Nonparametric estimates of option prices via Hermite basis functions
Tóm tắt
We consider approximate pricing formulas for European options based on approximating the logarithmic return’s density of the underlying by a linear combination of rescaled Hermite polynomials. The resulting models, that can be seen as perturbations of the classical Black-Scholes one, are nonpararametric in the sense that the distribution of logarithmic returns at fixed times to maturity is only assumed to have a square-integrable density. We extensively investigate the empirical performance, defined in terms of out-of-sample relative pricing error, of this class of approximating models, depending on their order (that is, roughly speaking, the degree of the polynomial expansion) as well as on several ways to calibrate them to observed data. Empirical results suggest that such approximate pricing formulas, when compared with simple nonparametric estimates based on interpolation and extrapolation on the implied volatility curve, perform reasonably well only for options with strike price not too far apart from the strike prices of the observed sample.
Tài liệu tham khảo
Ait-Sahalia, Y., Lo, A.W.: Nonparametric estimation of state-price densities implicit in financial asset prices. J. Finance 53(2), 499–547 (1998)
Askey, R., Wainger, S.: Mean convergence of expansions in Laguerre and Hermite series. Am. J. Math. 87, 695–708 (1965)
Breeden, D.T., Litzenberger, R.H.: Prices of state-contingent claims implicit in option prices. J. Bus. 51(4), 621–651 (1978)
Dobrushin, R.L., Major, P.: Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50(1), 27–52 (1979)
Föllmer, H., Schied, A.: Stochastic Finance. Walter de Gruyter & Co., Berlin (2004)
Grith, M., Härdle, W.K., Schienle, M.: Nonparametric estimation of risk-neutral densities. In: Duan, J.C., Härdle, W.K., Gentle, J.E. (eds.) Handbook of Computational Finance, pp. 277–305. Springer Verlag, Berlin, Heidelberg (2012)
Jameson, G.J.O.: The incomplete gamma functions. Math. Gaz. 100(548), 298–306 (2016)
Janson, S.: Gaussian Hilbert spaces. Cambridge University Press (1997)
Kolassa, J.E.: Series approximation methods in statistics, 2nd ed., Lecture Notes in Statistics, vol. 88, Springer-Verlag, New York, (1997)
Malliavin, P.: Stochastic analysis. Springer Verlag, Berlin (1997)
Marinelli, C.: On certain representations of pricing functionals (2021), arXiv:2109.05564
Marinelli, C., d’Addona, S.: Nonparametric estimates of pricing functionals. J. Empir. Finance 44, 19–35 (2017)
Muckenhoupt, B.: Mean convergence of Hermite and Laguerre series I II. Trans. Amer. Math. Soc. 147, 419-431; ibid. 147 (1970), 433–460. MR 0256051
Olver, F.W.J. et al (eds.): NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov, Release 1.1.6 of (2022)
Stoyanov, S.V., Rachev, S.T., Mittnik, S., Fabozzi, F.J.: Pricing derivatives in Hermite markets. Int. J. Theor. Appl. Finance 22(6), 1950031 (2019)
Taqqu, M.S.: Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50(1), 53–83 (1979)
Xiu, D.: Hermite polynomial based expansion of European option prices. J. Econom. 179(2), 158–177 (2014)