Crooked volatility smiles: Evidence from leveraged and inverse ETF options
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Abstract
We find that leverage in exchange traded funds (ETFs) can affect the ‘crookedness’ of volatility smiles. This observation is consistent with the intuition that return shocks are inversely correlated with volatility shocks – resulting in more expensive outofthemoney put options and less expensive outofthemoney call options. We show that the prices of options on leveraged and inverse ETFs can be used to better calibrate models of stochastic volatility. In particular, we study a sextet of leveraged and inverse ETFs based on the S&P 500 index. We show that the Heston model can reproduce the crooked smiles observed in the market price of options on leveraged and inverse leveraged ETFs. We show further that the model predicts a leveragedependent moneyness, consistent with empirical data, at which options on positively and negatively leveraged ETFs (LETF) have the same price. Finally, by analyzing the asymptotic behavior for the implied variances at extreme strikes, we observe an approximate symmetry between pairs of LETF smiles empirically consistent with the predictions of the Heston model.
Keywords
exchange traded funds (ETFs) leveraged ETFs stochastic volatility volatility smiles optionsINTRODUCTION
Empirical motivation
By relating option prices to observables (such as strike prices, timetoexpiration, spot price and so on) and one unknown parameter (implied volatility of the underlying asset return dynamics), Black and Scholes (1973) allowed practitioners to more systematically study observed option prices. Furthermore, this unknown parameter allowed researchers to normalize comparisons between options with different observables.
For all of the success of the BlackScholes model and for its nearubiquity today, there are several shortcomings of the model. For example, Rubinstein (1985) pointed out strike price biases and closetomaturity biases by studying the prices of the most active option on the CBOE.^{1} In subsequent works, Rubinstein (1994) and Jackwerth and Rubinstein (1996) showed that options on stocks and stock options exhibit volatility skew and that foreign currency options exhibit a volatility smile (atthemoney options are cheaper than inthemoney or outofthemoney options).
Sextet of LETFs tracking the S&P 500 index
Fund name  Ticker  ℓ 

