Mass at zero in the uncorrelated SABR model and implied volatility asymptotics

References

• Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, 1964 (Dover Publications: Washington, DC).
   Google Scholar
• Andreasen, J. and Huge, B., ZABR–Expansions for the masses. Preprint, 2011. Available online at: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1980726.
   Google Scholar
• Antonov, A. , Konikov, M. and Spector, M., SABR spreads its wings. Risk, 2013.
   Google Scholar
• Antonov, A. , Konikov, M. and Spector, M., The free boundary SABR: Natural extension to negative rates. Risk, 2015a, August .
   Google Scholar
• Antonov, A. , Konikov, M. and Spector, M., Mixing SABR models for negative rates, 2015b. Available online at: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2653682.
   Google Scholar
• Antonov, A. and Spector, M., Advanced analytics for the SABR model. Preprint, 2012. Available online at: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2026350.
   Google Scholar
• Balland, P. and Tran, Q., SABR goes normal. Risk, 2013, June , 76–81.
   Google Scholar
• Barrieu, P. , Rouault, A. and Yor, M., A study of the Hartman-Watson distribution motivated by numerical problems related to Asian option pricing. J. Appl. Probab., 2004, 41 , 1049–1058.
   Web of Science ®Google Scholar
• Benaim, S. and Friz, P., Regular variation and smile asymptotics. Math. Finance, 2009, 19 , 1–12.
   Web of Science ®Google Scholar
• Borodin, A.N. and Salminen, P., Handbook of Brownian Motion—Facts and Formulae, 2nd ed., 1996 (Birkhäuser: Basel).
   Google Scholar
• Chen, B. , Oosterlee, C.W. and van der Weide, H., A low-bias simulation scheme for the SABR stochastic volatility model. Int. J. Theor. Appl. Finance, 2012, 15 (2), 1250016.
   Google Scholar
• De Marco, S. , Hillairet, C. and Jacquier, A., Shapes of implied volatility with positive mass at zero. SIAM J. Financ. Math., 2017, 8 (1), 709–737.
   Google Scholar
• Döring, L. , Horvath, B. and Teichmann, J., Functional analytic (ir-)regularity properties of SABR-type processes. Int. J. Theor. Appl. Finance, 2017, 20 (3), 1–48.
   Web of Science ®Google Scholar
• Gatheral, J., The Volatility Surface: A Practitioner’s Guide, 2006 (Wiley: New York).
   Google Scholar
• Gerhold, S., The Hartman-Watson distribution revisited: Asymptotics for pricing Asian options. J. Appl. Probab., 2011, 48 (3), 892–899.
   Web of Science ®Google Scholar
• Grzelak, L.A. and Oosterlee, C.W., From arbitrage to arbitrage-free implied volatilities. J. Comput. Finance, 2016, 20 (3), 31–49.
   Web of Science ®Google Scholar
• Gulisashvili, A., Asymptotic formulas with error estimates for call pricing functions and the implied volatility at extreme strikes. SIAM J. Financ. Math., 2010, 1 , 609–641.
   Google Scholar
• Gulisashvili, A., Analytically Tractable Stochastic Stock Price Models, 2012 (Springer: New York).
   Google Scholar
• Gulisashvili, A., Left-wing asymptotics of the implied volatility in the presence of atoms. Int. J. Theor. Appl. Finance, 2015, 18 (2). Available online at: http://www.worldscientific.com/doi/abs/10.1142/S0219024915500132
   Web of Science ®Google Scholar
• Gulisashvili, A. , Horvath, B. and Jacquier, A., On the probability of hitting the boundary for Brownian motions on the SABR plane. Electron. Commun. Probab., 2016, 21 (75), 1–13.
   Google Scholar
• Hagan, P. , Kumar, D. , Lesniewski, A. and Woodward, D. Managing smile risk. Wilmott Mag., 2002, 84–108.
   Google Scholar
• Hagan, P. , Kumar, D. , Lesniewski, A. , and Woodward, D. Arbitrage-free SABR. Wilmott Mag., 2014, 60–75.
   Google Scholar
• Hagan, P. , Lesniewski, A. and Woodward, D., Probability distribution in the SABR model of stochastic volatility. In Large Deviations and Asymptotic Methods in Finance. Springer proceedings in mathematics and statistics. Vol. 110, edited by P. Friz , J. Gatheral , A. Gulisashvili , A. Jacquier and J. Teichmann, 2015 (Springer-Verlag: New York).
   Google Scholar
• Hobson, D., Comparison results for stochastic volatility models via coupling. Finance Stoch., 2010, 14 , 129–152.
   Web of Science ®Google Scholar
• Islah, O., Solving SABR in exact form and unifying it with LIBOR market model, 2009. Available online at: http://papers.ssrn.com/sol3/papers.cfm?astract-id=1489428.
   Google Scholar
• Jacod, J. and Shiryaev, A.N., Limit Theorems for Stochastic Processes, 2nd ed., 2003 (Springer: Berlin).
   Google Scholar
• Jeanblanc, M. , Yor, M. and Chesney, M., Mathematical Methods for Financial Markets, 2009 (Springer-Verlag: London).
   Google Scholar
• Jourdain, B., Loss of martingality in asset price models with lognormal stochastic volatility. Int. J. Theor. Appl. Finance, 2004, 13 , 767–787.
   Google Scholar
• Lee, R.W., The moment formula for implied volatility at extreme strikes. Math Finance, 2004, 14 , 469–480.
   Web of Science ®Google Scholar
• Leitao, A. , Grzelak, L.A. and Oosterlee, C.W., On a one time-step Monte Carlo simulation approach of the SABR model: Application to European options. Appl. Math. Comput., 2017, 293 (15), 461–479.
   Google Scholar
• Lorig, M. , Pagliarani, S. and Pascucci, A., Explicit implied volatilities for multifactor local-stochastic volatility models. Math Finance, forthcoming.
   Web of Science ®Google Scholar
• Matsumoto, H. and Yor, M., Exponential functionals of Brownian motion, I: Probability laws at fixed time. Probab. Surv., 2005a, 2 , 312–347.
   Google Scholar
• Matsumoto, H. and Yor, M., Exponential functionals of Brownian motion, II: Some related diffusion processes. Probab. Surv., 2005b, 2 , 348–384.
   Google Scholar
• Miller, P.D., Applied Asymptotic Analysis. Graduate Studies in Mathematics, Vol. 75, 2006 (American Mathematical Society).
   Google Scholar
• Obłój, J., Fine-tune your smile: Correction to Hagan et al. Wilmott Mag., 2008.
   Google Scholar
• Pagliarani, S. and Pascucci, A., The exact Taylor formula of the implied volatility. Finance Stoch., 2017, 21 (3), 661–718.
   Web of Science ®Google Scholar
• Paulot, L., Asymptotic implied volatility at the second order with application to the SABR model. In Large Deviations and Asymptotic Methods in Finance. Springer proceedings in mathematics and statistics. Vol. 110, edited by P. Friz , J. Gatheral , A. Gulisashvili , A. Jacquier and J. Teichmann, 2015 (Springer-Verlag: New York).
   Google Scholar
• Rebonato, R., A simple approximation for the no-arbitrage drifts for LMM-SABR-family interest-rate models. J. Comput. Finance 2015, 19 (1).
   Web of Science ®Google Scholar
• Veraart, A. and Winkel, M., Time change. In Encyclopedia of Quantitative Finance, edited by R. Cont, pp. 1812–1816, 2010 (Wiley).
   Google Scholar