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A framework for robust measurement of implied correlation. (English) Zbl 1319.91159

Summary: In this paper we consider the problem of deriving correlation estimates from observed option data. An implied correlation estimate arises when we match the observed index option price with a corresponding model price. The underlying model assumes that stock prices can be described using a lognormal distribution, while a Gaussian copula describes the dependence structure. Within this multivariate stock price model, the index option price is not given in a closed form and has to be approximated. Different methods exist and each choice leads to another implied correlation estimate.{ }We show that the traditional approach for determining implied correlations is a member of our more general framework. It turns out that the traditional implied correlation underestimates the real correlation. This error is more pronounced when some stock volatilities are large compared to the other volatility levels. We propose a new approach to measure implied correlation which does not have this drawback. However, our numerical illustrations show that determining implied correlations with the traditional approach may be justified for strike prices which are close to the at-the-money strike price.{ }We also show that implied correlation estimates can be used to define an index, called the Implied Correlation Index (ICX), which reflects the market’s perception about future (short-term) co-movement between stock prices. Using a volatility index together with the ICX gives an accurate description of the current level of market fear.

MSC:

91G70 Statistical methods; risk measures
62P05 Applications of statistics to actuarial sciences and financial mathematics
62H20 Measures of association (correlation, canonical correlation, etc.)
91G20 Derivative securities (option pricing, hedging, etc.)
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[1] Skintzi, V. D.; Skiadopoulos, G.; Refenes, A.-P. N., The effect of misestimating correlation on value at risk, J. Alternat. Invest., 7, 4, 66-82 (2005)
[2] Wong, A. S., ‘Quantitative impact of correlation errors on basket options with time-varying correlations’, J. Futures Markets, 32, 2, 152-165 (2012)
[3] Moskowitz, T., ‘An analysis of covariance risk and pricing anomalies’, Rev. Financ. Stud., 16, 2, 417-457 (2003)
[4] James, J.; Kasikov, K.; Edwards, K.-A., ‘The end of diversification’, Quant. Finance, 12, 11, 1629-1636 (2012)
[5] Skintzi, V. D.; Refenes, A. N., ‘Implied correlation index: A new measure of diversification’, J. Futures Markets, 25, 171-197 (2005)
[6] Cont, R.; Deguest, R., ‘Equity correlations implied by index options: Estimation and model uncertainty analysis’, Math. Finance, 23, 3, 496-530 (2013) · Zbl 1280.91167
[8] Kaas, R.; Dhaene, J.; Goovaerts, M. J., ‘Upper and lower bounds for sums of random variables’, Insurance Math. Econom., 27, 2, 151-168 (2000) · Zbl 0989.60019
[9] Dhaene, J.; Denuit, M.; Goovaerts, M.; Kaas, R.; Vyncke, D., ‘The concept of comonotonicity in actuarial science and finance: applications’, Insurance: Math. Econom., 31, 2, 133-161 (2002) · Zbl 1037.62107
[10] Dhaene, J.; Kukush, A.; Linders, D., ‘The multivariate black & scholes market: conditions for completeness and no-arbitrage’, Theory Probab. Math. Statist., 88, 1-14 (2013) · Zbl 1304.91215
[11] Björk, T., Arbitrage Theory in Continuous Time, 311 (1998), Oxford University Press
[12] Black, F.; Scholes, M., ‘The pricing of options and corporate liabilities’, J. Polit. Econ., 81, 3, 637-654 (1973) · Zbl 1092.91524
[13] Brooks, R.; Corson, J.; Wales, J. D., ‘The pricing of index options when the underlying assets all follow a lognormal diffusion’, Adv. Futures Options Res., 7 (1994)
[14] Milevsky, M.; Posner, S., ‘A closed-form approximation for valuing basket options’, J. Deriv., 5, 4, 54-61 (1998)
[15] Brigo, D.; Mercurio, F.; Rapisarda, F.; Scotti, R., ‘Approximated moment-matching dynamics for basket-options pricing’, Quant. Finance, 4, 1, 1-16 (2004) · Zbl 1405.91592
[16] Deelstra, G.; Liinev, J.; Vanmaele, M., ‘Pricing of arithmetic basket options by conditioning’, Insurance Math. Econom., 34, 1, 55-77 (2004) · Zbl 1068.91030
[17] Carmona, R.; Durrleman, V., ‘Generalizing the black-scholes formula to multivariate contingent claims’, J. Comput. Finance, 9, 43-67 (2006)
[18] Korn, R.; Zeytun, S., ‘Efficient basket monte carlo option pricing via a simple analytical approximation’, J. Comput. Appl. Math., 243, 1, 48-59 (2013) · Zbl 1258.91220
[19] Krekel, M.; de Kock, J.; Korn, R.; Man, T.-K., ‘An analysis of pricing methods for baskets options’, Wilmott Mag., April, 82-89 (2007)