ProShares UltraPro Short S&P 500 ETF  SPXU  −3 
ProShares UltraShort S&P 500 ETF  SDS  −2 
ProShares Short S&P 500 ETF  SH  −1 
SPDR S&P 500 ETF  SPY  1 
ProShares Ultra S&P 500 ETF  SSO  2 
ProShares UltraPro S&P 500 ETF  UPRO  3 
According to the BlackScholes model, an option on an LETF with leverage factor ℓ⩾1 should have the same price as an option on an ILETF with leverage factor (−ℓ) if both options share the same strike price and expiration – if these ETFs share the same underlying index.^{5} This is intuitive because, in riskneutral valuation, the only difference between the evolution of the LETF or ILETF is the leverage factor that proportionally increases the volatility. Since the option price is dependent on the square of the volatility, the price of options on LETFs and ILETFs with the same strike and expiration should have the same price.
Figure 2(b) shows the price of call options on these LETFs as a function of the normalized strike price. These observations suggest that the crookedness of the volatility smile is dependent on the leverage factor ℓ.
The prices of inthemoney call options on positively LETFs (ℓ_{1}⩾1) imply lower volatilities than the prices of inthemoney call options on ILETFs (ℓ_{2}⩽0) for the same level of leverage (ℓ_{1}=ℓ_{2}) and moneyness. Similarly, the prices of outofthemoney call options on positively LETFs imply lower volatilities than the prices of outofthemoney call options on ILETFs for the same level of leverage and moneyness.^{6}
Intuitively, we expect stock returns to be negatively correlated with volatility shocks – for example, large price decline and an increase in volatility. As a result, far from the money put options are more expensive than one may naively expect. As a result, the volatility implied by option market prices is increasing with decreasing moneyness, as shown in Figure 2(a). For an inverse ETF, stock returns would be positively correlated with volatility shocks – for example, large price declines in the underlying lead to large price increases in the inverse ETF. As a result, the volatility implied by option market prices on inverse ETFs should be increasing with increasing moneyness. The combination of these two arguments implies that low moneyness options on positively LETFs will be more expensive than similar options on negatively LETFs.
We explore this phenomenon, as well as the crossing phenomena shown in Figure 2(b), both empirically and theoretically in this article.^{7} Although this phenomenon is intuitive, there has been no study of the crossing behavior depicted in Figure 2(b) and this article seeks to fill this gap.
Observable implications
We use the stochastic volatility model in Heston (1993) to explain the ‘crooked smiles’ in the options market. This Heston model, similar to the BlackScholes model, has closedform solutions for the valuation of European put and call options but incorporates stochastic volatility.
Unlike the constant implied volatility in the BlackScholes model, the dynamics of the volatility is correlated with the dynamics of the stock price or index level in the Heston model. This stochastic volatility model is better equipped to account for observed skewness and kurtosis of return distributions.^{8}
Volatility smiles have often been explained either by jumpdiffusion models as suggested by Merton (1976) or with stochastic volatility as in models proposed by Hull and White (1987) and Heston (1993). We consider the Heston model here because of the convenient analytic properties.
Option pricing in the context of LETFs has recently gained interest in the literature. Ahn et al (2012) and Zhang (2010) use standard transformation methods to price options on ETFs and LETFs within the Heston stochastic volatility framework. We extended their research by deriving the dynamics of LETFs. The closedform solutions use a Green function approach following the method in Lipton (2001). Our numerical simulations show that the Heston model can explain the patterns in option prices on LETFs and ILETFs. We observe that the crooked smile is present only when there exists nonzero correlation between the Brownian motions for the asset returns and the variance.^{9} Furthermore, we show that the model predicts a moneyness – dependent on the level of leverage – for which the prices of positively and negatively LETF options have the same price and that this prediction is consistent with empirical data.
We also study the asymptotic behavior of implied variance curves for options on LETF and ILETF. This analysis relies heavily on the moment expansion framework in Lee (2004) and Rollin et al (2009). Subsequent analyses – Friz et al (2011) and de Marco and Martini (2012) – refined the volatility smile expansion within the Heston model. Building on Lee’s work, Benaim and Friz (2009) showed how the tail asymptotics of riskneutral returns can be directly related to the asymptotics of the volatility smile and they applied this approach to timechanged Lévy models and the Heston model (Benaim and Friz, 2008).
We provide empirical evidence that the asymptotic slopes of the implied variance curves for options on LETF and ILETF show an approximate symmetry and that this approximate symmetry is consistent with the Heston model. In particular, we show that the largestrike asymptotic slopes of the implied variance curves for LETFs are related to the smallstrike asymptotic slopes of the implied variance curves for ILETFs. Both simulated examples and market data are presented.
The rest of the article is organized as follows: In the second section, we review the BlackScholes model and Heston stochastic volatility model. The closedform solution of the call options of LETF and ILETF is derived under Heston dynamics. In the third section, the asymptotic behaviors of the option prices are studied as the moneyness goes extremely large and small. In the fourth section, we simulate the option prices under Heston dynamics, and compare the prices across different leverage numbers ℓ. In addition, we calibrate our model using the empirical data in the fifth section. The article is concluded in the sixth section.
LEVERAGED AND INVERSE ETF OPTION PRICES
The BlackScholes model
where μ_{ℓ}=ℓμ and σ_{ℓ}=ℓσ. This implies that the volatilities of the returns are leveraged ℓ times.
where r is the constant riskfree rate.
In this model, the volatility (the standard deviation for the instantaneous ETF return dL_{t}/L_{t}) is always ℓσ, and is independent of the strike price of the option contract, as well as the sign of the leverage number ℓ. If both LETF and ILETF have the same leverage factor in absolute value and the same dividend yield q_{ℓ}, we should observe the same implied volatility and same option price under the BlackScholes model. Figure 2(b) shows that this theoretical prediction is inconsistent with observed market option prices.
Heston’s stochastic volatility model
Since the BlackScholes model is not able to adequately describe the phenomena observed in the option prices on the sextet of LETFs in the section ‘The BlackScholes Model’, we turn to alternative models of asset dynamics. We choose to discuss the Heston model (Heston, 1993) in particular because of the closedform solutions for European option prices and the depth of literature analyzing the details of the model.
The equation implies that a higher realized variance implies a larger deviation between the holding period returns of LETFs and those of the underlying stock. Although the equation cannot be directly used in valuing options on LETFs, it shows the necessity of using stochastic volatility models since dynamic variance is an important component of the return of LETFs.^{11}
Theorem 1

