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A tractable model for indices approximating the growth optimal portfolio. (English) Zbl 1283.91198
Summary: The growth optimal portfolio (GOP) plays an important role in finance, where it serves as the numéraire portfolio, with respect to which contingent claims can be priced under the real world probability measure. This paper models the GOP using a time dependent constant elasticity of variance (TCEV) model. The TCEV model has high tractability for a range of derivative prices and fits well the dynamics of a global diversified world equity index. This is confirmed when pricing and hedging various derivatives using this index.

MSC:
91G70 Statistical methods; risk measures
91G10 Portfolio theory
62G07 Density estimation
91G20 Derivative securities (option pricing, hedging, etc.)
Software:
KernSmooth
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References:
[1] Becherer, D. 2001. “The Numeraire Portfolio for Unbounded Semimartingales.” Finance and Stochastics 5(3): 327-341. · Zbl 0978.91038
[2] Black, F., and M. Scholes. 1973. “The Pricing of Options and Corporate Liabilities.” The Journal of Political Economy 81(3): 637-654. · Zbl 1092.91524
[3] Botev, Z., J. F. Grotowski, and D. P. Kroese. 2010. “Kernel Density Estimation Via Diffusion.” The Annals of Statistics 38(5): 2916-2957. · Zbl 1200.62029
[4] Breiman, L. 1960. “Investment Policies for Expanding Business Optimal in a Long Run Sense.” Naval Research Logistics Quarterly 7(4): 647-651. · Zbl 0114.12401
[5] Brown, R., and S. Schaefer. 1994a. “Interest Rate Volatility and the Shape of the Term Structure.” Philosophical Transactions of the Royal Society of London A 347: 563-576. · Zbl 0822.90014
[6] Brown, R., and S. Schaefer. 1994b. “The Term Structure of Real Interest Rates and the Cox, Ingersoll, and Ross Model.” Journal of Financial Econometrics 35: 3-42.
[7] Carr, P., and V. Linetsky. 2006. “A Jump to Default Extended CEV Model: An Application of Bessel Processes.” 10: 303-330. · Zbl 1101.60057
[8] Cox, J. 1975. “Notes on Option Pricing I: Constant Elasticity of Variance Diffusions.” Working paper, Stanford University unpublised.
[9] Delbaen, F., and W. Schachermayer. 1994. “A General Version of the Fundamental Theorem of Asset Pricing.” Mathematische Annalen 300: 463-520. · Zbl 0865.90014
[10] Delbaen, F., and W. Schachermayer. 1998. “The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes.” Mathematische Annalen 312: 215-250. · Zbl 0917.60048
[11] Derman, E., and I. Kani. 1994. “The Volatility Smile and Its Implied Tree.” Goldman Sachs Quantitative Strategies Research Notes.
[12] Duffie, D., and N. Gârleanu. 2001. “Risk and Valuation of Collateralized Debt Obligations.” Financial Analysts Journal 57: 41-59.
[13] Dupire, B. 1994. “Pricing with a Smile.” Risk Magazine 7: 18-20.
[14] Epanechnikov, V. 1969. “Nonparametric Estimates of Multivariate Probability Density.” Theory of Probability and Applications 14: 153-158.
[15] Florens-Zmirou, D. 1993. “On Estimating the Diffusion Coeffcient from Discrete Observations.” Journal of Applied Probability 30(4): 790-804. · Zbl 0796.62070
[16] Fontana, C., and W. Runggaldier. 2012. “Diffusion-Based Models for Financial Markets Without Martingale Measures.” In Risk Measures and Attitudes, edited by F. Biagini, A. Richter, and H. Schlesinger. An LMU Excellent Symposium. · Zbl 1306.91125
[17] Heath, D., and E. Platen. 2002. “Consistent Pricing and Hedging for a Modified Constant Elasticity of Variance Model.” Quantitative Finance 2: 459-467.
[18] Heston, S. 1997. “A Simple New Formula for Options with Stochastic Volatility.” Working paper, University of St. Louis.
[19] Hulley, H., and E. Platen. 2012. “Hedging for the Long Run.” Mathematics and Financial Economics 6(2): 105-124. · Zbl 1264.91147
[20] Hulley, H., and M. Schweizer. 2010. “M6 - On Minimal Market Models and Minimal Martingale Measures.” edited by C. Chiarella, and A. Novikov, Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen. · Zbl 1229.91376
[21] Hulley, H. 2009. Strict local Martingales in Continuous Financial Market Models. Ph.D. thesis, UTS, Sydney.
