# zbMATH — the first resource for mathematics

Spectral calibration of exponential Lévy models. (English) Zbl 1126.91022
The goal of this paper is to investigate the problem of nonparametric inference for the Lévy triplet when the asset price follows an exponential Lévy model. It is supposed that at time $$t=0$$ one disposes of prices for vanilla European call and put option on this asset with different strike prices and possibly different maturities. By basing the estimates on option data, the authors draw inference on the underlying risk neutral price process, which in general cannot be determined from the historical price data due to the incompleteness of the Lévy market. It is unrealistic to determine the triplet correctly, so they try to provide an estimator which is as good as possible for the given accuracy of the data. This optimality property is assessed by the minimax paradigm which quantifies the error in the worst case scenario. The lower bound is established and it is demonstrated that already in the simple exponential Lévy model the estimation problem is in general severely ill-posed. It means that the estimation error as a function of the accuracy of observations converges with a logarithmic rate. An explicit construction of an estimator is proposed that attains this optimal minimax rate.

##### MSC:
 91B28 Finance etc. (MSC2000) 60G51 Processes with independent increments; Lévy processes 62G20 Asymptotic properties of nonparametric inference
Full Text:
##### References:
 [1] Aït-Sahalia Y., Duarte J. Nonparametric option pricing under shape restrictions. J. Econom. 116(1–2), 9–47 (2003) · Zbl 1016.62121 [2] Aït-Sahalia Y., Jacod, J.: Volatility estimators for discretely sampled Lévy processes. Ann. Stat. (in press) (2006) · Zbl 1114.62109 [3] Belomestny, D., Reiß, M.: Optimal calibration of exponential Lévy models, Preprint 1017, Weierstraß Institute, Berlin. http://www.wias-berlin.de (2005) [4] Belomestny, D., Reiß, M.: Spectral calibration of exponential Lévy models [2]. Discussion Paper 35, Collaborative Research Center 649 Economic Risk, Berlin. http://sfb649.wiwi.hu-berlin.de (2006) · Zbl 1126.91022 [5] Breeden D., Litzenberger R. (1978) Prices of state-contingent claims implicit in options prices. J. Business 51, 621–651 [6] Brown L.D., Low M.G. (1996) Asymptotic equivalence of nonparametric regression and white noise. Ann. Stat. 24, 2384–2398 · Zbl 0867.62022 [7] Butucea C., Matias C. (2005) Minimax estimation of the noise level and of the deconvolution density in a semiparametric convolution model. Bernoulli 11, 309–340 · Zbl 1063.62044 [8] Carr P., Geman H., Madan D.B., Yor M. (2002) The fine structure of asset returns: An empirical investigation. J. Business 75, 305–332 [9] Carr P., Madan D. (1999) Option valuation using the fast Fourier transform. J. Comput. Financ. 2, 61–73 [10] Cont R., Tankov P. Financial Modelling With Jump Processes. In: Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton (2004) · Zbl 1052.91043 [11] Cont R., Tankov P. (2004) Nonparametric calibration of jump-diffusion option pricing models. J. Comput. Financ. 7(3): 1–49 · Zbl 1052.91043 [12] Cont R., Tankov P. Retrieving Lévy processes from option prices: regularization of an ill-posed inverse problem. SIAM J. Numer. Opt. Control (in press) (2005) · Zbl 1110.49033 [13] Cont R., Voltchkova E. (2005) Integro-differential equations for option prices in exponential Lévy models. Financ. Stoch. 9, 299–325 · Zbl 1096.91023 [14] Crépey S. (2003) Calibration of the local volatility in a generalized Black–Scholes model using Tikhonov regularization. SIAM J. Math. Anal. 34: 1183–1206 · Zbl 1126.35373 [15] Duffie D., Filipović D., Schachermayer W. (2003) Affine processes and applications in finance. Ann. Appl. Probab. 13, 984–1053 · Zbl 1048.60059 [16] Eberlein E., Keller U., Prause K. (1998) New insights into smile, mispricing, and value at risk: the hyperbolic model. J. Business 71, 371–405 [17] Emmer S., Klüppelberg C. (2004) Optimal portfolios when stock prices follow an exponential Lévy process. Financ. Stoch. 8, 17–44 · Zbl 1051.60049 [18] Fengler, M.: Semiparametric modeling of implied volatility. In: Springer Finance Series (2005) · Zbl 1084.62109 [19] Goldenshluger A., Tsybakov A., Zeevi A. (2006) Optimal change-point estimation from indirect observations. Ann. Stat. 34, 350–372 · Zbl 1091.62021 [20] Jackson N., Süli E., Howison S. (1999) Computation of deterministic volatility surfaces. J. Comput. Financ. 2(2): 5–32 [21] Kallsen J. (2000) Optimal portfolios for exponential Lévy processes. Math. Meth. Oper. Res. 51, 357–374 · Zbl 1054.91038 [22] Korostelev, A., Tsybakov, A.: Minimax Theory of Image Reconstruction. Lecture Notes in Statistics vol. 82. Springer, Berlin Heidelberg New York (1993) · Zbl 0833.62039 [23] Kou S. (2002) A jump diffusion model for option pricing. Manag. Sci. 48: 1086–1101 · Zbl 1216.91039 [24] Merton R. (1976) Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 125–144 · Zbl 1131.91344 [25] Mordecki E. (2002) Optimal stopping and perpetual options for Lévy processes. Financ. Stoch. 6, 473–493 · Zbl 1035.60038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.