Geometrical Considerations on Heston's Market Model

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

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 Black-Scholes model. The solution technique is based upon coordinate transformations that reduce the initial PDE in a straightforward one-dimensional heat equation.

• Item Type: MPRA Paper
• 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

References

Andersen, L,. and J. Andreasen (2002), Volatile Volatilities, Risk Magazine, December.

Andersen, L. and R. Brotherton-Ratcliffe (2005), Extended LIBOR market models with stochastic volatility, Journal of Computational Finance, vol. 9, no.1, pp. 1-40.

Andersen, L. and V. Piterbarg (2005), “Moment explosions in stochastic volatility models, Finance and Stochastics, forthcoming.

Andreasen, J. (2006), Long-dated 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. 61-73.

Cox, J., J. Ingersoll and S.A. Ross (1985), A theory of the term structure of interest rates, Econometrica, vol. 53, no. 2, pp. 385-407.

Duffie, D. and P. Glynn (1995), Efficient Monte Carlo simulation of security prices, Annals of Applied Probability, 5, pp. 897-905

Duffie, D., J. Pan and K. Singleton (2000), Transform analysis and asset pricing for affine jump diffusions, Econometrica, vol. 68, pp. 1343-1376.

Dufresne, D. (2001), The integrated square-root 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), Arbitrage-free discretization of log-normal forward LIBOR and swap rate models, Finance and Stochastics, 4, pp. 35-68

Heston, S.L. (1993), A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, vol. 6, no. 2, pp. 327-343.

Johnson, N., S. Kotz, and N. Balakrishnan (1995), Continuous univariate distributions, vol. 2, Wiley Interscience.

Kahl, C. and P. Jackel (2005), Fast strong approximation Monte-Carlo 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. 51-86

Lewis, A. (2001), Option valuation under stochastic volatility, Finance Press, Newport Beach.

Lipton, A. (2002), The vol-smile problem, Risk Magazine, February, pp. 61-65.

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. 57-58. Patnaik, P. (1949), The non-central $\chi^{2}$ and F-distributions and their applications, Biometrika, 36, pp. 202-232.

Pearson, E. (1959), Note on an approximation to the distribution of non-central $\chi^{2}$, 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 time-dependent skew, Applied Mathematical Finance.

Press, W., S. Teukolsky, W. Vetterling, and B. Flannery (1992), Numerical recipes in C, Cambridge University Press, New York