The value of an European call option with strike K for an LETF in the Heston model is in the form
where v_{0} is the initial variance value at time 0 and τ is the time to maturity τ=T−t. X_{t} is defined as ln(F_{t})/(K), where F_{t} is the forward price of LETF and is defined as F_{t}=L_{t}e^{(r−q)τ}. The function U(τ, X_{t}, v_{0}) is given by equation (A.6) in Appendix.
Proof: See proof in Appendix. □
Note that the new equations are free of the leverage factor ℓ thus we can use the Lipton solution or Heston solution given ℓ=1. Having derived the dynamics of leveraged and inverse ETFs based on the dynamics of the underlying index, we move on to consider the pricing of options on leveraged and inverse ETFs.
SYMMETRY IN IMPLIED VARIANCES FOR LETF OPTIONS
In this section, we study the implications of Andersen and Piterbarg (2007) to the extreme strike behavior of option prices on leveraged and inverse ETFs.^{13} Our results build heavily on the work by Lee (2004), wherein the author showed that the extreme strike behavior is related to the number of finite moments of the return distribution.^{14}
In particular, at large strikes the BlackScholes implied variance becomes a linear function of the logarithm of the strike price. Lee (2004) showed that the smallstrike asymptotic behavior is also a linear function of the logarithm of the strike price using the properties of the inverse price process.
Following Friz and KellerRessel (2009), a model is said to exhibit moment explosion at order α>1 if there exists some finite T*(α) such that Open image in new window . The time of moment explosion T*(α) depends on the moment under consideration and is the smallest time such that Open image in new window If, for some β, Open image in new window is finite for all t then T*(β)=∞.
Theorem 2

If Open image in new window and β_{R}=limsup_{x→∞}(σ_{BS}^{2}(x))/(x/T), then
Theorem 3

If Open image in new window and β_{L}=lim sup_{x→∞}(σ_{BS}^{2}(x))/(x/T), then
When β_{L} or β_{R} is 0, the underlying model does not exhibit moment explosion. Note that p*=α_{+}^{*}−1, where α_{+}^{*} is the degree of moment explosion or the critical moment for the process S_{t} at time T and q*=α_{−}^{*}, where α_{−}^{*} is the critical moment for the inverse price process S_{T}^{−1} at time T.
Andersen and Piterbarg (2007), in their Proposition 3.1, show how to relate the degree of moment explosion α_{+} to the parameters in the Heston model as well as the time of moment explosion T*(α_{+}). Following Corollary 6.2 in Andersen and Piterbarg (2007), the critical moment α_{+}^{*} is obtained by solving T*(α_{+})=T.
where the parameters Open image in new window represent the analogous parameters in the inverse spot price process to those in the noninverted price process. Using these results, we can study the symmetry of BlackScholes implied variances at extreme strikes for options on LETFs and ILETFs.
Comparing the parameters used in the Heston model for the LETF price process L_{t, ℓ} and the inverse of ILETF price process L_{t, −ℓ}^{−1}
Parameters  L _{ t, ℓ}  L _{ t, − ℓ} ^{−1} 

