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Author (up) Contreras, G.M.
Title Stochastic volatility models at rho = +/- 1 as second class constrained Hamiltonian systems Type
Year 2014 Publication Physica A-Statistical Mechanics And Its Applications Abbreviated Journal Physica A
Volume 405 Issue Pages 289-302
Keywords Fokker-Planck equation; Stochastic volatility models; Option pricing; Singular Lagrangian systems; Dirac's method; Constrained Hamiltonian path integrals
Abstract The stochastic volatility models used in the financial world are characterized, in the continuous-time case, by a set of two coupled stochastic differential equations for the underlying asset price S and volatility sigma. In addition, the correlations of the two Brownian movements that drive the stochastic dynamics are measured by the correlation parameter rho (-1 <= rho <= 1). This stochastic system is equivalent to the Fokker-Planck equation for the transition probability density of the random variables S and sigma. Solutions for the transition probability density of the Heston stochastic volatility model (Heston, 1993) were explored in Dragulescu and Yakovenko (2002), where the fundamental quantities such as the transition density itself, depend on rho in such a manner that these are divergent for the extreme limit rho = +/- 1. The same divergent behavior appears in Hagan et al. (2002), where the probability density of the SABR model was analyzed. In an option pricing context, the propagator of the bi-dimensional Black-Scholes equation was obtained in Lemmens et al. (2008) in terms of the path integrals, and in this case, the propagator diverges again for the extreme values rho = +/- 1. This paper shows that these similar divergent behaviors are due to a universal property of the stochastic volatility models in the continuum: all of them are second class constrained systems for the most extreme correlated limit rho = +/- 1. In this way, the stochastic dynamics of the rho = +/- 1 cases are different of the rho (1 <= rho <= 1) case, and it cannot be obtained as a continuous limit from the rho not equal +/- 1 regimen. This conclusion is achieved by considering the Fokker-Planck equation or the bi-dimensional Black-Scholes equation as a Euclidean quantum Schrodinger equation. Then, the analysis of the underlying classical mechanics of the quantum model, implies that stochastic volatility models at rho = +/- 1 correspond to a constrained system. To study the dynamics in an appropriate form, Dirac's method for constrained systems (Dirac, 1958, 1967) must be employed, and Dirac's analysis reveals that the constraints are second class. In order to obtain the transition probability density or the option price correctly, one must evaluate the propagator as a constrained Hamiltonian path-integral (Henneaux and Teitelboim, 1992), in a similar way to the high energy gauge theory models. In fact, for all stochastic volatility models, after integrating over momentum variables, one obtains an effective Euclidean Lagrangian path integral over the volatility alone. The role of the second class constraints is determining the underlying asset price S completely in terms of volatility, so it plays no role in the path integral. In order to examine the effect of the constraints on the dynamics for both extreme limits, the probability density function is evaluated by using semi-classical arguments, in an analogous manner to that developed in Hagan et al. (2002), for the SABR model. (C) 2014 Elsevier B.V. All rights reserved.
Address Univ Adolfo Ibanez, Fac Ingn & Ciencias, Santiago, Chile, Email: mauricio.contreras@uai.cl
Corporate Author Thesis
Publisher Elsevier Science Bv Place of Publication Editor
Language English Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 0378-4371 ISBN Medium
Area Expedition Conference
Notes WOS:000337854200030 Approved
Call Number UAI @ eduardo.moreno @ Serial 382
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Author (up) Miranda, A.; Mentler, R.; Moletto-Lobos, I.; Alfaro, G.; Aliaga, L.; Balbontin, D.; Barraza, M.; Baumbach, S.; Calderon, P.; Cardenas, F.; Castillo, I.; Contreras, G.; de la Barra, F.; Galleguillos, M.; Gonzalez, M.E.; Hormazabal, C.; Lara, A.; Mancilla, I.; Munoz, F.; Oyarce, C.; Pantoja, F.; Ramirez, R.; Urrutia, V.
Title The Landscape Fire Scars Database: mapping historical burned area and fire severity in Chile Type
Year 2022 Publication Earth System Science Data Abbreviated Journal Earth Syst. Sci. Data
Volume 14 Issue 8 Pages 3599-3613
Abstract Achieving a local understanding of fire regimes requires high-resolution, systematic and dynamic databases. High-quality information can help to transform evidence into decision-making in the context of rapidly changing landscapes, particularly considering that geographical and temporal patterns of fire regimes and their trends vary locally over time. Global fire scar products at low spatial resolutions are available, but high-resolution wildfire data, especially for developing countries, are still lacking. Taking advantage of the Google Earth Engine (GEE) big-data analysis platform, we developed a flexible workflow to reconstruct individual burned areas and derive fire severity estimates for all reported fires. We tested our approach for historical wild-fires in Chile. The result is the Landscape Fire Scars Database, a detailed and dynamic database that reconstructs 8153 fires scars, representing 66.6% of the country's officially recorded fires between 1985 and 2018. For each fire event, the database contains the following information: (i) the Landsat mosaic of pre- and post-fire images; (ii) the fire scar in binary format; (iii) the remotely sensed estimated fire indexes (the normalized burned ratio, NBR, and the relative delta normalized burn ratio, RdNBR); and two vector files indicating (iv) the fire scar perimeter and (v) the fire scar severity reclassification, respectively. The Landscape Fire Scars Database for Chile and GEE script (JavaScript) are publicly available. The framework developed for the database can be applied anywhere in the world, with the only requirement being its adaptation to local factors such as data availability, fire regimes, land cover or land cover dynamics, vegetation recovery, and cloud cover. The Landscape Fire Scars Database for Chile is publicly available in https://doi.org/10.1594/PANGAEA.941127 (Miranda et al., 2022).
Corporate Author Thesis
Publisher Place of Publication Editor
Language Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 1866-3508 ISBN Medium
Area Expedition Conference
Notes WOS:000838024900001 Approved
Call Number UAI @ alexi.delcanto @ Serial 1667
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