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ASYMPTOTICS OF THE TIME-DISCRETIZED LOG-NORMAL SABR MODEL: THE IMPLIED VOLATILITY SURFACE

Published online by Cambridge University Press:  22 May 2020

Dan Pirjol Open the ORCID record for Dan Pirjol [Opens in a new window]  and
Lingjiong Zhu
Dan Pirjol
Affiliation:
School of Business, Stevens Institute of Technology, Hoboken, NJ07030, USA E-mail: dpirjol@gmail.com
Lingjiong Zhu
Affiliation:
Department of Mathematics, Florida State University, 1017 Academic Way, Tallahassee, FL32306, USA
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Abstract

We propose a novel time discretization for the log-normal SABR model which is a popular stochastic volatility model that is widely used in financial practice. Our time discretization is a variant of the Euler–Maruyama scheme. We study its asymptotic properties in the limit of a large number of time steps under a certain asymptotic regime which includes the case of finite maturity, small vol-of-vol and large initial volatility with fixed product of vol-of-vol and initial volatility. We derive an almost sure limit and a large deviations result for the log-asset price in the limit of a large number of time steps. We derive an exact representation of the implied volatility surface for arbitrary maturity and strike in this regime. Using this representation, we obtain analytical expansions of the implied volatility for small maturity and extreme strikes, which reproduce at leading order known asymptotic results for the continuous time model.

Keywords

applied probabilitymathematical financestochastic modelling
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 4 , October 2021 , pp. 942 - 974
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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References