ℓ ^{2} v _{ t}  ℓ ^{2} v _{ t}  
ℓε  ℓε  
Sign(ℓ)ρ  −Sign(ℓ)ρ  
κ  κ−ℓρɛ  
ℓ ^{2} θ  ℓ^{2}θκ/κ−lρɛ 
The Heston model parameters Open image in new window , Open image in new window and Open image in new window are the same for both processes. In practice, the Heston model parameters Open image in new window and Open image in new window are not exactly the same, but they are close. Therefore, the critical moments for the LETF price process α*_{+, ℓ} approximates the critical moment for the inverse of ILETF process α*_{−, −ℓ}. The theorems above predict that β_{R, ℓ}≈β_{L, −ℓ}. In other words, the ratio of the BlackScholes implied variance to the moneyness of deepinthemoney call options on LETFs equals the ratio of the BlackScholes implied variance to the moneyness of deepoutofthemoney call options on ILETFs of the same leveraged amplitude. The asymmetry between these two slopes is related to the volatility of variance (ε) and increases approximately linearly with the degree of leverage for small values of the volatility of variance. In the fourth and fifth sections, we will give two examples of critical moments and asymptotic slopes highlighting the relationships.
NUMERICAL SIMULATIONS
Simulation parameters
In what follows, we vary the correlation ρ between returns and return variance, the leverage number ℓ, and the strike price K (or equivalently, the moneyness). By varying these parameters, we analyze the model and explain the pattern implied by market prices of options on leveraged and inverse ETFs. The model calibration on the empirical data will be discussed in the fifth section.
The Heston model with the current set of parameters is ready to explain the volatility smile. Hull (2011) shows that the call option prices are a monotone increasing functions of the implied volatility. Hence, the higher the implied volatility, the higher the call option price.
Breakeven strike prices
Summary of the breakeven strike prices that lead to the same option values for leveraged ETFs within the same doublet (ℓ=±ℓ). For these simulations, we use ρ=−0.5
Leverage number  K*(ℓ) (in dollars)  Option price (in dollars) 

ℓ=±1  99.944  4.459 
ℓ=±2  98.289  9.414 
ℓ=±3  95.592  14.794 

If the correlation coefficient is 0, the difference between the call option prices is 0. In this case, the changes in the volatility are independent of the changes in the stock price. Hence the Heston model essentially becomes a BlackScholes model with implied volatility being the expected mean volatility in the Heston model.

As ℓ increases, the difference between the prices of options on LETF and ILETF increases in magnitude. That can be partially explained that the higher volatility of the leverage funds, the higher option premium.

Upon changing the sign of ρ, the cheaper option becomes the more expensive option for fixed strike within the same doublet of ETFs.^{17}

The breakeven strike price within each doublet is independent of ρ. Although we present only a handful of correlation examples in Figure 4, this conclusion holds for all values of ρ.
Note that deepoutofthemoney call options on ILETFs are more expensive than deepoutofthemoney call options on LETFs when ρ<0. If the correlation coefficient ρ is negative, the volatility and the price of ILETF tend to increase simultaneously. With the same parameters, the price of LETF increases, while the volatility tends to decrease. Intuitively, the ILETF option has a higher probability of ending inthemoney than the analogous LETF. As a result, the call option prices on an ILETF with high strike are often more expensive than call option prices on an LETF when stock price and implied volatility are negatively correlated.
Asymptotic implied variance slope
Calculated extremestrike asymptotic behavior of Heston model option prices for leveraged and inverse ETFs
α*_{+}  p*  β _{ R}  α*_{−}  q*  β _{ L}  

