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Asymptotics for exponential Lévy processes and their volatility smile: survey and new results. (English) Zbl 1275.91101

Summary: Exponential Lévy processes can be used to model the evolution of various financial variables such as FX rates, stock prices, and so on. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes, and the corresponding implied volatility surfaces have been analyzed in some detail. In the non-asymptotic regimes, option prices are described by the Lewis-Lipton formula, which allows one to represent them as Fourier integrals, and the prices can be trivially expressed in terms of their implied volatility. Recently, attempts at calculating the asymptotic limits of the implied volatility have yielded several expressions for the short-time, long-time, and wing asymptotics.
In order to study the volatility surface in required detail, in this paper we use the FX conventions and describe the implied volatility as a function of the Black-Scholes delta. Surprisingly, this convention is closely related to the resolution of singularities frequently used in algebraic geometry. In this framework, we survey the literature, reformulate some known facts regarding the asymptotic behavior of the implied volatility, and present several new results. We emphasize the role of fractional differentiation in studying the tempered stable exponential Lévy processes and derive novel numerical methods based on judicious finite-difference approximations for fractional derivatives. We also briefly demonstrate how to extend our results in order to study important cases of local and stochastic volatility models, whose close relation to the Lévy process based models is particularly clear when the Lewis-Lipton formula is used.
Our main conclusion is that studying asymptotic properties of the implied volatility is not always practically useful because the domain of validity of many asymptotic expressions is small.

MSC:

91B70 Stochastic models in economics
91G20 Derivative securities (option pricing, hedging, etc.)
91G60 Numerical methods (including Monte Carlo methods)
60G51 Processes with independent increments; Lévy processes
91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance
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References:

