New Approach to Density Estimation and Application to Value-at-Risk

The key contribution in this paper is to provide a new approach in estimating the physical distribution of the underlying asset return by using a quadratic Radon-Nikodym derivative function. The latter function transforms a fitted Variance Gamma risk-neutral distribution that is obtained from traded option prices. The generality of the VG distribution helps to avoid unnecessary mis-specification bias. The estimated empirical distribution is then used to find the risk measure of VaR. We show that possible underestimation of VaR risk using existing methods is largely not due to VaR itself but perhaps due to mis-specification errors which we minimize in our approach. Our method of measuring VaR clearly captures large tail risk in the empirical examples on S&P 500 index.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Lim, K. , Cheng, H. and L. Yap, N. (2015) New Approach to Density Estimation and Application to Value-at-Risk. Journal of Mathematical Finance, 5, 423-432. doi: 10.4236/jmf.2015.55036.

 [1] Rubinstein, M. (1996) Implied Binomial Trees. Journal of Finance, 49, 771-818. http://dx.doi.org/10.1111/j.1540-6261.1994.tb00079.x [2] Jackwerth, R. and M. Rubinstein (1996) Recovering Probability Distributions from Option Prices. Journal of Finance, 51, 1611-1631. http://dx.doi.org/10.1111/j.1540-6261.1996.tb05219.x [3] Breeden, D.T. and Robert H. Litzenberger (1978) Prices of State-Contingent Claims Implicit in Option Prices. The Journal of Business, 51, 621-651. http://dx.doi.org/10.1086/296025 [4] Longstaff, F.A. (1995) Option Pricing and the Martingale Restriction. Review of Financial Studies, 8, 1091-1124. http://dx.doi.org/10.1093/rfs/8.4.1091 [5] Chernov, M. and Ghysels, E. (2000) A Study towards a Unified Approach to the Joint Estimation of Objective and Risk Neutral Measures for the Purpose of Options Valuation. Journal of Financial Economics, 56, 407-458. http://dx.doi.org/10.1016/S0304-405X(00)00046-5 [6] Bakshi G., Kapadia, N. and Madan, D. (2003) Stock Return Characteristics, Skew Laws and the Differential Pricing of Individual Equity Options. Review of Financial Studies, 16, 101-143. http://dx.doi.org/10.1093/rfs/16.1.0101 [7] Carr, P. and Madan, D. (2001) Optimal Positioning in Derivative Securities. Quantitative Finance, 1, 19-37. http://dx.doi.org/10.1080/713665549 [8] Barndorff-Nielsen (1977) Exponentially Decreasing Distributions for the Logarithm of Particle Size. Proceedings of the Royal Society of London, 353, 401-419. http://dx.doi.org/10.1098/rspa.1977.0041 [9] Eberlein, E. and Keller, U. (1995) Hyperbolic Distributions in Finance. Bernoulli, 1, 281-299. http://dx.doi.org/10.2307/3318481 [10] Eberlein, E. and Hammerstein, E. (2004) Generalized Hyperbolic and Inverse Gaussian Distributions: Limiting Cases and Approximation of Processes. Seminar on Stochastic Analysis, Random Fields, and Applications IV. [11] Jiang, G.J. and Tian, Y.S. (2005) The Model-Free Implied Volatility and Its Information Content. Review of Financial Studies, 18, 1305-1342. http://dx.doi.org/10.1093/rfs/hhi027 [12] Jorion, P. (2000) Value at Risk. 2nd Edition, McGraw-Hill, North-America. [13] Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999) Coherent Measures of Risk. Mathematical Finance, 9, 203-228. http://dx.doi.org/10.1111/1467-9965.00068 [14] Basel Committee on Banking Supervision (2012) Fundamental Review of the Trading Book. [15] Basel Committee on Banking Supervision (2013) Fundamental Review of the Trading Book: A Revised Market Risk Framework. [16] Danielsson, J. (2013) An Evaluation of Basel III VaR and ES Probabilities. http://www.modelsandrisk.org/basel/ [17] Danielsson, J., de Vries, C., Jorgensen, B., Mandira, S. and Samorodnitsky, G. (2013) Fat Tails, VaR and Subadditivity. Journal of Econometrics, 172, 283-291. http://dx.doi.org/10.1016/j.jeconom.2012.08.011 [18] Yasuhiro, Y. and Yoshiba, T. (2002) On the Validity of Value-at-Risk: Comparative Analyses with Expected Shortfall. Monetary and Economic Studies, Bank of Japan. [19] Yasuhiro, Y. and Yoshiba, T. (2005) Value-at-Risk versus Expected Shortfall: A Practical Perspective. Journal of Banking and Finance, 29, 997-1015. http://dx.doi.org/10.1016/j.jbankfin.2004.08.010 [20] Steven, K., Peng, X. and Heyde, C.C. (2013) External Risk Measures and Basel Accords. Mathematics of Operations Research, 38, 393-417. http://dx.doi.org/10.1287/moor.1120.0577