Local Parametric Analysis of Hedging in Discrete Time

When continuous-time portfolio weights are applied to a discrete-time hedging problem, errors are likely to occur. This paper evaluates the overall importance of the discretization-induced tracking error. It does so by comparing the performance of Black-Scholes hedge ratios against those obtained from a novel estimation procedure, namely local parametric estimation. In the latter, the weights of the duplicating portfolio are estimated by fitting parametric models (in this paper, Black-Scholes) in the neighborhood of the derivative's moneyness and maturity. Local parametric estimation directly incorporates the error from hedging in discrete time. Results are shown where the root mean square tracking error is reduced up to 41% for short-maturity options. The performance can still be improved by combining locally estimated hedge portfolio weights with standard analysis based on historically estimated parameters. The root mean square tracking error is thereby reduced by about 18% for long-maturity options. Plots of the locally estimated volatility parameter against moneyness and maturity reveal the biases of the Black-Scholes model when hedging in discrete time. In particular, there is a sharp ``smile'' effect in the relation between estimated volatility and moneyness for short-maturity options, as well as a significant ``wave'' effect in the relation with maturity for deep out-of-the-money options.

Perception of intentionality in investor attitudes towards financial risks

Traditionally, financial market participation has been treated as analogous to playing games of chance with a physical device such as roulette. Here, we propose that humans treat financial markets as intentional agents, with own beliefs and aspirations. As a result, the capacity to infer the intentions of others, Theory of Mind, explains behaviour. As evidence, we appeal to results from recent studies of: (i) forecasting in the presence of insiders, (ii) trading in markets with bubbles, and (iii) financial contagion. Intensity of, and skill in, Theory of Mind explains heterogeneity, not only in choices but also in neural activation

Encoding of Marginal Utility across Time in the Human Brain

Marginal utility theory prescribes the relationship between the objective property of the magnitude of rewards and their subjective value. Despite its pervasive influence, however, there is remarkably little direct empirical evidence for such a theory of value, let alone of its neurobiological basis. We show that human preferences in an intertemporal choice task are best described by a model that integrates marginally diminishing utility with temporal discounting. Using functional magnetic resonance imaging, we show that activity in the dorsal striatum encodes both the marginal utility of rewards, over and above that which can be described by their magnitude alone, and the discounting associated with increasing time. In addition, our data show that dorsal striatum may be involved in integrating subjective valuation systems inherent to time and magnitude, thereby providing an overall metric of value used to guide choice behavior. Furthermore, during choice, we show that anterior cingulate activity correlates with the degree of difficulty associated with dissonance between value and time. Our data support an integrative architecture for decision making, revealing the neural representation of distinct subcomponents of value that may contribute to impulsivity and decisiveness

Role of the Ventromedial Prefrontal Cortex in Abstract State-Based Inference during Decision Making in Humans

Many real-life decision-making problems incorporate higher-order structure, involving interdependencies between different stimuli, actions, and subsequent rewards. It is not known whether brain regions implicated in decision making, such as the ventromedial prefrontal cortex (vmPFC), use a stored model of the task structure to guide choice (model-based decision making) or merely learn action or state values without assuming higher-order structure as in standard reinforcement learning. To discriminate between these possibilities, we scanned human subjects with functional magnetic resonance imaging while they performed a simple decision-making task with higher-order structure, probabilistic reversal learning. We found that neural activity in a key decision-making region, the vmPFC, was more consistent with a computational model that exploits higher-order structure than with simple reinforcement learning. These results suggest that brain regions, such as the vmPFC, use an abstract model of task structure to guide behavioral choice, computations that may underlie the human capacity for complex social interactions and abstract strategizing

Evidence for Model-based Computations in the Human Amygdala during Pavlovian Conditioning

Contemporary computational accounts of instrumental conditioning have emphasized a role for a model-based system in which values are computed with reference to a rich model of the structure of the world, and a model-free system in which values are updated without encoding such structure. Much less studied is the possibility of a similar distinction operating at the level of Pavlovian conditioning. In the present study, we scanned human participants while they participated in a Pavlovian conditioning task with a simple structure while measuring activity in the human amygdala using a high-resolution fMRI protocol. After fitting a model-based algorithm and a variety of model-free algorithms to the fMRI data, we found evidence for the superiority of a model-based algorithm in accounting for activity in the amygdala compared to the model-free counterparts. These findings support an important role for model-based algorithms in describing the processes underpinning Pavlovian conditioning, as well as providing evidence of a role for the human amygdala in model-based inference

Behavioral contagion during learning about another agent’s risk-preferences acts on the neural representation of decision-risk

