Valuing European Put Options under Skewness and Increasing [Excess] Kurtosis
Author(s)
John-Peter D. Chateau*
ABSTRACT
To capture the impact of skewness and increase kurtosis on Black’s [1] European put values, we first substitute a Gram-Charlier (GC) distribution and next a Johnson distribution for Black’s Gaussian one. We introduce next each distribution in the option payoff and develop until the closedform expression of each put is arrived at. Finally, we estimate by simulations GC, Johnson and Black put options, choosing the latter one as benchmark. Simulation estimates encompassing both skewness and kurtosis show that, for at-the-money (ATM) or slightly in-the-money put values, 1) Black’s overvaluation with respect to Johnson puts is very significant and 2) its undervaluation with respect to GC ones remains moderate. Yet, by using the same skewness values for both GC and Johnson puts, we highlight the differences induced by increasing kurtosis between the two models. In this case, the GC overvaluation for ATM values is explained by value differences in the put time component. Yet, while both Black and GC values exhibit significant time decay close to expiry, Johnson’s ones remain stable up to maturity.
KEYWORDS
Skewness and Increasing Kurtosis in Gram-Charlier (GC) and Johnson Distributions, At-the-Money Overvaluation of Black’s but Undervaluation of GC European Futures Put Values in Terms of Johnson’s Ones, Value Differences Explained by the Put Time Component
Skewness and Increasing Kurtosis in Gram-Charlier (GC) and Johnson Distributions, At-the-Money Overvaluation of Black’s but Undervaluation of GC European Futures Put Values in Terms of Johnson’s Ones, Value Differences Explained by the Put Time Component
Cite this paper
Chateau, J. (2014) Valuing European Put Options under Skewness and Increasing [Excess] Kurtosis. Journal of Mathematical Finance, 4, 160-177. doi: 10.4236/jmf.2014.43015.
Chateau, J. (2014) Valuing European Put Options under Skewness and Increasing [Excess] Kurtosis. Journal of Mathematical Finance, 4, 160-177. doi: 10.4236/jmf.2014.43015.
References
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http://dx.doi.org/10.1016/0304-405X(76)90024-6 [2] Barton, D.E. and Dennis, K.E.R. (1952) The Conditions under Which Gram-Charlier and Edgeworth Curves are Positive Definite and Unimodal. Biometrika, 39, 425-427. [3] Schlogl, E. (2013) Option Pricing Where the Underlying Assets Follow a Gram-Charlier Density of Arbitrary Order. Journal of Economic Dynamics and Control, 37, 611-632.
http://dx.doi.org/10.1016/j.jedc.2012.10.001 [4] Jondeau, E. and Rockinger, R. (2001) Gram-Charlier Densities. Journal of Economic Dynamics and Control, 25, 1457-1483.
http://dx.doi.org/10.1016/S0165-1889(99)00082-2 [5] Jha, R. and Kalimipalli, M. (2010) The Economic Significance of the Conditional Skewness in Index Option Markets. Journal of Futures Markets, 30, 378-406. [6] Polanski, A. and Stoja, E. (2010) Incorporating Higher Moments into Value-at-Risk Forecasting. Journal of Forecasting, 29, 523-535.
http://dx.doi.org/10.1002/for.1155 [7] Chalamandrias, G. and Rompolis, L.S. (2012) Exploring the Role of the Realized Return Distribution in the Formation of the Implied Volatility Smile. Journal of Banking and Finance, 36, 1028-1044.
http://dx.doi.org/10.1016/j.jbankfin.2011.10.016 [8] Del Brio, E.B. and Perote, J. (2012) Gram-Charlier Densities: Maximum Likelihood versus the Method of Moments. Insurance: Mathematics and Economics, 51, 531-537.
http://dx.doi.org/10.1016/j.insmatheco.2012.07.005 [9] Johnson, N.L. (1949) Systems of Frequency Curves Generated by Methods of Translation. Biometrika, 36, 149-176. [10] Cayton, P.J. and Mapa, D.S. (2012) Time-Varying Conditional Johnson SU Density in Value-at-Risk (VaR) Methodology. MPRA Paper No 36206, University of the Philippines at Diliman, Philippines. [11] Guldiman, T. (1997) Risk Metrics Technical Document. Morgan Guarantee Trust Company, New York. [12] Longerstaey, J. and Spencer, M. (1996) Risk MetricsTM. Technical Document, J.P. Morgan/Reuters, New York. [13] Simonato, J.-G. (2011) The Performance of Johnson Distributions for Computing Value at Risk. Journal of Derivatives, 19, 7-24.
