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Technical report : Risk-neutral density recovery via spectral analysis

  • Jean-Baptiste Monnier
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    In this paper, we propose a new method for estimating the conditional risk-neutral density (RND) directly from a cross-section of put option bid-ask quotes. More precisely, we propose to view the RND recovery problem as an inverse problem. We first show that it is possible to define restricted put and call operators that admit a singular value decomposition (SVD), which we compute explicitly. We subsequently show that this new framework allows us to devise a simple and fast quadratic programming method to recover the smoothest RND whose corresponding put prices lie inside the bid-ask quotes. This method is termed the spectral recovery method (SRM). Interestingly, the SVD of the restricted put and call operators sheds some new light on the RND recovery problem. The SRM improves on other RND recovery methods in the sense that: - it is fast and simple to implement since it requires solution of a single quadratic program, while being fully nonparametric; - it takes the bid ask quotes as sole input and does not require any sort of calibration, smoothing or preprocessing of the data; - it is robust to the paucity of price quotes; - it returns the smoothest density giving rise to prices that lie inside the bid ask quotes. The estimated RND is therefore as well-behaved as can be; - it returns a closed form estimate of the RND on the interval [0,B] of the positive real line, where B is a positive constant that can be chosen arbitrarily. We thus obtain both the middle part of the RND together with its full left tail and part of its right tail. We confront this method to both real and simulated data and observe that it fares well in practice. The SRM is thus found to be a promising alternative to other RND recovery methods.

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    File URL: http://arxiv.org/pdf/1302.2567
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    Paper provided by arXiv.org in its series Papers with number 1302.2567.

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    Date of creation: Feb 2013
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    Handle: RePEc:arx:papers:1302.2567
    Contact details of provider: Web page: http://arxiv.org/

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    1. Jackwerth, Jens Carsten & Rubinstein, Mark, 1996. " Recovering Probability Distributions from Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1611-32, December.
    2. Bondarenko, Oleg, 2003. "Estimation of risk-neutral densities using positive convolution approximation," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 85-112.
    3. Jarrow, Robert & Rudd, Andrew, 1982. "Approximate option valuation for arbitrary stochastic processes," Journal of Financial Economics, Elsevier, vol. 10(3), pages 347-369, November.
    4. Mark H. A. Davis & David G. Hobson, 2007. "The Range Of Traded Option Prices," Mathematical Finance, Wiley Blackwell, vol. 17(1), pages 1-14.
    5. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
    6. Ruijun Bu & Kaddour Hadri, 2007. "Estimating option implied risk-neutral densities using spline and hypergeometric functions," Econometrics Journal, Royal Economic Society, vol. 10(2), pages 216-244, 07.
    7. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-51, October.
    8. Hentschel, Ludger, 2003. "Errors in Implied Volatility Estimation," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 38(04), pages 779-810, December.
    9. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
    10. Ait-Sahalia, Yacine & Duarte, Jefferson, 2003. "Nonparametric option pricing under shape restrictions," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 9-47.
    11. Buchen, Peter W. & Kelly, Michael, 1996. "The Maximum Entropy Distribution of an Asset Inferred from Option Prices," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 31(01), pages 143-159, March.
    12. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
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