Abstract
The implied volatility skew has received relatively little attention in the literature on short-term asymptotics for financial models with jumps, despite its importance in model selection and calibration. We rectify this by providing high order asymptotic expansions for the at-the-money implied volatility skew, under a rich class of stochastic volatility models with independent stable-like jumps of infinite variation. The case of a pure-jump stable-like Lévy model is also considered under the minimal possible conditions for the resulting expansion to be well defined. Unlike recent results for “near-the-money” option prices and implied volatility, the results herein aid in understanding how the implied volatility smile near expiry is affected by important features of the continuous component, such as the leverage and vol-of-vol parameters. As intermediary results, we obtain high order expansions for at-the-money digital call option prices, which furthermore allow us to infer analogous results for the delta of at-the-money options. Simulation results indicate that our asymptotic expansions give good fits for options with maturities up to one month, underpinning their relevance in practical applications, and an analysis of the implied volatility skew in recent S&P 500 options data shows it to be consistent with the infinite variation jump component of our models.
Similar content being viewed by others
Notes
Practitioners commonly use the terms “skew” and “implied volatility skew” for the ATM slope of the implied volatility curve for a given expiration date (see e.g. [34]). We use the terms interchangeably.
For a Lévy process \(X\) with Lévy measure \(\nu\), the Blumenthal–Getoor index is defined as \(\inf\{p\geq 0:\int_{\{|x|\leq 1\}}|x|^{p}\nu(dx)<\infty\}\).
Equivalently, \(Z_{t}^{(p)}\) and \(Z_{t}^{(n)}\) are \(Y\)-stable random variables with location parameter 0, skewness parameters 1 and −1, and respective scale parameters \((tC(1)|\cos(\pi Y/2)|\Gamma(-Y))^{1/Y}\) and \((tC(-1)|\cos(\pi Y/2)|\Gamma(-Y))^{1/Y}\).
In addition to traditional S&P 500 index options (SPX), our dataset includes SPXQ (quarterly) and SPXW (weekly) options. The latter class was first introduced in 2005, and by the end of 2014, it accounted for over 40 % of the overall trading of S&P 500 options on the CBOE (see Fig. 2 in [6]).
The ATM strike is taken to be the strike price at which the call and put options prices are closest in value. We also set the risk-free interest rate to zero, but using a nonzero rate based on U.S. treasury yields did not change the results of our analysis since the rate is close to zero over the sample period and the time-to-maturity is small.
The 25-delta put (resp. call) is the option whose strike price has been chosen such that the option’s delta is −0.25 (resp. 0.25). For each maturity, we choose the put (resp. call) whose delta is closest in value to −0.25 (resp. 0.25).
Repeating the analysis using 10-delta options did not have a qualitative effect on the outcome.
Pooling data also makes this estimation procedure viable for indices with fewer liquid maturities than S&P 500, as well as individual equity names.
References
Aït-Sahalia, Y., Jacod, J.: Estimating the degree of activity of jumps in high-frequency data. Ann. Stat. 37, 2202–2244 (2009)
Aït-Sahalia, Y., Jacod, J.: Is Brownian motion necessary to model high-frequency data? Ann. Stat. 38, 3093–3128 (2010)
Aït-Sahalia, Y., Lo, A.: Nonparametric estimation of state-price densities implicit in financial asset prices. J. Finance 53, 499–547 (1998)
Alòs, E., León, J., Vives, J.: On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility. Finance Stoch. 11, 571–589 (2007)
Andersen, L., Lipton, A.: Asymptotics for exponential Lévy processes and their volatility smile: survey and new results. Int. J. Theor. Appl. Finance 16, 1–98 (2013)
Andersen, T., Fusari, N., Todorov, V.: Pricing short-term market risk: evidence from weekly options. NBER working paper no. 2149 (2015). Available online at: http://nber.org/papers/w21491
Bakshi, G., Kapadia, N., Madan, D.: Stock return characteristics, skew laws, and the differential pricing of individual equity options. Rev. Financ. Stud. 16, 101–143 (2003)
Bates, D.S.: The crash of ’87: was it expected? The evidence from options markets. J. Finance 46, 1009–1044 (1991)
Bergomi, L.: Smile dynamics. Risk Mag. 9, 117–123 (2004)
Bergomi, L.: Stochastic Volatility Modeling. Chapman & Hall, London (2016)
Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1998)
Carr, P., Geman, H., Madan, D., Yor, M.: The fine structure of asset returns: an empirical investigation. J. Bus. 75, 303–325 (2002)
Carr, P., Wu, L.: What type of process underlies options? A simple robust test. J. Finance 58, 2581–2610 (2003)
Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall, London (2004)
De Leo, L., Vargas, V., Ciliberti, S., Bouchaud, J.-P.: We’ve walked a million miles for one of these smiles. Preprint (2012). Available online at: http://arxiv.org/abs/1203.5703
Dupire, B.: Pricing with a smile. Risk 7, 18–20 (1994)
Fajardo, J., Mordecki, E.: Skewness premium with Lévy processes. Quant. Finance 14, 1619–1626 (2014)
Figueroa-López, J.E., Forde, M.: The small-maturity smile for exponential Lévy models. SIAM J. Financ. Math. 3, 33–65 (2012)
Figueroa-López, J.E., Gong, R., Houdré, C.: High-order short-time expansions for ATM option prices of exponential Lévy models. Math. Finance 26, 516–557 (2016)
Figueroa-López, J.E., Houdré, C.: Small-time expansions for the transition distribution of Lévy processes. In: Stochastic Processes and Their Applications, vol. 119, pp. 3862–3889 (2009)
Figueroa-López, J.E., Ólafsson, S.: Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility. Finance Stoch. 20, 219–265 (2016)
Fukasawa, M.: Short-time at-the-money skew and rough fractional volatility. Quant. Finance (2016, to appear). Available online at: https://arxiv.org/abs/1501.06980. doi:10.1080/14697688.2016.1197410
Gao, K., Lee, R.: Asymptotics of implied volatility to arbitrary order. Finance Stoch. 18, 349–392 (2014)
Gatheral, J.: The Volatility Surface: A Practitioner’s Guide. Wiley Finance Series (2006)
Gatheral, J., Jaisson, T., Rosenbaum, M.: Volatility is rough. Working paper (2014). Available online at: https://arxiv.org/abs/1410.3394
Gerhold, S., Gülüm, I.C., Pinter, A.: The small-maturity implied volatility slope for Lévy models. Appl. Math. Finance 23, 135–157 (2016)
Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, Berlin (1997)
Kawai, R.: On sequential calibration for an asset price model with piecewise Lévy processes. IAENG Int. J. Appl. Math. 40, 239–246 (2010)
Konikov, M., Madan, D.: Stochastic volatility via Markov chains. Rev. Deriv. Res. 5, 81–115 (2002)
Koponen, I.: Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. Phys. Rev. E 52, 1197–1199 (1995)
Lee, R.: Implied volatility: statics, dynamics, and probabilistic interpretation. In: Baeza-Yates, R., et al. (eds.) Recent Advances in Applied Probability, pp. 241–268. Springer, New York (2005)
Medvedev, A., Scailllet, O.: Approximation and calibration of short-term implied volatility under jump-diffusion stochastic volatility. Rev. Financ. Stud. 20, 427–459 (2007)
Mijatović, A., Tankov, P.: A new look at short-term implied volatility in asset price models with jumps. Math. Finance 26, 149–183 (2016)
Mixon, S.: What does implied volatility skew measure? J. Deriv. 18(4), 9–25 (2011)
Muhle-Karbe, J., Nutz, M.: Small-time asymptotics of option prices and first absolute moments. J. Appl. Probab. 48, 1003–1020 (2011)
Ólafsson, S.: Applications of short-time asymptotic methods to option pricing and change-point detection for Lévy processes. Ph.D. thesis, Purdue University (2015). Available online at: http://docs.lib.purdue.edu/dissertations/AAI3734543/
Pan, J.: The jump-risk premia implicit in options: evidence from an integrated time-series study. J. Financ. Econ. 63, 3–50 (2002)
Roper, M., Rutkowski, M.: On the relationship between the call price surface and the implied volatility surface close to expiry. Int. J. Theor. Appl. Finance 12, 427–441 (2009)
Rosenbaum, M., Tankov, P.: Asymptotic results for time-changed Lévy processes sampled at hitting times. In: Stochastic Processes and their Applications, vol. 121, pp. 1607–1633 (2011)
Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)
Schoutens, W.: Lévy Processes in Finance. Wiley, New York (2003)
Tankov, P.: Pricing and hedging in exponential Lévy models: review of recent results. In: Carmona, R., et al. (eds.) Paris–Princeton Lectures on Mathematical Finance. Lecture Notes in Mathematics, vol. 2003, pp. 319–359. Springer, Berlin (2010)
Xing, Y., Zhang, X., Zhao, R.: What does individual option volatility smirk tell us about future equity returns? J. Financ. Quant. Anal. 45, 641–662 (2010)
Yan, S.: Jump risk, stock returns, and slope of implied volatility smile. J. Financ. Econ. 99, 216–233 (2011)
Zhang, J.E., Xiang, Y.: The implied volatility smirk. Quant. Finance 8, 263–284 (2008)
Zolotarev, V.M.: One-Dimensional Stable Distributions. Am. Math. Soc., Providence (1996)
Acknowledgements
The authors gratefully acknowledge two anonymous reviewers and the editor for providing constructive and insightful comments, which improved significantly the quality of the manuscript. The authors would also like to thank Christian Houdré and Frederi Viens for useful suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by the NSF Grant: DMS-1149692.
