## Final Report Summary - HETEROVOL (Heterogeneity and the Volatility of Financial Assets)

The scientific objectives of the project consisted in exploring a new, more realistic, yet parsimonious formulation for modeling heterogeneity in economics and finance and in investigating its implications for analyzing the volatility of financial assets. The training part of the project was concerned with both scientific and transferable skills. The objectives of the scientific training consisted in acquainting the fellow with the front line of research in Economics and in instructing him with respect to advanced methods connected to the scientific objectives of the project. The objectives of the professional training consisted in cultivating transferable skills, crucial in an academic context, with special focus on academic writing, grant proposal writing, presenting and communicating scientific findings and project management.

Regarding the research part of the work plan, the project started with the development of a modeling framework for heterogeneity in economics based on measure-valued stochastic processes. Such a stochastic process can be imagined as a sequence of random variables having as value a measure, instead of a number as in the case of classical stochastic processes. A measure is a mathematical object that is usually employed to quantify the distribution among a population of a given characteristic (for example, in economics, the distribution of income, the distribution of the propensity for consumption, the distribution of consumer confidence, etc.). The fellow conducted an extensive literature review on the topic of measure-valued processes and analyzed the properties of several kinds of such processes. He explored various implications for economics and settled on applying the new framework in the field of asset pricing. The result is a research paper that develops a novel methodology, based on measure-valued stochastic processes, for analyzing the evolution of heterogeneity in a tractable manner and studying its impact on asset prices. The agents in the economy analyzed in the paper differ with respect to a series of features such as their impatience with respect to future consumption, their risk attributes, their beliefs with respect to the dynamics of the fundamental variable driving this economy, as well as to the manner they update these believes. The key innovation consists in the fact that the heterogeneity itself is described by a single object, a measure or distribution, and its dynamics by a measure-valued stochastic process. To have a better grasp on the advantages of this novel approach, imagine a comparable situation in physics regarding the analysis of a system composed of molecules. If the number of molecules in the system is small, one can employ efficiently classical tools. However, when the system is large, tracking the dynamics of each molecule is virtually impossible, and new tools have to be used for analyzing the system as a whole. By focusing on the analogy, when the number of types of agents interacting in the economy is small, one can employ efficiently classical tools from the theory of stochastic processes. However, in the more realistic situation of a large number of types, the classical tools might no longer be appropriate and new ones can be more efficiently employed, tools based on the theory of measure-valued stochastic processes. The paper highlights the importance as a key ingredient in driving the equilibrium variables of a measure quantifying the heterogeneity of the agents interacting in this economy. This measure is itself endogenously determined meaning that its dynamics is derived in equilibrium. The paper focuses on obtaining a tractable formula for the stock price as a function of the fundamental variable (i.e. aggregate dividend) and of the measure quantifying the heterogeneity in the economy. As future research in this direction, it would be interesting to embed this methodology based on measure-valued stochastic processes in macroeconomic models in order to capture heterogeneity in the economy and in this respect to have a more realistic instrument for the support of macroeconomic policy decisions.

At the same time, research was conducted for analyzing the impact of heterogeneity on the volatility of financial assets in the context of continuous time stochastic volatility models (i.e. models in which volatility of the asset fluctuates in a stochastic manner over time). In this respect, a second research paper explores a stochastic volatility model where the interaction and herding of the heterogeneous population of agents trading the financial asset induce an amplification of the volatility of the asset over the volatility of the fundamentals. The focus is on pricing European style financial options in such a model. Most stochastic volatility models are affine, a concept of linearity. The affine structure is, in general, chosen exogenously in order to obtain a tractable option pricing formula. In contrast, in the model explored in the paper the dynamics of the asset price is endogenously determined from micro-foundations being derived from an equilibrium perspective. Moreover, the model has a more realistic non-affine structure, but at the same time one can use a tractable methodology for pricing options. A vast empirical study was implemented to analyze the option pricing performance of the model. When its results are compared to some benchmark models we find that the new non-affine model outperforms other existing affine and non-affine models. Therefore, practitioners in the financial industry interested in financial options can use the novel non-affine stochastic volatility model to price options efficiently in a more realistic context. The methodology employed for pricing options in this non-affine model is based on a more general framework for pricing options using expansion-based methods. In this context, we conducted an extensive empirical study regarding the performance of an option pricing formula derived using a specific expansion, known as the Gauss-Hermite expansion. This empirical study was included into another research paper. At the same time, given the need in the financial industry of tractable methodologies for pricing various types of financial options in both affine and non-affine models, two other research papers dealing with such methodologies were developed: a paper detailing a pricing formula for European style basket and spread options and a paper focused on American type options that develops a numerical procedure to approximate the payoff of a European type option that generates prices (which can be efficiently computed) that are equal to the prices of the American put option in the so-called continuation region.

Regarding the training part of the work plan, the fellow attended research seminars as well as several top conferences and was instructed with respect to advanced methods related to the theory of measure-valued stochastic processes. At the same time, he participated in several formal courses designed for developing transferable skills and offered for Post-Doctoral researchers by the specialized office from the University of Zurich. Moreover, the fellow contributed to writing several scientific papers, prepared and gave presentations at internal seminars and at international conferences, assisted the supervisor in preparing two grant proposals, prepared a proposals of his own and (co-)tutored several master and PhD students.