[1] DOI: 10.1007/s00780-007-0049-1 · Zbl 1145.91020 · doi:10.1007/s00780-007-0049-1
[2] DOI: 10.1007/s00780-010-0142-8 · Zbl 1303.91165 · doi:10.1007/s00780-010-0142-8
[3] DOI: 10.1023/A:1011354913068 · Zbl 1274.91398 · doi:10.1023/A:1011354913068
[4] Andersen L., Interest Rate Modeling (2010)
[5] Applebaum D., Notices of the AMS 51 pp 1336– (2004)
[6] DOI: 10.1017/CBO9780511755323 · Zbl 1073.60002 · doi:10.1017/CBO9780511755323
[7] DOI: 10.1007/978-3-642-58009-3 · doi:10.1007/978-3-642-58009-3
[8] DOI: 10.1239/jap/996986757 · Zbl 0989.60047 · doi:10.1239/jap/996986757
[9] DOI: 10.1007/s007800050032 · Zbl 0894.90011 · doi:10.1007/s007800050032
[10] DOI: 10.1093/rfs/9.1.69 · doi:10.1093/rfs/9.1.69
[11] DOI: 10.1239/jap/1208358948 · Zbl 1151.62079 · doi:10.1239/jap/1208358948
[12] DOI: 10.1111/j.1467-9965.2008.00354.x · Zbl 1155.91377 · doi:10.1111/j.1467-9965.2008.00354.x
[13] DOI: 10.1088/1469-7688/2/1/305 · doi:10.1088/1469-7688/2/1/305
[14] DOI: 10.1002/cpa.20039 · Zbl 1181.91356 · doi:10.1002/cpa.20039
[15] Bertoin J., Lévy Processes (1996)
[16] DOI: 10.1086/260062 · Zbl 1092.91524 · doi:10.1086/260062
[17] DOI: 10.1142/S0219024900000541 · Zbl 0973.91037 · doi:10.1142/S0219024900000541
[18] DOI: 10.1142/9789812777485 · doi:10.1142/9789812777485
[19] DOI: 10.1098/rspa.2011.0670 · Zbl 1364.91148 · doi:10.1098/rspa.2011.0670
[20] DOI: 10.1086/338705 · doi:10.1086/338705
[21] DOI: 10.21314/JCF.1999.043 · doi:10.21314/JCF.1999.043
[22] DOI: 10.1111/1540-6261.00544 · doi:10.1111/1540-6261.00544
[23] DOI: 10.1046/j.1540-6261.2003.00616.x · doi:10.1046/j.1540-6261.2003.00616.x
[24] DOI: 10.1016/S0375-9601(97)00947-X · Zbl 1026.82524 · doi:10.1016/S0375-9601(97)00947-X
[25] DOI: 10.1016/j.jcp.2007.05.012 · Zbl 1165.65053 · doi:10.1016/j.jcp.2007.05.012
[26] Clark I. J., Foreign Exchange Option Pricing, A Practitioner’s Guide (2011)
[27] Cont R., Financial Modelling with Jump Processes (2004) · Zbl 1052.91043
[28] De Bruijn N. G., Asymptotic Methods in Analysis (1970) · Zbl 0082.04202
[29] DOI: 10.1093/imanum/drh011 · Zbl 1134.91405 · doi:10.1093/imanum/drh011
[30] DOI: 10.1007/978-1-4612-5320-4 · doi:10.1007/978-1-4612-5320-4
[31] DOI: 10.1016/j.cma.2004.06.006 · Zbl 1119.65352 · doi:10.1016/j.cma.2004.06.006
[32] DOI: 10.1239/jap/1214950366 · Zbl 1152.91682 · doi:10.1239/jap/1214950366
[33] DOI: 10.2307/3318481 · Zbl 0836.62107 · doi:10.2307/3318481
[34] Fedoryuk M. V., The Saddle-Point Method (1977) · Zbl 0463.41020
[35] DOI: 10.1016/j.spl.2008.07.012 · Zbl 1489.60074 · doi:10.1016/j.spl.2008.07.012
[36] DOI: 10.1142/S021902490900549X · Zbl 1203.91290 · doi:10.1142/S021902490900549X
[37] DOI: 10.1098/rspa.2009.0610 · Zbl 1211.91253 · doi:10.1098/rspa.2009.0610
[38] DOI: 10.1093/imamat/26.3.209 · Zbl 0455.35069 · doi:10.1093/imamat/26.3.209
[39] Gerber H. U., Transactions of the Society of Actuaries 46 pp 99– (1994)
[40] DOI: 10.1137/090762713 · Zbl 1284.91545 · doi:10.1137/090762713
[41] Henry-Labordere P., Analysis, Geometry, and Modeling in Finance (2009)
[42] DOI: 10.1093/rfs/6.2.327 · Zbl 1384.35131 · doi:10.1093/rfs/6.2.327
[43] DOI: 10.1007/s10614-011-9269-8 · Zbl 1254.91747 · doi:10.1007/s10614-011-9269-8
[44] Janek A., Statistical Tools for Finance and Insurance (2011)
[45] Jensen J. L., Saddlepoint Approximations (1995) · Zbl 1274.62008
[46] DOI: 10.1103/PhysRevE.52.1197 · doi:10.1103/PhysRevE.52.1197
[47] DOI: 10.1287/mnsc.48.8.1086.166 · Zbl 1216.91039 · doi:10.1287/mnsc.48.8.1086.166
[48] DOI: 10.1007/BFb0077628 · doi:10.1007/BFb0077628
[49] DOI: 10.21314/JCF.2004.121 · doi:10.21314/JCF.2004.121
[50] DOI: 10.1111/j.0960-1627.2004.00200.x · Zbl 1134.91443 · doi:10.1111/j.0960-1627.2004.00200.x
[51] Lewis A., Option Valuation Under Stochastic Volatility (2000) · Zbl 0937.91060
[52] DOI: 10.1142/4694 · doi:10.1142/4694
[53] DOI: 10.1093/oxfordhb/9780199546787.001.0001 · doi:10.1093/oxfordhb/9780199546787.001.0001
[54] DOI: 10.2307/1426607 · Zbl 0425.60042 · doi:10.2307/1426607
[55] Lukacs E., Characteristic Functions (1970)
[56] DOI: 10.1086/296519 · doi:10.1086/296519
[57] DOI: 10.1086/294632 · doi:10.1086/294632
[58] DOI: 10.1214/ECP.v14-1452 · Zbl 1192.60071 · doi:10.1214/ECP.v14-1452
[59] Marchaud A., Journal de mathématiques pures et appliquées 9e série 6 pp 337– (1927)
[60] Martin R., The Oxford Hanbook of Credit Derivatives (2011)
[61] DOI: 10.1007/978-94-009-8410-3 · doi:10.1007/978-94-009-8410-3
[62] DOI: 10.1142/S0219024900000073 · Zbl 1154.91465 · doi:10.1142/S0219024900000073
[63] Matytsin A., Modelling Volatility and Volatility Derivatives (1999)
[64] DOI: 10.1016/j.cam.2004.01.033 · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033
[65] DOI: 10.2307/3003143 · doi:10.2307/3003143
[66] DOI: 10.1016/0304-405X(76)90022-2 · Zbl 1131.91344 · doi:10.1016/0304-405X(76)90022-2
[67] Miller K., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993) · Zbl 0789.26002
[68] Nolan J. P., Stable Distributions. Models for Heavy Tailed Data (2011)
[69] DOI: 10.1051/ps:1997114 · Zbl 0899.60065 · doi:10.1051/ps:1997114
[70] Podlubny I., Fractional Differential Equations (1999) · Zbl 0924.34008
[71] DOI: 10.3905/jod.2010.18.2.058 · doi:10.3905/jod.2010.18.2.058
[72] DOI: 10.1007/s00780-008-0081-9 · Zbl 1224.91197 · doi:10.1007/s00780-008-0081-9
[73] DOI: 10.1214/aoap/1029962752 · Zbl 0963.91054 · doi:10.1214/aoap/1029962752
[74] DOI: 10.1016/j.spa.2006.10.003 · Zbl 1118.60037 · doi:10.1016/j.spa.2006.10.003
[75] Ruschendorf L., Bernoulli 8 pp 81– (2002)
[76] Samorodnitsky G., Stable Non-Gaussian Random Processes (1994) · Zbl 0925.60027
[77] Sato K., Lévy Processes and Infinitely Divisible Distributions (1999) · Zbl 0973.60001
[78] DOI: 10.1002/0470870230 · doi:10.1002/0470870230
[79] DOI: 10.1137/1.9781611971408 · doi:10.1137/1.9781611971408
[80] DOI: 10.1016/j.jcp.2005.08.008 · Zbl 1089.65089 · doi:10.1016/j.jcp.2005.08.008
[81] DOI: 10.1007/978-3-642-14660-2_5 · Zbl 1205.91161 · doi:10.1007/978-3-642-14660-2_5
[82] DOI: 10.1239/jap · doi:10.1239/jap
[83] DOI: 10.1002/cpa.3160200210 · doi:10.1002/cpa.3160200210
[84] DOI: 10.1002/cpa.3160200404 · doi:10.1002/cpa.3160200404
[85] Zolotarev V. M., Translation of Mathematical Monographs 65, in: One-Dimensional Stable Distributions (1986) · Zbl 0589.60015
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