Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T12:19:53.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Small-Time Asymptotics of Option Prices and First Absolute Moments

Part of: Stochastic processes Mathematical economics

Published online by Cambridge University Press:  14 July 2016

Johannes Muhle-Karbe  and
Marcel Nutz
Johannes Muhle-Karbe*
Affiliation:
Universität Wien
Marcel Nutz*
Affiliation:
ETH Zürich
*
∗ Current address: Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland. Email address: [email protected]
∗∗ Current address: Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the leading term in the small-time asymptotics of at-the-money call option prices when the stock price process S follows a general martingale. This is equivalent to studying the first centered absolute moment of S. We show that if S has a continuous part, the leading term is of order √T in time T and depends only on the initial value of the volatility. Furthermore, the term is linear in T if and only if S is of finite variation. The leading terms for pure-jump processes with infinite variation are between these two cases; we obtain their exact form for stable-like small jumps. To derive these results, we use a natural approximation of S so that calculations are necessary only for the class of Lévy processes.

Keywords

Option priceabsolute momentsmall-time asymptoticsapproximation by Lévy processes

MSC classification

Secondary: 91B25: Asset pricing models 60G44: Martingales with continuous parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 1003 - 1020
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. US Government Printing Office, Washington, DC.Google Scholar
[2] Alòs, E., Léon, J. A. and Vives, J. (2007). On the short-time behavior of the implied volatility for Jump-diffusion models with stochastic volatility. Finance Stoch. 11, 571–589.Google Scholar
[3] Andersen, L. B. G. and Piterbarg, V. V. (2007). Moment explosions in stochastic volatility models. Finance Stoch. 11, 29–50.Google Scholar
[4] Barndorff-Nielsen, O. E. and Stelzer, R. (2005). Absolute moments of generalized hyperbolic distributions and approximate scaling of normal inverse Gaussian Lévy processes. Scand. J. Statist. 32, 617–637.Google Scholar
[5] Berestycki, H., Busca, J. and Florent, I. (2002). Asymptotics and calibration of local volatility models. Quant. Finance 2, 61–69.Google Scholar
[6] Berestycki, H., Busca, J. and Florent, I. (2004). Computing the implied volatility in stochastic volatility models. Commun. Pure Appl. Math. 57, 1352–1373.Google Scholar
[7] Carr, P. and Wu, L. (2003). What type of process underlies options? A simple robust test. J. Finance 58, 2581–2610.Google Scholar
[8] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
[9] Cox, J. C., Ingersoll, J. E. Jr. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53, 385–407.Google Scholar
[10] Durrleman, V. (2008). Convergence of at-the-money implied volatilities to the spot volatility. J. Appl. Prob. 45, 542–550.Google Scholar
[11] Durrleman, V. (2010). From implied to spot volatilities. Finance Stoch. 14, 157–177.Google Scholar
[12] Feng, J., Forde, M. and Fouque, J.-P. (2010). Short-maturity asymptotics for a fast mean-reverting Heston stochastic volatility model. SIAM J. Financial Math. 1, 126–141.Google Scholar
[13] Figueroa-López, J. E. and Forde, M. (2011). The small-maturity smile for exponential Lévy models. To appear in SIAM J. Financial Math. Available at http://arxiv.org/abs/1105.3180v1.Google Scholar
[14] Forde, M. (2009). Small-time asymptotics for a general local-stochastic volatility model, using the heat kernel expansion. Preprint.Google Scholar
[15] Forde, M. and Jacquier, A. (2009). Small-time asymptotics for implied volatility under the Heston model. Internat. J. Theoret. Appl. Finance 12, 861–876.Google Scholar
[16] Forde, M., Jacquier, A. and Lee, R. (2011). The small-time smile and term structure of implied volatility under the Heston model. Preprint.Google Scholar
[17] Gatheral, J., Hsu, E. P., Laurence, P. M., Ouyang, C. and Wang, T.-H. (2011). Asymptotics of implied volatility in local volatility models. To appear in Math. Finance.Google Scholar
[18] Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales (Lecture Notes Math. 714). Springer, Berlin.Google Scholar
[19] Jacod, J. (2007). Asymptotic properties of power variations of Lévy processes. ESAIM Prob. Statist. 11, 173–196.Google Scholar
[20] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.Google Scholar
[21] Luschgy, H. and Pagès, G. (2008). Moment estimates for Lévy processes. Electron. Commun. Prob. 13, 422–434.Google Scholar
[22] Medvedev, A. and Scaillet, O. (2007). Approximation and calibration of short-term implied volatilities under Jump-diffusion stochastic volatility. Rev. Financial Studies 20, 427–459.Google Scholar
[23] Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.Google Scholar
[24] Roper, M. (2008). Implied volatility explosions: European calls and implied volatilities close to expiry in exponential Lévy models. Preprint.Google Scholar
[25] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New York.Google Scholar
[26] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
[27] Tankov, P. (2010). Pricing and hedging in exponential Lévy models: Review of recent results. In Paris-Princeton Lectures on Mathematical Finance 2010 (Lecture Notes Math. 2003), Springer, Berlin, pp. 319–359.Google Scholar