ℓ=−3  19.5  18.5  0.0263  42.0  42.0  0.0118 
ℓ=−2  29.1  28.1  0.0175  62.5  62.5  0.0079 
ℓ=−1  57.8  56.8  0.0087  124.2  124.2  0.0040 
ℓ=+1  123.5  122.5  0.0041  58.0  58.0  0.0085 
ℓ=+2  62.2  61.2  0.0081  29.2  29.2  0.0168 
ℓ=+3  41.72  40.7  0.0121  19.6  19.6  0.0249 
Different models will have different predictions concerning the asymptotic slopes of implied variance curves. By studying a crosssection of option prices, one can determine the empirical validity of a given model by simply observing these slopes and comparing the observations to the predictions of a calibrated model. We take this approach in the following section to show the emprical validity of the Heston model.
EMPIRICAL OPTIONS DATA
In the previous section, we relied on simulated data to test the Heston model’s sensitivity to model parameters. In this section, we calibrate our model to options data for all LETFs and ILETFs listed in Table 1.
There are five parameters to calibrate: mean reversion rate κ, longterm variance θ, volatility of variance ε, correlation ρ and initial volatility v_{0}. Since all the ETFs track the same underlying index, their underlying processes are related through the leverage factor ℓ. The five unknown parameters to be calibrated are therefore common to all members of the sextet.
with the weight function ω_{T, l}. Since the call options values are proportional to the initial price L_{0}^{ℓ} for each ETF, we normalize the absolute difference of the market price and the model price by dividing by the initial LETF price. Other calibration approaches have been proposed according to the onetoone relationship between the option price and the implied volatility. Instead of fitting the option prices directly, one could fit the model volatility/variance to the market volatility/variance implied by the BlackScholes model in the least square sense (see Kjellin and Lövgren, 2006). For the current article, we discuss the optimization problem in equation (13) with equal weight ω_{T, l}=1.
Using the procedure outlined above, we solved for the model parameters for over 100 trading days between July and December 2009. The bestfit parameters varied each day, but over 90 per cent of the days have a volatility of variance parameter (ɛ) in the range [0.5, 3.5] and a correlation parameter (ρ) in the range [−0.45,−0.75].
As an example, we again focus on the LETF option data as of 19 October 2009. The optimization procedure is implemented in MATLAB using the routine lsqnonlin. Most optimization procedures are very efficient and require less than 200 function evaluations.
Since inthemoney call options on negatively LETFs have higher prices than the same option on a positively LETFs, we expect the correlation between the asset price and variance to be negative. Indeed, our results indicate the bestfit value for the correlation is −0.565.
Since our model uses more options data (option data over a sextet of LETFs), the calibrated parameters are more precisely determined and stable when compared to a calibration using only options on the unleveraged member of the sextet (SPY, ℓ=1).
Heston model calibration using market prices of options on the sextet of LETFs and ILETFs
κ  θ  ε  ρ  ν _{0} 

6.8791  0.0669  1.3357  −0.5649  0.0359 
Calculated extremestrike asymptotic behavior of Heston model after calibration to market prices of options on the sextet of ETFs with the S&P 500 as the underlying index on 19 October 2009
α*_{+}  p*  β _{R}  α*_{−}  q*  β _{ L}  