[22] Ignatieva, K., and E. Platen. 2012. “Estimating the Diffusion Coefficient Function for a Diversified World Stock Index.” Computational Statistics & Data Analysis 56(6): 1333-1349. · Zbl 1242.91215
[23] Jacod, J. 2000. “Non-Parametric Kernel Estimation of the Coeffcient of a Diffusion.” Scandinavian Journal of Statistics 27(1): 83-96. · Zbl 0938.62085
[24] Jiang, G., and J. Knight. 1997. “A Nonparametric Approach to the Estimation of Diffusion Processes, with an Application to a Short-term Interest Rate Model.” Econometric Theory 13(7): 615-645.
[25] Johnson, N., S. Kotz, and N. Balakrishnan. 1994. Continuous Univariate Distributions, Vol. 1, 2nd ed. New York: John Wiley. · Zbl 0811.62001
[26] Karatzas, I., and S. E. Shreve. 1991. Brownian Motion and Stochastic Calculus, 2nd ed. New York: Springer. · Zbl 0734.60060
[27] Kardaras, C. 2010. “Finitely Additive Probabilities and the Fundamental Theorem of Asset Pricing.” In Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen, edited by C. Chiarella and A. Novikov, 19-34. New York: Springer. · Zbl 1217.91221
[28] Kelly, J. R. 1956. “A New Interpretation of Information Rate.” The Bell Journal of Economics and Management Science 35: 917-926.
[29] Künsch, H. R. 1989. “The Jackknife and the Bootstrap for General Stationary Observations.” The Annals of Statistics 17: 1217-1241. · Zbl 0684.62035
[30] Latané, H. 1959. “Criteria for Choice Among Risky Ventures.” Journal of Political Economy 38: 145-155.
[31] Lewis, A. 2000. Option Valuation under Stochastic Volatility. New York: Finance Press. · Zbl 0937.91060
[32] Loewenstein, M., and G. Willard. 2000. “Local Martingales, Arbitrage, and Viability: Free Snacks and Cheap Thrills.” Econometric Theory 16(1): 135-161. · Zbl 1034.91043
[33] Long, J. B. 1990. “The Numeraire Portfolio.” Journal of Financial Economics 26: 29-69.
[34] MacLean, L., E. Thorp, and W. Ziemba. 2011. “The Kelly Capital Growth Investment Criterion.” World Scientific Handbook in Financial Economic Series, Vol. 3.
[35] Markowitz, H. 1976. “Investment for the Long Run: New Evidence for An Old Rule.” Journal of Finance XXXI(5): 1273-1286.
[36] Merton, R. 1973. “Theory of Rational Option Pricing.” The Bell Journal of Economics and Management Science 4: 141-183. · Zbl 1257.91043
[37] Pitman, J., and M. Yor. 1982. “A Decomposition of Bessel Bridges.” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 59(4): 425-457. · Zbl 0484.60062
[38] Platen, E., and D. Heath. 2010. A Benchmark Approach to Quantitative Finance. New York: Springer-Verlag. · Zbl 1104.91041
[39] Platen, E., and R. Rendek. 2011. “Approximating the Numeraire Portfolio by Naive Diversification.” Journal of Asset Management 13(1): 34-50.
[40] Platen, E. 1997. “A Non-Linear Stochstic Volatility Model.” Working paper, Australian National University unpublised.
[41] Platen, E. 2001. A Minimal Financial Market Model. Birkhauser Verlag. · Zbl 1004.91029
[42] Platen, E. 2002. “Arbitrage in Continuous Complete Markets.” Advances in Applied Probability 34(3): 540-558. · Zbl 1055.91033
[43] Platen, E. 2005. “On the Role of the Growth Optimal Portfolio in Finance.” Australian Economic Papers, Blackwell Publishing 44(4): 365-388.
[44] Revuz, D. and M. Yor. 1999. Continous Martingales and Brownian Motion. New York: Springer. · Zbl 0917.60006
[45] Samuelson, P. A. 1979. “Why We Should Not Make Mean Log of Wealth Big Though Years to Act are Long.” Journal of Banking and Finance 3: 305-307.
[46] Schroder, M. 1989. “Computing the Constant Elasticity of Variance Option Pricing Formula.” Journal of Finance 44(1): 211-219.
[47] Scott, D. 1992. Multivariate Density Estimation: Theory, Practice and Visualization. New York: John Wiley. · Zbl 0850.62006
[48] Silverman, B. W. 1986. Density Estimation. Chapman and Hall. · Zbl 0617.62042
[49] Soulier, P. 1998. “Nonparametric Estimation of the Diffusion Coeffcient of a Diffusion Process.” Stochastic Analysis and Applications 16: 185-200. · Zbl 0894.62093
[50] Stanton, R. 1997. “A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk.” Journal of Finance 52(5): 1973-2002.
[51] Thorp, E. O. 1961. “A Favourable Strategy for Twenty-One.” In Proceedings of the National Academy of Sciences. Vol. 47. pp. 110-112. · Zbl 0099.36601
[52] Wand, M. P., and M. C. Jones. 1995. Kernel Smoothing. London: Chapman and Hall. · Zbl 0854.62043
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