Appendix: Additional proofs
Appendix: Additional proofs
Lemma A.1
Let \(V\) be as in (1.8), with \(\mu(Y_{t})\) and \(\sigma(Y_{t})\) replaced by \(\bar{\mu}_{t}\) and \(\bar{\sigma}_{t}\) defined in (4.8). Also let \(\bar{\sigma}'\), \(\bar{\sigma}''\), \(\bar{\alpha}\) and \(\bar{\gamma}\) be the stopped processes in (4.20) and \(\bar{\sigma}_{t}^{*}:=\sqrt{\frac{1}{t}\int_{0}^{t}\bar{\sigma}_{s}^{2}ds}\). Then the following relations hold for any \(p\geq1\):
-
(i)
\({\mathbb {E}}[|\bar{\mu}_{t}-\mu_{0}|^{p}] = O(t^{\frac{p}{2}}),\quad t\to0\).
-
(ii)
\({\mathbb {E}}[|\bar{\sigma}_{t}-\sigma_{0}|^{p}] = O(t^{\frac{p}{2}}),\quad t\to0\).
-
(iii)
\({\mathbb {E}}[|\bar{\sigma}^{*}_{t}-\sigma_{0}|^{p}] = O(t^{\frac {p}{2}}),\quad t\to0\).
-
(iv)
\({\mathbb {E}}[\bar{\sigma}_{t}^{*}-\sigma_{0}] = O(t), \quad t\to0\).
-
(v)
For \(\xi_{t}^{1},\xi_{t}^{2}\) and \(\xi_{t}^{1,0}\) defined as in (4.20) and (4.21), we have \({\mathbb {E}}[|\xi_{t}^{1}|]=O(t)\) and \({\mathbb {E}}[|\xi _{t}^{2}|]+{\mathbb {E}}[|\xi_{t}^{1}-\xi_{t}^{1,0}|] = O(t^{\frac{3}{2}})\), \(t\to0\).
-
(vi)
\({\mathbb {E}}[|\bar{\sigma}_{t}^{*}-\sigma_{0}-\sigma_{0}'\gamma_{0}\frac {1}{t}\int_{0}^{t}W_{s}^{1}ds|]=O(t),\quad t\to0\).
Proof
Let \(L\) be a common Lipschitz constant for \(\mu\), \(\sigma\) and \(\gamma\).
(i) By the Lipschitz-continuity of \(\mu\) and the Burkholder–Davis–Gundy (BDG) inequality, we can find a constant \(C_{p}\) such that
since \(\alpha\) and \(\gamma\) are bounded.
(ii) is proved in a similar way, and for (iii) we use the boundedness of \(\sigma\), Jensen’s inequality and (ii) to write
(iv) We can write
where the second term is of order \(O(t)\) by (iii), while for the first term we have by Itô’s lemma
since the expected value of the stochastic integral is zero.
(v) By Cauchy’s inequality and Itô’s isometry, we have
Similarly,
because the boundedness of \(\sigma'\) and \(\gamma\) allows us to find a constant \(K\) such that
where in the last step we again used the BDG inequality. Similarly, Cauchy’s inequality and Itô’s isometry yield \({\mathbb {E}}[|\xi _{t}^{1}|]=O(t)\), as \(t\to0\).
(vi) follows from the triangle inequality and the following three identities. First, by (iii) above, we have
Second, by Itô’s lemma,
since the integrand in the last integral is bounded. Third, Cauchy’s inequality and Itô’s isometry can be used to show
by following similar steps as in the proof of (v). □
Rights and permissions
About this article
Cite this article
Figueroa-López, J.E., Ólafsson, S. Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps. Finance Stoch 20, 973–1020 (2016). https://doi.org/10.1007/s00780-016-0313-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00780-016-0313-3
Keywords
- Exponential Lévy models
- Stochastic volatility models
- Short-term asymptotics
- ATM implied volatility slope
- ATM digital call option prices