Empirical performance of a spline-based implied volatility surface
Tóm tắt
Since the crash of 1987, it has been observed by option market participants that implied volatilities for out-of-the-money options are higher than predicted by the constant volatility Black-Scholes (1973) model. Option prices also exhibit dependence on time to expiry. The collection of these implied volatilities across strike and maturity is known as the implied volatility surface (IVS). We propose a nonparametric spline-based representation of the IVS and evaluate its empirical performance. Our findings indicate that the proposed model significantly outperforms the best performing implied volatility model reported in the current literature for the purpose of pricing European-style S&P500 index options. We further contribute to the empirical finance literature by choosing a proper model evaluation criterion. By measuring the leave-one-out cross-validation model pricing error of the thin-plate spline-based model, we demonstrate that this superior performance is not the result of overfitting. Although we have previously shown that spline-based models have superior empirical performance, the models considered in this study have advantages over the previously considered models.
Tài liệu tham khảo
Achdou, Y., Indragoby, G. and Pironneau, O. (2004) Volatility calibration with American options. Methods and Applications of Analysis 11 (3): 001–024.
Bakshi, G., Cao, C. and Chen, Z. (1997) Empirical performance of alternative option pricing models. Journal of Finance 52 (5): 2003–2050.
Bates, D.S. (2003) Empirical option pricing: A retrospection. Journal of Econometrics 116 (1–2): 387–404.
Black, F. and Scholes, M.S. (1973) The pricing of options and corporate liabilities. Journal of Political Economy 81 (3): 637–659.
Carr, P. and Wu, L. (2003) Finite moment log stable process and option pricing. Journal of Finance 58 (2): 753–777.
Carr, P. and Wu, L. (2004) Time-changed levy processes and option pricing. Journal of Financial Economics 17 (1): 113–141.
Christoffersen, P., Heston, S. and Jacobs, K. (2009) The shape and term structure of the index option smirk: Why multifactor stochastic volatility models work so well. Management Science 55 (12): 1914–1932.
Christoffersen, P. and Jacobs, K. (2004) The importance of the loss function in option valuation. Journal of Financial Economics 72 (2): 291–318.
Cont, R. and Fonseca, J. (2002) Dynamics of implied volatility surfaces. Quantitative Finance 2 (2): 45–60.
Crépey, S. (2003) Calibration of the local volatility in a trinomial tree using tikhonov regularization. Inverse Problems 19 (1): 91–127.
Derman, E. and Kani, I. (1994) Riding on a smile. Risk 7 (2): 32–39.
Duan, J. and Wei, J. (2009) Systematic risk and the price structure of individual equity options. Review of Financial Studies 22 (5): 1981–2006.
Dumas, B., Fleming, J. and Whaley, R.E. (1997) Implied volatility functions: Empirical tests. Journal of Finance 53 (6): 2059–2106.
Dupire, B. (1994) Pricing with a smile. Risk 7 (1): 18–20.
Egger, H. and Engl, H. (2005) Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates. Inverse Problems 21 (3): 1027–1045.
Fengler, M.R. (2009) Arbitrage-free smoothing of the implied volatility surface. Quantitative Finance 9 (4): 417–428.
Figlewski, S. (2009) Estimating the Implied Risk-Neutral Density for the US Market Portfolio, Volatility and Time Series Econometrics: Essays in Honor of Robert F. Engle, Oxford University Press.
Goncalves, S. and Guidolin, M. (2006) Predictable dynamics in the S&P 500 index options volatility vurface. Journal of Business 79 (3): 1591–1635.
Heston, S.L. (1993) A closed-form solution for options with stochastic volatility applications to bond and currency options. Review of Financial Studies 6 (2): 327–343.
Jiang, G. and Tian, Y. (2007) Extracting model-free volatility from option prices: An examination of the VIX index. Journal of Derivatives 14 (3): 35–60.
Lagnado, R. and Osher, S. (1997) A technique for calibrating derivative security pricing models: Numerical solution of an inverse problem. Journal of Computational Finance 1 (1): 13–25.
Merton, R. (1976) Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3 (1–2): 125–144.
Orosi, G. (2010) Improved implementation of local volatility and its application to S&P 500 index options. Journal of Derivatives 17 (3): 53–64.
Stone, M. (1974) Cross-validatory choice and assessment of statistical predictions. Journal of the Royal Statistical Society. B 36 (1): 111–147.
Wahba, G. (1990) Splines Models for Observational Data, Series in Applied Mathematics. CBMS-NSF Regional Conference Series in Applied Mathematics, p. 56.
Yang, Y. (2007) Consistency of cross validation for comparing regression procedures. Annals of Statistics 35 (6): 1564–1599.