http://dx.doi.org/10.3905/jod.2011.19.1.007 [14] Posner, S.E. and Milesky, M.A. (1998) Valuing Exotic Options by Approximating the SPD with Higher Moments. Journal of Financial Engineering, 7, 109-125. [15] Chateau, J.-P. (2009) Marking-to-Model Credit and Operational Risks of Loan Commitments: A Basel-2 Advanced Internal-Ratings Based Approach. International Review of Financial Analysis, 18, 260-270.
http://dx.doi.org/10.1016/j.irfa.2009.07.001 [16] Bakshi, G., Cao, C. and Chen, Z. (1997) Empirical Performance of Alternative Option Pricing Models. Journal of Finance, 52, 2003-2049.
http://dx.doi.org/10.1111/j.1540-6261.1997.tb02749.x [17] Heston, S.L. and Nandi, S. (2000) A Close-Form GARCH Option Valuation Model. Review of Financial Studies, 13, 585-625.
http://dx.doi.org/10.1093/rfs/13.3.585 [18] Backus, D., Foresi, S., Li, K. and Wu, L. (2004) Accounting for Biases in Black-Scholes. Working Paper, New York University, New York. [19] Bakshi, G. and Madan, D. (2000) Spanning and Derivative-Security Valuation. Journal of Financial Economics, 55, 205-238.
http://dx.doi.org/10.1016/S0304-405X(99)00050-1 [20] Corrado, C.J. (2007) The Hidden Martingale Restriction in Gram-Charlier Option Prices. Journal of Futures Markets, 27, 517-534.
http://dx.doi.org/10.1002/fut.20255 [21] Corrado, C.J. and Su, T. (1996) Skewness and Kurtosis in S&P 500 Index Returns Implied by Option Prices. Journal of Financial Research, 19, 175-192. [22] Jurczenko, E., Maillet, B. and Negrea, B. (2004) A Note on Skewness and Kurtosis Adjusted Option Pricing Models Under the Martingale Measure. Quantitative Finance, 4, 479-488. [23] Tanaka, K., Yamada, T. and Watanabe, T. (2010) Applications of Gram-Charlier Expansions and Bond Moments for Pricing of Interest Rates and Credit Risk. Quantitative Finance, 10, 645-662.
http://dx.doi.org/10.1080/14697680903193371 [24] Harrison, J.M. and Pliska, S. (1981) Martingales and Stochastic Integrals in Theory of Continuous Trading. Stochastic Processes and Their Applications, 11, 215-260.
http://dx.doi.org/10.1016/0304-4149(81)90026-0 [25] Leon, A., Mencia, J. and Sentana, E. (2009) Parametric Properties of the Semi-Nonparametric Distributions, with Applications to Option Valuation. Journal of Business and Economic Statistics, 27, 176-192.
http://dx.doi.org/10.1198/jbes.2009.0013 [26] Thakor, A.V., Hong, H. and Greenbaum, S.I. (1981) Bank Loan Commitments and Interest Rate Volatility. Journal of Banking and Finance, 5, 497-510.
http://dx.doi.org/10.1016/S0378-4266(81)80004-0 [27] Basel Committee on Banking Supervision (2013) Progress Report on Basel III Implementation. Bank for International Settlements, Basel, April 2013. [28] Chava, S. and Jarrow, R. (2007) Modeling Loan Commitments. Finance Research Letters, 5, 11-20.
http://dx.doi.org/10.1016/j.frl.2007.11.002 [29] Saunders, A. and Cornett, M.M. (2011) Financial Institutions Management: A Risk Management Approach. 7th Edition, Irwin/McGraw-Hill, New York. [30] Standhouse, B., Schwarzkopf, A. and Ingram, M. (2011) A Computational Approach to Pricing a Bank Credit Line. Journal of Banking and Finance, 35, 1341-1351.
http://dx.doi.org/10.1016/j.jbankfin.2010.10.002 [31] Thakor, A.V. (1982) Toward a Theory of Bank Loan Commitments. Journal of Banking and Finance, 6, 55-84.
http://dx.doi.org/10.1016/0378-4266(82)90022-X [32] Hull, J.C. (2012) Options, Futures and Other Derivatives. 8th Edition, Prentice-Hall, Upper Saddle River. [33] Merton, R.C. (1977) An Analytic Derivation of the Cost of the Deposit Insurance and Loan Guarantees. Journal of Banking and Finance, 1, 3-11.
