Li, Minqiang (2008): A Damped Diffusion Framework for Financial Modeling and Closed-form Maximum Likelihood Estimation.
Asset price bubbles can arise unintentionally when one uses continuous-time diffusion processes to model financial quantities. We propose a flexible damped diffusion framework that is able to break many types of bubbles and preserve the martingale pricing approach. Damping can be done on either the diffusion or drift function. Oftentimes, certain solutions to the valuation PDE can be ruled out by requiring the solution to be a limit of martingale prices for damped diffusion models. Monte Carlo study shows that with finite time-series length, maximum likelihood estimation often fails to detect the damped diffusion function while fabricates nonlinear drift function.
|Item Type:||MPRA Paper|
|Original Title:||A Damped Diffusion Framework for Financial Modeling and Closed-form Maximum Likelihood Estimation|
|Keywords:||Damped diffusion, asset price bubbles, martingale pricing, maximum likelihood estimation|
|Subjects:||G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing; Trading volume; Bond Interest Rates
G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing; Futures Pricing
C - Mathematical and Quantitative Methods > C6 - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling > C60 - General
|Depositing User:||Minqiang Li|
|Date Deposited:||19. Oct 2008 08:33|
|Last Modified:||11. Feb 2013 10:20|
Andersen, Leif, and Jesper Andreasen, 2000, Volatility skews and extensions of the LIBOR market model, Applied Mathematical Finance 7, 1-32.
Ait-Sahalia, Yacine, 1996a, Testing continuous-time models of the spot interest rate, Review of Financial Studies 9, 385-426.
Ait-Sahalia, Yacine, 1996b, Nonparametric pricing of interest rate derivative securities, Econometrica 64, 527-560.
Ait-Sahalia, Yacine, 1999, Transition densities for interest rate and other nonlinear diffusions, Journal of Finance 54, 1361-1395.
Ait-Sahalia, Yacine, 2002, Maximum-likelihood estimation of discretely-sampled diffusions: A closed-form approximation approach, Econometrica 70, 223-262.
Ait-Sahalia, Yacine, 2008, Closed-form likelihood expansions for multivariate di®usions, Annals of Statis- tics 36, 906-937.
Ait-Sahalia, Yacine and Robert L. Kimmel, 2007, Maximum likelihood estimation of stochastic volatility models, Journal of Financial Economics 83, 413-452.
Arapis, Manuel, and Jiti Gao, 2006, Empirical comparisons in short-term interest rate models using nonparametric methods, Journal of Financial Econometrics 4, 310-345.
Bachelier, Louis, 1900, Th¶eorie de la sp¶eculation, Annales de l'Ecole Normale Superiure 17, 21-86. Reprinted in Cootner (1964).
Bakshi, Gurdip, Nengjiu Ju, and Hui Ou-Yang, 2006, Estimation of continuous-time models with an application to equity volatility dynamics, Journal of Financial Economics 82, 227-249.
Bali, Turan, and Liuren Wu, 2006, A comprehensive analysis of the short-term interest-rate dynamics, Journal of Banking and Finance 30, 1269-1290.
Beckers, Stan, 1980, The constant elasticity of variance model and its implications for option pricing, Journal of Finance 35, 661-673.
Boyle, Phelim P., and Yisong (Sam) Tian, 1999, Pricing lookback and barrier options under the CEV process, Journal of Financial and Quantitative Analysis 34, 241-264.
Chacko, George, and Luis M. Viceira, 2003, Spectral GMM estimation of continuous-time processes, Journal of Econometrics 116, 2590-292.
Chan, K. C., G. Andrew Karolyi, Francis A. Longsta®, and Anthony B. Sanders, 1992. An empirical comparison of alternative models of the short-term interest rate, Journal of Finance 47, 1209-1227.
Chapman, David, and Neil D. Pearson, 2000, Is the short rate drift actually nonlinear? Journal of Finance 55, 355-399.