1Aït-Sahalia, Y., Li, C., & Li, C.X. (2020). Implied stochastic volatility models. Review of Financial Studies. https://doi.org/10.1093/rfs/hhaa041Google Scholar
2Antonov, A., Konikov, M., & Spector, M. (2013). SABR spreads its wings. Risk 26: 58–63.Google Scholar
3Antonov, A., Konikov, M., & Spector, M. (2019). Modern SABR analytics. New York: Springer.CrossRefGoogle Scholar
4Berestycki, H., Busca, J., & Florent, I. (2004). Computing the implied volatility in stochastic volatility models. Communications in Pure and Applied Mathematics 57: 1352–1373.CrossRefGoogle Scholar
5Bernard, C., Cui, Z., & McLeish, D. (2017). On the martingale property in stochastic volatility models based on time-homogeneous diffusions. Mathematical Finance 27(1): 194–223.CrossRefGoogle Scholar
6Cai, N., Song, Y., & Chen, N. (2017). Exact simulation of the SABR model. Operations Research 65(4): 931–951.CrossRefGoogle Scholar
7Constantinescu, R., Costanzino, N., Mazzucato, A.L., & Nistor, V. (2010). Approximate solutions to second order parabolic equations: I. Analytical estimates. Journal of Mathematical Physics 51: 103502.CrossRefGoogle Scholar
8Dembo, A. & Zeitouni, O. (1998). Large deviations techniques and applications, 2nd ed. New York: Springer.CrossRefGoogle Scholar
9Deuschel, J.D., Friz, P.K., Jacquier, A., & Violante, S. (2014). Marginal density expansions for diffusions and stochastic volatility, part I: Theoretical Foundations. Communications in Pure and Applied Mathematics 67: 40–82.CrossRefGoogle Scholar
10Donsker, M.D. & Varadhan, S.R.S. (1975). Asymptotic evaluation of certain Markov process expectations for large time. I. Communications in Pure and Applied Mathematics 28: 1–47.CrossRefGoogle Scholar
11Forde, M. (2011). Large-time asymptotics for an uncorrelated stochastic volatility model. Statistics and Probability Letters 81: 1230–1232.CrossRefGoogle Scholar
12Forde, M. & Jacquier, A. (2009). Small-time asymptotics for implied volatility under the Heston model. International Journal of Theoretical and Applied Finance 12: 861–876.CrossRefGoogle Scholar
13Forde, M. & Jacquier, A. (2011). The large-maturity smile for the Heston model. Finance and Stochastics 15: 755–780.CrossRefGoogle Scholar
14Forde, M. & Kumar, R. (2016). Large-time option pricing using the Donsker-Varadhan LDP-correlated stochastic volatility with stochastic interest rates and jumps. Annals of Applied Probability 6: 3699–3726.Google Scholar
15Forde, M. & Pogudin, A. (2013). The large-maturity smile for the SABR and CEV-Heston models. International Journal of Theoretical and Applied Finance 16(8): 1350047.CrossRefGoogle Scholar
16Forde, M., Jacquier, A., & Lee, R. (2012). The small-time smile and term structure of implied volatility under the Heston model. SIAM Journal of Financial Mathematics 3: 690–708.CrossRefGoogle Scholar
17Fouque, J.-P., Lorig, M., & Sircar, R. (2016). Second order multiscale stochastic volatility asymptotics: Stochastic terminal layer analysis and calibration. Finance and Stochastics 63: 1648–1665.Google Scholar
18Fukasawa, M. (2011). Asymptotic analysis for stochastic volatility: Martingale expansion. Finance and Stochastics 15: 635–654.CrossRefGoogle Scholar
19Gao, K. & Lee, R. (2014). Asymptotics of implied volatility to arbitrary order. Finance and Stochastics 18(2): 349–392.CrossRefGoogle Scholar
20Gatheral, J., Hsu, E.P., Laurence, P., Ouyang, C., & Wang, T.H. (2012). Asymptotics of implied volatility in local volatility models. Mathematical Finance 22(4): 591–620.CrossRefGoogle Scholar
21Grischenko, O., Han, X., & Nistor, V. (2019). A volatility-of-volatility expansion of the option prices in the SABR stochastic volatility model. IJTAF 23(3): 205 0018.Google Scholar
22Gulisashvili, A. (2014). Analytically tractable stochastic stock price models. Springer Finance. New York: Springer.Google Scholar
23Gulisashvili, A. & Stein, E.M. (2006). Asymptotic behavior of the distribution of the stock price in models with stochastic volatility: The Hull-White model. Compte Rendus Academie Sciences Paris, Series I 343: 519–523.Google Scholar
24Gulisashvili, A. & Stein, E.M. (2009). Implied volatility in the Hull-White model. Mathematical Finance 19: 303–327.CrossRefGoogle Scholar
25Gulisashvili, A. & Stein, E.M. (2010). Asymptotic behavior of the distribution of the stock price in models with stochastic volatility, I. Mathematical Finance 20: 447.CrossRefGoogle Scholar
26Guyon, J. (2006). Euler scheme and tempered distributions. Stochastic Processes and their Applications 116: 877–904.CrossRefGoogle Scholar
27Hagan, P.S., Kumar, D., Lesniewski, A.S., Woodward, D.E. (2002). Managing smile risk. Wilmott Magazine, September 2002.Google Scholar
28Henry-Labordere, P. (2009). Analysis, geometry and modeling in finance: Advanced methods in option pricing. New York: Chapman & Hall.Google Scholar
29Hull, J. & White, A. (1987). Pricing of options on assets with stochastic volatilities. Journal of Finance 42: 281–300.CrossRefGoogle Scholar
30Islah, O. (2009). Solving SABR in exact form and unifying it with LIBOR market model. http://dx.doi.org/10.2139/ssrn.1489428.CrossRefGoogle Scholar
31Jourdain, B. (2004). Loss of martingality in asset price models with lognormal stochastic volatility, preprint Cermics, vol. 267.Google Scholar
32Lee, R. (2001). Implied and local volatilities under stochastic volatility. International Journal of Theoretical and Applied Finance 4: 45–89.CrossRefGoogle Scholar
33Lee, R. (2004). The moment formula for implied volatility at extreme strikes. Mathematical Finance 14: 469–480.CrossRefGoogle Scholar
34Leitao, A., Grzelak, L.A., & Oosterlee, C.W. (2017). On an efficient multiple time step Monte Carlo simulation of the SABR model. Quantitative Finance 8: 1–17.Google Scholar
35Lewis, A. Unpublished results (personal communication).Google Scholar
36Lewis, A. (2000). Option valuation under stochastic volatility: with Mathematica code. Newport Beach, California, USA: Finance Press.Google Scholar
37Lewis, A. (2002). The mixing approach to stochastic volatility and jump models. Wilmott Magazine, March 2002.Google Scholar
38Lewis, A. (2016). Option valuation under stochastic volatility, vol. 2. Newport Beach, California, USA: Finance Press.Google Scholar
39Lions, P.L. & Musiela, M. (2007). Correlations and bounds for stochastic volatility models. Annales de l'Institut Henri Poincaré (C) Non Linear Analysis 24: 1–16.CrossRefGoogle Scholar
40Lorig, M., Pagliarani, S., & Pascucci, A. (2007). Explicit implicit volatilities for multifactor local-stochastic volatility models. Mathematical Finance 27: 927–960.Google Scholar
41Olver, F.W.J. (1974). Introduction to asymptotics and special functions. New York: Academic Press.Google Scholar
42Paulot, L. (2009). Asymptotic implied volatility at the second order with application to the SABR model. arXiv:0906.0658[q-fin.PR].Google Scholar
43Paulot, L. (2013). Arbitrage-free pricing before and beyond probabilities. arXiv:1310.1102[q-fin].Google Scholar
44Pirjol, D. (2020). Asymptotic expansion for the Hartman-Watson distribution. arXiv:2001.09579[math.PR].Google Scholar
45Pirjol, D. & Zhu, L. (2016). Discrete sums of geometric Brownian motions, annuities and Asian options. Insurance: Mathematics and Economics 70: 19–37.Google Scholar
46Pirjol, D. & Zhu, L. (2016). Short maturity Asian options in local volatility models. SIAM Journal of Financial Mathematics 7(1): 947–992.CrossRefGoogle Scholar
47Pirjol, D. & Zhu, L. (2017). Asymptotics for the average of the geometric Brownian motion and Asian options. Advances in Applied Probability 49: 446–480.CrossRefGoogle Scholar
48Pirjol, D. & Zhu, L. (2018). Asymptotics for the Euler-discretized Hull-White stochastic volatility model. Methodology and Computing in Applied Probability 20: 289–331.CrossRefGoogle Scholar
49Renault, E. & Touzi, N. (1996). Option hedging and implied volatilities in a stochastic volatility model. Mathematical Finance 6: 279–302.CrossRefGoogle Scholar
50Rogers, L.C. & Tehranchi, M. (2010). Can the implied volatility surface move by parallel shifts? Finance and Stochastics 14: 235–248.CrossRefGoogle Scholar
51Wang, T.H., Laurence, P., & Wang, S.L. (2010). Generalized uncorrelated SABR models with a high degree of symmetry. Quantitative Finance 10: 663–679.CrossRefGoogle Scholar
52Yor, M. (1982). On some exponential functionals of the Brownian motion. Journal of Applied Probability 24(3): 509–531.CrossRefGoogle Scholar