http://dx.doi.org/10.1016/0378-4266(77)90015-2 [34] Tuenter, H. (2001) An Algorithm to Determine the Parameters of SU-Curves in the Johnson System of Probability Distributions by Moment Matching. Journal of Statistical Computation and Simulation, 70, 325-347.
http://dx.doi.org/10.1080/00949650108812126 [35] Chateau, J.-P. and Dufresne, D. (2012) Gram-Charlier Processes and Equity-Indexed Annuities. Working Paper 227, Centre for Actuarial Studies, University of Melbourne, Australia, April 2012. [36] Harrison, J.M. and Kreps, D.M. (1979) Martingales and Arbitrage in Multiperiod Securities Markets. Journal of Economic Theory, 20, 381-408.
http://dx.doi.org/10.1016/0022-0531(79)90043-7 [37] Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions. Vol. 1, 2nd Edition, John Wiley & Sons Ltd., New York. [38] Pearson, E.S. and Hartley, H.O. (1972) Biometrika Tables for Statisticians. Vol. 2, University Press, Cambridge.
[1] Black, F. (1976) The Pricing of Commodity Contracts. Journal of Financial Economics, 3, 167-179.
http://dx.doi.org/10.1016/0304-405X(76)90024-6 [2] Barton, D.E. and Dennis, K.E.R. (1952) The Conditions under Which Gram-Charlier and Edgeworth Curves are Positive Definite and Unimodal. Biometrika, 39, 425-427. [3] Schlogl, E. (2013) Option Pricing Where the Underlying Assets Follow a Gram-Charlier Density of Arbitrary Order. Journal of Economic Dynamics and Control, 37, 611-632.
http://dx.doi.org/10.1016/j.jedc.2012.10.001 [4] Jondeau, E. and Rockinger, R. (2001) Gram-Charlier Densities. Journal of Economic Dynamics and Control, 25, 1457-1483.
http://dx.doi.org/10.1016/S0165-1889(99)00082-2 [5] Jha, R. and Kalimipalli, M. (2010) The Economic Significance of the Conditional Skewness in Index Option Markets. Journal of Futures Markets, 30, 378-406. [6] Polanski, A. and Stoja, E. (2010) Incorporating Higher Moments into Value-at-Risk Forecasting. Journal of Forecasting, 29, 523-535.
http://dx.doi.org/10.1002/for.1155 [7] Chalamandrias, G. and Rompolis, L.S. (2012) Exploring the Role of the Realized Return Distribution in the Formation of the Implied Volatility Smile. Journal of Banking and Finance, 36, 1028-1044.
http://dx.doi.org/10.1016/j.jbankfin.2011.10.016 [8] Del Brio, E.B. and Perote, J. (2012) Gram-Charlier Densities: Maximum Likelihood versus the Method of Moments. Insurance: Mathematics and Economics, 51, 531-537.
http://dx.doi.org/10.1016/j.insmatheco.2012.07.005 [9] Johnson, N.L. (1949) Systems of Frequency Curves Generated by Methods of Translation. Biometrika, 36, 149-176. [10] Cayton, P.J. and Mapa, D.S. (2012) Time-Varying Conditional Johnson SU Density in Value-at-Risk (VaR) Methodology. MPRA Paper No 36206, University of the Philippines at Diliman, Philippines. [11] Guldiman, T. (1997) Risk Metrics Technical Document. Morgan Guarantee Trust Company, New York. [12] Longerstaey, J. and Spencer, M. (1996) Risk MetricsTM. Technical Document, J.P. Morgan/Reuters, New York. [13] Simonato, J.-G. (2011) The Performance of Johnson Distributions for Computing Value at Risk. Journal of Derivatives, 19, 7-24.
http://dx.doi.org/10.3905/jod.2011.19.1.007 [14] Posner, S.E. and Milesky, M.A. (1998) Valuing Exotic Options by Approximating the SPD with Higher Moments. Journal of Financial Engineering, 7, 109-125. [15] Chateau, J.-P. (2009) Marking-to-Model Credit and Operational Risks of Loan Commitments: A Basel-2 Advanced Internal-Ratings Based Approach. International Review of Financial Analysis, 18, 260-270.
http://dx.doi.org/10.1016/j.irfa.2009.07.001 [16] Bakshi, G., Cao, C. and Chen, Z. (1997) Empirical Performance of Alternative Option Pricing Models. Journal of Finance, 52, 2003-2049.
http://dx.doi.org/10.1111/j.1540-6261.1997.tb02749.x [17] Heston, S.L. and Nandi, S. (2000) A Close-Form GARCH Option Valuation Model. Review of Financial Studies, 13, 585-625.
