Dell'Era, Mario (2010): Geometrical Considerations on Heston's Market Model. Forthcoming in: Quantitative Finanace

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Abstract
We propose to discuss a new technique to derive an good approximated solution for the price of a European call and put options, in a market model with stochastic volatility. In particular, the model that we have considered is the Heston's model. This allows 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 Considerations on Heston's Market Model 
English Title:  Geometrical Considerations on Heston's Market Model 
Language:  English 
Keywords:  Quantitative methods in Finance 
Subjects:  D  Microeconomics > D5  General Equilibrium and Disequilibrium > D53  Financial Markets D  Microeconomics > D4  Market Structure, Pricing, and Design > D46  Value Theory C  Mathematical and Quantitative Methods > C0  General C  Mathematical and Quantitative Methods > C0  General > C02  Mathematical Methods 
Item ID:  21523 
Depositing User:  Mario Dell'Era 
Date Deposited:  31. Mar 2010 06:46 
Last Modified:  22. Feb 2013 06:02 
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:  https://mpra.ub.unimuenchen.de/id/eprint/21523 