Top

Finance and Stochastics

Published in:

01-10-2016

Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps

Authors: José E. Figueroa-López, Sveinn Ólafsson

Published in: Finance and Stochastics | Issue 4/2016

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

previous article Polynomial diffusions and applications in finance

next article A Feynman–Kac-type formula for Lévy processes with discontinuous killing rates

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

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.

Aït-Sahalia, Y., Jacod, J.: Estimating the degree of activity of jumps in high-frequency data. Ann. Stat. 37, 2202–2244 (2009) MathSciNetCrossRefMATH

Aït-Sahalia, Y., Jacod, J.: Is Brownian motion necessary to model high-frequency data? Ann. Stat. 38, 3093–3128 (2010) MathSciNetCrossRefMATH

Aït-Sahalia, Y., Lo, A.: Nonparametric estimation of state-price densities implicit in financial asset prices. J. Finance 53, 499–547 (1998) CrossRef

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) MathSciNetCrossRefMATH

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) MathSciNetCrossRefMATH

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) CrossRef

Bates, D.S.: The crash of ’87: was it expected? The evidence from options markets. J. Finance 46, 1009–1044 (1991) CrossRef

Bergomi, L.: Smile dynamics. Risk Mag. 9, 117–123 (2004)

10.

Bergomi, L.: Stochastic Volatility Modeling. Chapman & Hall, London (2016) MATH

11.

Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1998) MATH

12.

Carr, P., Geman, H., Madan, D., Yor, M.: The fine structure of asset returns: an empirical investigation. J. Bus. 75, 303–325 (2002)

13.

Carr, P., Wu, L.: What type of process underlies options? A simple robust test. J. Finance 58, 2581–2610 (2003) CrossRef

14.

Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall, London (2004) MATH

15.

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

16.

Dupire, B.: Pricing with a smile. Risk 7, 18–20 (1994)

17.

Fajardo, J., Mordecki, E.: Skewness premium with Lévy processes. Quant. Finance 14, 1619–1626 (2014) MathSciNetCrossRefMATH

18.

Figueroa-López, J.E., Forde, M.: The small-maturity smile for exponential Lévy models. SIAM J. Financ. Math. 3, 33–65 (2012) CrossRefMATH

19.

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) MathSciNetCrossRefMATH

20.

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)

21.

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) MathSciNetCrossRefMATH

22.

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

23.

Gao, K., Lee, R.: Asymptotics of implied volatility to arbitrary order. Finance Stoch. 18, 349–392 (2014) MathSciNetCrossRefMATH

24.

Gatheral, J.: The Volatility Surface: A Practitioner’s Guide. Wiley Finance Series (2006)

25.

Gatheral, J., Jaisson, T., Rosenbaum, M.: Volatility is rough. Working paper (2014). Available online at: https://arxiv.org/abs/1410.3394

26.

Gerhold, S., Gülüm, I.C., Pinter, A.: The small-maturity implied volatility slope for Lévy models. Appl. Math. Finance 23, 135–157 (2016) MathSciNetCrossRef

27.

Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, Berlin (1997) MATH

28.

Kawai, R.: On sequential calibration for an asset price model with piecewise Lévy processes. IAENG Int. J. Appl. Math. 40, 239–246 (2010) MathSciNetMATH

29.

Konikov, M., Madan, D.: Stochastic volatility via Markov chains. Rev. Deriv. Res. 5, 81–115 (2002) CrossRefMATH

30.

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) CrossRef

31.

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) CrossRef

32.

Medvedev, A., Scailllet, O.: Approximation and calibration of short-term implied volatility under jump-diffusion stochastic volatility. Rev. Financ. Stud. 20, 427–459 (2007) CrossRef

33.

Mijatović, A., Tankov, P.: A new look at short-term implied volatility in asset price models with jumps. Math. Finance 26, 149–183 (2016) MathSciNetCrossRefMATH

34.

Mixon, S.: What does implied volatility skew measure? J. Deriv. 18(4), 9–25 (2011) CrossRef

35.

Muhle-Karbe, J., Nutz, M.: Small-time asymptotics of option prices and first absolute moments. J. Appl. Probab. 48, 1003–1020 (2011) MathSciNetCrossRefMATH

36.

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/

37.

Pan, J.: The jump-risk premia implicit in options: evidence from an integrated time-series study. J. Financ. Econ. 63, 3–50 (2002) CrossRef

38.

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) MathSciNetCrossRefMATH

39.

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)

40.

Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) MATH

41.

Schoutens, W.: Lévy Processes in Finance. Wiley, New York (2003) CrossRef

42.

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)

43.

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) CrossRef

44.

Yan, S.: Jump risk, stock returns, and slope of implied volatility smile. J. Financ. Econ. 99, 216–233 (2011) CrossRef

45.

Zhang, J.E., Xiang, Y.: The implied volatility smirk. Quant. Finance 8, 263–284 (2008) MathSciNetCrossRefMATH

46.

Zolotarev, V.M.: One-Dimensional Stable Distributions. Am. Math. Soc., Providence (1996) MATH

Title: Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps
Authors: José E. Figueroa-López
Sveinn Ólafsson
Publication date: 01-10-2016
Publisher: Springer Berlin Heidelberg
Published in: Finance and Stochastics / Issue 4/2016
Print ISSN: 0949-2984
Electronic ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-016-0313-3

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 4/2016

A BSDE approach to fair bilateral pricing under endogenous collateralization

A Feynman–Kac-type formula for Lévy processes with discontinuous killing rates

Editorial: 20th anniversary of Finance and Stochastics

Polynomial diffusions and applications in finance

Another look at the integral of exponential Brownian motion and the pricing of Asian options

Liquidity management with decreasing returns to scale and secured credit line