http://dx.doi.org/10.1093/rfs/13.3.585 [18] Backus, D., Foresi, S., Li, K. and Wu, L. (2004) Accounting for Biases in Black-Scholes. Working Paper, New York University, New York. [19] Bakshi, G. and Madan, D. (2000) Spanning and Derivative-Security Valuation. Journal of Financial Economics, 55, 205-238.
http://dx.doi.org/10.1016/S0304-405X(99)00050-1 [20] Corrado, C.J. (2007) The Hidden Martingale Restriction in Gram-Charlier Option Prices. Journal of Futures Markets, 27, 517-534.
http://dx.doi.org/10.1002/fut.20255 [21] Corrado, C.J. and Su, T. (1996) Skewness and Kurtosis in S&P 500 Index Returns Implied by Option Prices. Journal of Financial Research, 19, 175-192. [22] Jurczenko, E., Maillet, B. and Negrea, B. (2004) A Note on Skewness and Kurtosis Adjusted Option Pricing Models Under the Martingale Measure. Quantitative Finance, 4, 479-488. [23] Tanaka, K., Yamada, T. and Watanabe, T. (2010) Applications of Gram-Charlier Expansions and Bond Moments for Pricing of Interest Rates and Credit Risk. Quantitative Finance, 10, 645-662.
http://dx.doi.org/10.1080/14697680903193371 [24] Harrison, J.M. and Pliska, S. (1981) Martingales and Stochastic Integrals in Theory of Continuous Trading. Stochastic Processes and Their Applications, 11, 215-260.
http://dx.doi.org/10.1016/0304-4149(81)90026-0 [25] Leon, A., Mencia, J. and Sentana, E. (2009) Parametric Properties of the Semi-Nonparametric Distributions, with Applications to Option Valuation. Journal of Business and Economic Statistics, 27, 176-192.
http://dx.doi.org/10.1198/jbes.2009.0013 [26] Thakor, A.V., Hong, H. and Greenbaum, S.I. (1981) Bank Loan Commitments and Interest Rate Volatility. Journal of Banking and Finance, 5, 497-510.
http://dx.doi.org/10.1016/S0378-4266(81)80004-0 [27] Basel Committee on Banking Supervision (2013) Progress Report on Basel III Implementation. Bank for International Settlements, Basel, April 2013. [28] Chava, S. and Jarrow, R. (2007) Modeling Loan Commitments. Finance Research Letters, 5, 11-20.
http://dx.doi.org/10.1016/j.frl.2007.11.002 [29] Saunders, A. and Cornett, M.M. (2011) Financial Institutions Management: A Risk Management Approach. 7th Edition, Irwin/McGraw-Hill, New York. [30] Standhouse, B., Schwarzkopf, A. and Ingram, M. (2011) A Computational Approach to Pricing a Bank Credit Line. Journal of Banking and Finance, 35, 1341-1351.
http://dx.doi.org/10.1016/j.jbankfin.2010.10.002 [31] Thakor, A.V. (1982) Toward a Theory of Bank Loan Commitments. Journal of Banking and Finance, 6, 55-84.
http://dx.doi.org/10.1016/0378-4266(82)90022-X [32] Hull, J.C. (2012) Options, Futures and Other Derivatives. 8th Edition, Prentice-Hall, Upper Saddle River. [33] Merton, R.C. (1977) An Analytic Derivation of the Cost of the Deposit Insurance and Loan Guarantees. Journal of Banking and Finance, 1, 3-11.
http://dx.doi.org/10.1016/0378-4266(77)90015-2 [34] Tuenter, H. (2001) An Algorithm to Determine the Parameters of SU-Curves in the Johnson System of Probability Distributions by Moment Matching. Journal of Statistical Computation and Simulation, 70, 325-347.
http://dx.doi.org/10.1080/00949650108812126 [35] Chateau, J.-P. and Dufresne, D. (2012) Gram-Charlier Processes and Equity-Indexed Annuities. Working Paper 227, Centre for Actuarial Studies, University of Melbourne, Australia, April 2012. [36] Harrison, J.M. and Kreps, D.M. (1979) Martingales and Arbitrage in Multiperiod Securities Markets. Journal of Economic Theory, 20, 381-408.
http://dx.doi.org/10.1016/0022-0531(79)90043-7 [37] Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions. Vol. 1, 2nd Edition, John Wiley & Sons Ltd., New York. [38] Pearson, E.S. and Hartley, H.O. (1972) Biometrika Tables for Statisticians. Vol. 2, University Press, Cambridge.