ℓ=−3  3.510  2.510  0.1672  9.246  9.246  0.0513 
ℓ=−2  5.055  4.055  0.1101  13.348  13.348  0.0361 
ℓ=−1  9.710  8.710  0.0543  25.679  25.679  0.0191 
ℓ=+1  24.711  23.711  0.0207  9.973  9.973  0.0478 
ℓ=+2  12.862  11.862  0.0405  5.185  5.185  0.0881 
ℓ=+3  8.919  7.919  0.0594  3.597  3.597  0.1225 
The variance implied by the market prices of options on UPRO and SPXU are plotted in bold in Figure 7. The asymptotic behavior of these empirical curves is explained by the calibrated Heston model.
CONCLUSION
This article was motivated by the empirical observations of robust and persistent crooked smiles implied by the option prices on LETFs and ILETFs. In particular, we note that deepinthemoney call options on the LETF with leverage number ℓ>0 are more expensive than call options with the same moneyness and expiration on the ILETF with leverage number −ℓ. Since the BlackScholes model cannot properly explain this phenomena, we turn to the Heston stochastic volatility model. Since the call option price of Heston model is given explicitly, it is convenient to analyze the Heston model than other stochastic volatility model, which in general need Monte Carlo simulations. Following Lipton (2001), we derived the option values of LETFs and ILETFs with the same underlying via the Green’s function approach. Here we used one important fact that the dynamics of LETFs and ILETFs with the same underlying index have related dynamics.
We gathered pricing data for LETF options on the sextet of ETFs (ℓ∈{−3, −2, −1, 1, 2, 3}) with the S&P 500 as the underlying index. We showed that the Heston model can reproduce the crooked smiles observed in the empirical data and, using numerical simulations, showed how different phenomena in the options data could be produced with different model parameters. In particular, we studied the asymptotic behavior of BlackScholes implied volatility curves and compared these to the predictions given by the Heston model calibrated to the market prices of options.
An important contribution of this article is that using options data of LETFs and ILETFs in addition to options data on the underlying index itself can better calibrate the pricing dynamics of the index. Using the closedform solutions for the option prices of LETFs and ILETFs derived in this article, practitioners can more effectively and accurately determine the model parameters.
Footnotes
 1.
Other early empirical studies of implied volatilities include Latane and Rendleman (1976), Beckers (1981), Canina and Figlewski (1993) and Derman and Kani (1994).
 2.
For a review, see Hull (2011).
 3.
For a recent discussion of the detrimental effect of daily rebalancing in LETFs and ILETFs on buyandhold investors, see Dulaney et al (2012).
 4.
SPY has leverage factor of 1, and is actually unleveraged. In an abuse of terminology, we refer to SPY as an LETF with leverage factor ℓ=1.
 5.
In the current article, we assume the spot prices of both LEFT and ILEFT are identical, and set them as US$100. For empirical data, we use the concept of moneyness, defined as the ratio of the strike price to the spot price of the ETF (see also Lee, 2004). For consistency, we also normalize the option prices by the spot price of the ETF.
 6.
Both of these statements make the implicit assumption that the prices are being compared on options with the same moneyness and maturity.
 7.
The strike price at which the prices of call options on LETFs and ILETFs are equal is another common feature of the options data.
 8.
Derman and Kani (1994) discuss the connection between the dynamics of the underlying stock price – which determines the moments of the return distribution – and the volatility smile. The Heston model has been studied before as an explanation for volatility smiles present in the BlackScholes model – see, for example, Sircar and Papanicolaou (1999).
 9.
When the correlation is 0, the Heston model is equivalent to the BlackScholes model.
 10.
For simplicity, we drop the subscript t for the price process L_{t}.
 11.
For a constant volatility model, the realized variance is simply ∫_{0}^{t}vdt=vt. For a more complete discussion of this topic, see Avellaneda and Zhang (2010).
 12.
This change of variables is consistent with that given by Proposition 1 in Ahn et al (2012).
 13.
Andersen and Piterbarg (2007) have studied the time of moment explosion within the Heston model.
 14.
See Leung and Sircar (2012) for a recent study of implied volatility surfaces implied by the market prices of options on LETFs.
 15.
Here we use the notation x=log(K/F_{0}) for the moneyness with respect to the stock’s forward price for contracts expiring T years from today.
 16.
Both of these statements rely on the implicit assumption that ρ<0.
 17.
For example, if the call option with strike K on the positively LETF with leverage number ℓ is more expensive than the call option with strike K on the negatively LETF with leverage number −ℓ when ρ>0 then, if ρ is changed to −ρ, the negatively leveraged option becomes more expensive.
 18.
For examples of other papers that have used this calibration procedure, see Bates (1996), Bakshi et al (1997), Carr et al (2003) and Schoutens et al (2004).
 19.
Again, for simplicity, we drop the subscript t for both L_{t} and v_{t}.
 20.
Where Open image in new window and Open image in new window .
 21.
Open image in new window denotes the complex conjugation of Z(τ, k, v).
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