Dell'Era, Mario (2010): Geometrical Approximation method and stochastic volatility market models. Forthcoming in: Wilmott Magazine

PDF
MPRA_paper_22568.pdf Download (191kB)  Preview 
Abstract
We propose to discuss a new technique to derive an good approximated solution for the price of a European Vanilla options, in a market model with stochastic volatility. In particular, the models that we have considered are the Heston and SABR(for beta=1). These models allow arbitrary correlation between volatility and spot asset returns. We are able to write the price of European call and put, in the same form in which one can see in the BlackScholes model. The solution technique is based upon coordinate transformations that reduce the initial PDE in a straightforward onedimensional heat equation.
Item Type:  MPRA Paper 

Original Title:  Geometrical Approximation method and stochastic volatility market models 
English Title:  Geometrical Approximation method and stochastic volatility market models 
Language:  English 
Keywords:  Financial pricing method 
Subjects:  C  Mathematical and Quantitative Methods > C0  General C  Mathematical and Quantitative Methods > C0  General > C02  Mathematical Methods I  Health, Education, and Welfare > I2  Education and Research Insititutions > I22  Educational Finance 
Item ID:  22568 
Depositing User:  Mario Dell'Era 
Date Deposited:  10. May 2010 12:53 
Last Modified:  13. Feb 2013 17:55 
References:  Andersen, L,. and J. Andreasen (2002), Volatile Volatilities, Risk Magazine, December. Andersen, L. and R. BrothertonRatcliffe (2005), Extended LIBOR market models with stochastic volatility, Journal of Computational Finance, vol. 9, no.1, pp. 140. Andersen, L. and V. Piterbarg (2005), “Moment explosions in stochastic volatility models, Finance and Stochastics, forthcoming. Andreasen, J. (2006), Longdated FX hybrids with stochastic volatility,Working paper, Bank of America. Broadie, M. and O¨ . Kaya (2006), Exact simulation of stochastic volatility and other affine jump diffusion processes, Operations Research, vol. 54, no. 2. Broadie, M. and O¨ . Kaya (2004), Exact simulation of option greeks under stochastic volatility and jump diffusion models,” in R.G. Ingalls, M.D. Rossetti, J.S. Smith and B.A. Peters (eds.), Proceedings of the 2004 Winter Simulation Conference. Carr, P. and D. Madan (1999), Option Pricing and the fast Fourier transform, Journal of Computational Finance, 2(4), pp. 6173. Cox, J., J. Ingersoll and S.A. Ross (1985), A theory of the term structure of interest rates, Econometrica, vol. 53, no. 2, pp. 385407. Duffie, D. and P. Glynn (1995), Efficient Monte Carlo simulation of security prices, Annals of Applied Probability, 5, pp. 897905 Duffie, D., J. Pan and K. Singleton (2000), Transform analysis and asset pricing for affine jump diffusions, Econometrica, vol. 68, pp. 13431376. Dufresne, D. (2001), The integrated squareroot process, Working paper, University of Montreal. Glasserman, P. (2003), Monte Carlo methods in financial engineering, Springer Verlag, New York. Glasserman, P. and X. Zhao (1999), Arbitragefree discretization of lognormal forward LIBOR and swap rate models, Finance and Stochastics, 4, pp. 3568 Heston, S.L. (1993), A closedform solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, vol. 6, no. 2, pp. 327343. Johnson, N., S. Kotz, and N. Balakrishnan (1995), Continuous univariate distributions, vol. 2, Wiley Interscience. Kahl, C. and P. Jackel (2005), Fast strong approximation MonteCarlo schemes for stochastic volatility models, Working Paper, ABN AMRO and University of Wuppertal. Lee, R. (2004), Option Pricing by Transform Methods: Extensions, Unification, and Error Control, Journal of Computational Finance, vol 7, issue 3, pp. 5186 Lewis, A. (2001), Option valuation under stochastic volatility, Finance Press, Newport Beach. Lipton, A. (2002), The volsmile problem, Risk Magazine, February, pp. 6165. Lord, R., R. Koekkoek and D. van Dijk (2006), A Comparison of biased simulation schemes for stochastic volatility models, Working Paper, Tinbergen Institute. Kloeden, P. and E. Platen (1999), Numerical solution of stochastic differential equations, 3rd edition, Springer Verlag, New York. Moro, B. (1995), The full Monte, Risk Magazine, Vol.8, No.2, pp. 5758. Patnaik, P. (1949), The noncentral $\chi^{2}$ and Fdistributions and their applications, Biometrika, 36, pp. 202232. Pearson, E. (1959), Note on an approximation to the distribution of noncentral $\chi^{2}$, Biometrika, 46, p. 364. Piterbarg, V. (2003), Discretizing Processes used in Stochastic Volatility Models, Working Paper, Bank of America. Piterbarg, V. (2005), Stochastic volatility model with timedependent skew, Applied Mathematical Finance. Press, W., S. Teukolsky, W. Vetterling, and B. Flannery (1992), Numerical recipes in C, Cambridge University Press, New York 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/22568 