Publication: Valoración de activos financieros por entropía máxima con programación lineal
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2017-01-16
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Universidad Complutense de Madrid
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Abstract
El objeto de este trabajo de tesis es la recuperación del vector de probabilidades neutrales al risgo para la valoración de activos financioneros a partir de los precios de los activos que se observan en el mercado con una metodología de minimización de la entropía relativa. La principal innovación de este trabajo es el uso de una divergencia diferente de la de Kullback-Leibler, resultando en la distancia de variación total (Ali & Silvey, 1966), (Denuit, Dhaene, Goovaerts, & Kaas, 2005). El proceso de optimización propuesto aplica técnicas de programación lineal, garantizando, de esta manera, una resolución rápida y fácil del problema numérico, que es una de las principales aportaciones de este trabajo de tesis. Este nuevo método es un método inverso para valorar activos ilíquidos y encontrar la probabilidad neutral al riesgo minimizando la entropía relativa a la distribución a priori uniforme sobre un conjunto finito de trayectorias de precios simulados y satisfaciendo la restricción dada por los precios objetivo a los que se calibra la distribución de probabilidad. En el caso de mercados incompletos, se puede obtener cualquier valor que pertenece al intervalo de precios en el que se cumple la no existencia de carteras de arbitraje (Luenberger, 2002), incrementando la entropía relativa de la probabilidad neutral al riesgo obtenida para el valor óptimo, donde este valor de entropía relativa es mínimo. Desde el punto de vista de la réplica de activos, este nuevo método permite el cálculo de las carteras infra y supra-replicante, correspondientes a las cotas inferior y superor del intervalo libre de arbitraje, y da una representación del activo dependiendo de estos límites (Luenberger, 2002). Además, el programa dual permite obtener las sensibilidades del valor del activo respecto a cambios en las probabilidades a priori, el nivel de entropía relativa del vector de probabilidad neutral al riesgo y, finalmente, a cambios en los precios observados en el mercado. Para estudiar el comportamiento del nuevo método propuesto, se han incluido cuatro ejemplos: uno sintético con opciones europeas, otro con datos del mundo real utilizando opciones sobre el índice STOXX50E, un tercer ejemplo basado en la valoración de opciones asiáticas y un ejemplo donde la distribución a priori, respecto a la cual se minimiza la distancia de variación total, se cambia, utilizando una distribución que no es la de máxima entropía en un intervalo. Se concluye que este nuevo método propuesto valora adecuadamente los activos que no están dentro de la muestra de calibración, mediante la comparación que se hace de este método con otros más tradicionales como el de la minimización de la entropía relativa de Kullback-Leibler. Por otra parte, en la calibración a precios reales de mercado, se observa que el método refleja las formas no lineales de la superficie de volatilidad que se observan en el mercado. Por último, utilizando el método con activos más complejos, como la valoración de opciones asiáticas, el método sigue dando buenos resultados de valoración.
This work is devoted to risk neutral probabilities (RNP) retrieval from asset prices using the relative entropy minimization methodology. The most important feature of this work is that a different divergence than the Kullback-Leibler one is used, resulting in the total-variation distance (Ali & Silvey, 1966), (Denuit, Dhaene, Goovaerts, & Kaas, 2005). The optimization process proposed applies linear programming, thus guaranteeing a fast and easy numerical resolution. Therefore this new method is an inverse method for valuing illiquid assets finding a RNP minimizing the relative entropy to the uniform distribution on a finite set of simulated price trajectories and satisfying the benchmarks prices constraints. In the case of an incomplete market, any value belonging to the prices interval fulfilling the non-existence of an arbitrage portfolio can be reached (Luenberger, 2002), by increasing the relative entropy of the resulting RNP from its optimal value. From the point of view of the asset replication, this new method calculates the super and infra-replicating portfolios corresponding to the two bounds of that interval, and gives a representation of the asset depending on them. It also provides the sensibility of the asset value with respect to changes in the prior probabilities, the relative entropy level of the risk neutral probabilities and finally, to changes in the benchmark prices. In order to study the performance of this new method presented, four examples have been included: one synthetic with European options, another from real world using options over the STOXX50E index, a third example based on Asian options rather than European options and an example where the a priori distribution, to which the total variation distance is minimized, is changed, using a distribution which is not the one with the maximum entropy in the interval considered. Conclusions In this work a new methodology is used, alternative to that of (Avellaneda, et al., 2001) for calculating a maximum entropy RNP by minimizing the relative entropy (divergence) to the prior uniform distribution on a finite sample space. This consists of using another convex function (2.12) (2.13) (also named total-variation distance), instead of the traditional Kullback-Leibler. An advantage of this new method seems clear, as it implies the numerical resolution by means of linear programs which can be solved very fast. Moreover, solving the dual linear program (3.10) we can finally get a representation of the asset in terms of the super-replicating portfolios opening the possibility to a sensibility analysis (3.11) of the illiquid asset value with respect to changes in the prices of the benchmarks, the relative entropy level and the prior probabilities...
This work is devoted to risk neutral probabilities (RNP) retrieval from asset prices using the relative entropy minimization methodology. The most important feature of this work is that a different divergence than the Kullback-Leibler one is used, resulting in the total-variation distance (Ali & Silvey, 1966), (Denuit, Dhaene, Goovaerts, & Kaas, 2005). The optimization process proposed applies linear programming, thus guaranteeing a fast and easy numerical resolution. Therefore this new method is an inverse method for valuing illiquid assets finding a RNP minimizing the relative entropy to the uniform distribution on a finite set of simulated price trajectories and satisfying the benchmarks prices constraints. In the case of an incomplete market, any value belonging to the prices interval fulfilling the non-existence of an arbitrage portfolio can be reached (Luenberger, 2002), by increasing the relative entropy of the resulting RNP from its optimal value. From the point of view of the asset replication, this new method calculates the super and infra-replicating portfolios corresponding to the two bounds of that interval, and gives a representation of the asset depending on them. It also provides the sensibility of the asset value with respect to changes in the prior probabilities, the relative entropy level of the risk neutral probabilities and finally, to changes in the benchmark prices. In order to study the performance of this new method presented, four examples have been included: one synthetic with European options, another from real world using options over the STOXX50E index, a third example based on Asian options rather than European options and an example where the a priori distribution, to which the total variation distance is minimized, is changed, using a distribution which is not the one with the maximum entropy in the interval considered. Conclusions In this work a new methodology is used, alternative to that of (Avellaneda, et al., 2001) for calculating a maximum entropy RNP by minimizing the relative entropy (divergence) to the prior uniform distribution on a finite sample space. This consists of using another convex function (2.12) (2.13) (also named total-variation distance), instead of the traditional Kullback-Leibler. An advantage of this new method seems clear, as it implies the numerical resolution by means of linear programs which can be solved very fast. Moreover, solving the dual linear program (3.10) we can finally get a representation of the asset in terms of the super-replicating portfolios opening the possibility to a sensibility analysis (3.11) of the illiquid asset value with respect to changes in the prices of the benchmarks, the relative entropy level and the prior probabilities...
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Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Departamento de Economía Financiera y Actuarial, leída el 04/12/2015