. The fitted lines are respectively y vl and 3 2 y k vl where vl is the variance of the Log Contract.
- 42 - Figure 4 The normalized slope of 3 month S&P 500 option contracts 1992- 2009
0
-0.2
-0.4 e p o l
S -0.6
d e z i l a -0.8 m r o N -1
-1.2
-1.4 92 94 96 98 00 02 04 06 08
The normalized slope is the slope divided by the cube of the implied volatility and the square of the time to maturity (3 months). The implied volatility used is the implied volatility of the 3 month Log Contract. The normalized slope can be interpreted as the beta of implied volatility on underlying returns.
- 43 - Figure 5 The difference in the estimated the 1 month log variance based on Bank of England pdfs and that based on the VIX index.
40%
30%
20%
10%
0%
-10%
-20%
-30%
-40% 91 93 95 97 99 01 03 05 07 09
The chart plots the ratio of the implied variance of the Log Contract, computed from the Bank of England’s pdfs to the square of the VIX index divided by 12, minus one. The VIX index is based on the new (post 2003) methodology and is downloaded from the CBOE website.
- 44 - Figure 6 The Normalized Slope and the Beta of the VIX on the S&P500 Index
0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 -1.6 -1.8 92 94 96 98 00 02 04 06 08
Beta of VIX Normalized 3 month slope
The Normalized 3 month slope is as in Figure 4. The beta is the slope of a rolling regression of daily changes in the VIX index (divided by 100) on the daily returns on the S&P 500 index computed over the preceding quarter.
- 45 - Figure 7 The normalized slope at 1 and 6 months compared with 3 months, on the S&P 500 index, 1992-2009
0
-0.4
-0.8
-1.2
-1.6 1 month 6 month -2
-2.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 Normalized Slope at 3 months
The 1 and 6 month normalized slope are plotted against the 3 month normalized slope on a daily basis. The slopes are normalized using the implied volatility of the Log Contract with the same horizon as the slope.
- 46 -