Cheridito, Patrick, Damir Filipovi¶c, Robert L. Kimmel, 2007, Market price of risk speci¯cations for a±ne models: Theory and evidence, Journal of Financial Economics 83, 123-170.
Conley Timothy G., Lars Peter Hansen, Erzo G. J. Luttmer, and Jos¶e A. Scheinkman, 1997, Short-term interest rates as subordinated di®usions, Review of Financial Studies 10, 525-577.
Cox, John C., 1975, Notes on option pricing I: Constant elasticity of variance diffusions. mimeo, Standard University.
Cox, John C., 1996, The constant elasticity of variance option pricing model, The Journal of Portfolio Management, Special Issue.
Cox, John C., and Stephen A. Ross, 1976, The valuation of options for alternative stochastic processes, Journal of Financial Economics 3, 145-166.
Cox, John C., Jonathan E. Ingersoll Jr., and Stephen A. Ross, 1985, A theory of the term structure of interest rates, Econometrica, Vol. 53, No. 2, 385-407.
Cox, Alexander M. G., and David G. Hobson, 2005, Local martingales, bubbles and option prices, Finance and Stochastics, Vol. 9, No. 4, 477-92.
D. Davydov, and V. Linetsky, 2001, The valuation and hedging of barrier and lookback options under the CEV process, Management Science 47, 949-965.
Durham, Garland B., 2003, Likelihood-based speci¯cation analysis of continuous-time models of the short-term interest rate, Journal of Financial Economics 70, 463-87.
Emanuel, David C., and James D. MacBeth, 1982, Further results on the constant elasticity of variance call option pricing model, Journal of Financial and Quantitative Analysis 17, 533-54.
Evans, Lawrence C., 1998. Partial Di®erential Equations (American Mathematical Society.).
Goldstein, Robert S., and Willaim P. Keirstead, 1997, On the term structure of interest rates in the presence of reflecting and absorbing boundary conditions, Working paper.
Hajek, Bruce, 1985, Mean stochastic comparison of di®usions, Probability Theory and Related Fields 68, No. 3, 315-329.
Heston, Steve, Mark Loewenstein, and Gregory A. Willard, 2007, Options and bubbles, Review of Finan- cial Studies, 20, 359-390.
Jensen, Bjarke, and Rolf Poulsen, 2002, Transition densities of di®usion processes: Numerical comparison of approximation techniques, Journal of Derivatives 9, 18-32.
Jones, Christopher S., 2003a, The dynamics of stochastic volatility evidence from underlying and option markets, Journal of Econometrics 116, 181-224.
Jones, Christopher S., 2003b, Nonlinear mean reversion in the short-term interest rate, Review of Finan- cial Studies 16, 793-843.
Karlin, Samuel, and Howard M. Taylor, 1981. A Second Course in Stochastic Processes (Academic Press, New York.).
Karatzas, Ioannis, and Steven E. Shreve, 1991. Brownian Motion and Stochastic Calculus (Springer.).
Kloeden, Peter E., and Eckhard Platen, 1992. Numerical Solution of Stochastic Di®erential Equations (Springer.).
Lewis, Alan L., 2000. Option Valuation Under Stochastic Volatility (Finance Press, Newport Beach, California.).
Li, Minqiang, Neil D. Pearson, and Allen M. Poteshman, 2004, Conditional estimation of di®usion pro- cesses, Journal of Financial Economics 74, 31-66.
Mao, Xuerong, 1997. Stochastic Di®erential Equations and Applications (Horwood Publishing.).
Schroder, Mark, 1989, Computing the constant elasticity of variance option pricing formula, Journal of Finance 44, 211-219.
Sin, Carlos A., 1998, Complications with stochastic volatility models, Advances in Applied Probability 30, 256-268.
Vasicek, Oldrich, 1977, An equilibrium characterization of the term structure, Journal of Financial Economics 5, 177-188.
Yamada, Toshio, and Shinzo Watanabe, 1971, On the uniqueness of solutions of stochastic di®erential equations, Journal of Mathematics of Kyoto University 11, 155-167.