[20] Linders, D., Pricing index options in a multivariate Black & Scholes model, (Research report afi-1383 feb (2013), KU Leuven - Faculty of Business and Economics: KU Leuven - Faculty of Business and Economics Leuven)
[21] Vanduffel, S.; Hoedemakers, T.; Dhaene, J., ‘Comparing approximations for risk measures of sums of non-independent lognormal random variables’, N. Am. Actuar. J., 9, 4, 71-82 (2005) · Zbl 1215.91038
[22] Dhaene, J.; Vanduffel, S.; Goovaerts, M. J.; Kaas, R.; Vyncke, D., ‘Comonotonic approximations for optimal portfolio selection problems’, J. Risk Insur., 72, 253-301 (2005)
[23] Van Weert, K.; Dhaene, J.; Goovaerts, M. J., ‘Comonotonic approximations for a generalized provisioning problem with application to optimal portfolio selection’, J. Comput. Appl. Math., 235, 3245-3256 (2011) · Zbl 1211.91228
[24] Van Weert, K.; Dhaene, J.; Goovaerts, M., ‘Optimal portfolio selection for general provisioning and terminal wealth problems’, Insurance Math. Econom., 47, 1, 90-97 (2010) · Zbl 1231.91422
[25] Dhaene, J.; Denuit, M.; Goovaerts, M.; Kaas, R.; Vyncke, D., ‘The concept of comonotonicity in actuarial science and finance: theory’, Insurance: Math. Econom., 31, 1, 3-33 (2002) · Zbl 1051.62107
[26] Deelstra, G.; Dhaene, J.; Vanmaele, M., An overview of comonotonicity and its applications in finance and insurance, (Oksendal, B.; Nunno, G., Advanced Mathematical Methods for Finance (2011), Springer: Springer Berlin, Heidelberg), 155-179 · Zbl 1233.60006
[27] Rubinstein, M., ‘Implied binomial trees’, J. Finance, 49, 3, 771-818 (1994)
[28] Chriss, N., Black Scholes and Beyond: Option Pricing Models (1996), McGraw-Hill Companies, Incorporated
[29] Gatheral, J., The Volatility Surface: A Practitioner’s Guide (2006), Wiley Finance: Wiley Finance Wiley
[30] Schmid, F.; Schmidt, R., ‘Multivariate extensions of Spearman’s rho and related statistics’, Statist. Probab. Lett., 77, 4, 407-416 (2007) · Zbl 1108.62056
[31] Dhaene, J.; Linders, D.; Schoutens, W.; Vyncke, D., ‘A multivariate dependence measure for aggregating risks’, J. Comput. Appl. Math., 263, 78-87 (2014) · Zbl 1386.91172
[32] Huang, H.; Milevsky, M.; Wang, J., Ruined moments in your life: How good are the approximations?, Insurance: Math. Econom., 3, 34, 421-447 (2004) · Zbl 1188.91233
[33] Embrechts, P.; McNeil, A.; Straumann, D., Correlation and dependence in risk management: properties and pitfalls, (‘Risk Management: Value at Risk and Beyond’ (1999), Cambridge University Press), 176-223
[34] Guillaume, F.; Schoutens, W., ‘Calibration risk: Illustrating the impact of calibration risk under the heston model’, Rev. Deriv. Res., 15, 1, 57-79 (2012)
[35] Cox, J. C.; Ross, S. A.; Rubinstein, M., ‘Option pricing: A simplified approach’, J. Financ. Econom., 7, 3, 229-263 (1979) · Zbl 1131.91333
[36] Carr, P.; Wu, L., ‘A tale of two indices’, J. Deriv., 13, 3, 13-29 (2006)
[37] Whaley, R., ‘The investor fear gauge’, J. Portfolio Manag., 26, 12-17 (2000)
[39] Moosbrucker, T., ‘Explaining the correlation smile using Variance Gamma distributions’, J. Fixed Income, 16, 1, 71-87 (2006)
[40] Albrecher, H.; Ladoucette, S.; Schoutens, W., A generic one-factor lévy model for pricing synthetic cdos, (Fu, M.; Jarrow, R.; Yen, J.-Y.; Elliott, R., Advances in Mathematical Finance’, Applied and Numerical Harmonic Analysis (2007), Birkhäuser: Birkhäuser Boston), 259-277 · Zbl 1154.91421
[41] Garcia, J.; Goossens, S.; Masol, V.; Schoutens, W., ‘Levy base correlation’, Wilmott J., 1, 95-100 (2009)
[42] Tavin, B., Hedging dependence risk with spread options via the power frank and power student t copulas, (Technical Report (2013), Université Paris I Panthéon-Sorbonne), Available at SSRN: http://ssrn.com/abstract=2192430
[43] Guillaume, F.; Linders, D., (Stochastic modeling of herd behavior indices, Research report AFI-1490 FEB (2014), KU Leuven)
[44] Dhaene, J.; Linders, D.; Schoutens, W.; Vyncke, D., ‘The herd behavior index: A new measure for the implied degree of co-movement in stock markets’, Insurance Math. Econom., 50, 3, 357-370 (2012) · Zbl 1237.91237
[46] Laurence, P.; Wang, T.-H., ‘Distribution-free upper bounds for spread options and market-implied antimonotonicity gap’, Eur. J. Finance, 14, 8, 717-734 (2008)
[47] Dhaene, J.; Dony, J.; Forys, M. B.; Linders, D.; Schoutens, W., Fix - the fear index: Measuring market fear, (Cummins, M.; etal., ‘Topics in Numerical Methods for Finance. ‘Topics in Numerical Methods for Finance, Springer Proceedings in Mathematics and Statistics’ (2012)) · Zbl 1296.91219
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