On dependence of volatility on return for stochastic volatility models

References

• Abramowitz , M. and Stegun , I.A. 1992 . Handbook of mathematical functions with formulas, graphs, and mathematical tables , Edited by: Abramowitz , M. and Stegun , I.A. New York : Dover Publications, Inc. .
   Google Scholar
• Albeverio , S. and Rozanova , O. 2009 . The non-viscous Burgers equation associated with random positions in coordinate space: A threshold for blow up behavior . Math. Models Methods Appl. Sci. , 19 : 1 – 19 .
   Web of Science ®Google Scholar
• Albeverio , S. and Rozanova , O. 2010 . Suppression of unbounded gradients in a SDE associated with the Burgers equation . Proc. Am. Math. Soc. , 138 : 241 – 251 .
   Web of Science ®Google Scholar
• Cairns , A.J.G. 2004 . Interest Rate Models – An Introduction , Princeton : Princeton University Press .
   Google Scholar
• Chorin , A.J. and Hald , O.H. 2006 . Stochastic Tools in Mathematics and Science , New York : Springer .
   Google Scholar
• Dragulescu , A.A. and Yakovenko , V.M. 2002 . Probability distribution of returns in the Heston model with stochastic volatility . Quant. Finance , 2 : 443 – 453 .
   Google Scholar
• Duffie , D. , Filipović , D. and Schachermayer , W. 2003 . Affine processes and applications in finance . Ann. Appl. Prob. , 13 : 984 – 1053 .
   Web of Science ®Google Scholar
• Feller , W. 1951 . Two singular diffusion problems . Ann. Math. , 54 : 173 – 182 .
   Web of Science ®Google Scholar
• Fouque , J.P. , Papanicolaou , G. and Sircar , K.R. 2000 . Derivatives in Financial Markets with Stochastic Volatility , Cambridge : Cambridge University Press .
   Google Scholar
• Gatheral , J. 2006 . The Volatility Surface , Hoboken, NJ : Wiley .
   Google Scholar
• Gulisashvili , A. and Stein , E.M. 2006 . Asymptotic behavior of the distribution of the stock price in models with stochastic volatility: The Hull-White model . C. R. Acad. Sci. Paris, Ser. I , 343 : 519 – 523 .
   Web of Science ®Google Scholar
• Gulisashvili , A. and Stein , E.M. 2010 . Asymptotic behavior of the stock price distribution density and implied volatility in stochastic volatility models . Math. Finance , 30 : 447 – 477 .
   Google Scholar
• Heston , S.L. 1993 . A closed-form solution for options with stochastic volatility with applications to bond and currency options . Rev. Financ. Stud. , 6 : 327 – 343 .
   Web of Science ®Google Scholar
• Hull , J. and White , A. 1987 . The pricing of options on asset with stochastic volatilities . J. Finance , 42 : 281 – 300 .
   Web of Science ®Google Scholar
• Kambarbaeva , G.S. 2010 . Some explicit formulas for calculation of conditional mathematical expectations of random variables and their applications . Moscow Univ. Math. Bull. , 65 : 186 – 191 .
   Google Scholar
• Kambarbaeva , G.S. 2011 . Composition of an efficient portfolio in the Bielecki and Pliska market model . Moscow Univ. Math. Bull. , 66 : 197 – 204 .
   Google Scholar
• Kambarbaeva , G.S. and Rozanova , O.S. to appear . On efficient portfolio depending on the Cox–Ingersoll–Ross interest rate . Moscow Univ. Math. Bull. ,
   Google Scholar
• Miccichè , S. , Bonanno , G. , Lillo , F. and Mantegna , R.N. 2002 . Volatility in financial markets: Stochastic models and empirical results . Phys. A , 314 : 756 – 761 .
   Web of Science ®Google Scholar
• S. Mitra, A review of volatility and option pricing, Available at http://arxiv.org/pdf/0904.1392
   Google Scholar
• Øksendal , B. 2002 . Stochastic Differential Equations: An Introduction with Applications , 5th ed. , Heidelberg : Springer .
   Google Scholar
• Piessens , R. , de Doncker-Kapenga , E. , Überhuber , C. and Kahaner , D. 1983 . QUADPACK, A Subroutine Package for Automatic Integration , Berlin : Springer-Verlag .
   Google Scholar
• Risken , H. 1989 . The Fokker–Planck Equation. Methods of Solution and Applications , 2nd ed. , New York : Springer .
   Google Scholar
• Schöbel , R. and Zhu , J. 1999 . Stochastic volatility with an Ornstein – Uhlenbeck process: An extension . Eur. Finance Rev. , 4 : 23 – 46 .
   Google Scholar
• Scott , L. 1987 . Option pricing when the variance changes randomly: Theory, estimaton and an applications . J. Financial Quant. Anal. , 22 : 419 – 438 .
   Web of Science ®Google Scholar
• Stein , E.M. and Stein , J.C. 1991 . Stock price distributions with stochastic volatility: An analytic approach . Rev. Financ. Stud. , 4 : 727 – 752 .
   Web of Science ®Google Scholar