Our attitude toward risk plays a crucial role in influencing our everyday decision-making. Despite its importance, little is known about how human risk-preference can be modulated by observing risky behavior in other agents at either the behavioral or the neural level. Using fMRI combined with computational modeling of behavioral data, we show that human risk-preference can be systematically altered by the act of observing and learning from others’ risk-related decisions. The contagion is driven specifically by brain regions involved in the assessment of risk: the behavioral shift is implemented via a neural representation of risk in the caudate nucleus, whereas the representations of other decision-related variables such as expected value are not affected. Furthermore, we uncover neural computations underlying learning about others’ risk-preferences and describe how these signals interact with the neural representation of risk in the caudate. Updating of the belief about others’ preferences is associated with neural activity in the dorsolateral prefrontal cortex (dlPFC). Functional coupling between the dlPFC and the caudate correlates with the degree of susceptibility to the contagion effect, suggesting that a frontal–subcortical loop, the so-called dorsolateral prefrontal–striatal circuit, underlies the modulation of risk-preference. Taken together, these findings provide a mechanistic account for how observation of others’ risky behavior can modulate an individual’s own risk-preference

Modeling the Evolution of Beliefs Using an Attentional Focus Mechanism

For making decisions in everyday life we often have first to infer the set of environmental features that are relevant for the current task. Here we investigated the computational mechanisms underlying the evolution of beliefs about the relevance of environmental features in a dynamical and noisy environment. For this purpose we designed a probabilistic Wisconsin card sorting task (WCST) with belief solicitation, in which subjects were presented with stimuli composed of multiple visual features. At each moment in time a particular feature was relevant for obtaining reward, and participants had to infer which feature was relevant and report their beliefs accordingly. To test the hypothesis that attentional focus modulates the belief update process, we derived and fitted several probabilistic and non-probabilistic behavioral models, which either incorporate a dynamical model of attentional focus, in the form of a hierarchical winner-take-all neuronal network, or a diffusive model, without attention-like features. We used Bayesian model selection to identify the most likely generative model of subjects’ behavior and found that attention-like features in the behavioral model are essential for explaining subjects’ responses. Furthermore, we demonstrate a method for integrating both connectionist and Bayesian models of decision making within a single framework that allowed us to infer hidden belief processes of human subjects

Neural correlates of mentalizing-related computations during strategic interactions in humans

Competing successfully against an intelligent adversary requires the ability to mentalize an opponent's state of mind to anticipate his/her future behavior. Although much is known about what brain regions are activated during mentalizing, the question of how this function is implemented has received little attention to date. Here we formulated a computational model describing the capacity to mentalize in games. We scanned human subjects with functional MRI while they participated in a simple two-player strategy game and correlated our model against the functional MRI data. Different model components captured activity in distinct parts of the mentalizing network. While medial prefrontal cortex tracked an individual's expectations given the degree of model-predicted influence, posterior superior temporal sulcus was found to correspond to an influence update signal, capturing the difference between expected and actual influence exerted. These results suggest dissociable contributions of different parts of the mentalizing network to the computations underlying higher-order strategizing in humans

Risk, Unexpected Uncertainty, and Estimation Uncertainty: Bayesian Learning in Unstable Settings

Recently, evidence has emerged that humans approach learning using Bayesian updating rather than (model-free) reinforcement algorithms in a six-arm restless bandit problem. Here, we investigate what this implies for human appreciation of uncertainty. In our task, a Bayesian learner distinguishes three equally salient levels of uncertainty. First, the Bayesian perceives irreducible uncertainty or risk: even knowing the payoff probabilities of a given arm, the outcome remains uncertain. Second, there is (parameter) estimation uncertainty or ambiguity: payoff probabilities are unknown and need to be estimated. Third, the outcome probabilities of the arms change: the sudden jumps are referred to as unexpected uncertainty. We document how the three levels of uncertainty evolved during the course of our experiment and how it affected the learning rate. We then zoom in on estimation uncertainty, which has been suggested to be a driving force in exploration, in spite of evidence of widespread aversion to ambiguity. Our data corroborate the latter. We discuss neural evidence that foreshadowed the ability of humans to distinguish between the three levels of uncertainty. Finally, we investigate the boundaries of human capacity to implement Bayesian learning. We repeat the experiment with different instructions, reflecting varying levels of structural uncertainty. Under this fourth notion of uncertainty, choices were no better explained by Bayesian updating than by (model-free) reinforcement learning. Exit questionnaires revealed that participants remained unaware of the presence of unexpected uncertainty and failed to acquire the right model with which to implement Bayesian updating

Perception of intentionality in investor attitudes towards financial risks

Traditionally, financial market participation has been treated as analogous to playing games of chance with a physical device such as roulette. Here, we propose that humans treat financial markets as intentional agents, with own beliefs and aspirations. As a result, the capacity to infer the intentions of others, Theory of Mind, explains behaviour. As evidence, we appeal to results from recent studies of: (i) forecasting in the presence of insiders, (ii) trading in markets with bubbles, and (iii) financial contagion. Intensity of, and skill in, Theory of Mind explains heterogeneity, not only in choices but also in neural activation