Abstract
Based upon an observation that it is too restrictive to assume a definite correlation of the underlying asset price and its volatility, we use a hybrid model of the constant elasticity of variance and stochastic volatility to study a portfolio optimization problem for pension plans. By using asymptotic analysis, we derive a correction to the optimal strategy for the constant elasticity of variance model and subsequently the fine structure of the corrected optimal strategy is revealed. The result is a generalization of Merton’s strategy in terms of the stochastic volatility and the elasticity of variance.
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M. Asch, W. Kohler, G. Papanicolaou, M. Postel, B. White: Frequency content of randomly scattered signals. SIAM Rev. 33 (1991), 519–625.
S. Beckers: The constant elasticity of variance model and its implications for option pricing. Journal of Finance 35 (1980), 661–673.
J.-F. Boulier, S. Huang, G. Taillard: Optimal management under stochastic interest rates: The case of a protected defined contribution pension fund. Insur. Math. Econ. 28 (2001), 173–189.
P. P. Boyle, Y. S. Tian: Pricing lookback and barrier options under the CEV process. Journal of Financial and Quantitative Analysis 34 (1999), 241–264.
S. Cerrai: A Khasminskii type averaging principle for stochastic reaction-diffusion equations. Ann. Appl. Probab. 19 (2009), 899–948.
S.-Y. Choi, J.-P. Fouque, J.-H. Kim: Option pricing under hybrid stochastic and local volatility. Quant. Finance 13 (2013), 1157–1165.
J. C. Cox: The constant elasticity of variance option pricing model. The Journal of Portfolio Management 23 (1996), 15–17.
J. C. Cox, C.-F. Huang: Optimal consumption and portfolio policies when asset prices follow a diffusion process. J. Econ. Theory 49 (1989), 33–83.
J. C. Cox, S. Ross: The valuation of options for alternative stochastic processes. Journal of Financial Economics 3 (1976), 145–166.
D. Davydov, V. Linetsky: The valuation and hedging of barrier and lookback options under the CEV process. Manage. Sci. 47 (2001), 949–965.
G. Deelstra, M. Grasselli, P.-F. Koehl: Optimal design of the guarantee for defined contribution funds. J. Econ. Dyn. Control 28 (2004), 2239–2260.
P. Devolder, M. Bosch Princep, I. Dominguez Fabian: Stochastic optimal control of annuity contracts. Insur. Math. Econ. 33 (2003), 227–238.
W. H. Fleming, H. M. Soner: Controlled Markov Processes and Viscosity Solutions. Stochastic Modelling and Applied Probability 25, Springer, New York, 2006.
J.-P Fouque, G. Papanicolaou, K. R. Sircar: Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press, Cambridge, 2000.
J.-P. Fouque, G. Papanicolaou, R. Sircar, K. Sølna: Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives. Cambridge University Press, Cambridge, 2011.
I. Fredholm: Sur une classe d’équations fonctionnelles. Acta Math. 27 (1903), 365–390. (In French.)
J. Gao: Optimal portfolios for DC pension plans under a CEV model. Insur. Math. Econ. 44 (2009), 479–490.
E. Ghysels, A. Harvey, E. Renault: Stochastic volatility. Statistical Methods in Finance. Handbook of Statistics 14, North-Holland, Amsterdam, 1996.
S. Haberman, E. Vigna: Optimal investment strategies and risk measures in defined contribution pension schemes. Insur. Math. Econ. 31 (2002), 35–69.
C. R. Harvey: The specification of conditional expectations. Journal of Empirical Finance 8 (2001), 573–637.
J. C. Jackwerth, M. Rubinstein: Recovering probability distributions from option prices. Journal of Finance 51 (1996), 1611–1631.
R. Z. Khas’minskii: On stochastic processes defined by differential equations with a small parameter. Theory Probab. Appl. 11 (1966), 211–228; translation from Teor. Veroyatn. Primen. 11 (1966), 240–259. (In Russian.)
J.-H. Kim: Asymptotic theory of noncentered mixing stochastic differential equations. Stochastic Processes Appl. 114 (2004), 161–174.
R. C. Merton: Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theory 3 (1971), 373–413.
E.-J. Noh, J.-H. Kim: An optimal portfolio model with stochastic volatility and stochastic interest rate. J. Math. Anal. Appl. 375 (2011), 510–522.
B. Øksendal: Stochastic Differential Equations. An Introduction with Applications. Universitext, Springer, Berlin, 2003.
G. C. Papanicolaou, D. W. Stroock, S. R. S. Varadhan: Martingale approach to some limit theorems. Proc. Conf. Durham, 1976. Duke Univ. Math. Ser., Vol. III, Duke Univ., Durham, 1977.
M. Rubinstein: Nonparametric tests of alternative option pricing models using CBOE reported trades. Journal of Finance 40 (1985), 455–480.
J. Xiao, Z. Hong, C. Qin: The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts. Insur. Math. Econ. 40 (2007), 302–310.
K. C. Yuen, H. Yang, K. L. Chu: Estimation in the constant elasticity of variance model. British Actuarial Journal 7 (2001), 275–292.
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Yang, SJ., Kim, JH. & Lee, MK. Portfolio optimization for pension plans under hybrid stochastic and local volatility. Appl Math 60, 197–215 (2015). https://doi.org/10.1007/s10492-015-0091-9
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DOI: https://doi.org/10.1007/s10492-015-0091-9
Keywords
- pension plan
- portfolio optimization
- constant elasticity of variance
- stochastic volatility
- asymptotic analysis