Asymptotic representation in stochastic volatility models
- Authors: Buslova K.V.1
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Affiliations:
- Lomonosov Moscow State University
- Issue: No 116 (2025)
- Pages: 203-231
- Section: Control of social-economic systems
- URL: https://journal-vniispk.ru/1819-2440/article/view/307006
- ID: 307006
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Abstract
This work is devoted to the study of the asymptotic behavior of the probability density function of stock prices within models with stochastic volatility. The main focus is on generalizing previously known results for the one-factor Heston model to more complex cases, including multivariate models. The research methodology is based on the use of affine principles, which allow analyzing the necessary aspects of asymptotics through solving corresponding Riccati equations near the critical point. The application of high-order Euler estimates ensures accuracy in calculations and robustness of conclusions. Additionally, the method of steepest descent combined with Tauber's principle is used, providing an opportunity to extract precise data about the asymptote of the original function from the analysis of its transformation near the critical point. This allows a deeper understanding of the structure of probability density functions in the context of complex stochastic systems. The obtained results are important for the development of the theory of stochastic differential equations and can find applications in various fields such as financial mathematics and econometrics.
About the authors
Kristina Vladimirovna Buslova
Lomonosov Moscow State University
Email: buslova.kristina@mail.ru
Moscow
References
- BRU M. Wishart processes // Journal of TheoreticalProbability. – 1991. – Vol. 4. – P. 725–743.
- CARR P., MADAN D.B. Option valuation using the fastFourier transform // Journal of Computational Finance. –1999. – Vol. 2, No. 4.
- CARR P., WU L. Stochastic Skew in Currency Options //Preprint, 2004.
- CONT R., DA FONSECA J. Dynamics of implied volatilitysurfaces // Quantitative Finance. – 2002. – Vol. 2, No. 1. –P. 45–60.
- DA FONSECA J., GRASSELLI M., TEBALDI C.A multifactor volatility Heston model // Quantitative Finance. –2008. – Vol. 8, No. 6. – P. 591–604.
- DAI Q., SINGLETON K. Specification Analysis of Affine TermStructure Models // Journal of Finance. – 2000. – Vol. 55. –P. 1943–1978.
- DUFFIE D., KAN R. A Yield-Factor Model of Interest Rates //Mathematical Finance. – 1996. – Vol. 6(4). – P. 379–406.
- DUFFIE D., PAN J., SINGLETON K. Transform analysisand asset pricing for affine jump-diffusions // Econometrica. –2000. – Vol. 68. – P. 1343–1376.
- FLAJOLET P., GOURDON X., DUMAS P. Mellin transformsand asymptotics: Harmonic sums // Theoretical ComputerScience. – 1995. – Vol. 144. – P. 3–58.
- FREILING G. A Survey of Nonsymmetric Riccati Equations //Linear Algebra and Its Applications. – 2002. – P. 243–270.
- FRIZ P., GERHOLD S., GULISASHVILI A. et al. On refinedvolatility smile expansion in the Heston model // QuantitativeFinance. – 2011. – Vol. 11. – P. 1151–1164.
- GOURIEROUX C., SUFANA R. Derivative Pricing withMultivariate Stochastic Volatility: Application to Credit Risk //Working paper. – CREST, 2004.
- GRASSELLI M., TEBALDI C. Solvable Affine Term StructureModels // Mathematical Finance. – 2004.
- GULISASHVILI A. Analytically Tractable Stochastic StockPrice Models // Springer Finance. – 2012. – P. 167–184.
- GULISASHVILI A., STEIN E.M. Asymptotic behavior ofthe stock price distribution density and impied volatilityin stochastic volatility models // Applied Mathematics andOptimization. – 2010. – Vol. 61. – P. 287–315.
- HESTON S. A Closed-Form Solution for Option with StochasticVolatility with Applications to Bond and Currency Options //Review of Financial Studies. – 1993. – Vol. 6. – P. 327–343.
- HORN R.A., JOHNSON C.R. Matrix analysis. – Cambridge,1990. – P. 464–469.
- LEWIS A.L. Option Valuation Under Stochastic Volatility. –Finance Press, 2000.
- LUCIC V. On singularities in the Heston models // LargeDeviations and Asymptotic Methods in Finance, 2007. –P. 439–448.
- WONG G. Forward Smile and Derivative Pricing // EquityQuantitative. Strategists Working Paper, UBS, 2004.
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