Regarding dissemination, the fellow gave presentations at several international conferences and 5 research papers benefited from the support of the project. Moreover, in order to increase the visibility of the research conducted in the project to practitioners, the fellow gave a talk at an international forum where academics in the field of quantitative finance meet practitioners from financial institutions.

Regarding the research part of the work plan, the project started with the development of a modeling framework for heterogeneity in economics based on measure-valued stochastic processes. Such a stochastic process can be imagined as a sequence of random variables having as value a measure, instead of a number as in the case of classical stochastic processes. A measure is a mathematical object that is usually employed to quantify the distribution among a population of a given characteristic (for example, in economics, the distribution of income, the distribution of the propensity for consumption, the distribution of consumer confidence, etc.). The fellow conducted an extensive literature review on the topic of measure-valued processes and analyzed the properties of several kinds of such processes. He explored various implications for economics and settled on applying the new framework in the field of asset pricing. The result is a research paper that develops a novel methodology, based on measure-valued stochastic processes, for analyzing the evolution of heterogeneity in a tractable manner and studying its impact on asset prices. The agents in the economy analyzed in the paper differ with respect to a series of features such as their impatience with respect to future consumption, their risk attributes, their beliefs with respect to the dynamics of the fundamental variable driving this economy, as well as to the manner they update these believes. The key innovation consists in the fact that the heterogeneity itself is described by a single object, a measure or distribution, and its dynamics by a measure-valued stochastic process. To have a better grasp on the advantages of this novel approach, imagine a comparable situation in physics regarding the analysis of a system composed of molecules. If the number of molecules in the system is small, one can employ efficiently classical tools. However, when the system is large, tracking the dynamics of each molecule is virtually impossible, and new tools have to be used for analyzing the system as a whole. By focusing on the analogy, when the number of types of agents interacting in the economy is small, one can employ efficiently classical tools from the theory of stochastic processes. However, in the more realistic situation of a large number of types, the classical tools might no longer be appropriate and new ones can be more efficiently employed, tools based on the theory of measure-valued stochastic processes. The paper highlights the importance as a key ingredient in driving the equilibrium variables of a measure quantifying the heterogeneity of the agents interacting in this economy. This measure is itself endogenously determined meaning that its dynamics is derived in equilibrium. The paper focuses on obtaining a tractable formula for the stock price as a function of the fundamental variable (i.e. aggregate dividend) and of the measure quantifying the heterogeneity in the economy. As future research in this direction, it would be interesting to embed this methodology based on measure-valued stochastic processes in macroeconomic models in order to capture heterogeneity in the economy and in this respect to have a more realistic instrument for the support of macroeconomic policy decisions.

At the same time, research was conducted for analyzing the impact of heterogeneity on the volatility of financial assets in the context of continuous time stochastic volatility models (i.e. models in which volatility of the asset fluctuates in a stochastic manner over time). In this respect, a second research paper explores a stochastic volatility model where the interaction and herding of the heterogeneous population of agents trading the financial asset induce an amplification of the volatility of the asset over the volatility of the fundamentals. The focus is on pricing European style financial options in such a model. Most stochastic volatility models are affine, a concept of linearity. The affine structure is, in general, chosen exogenously in order to obtain a tractable option pricing formula. In contrast, in the model explored in the paper the dynamics of the asset price is endogenously determined from micro-foundations being derived from an equilibrium perspective. Moreover, the model has a more realistic non-affine structure, but at the same time one can use a tractable methodology for pricing options. A vast empirical study was implemented to analyze the option pricing performance of the model. When its results are compared to some benchmark models we find that the new non-affine model outperforms other existing affine and non-affine models. Therefore, practitioners in the financial industry interested in financial options can use the novel non-affine stochastic volatility model to price options efficiently in a more realistic context. The methodology employed for pricing options in this non-affine model is based on a more general framework for pricing options using expansion-based methods. In this context, we conducted an extensive empirical study regarding the performance of an option pricing formula derived using a specific expansion, known as the Gauss-Hermite expansion. This empirical study was included into another research paper. At the same time, given the need in the financial industry of tractable methodologies for pricing various types of financial options in both affine and non-affine models, two other research papers dealing with such methodologies were developed: a paper detailing a pricing formula for European style basket and spread options and a paper focused on American type options that develops a numerical procedure to approximate the payoff of a European type option that generates prices (which can be efficiently computed) that are equal to the prices of the American put option in the so-called continuation region.

Regarding the training part of the work plan, the fellow attended research seminars as well as several top conferences and was instructed with respect to advanced methods related to the theory of measure-valued stochastic processes. At the same time, he participated in several formal courses designed for developing transferable skills and offered for Post-Doctoral researchers by the specialized office from the University of Zurich. Moreover, the fellow contributed to writing several scientific papers, prepared and gave presentations at internal seminars and at international conferences, assisted the supervisor in preparing two grant proposals, prepared a proposals of his own and (co-)tutored several master and PhD students.

Regarding dissemination, the fellow gave presentations at several international conferences and 5 research papers benefited from the support of the project. Moreover, in order to increase the visibility of the research conducted in the project to practitioners, the fellow gave a talk at an international forum where academics in the field of quantitative finance meet practitioners from financial institutions.