Forecasting realized densitiesa comparison of historical, risk-neutral, risk-adjusted, and sentiment-based transformations (Resumen)

Resumen

This thesis deals with the mathematical models used to forecast future asset prices. Estimating asset prices is arguably one of the most relevant problems for risk managers, central bankers, and investors. Traditional statistical methods rely on point estimates or confidence intervals to estimate future realizations. However, when it comes to analyzing asset prices at a future date, obtaining the full price distribution significantly improves the information available for decision-making. This is particularly relevant in financial prices, which typically exhibit asymmetries, fat tails and other non-normal features. Consequently, estimation methods relying on mean-variance approximations do not appropriately reproduce the real-world characteristics of financial asset prices, leading to biased predictions and inappropriate model choices. However, while the rationale to move beyond a Gaussian framework is clear, there is a remarkable lack of consensus on which mathematical models are best suited to estimate future asset prices. The disagreement remains despite the continuous improvements and extensive use that price evolution models have experienced over the last decades. Although different approaches have been proposed to forecast future asset prices, the lack of consensus is driven by three main reasons: First, estimating and calibrating the mathematical models used to forecast future asset prices has become increasingly complex, posing technical challenges that often lead to mispricing. Biased prices can invalidate the outputs of asset pricing models, leading to probabilistic paths that are not representative of real-world dynamics. Given these challenges, an appropriate implementation of asset evolution models requires that all mathematical nuisances and limitations of each forecast scheme are properly understood and considered in the implementation process. Technical challenges are particularly relevant in the stochastic processes that have been developed in the last decades, where the new features included to better approximate the asset price dynamics complicate the estimation and calibration process. Second, within the mathematical finance community, most stakeholders already hold notably dogmatic views about the types of models that should or should not be used in asset price forecasting. However, these views are generally supported by subjective beliefs about the relative merits of each competing approach instead of proper empirical validation. In the academic literature, comprehensive analyses of alternative price forecast models are scarce and mainly devoted to return and variance comparisons. In contrast, much fewer analyses consider the full price distribution, and generalization to other datasets is hindered by small sample problems, limited model choices, or non-holistic evaluations. Third, beyond modelling aspects, there is also a need for an evaluation framework that consistently assesses the predictive power of probabilistic forecasts. Although different metrics have been proposed to evaluate specific angles of probabilistic forecasts, there is no framework that jointly considers such partial measures to obtain a global evaluation. The lack of a global 7 framework complicates the comparison of alternative modeling approaches, leading to contradictory results in different studies and preventing consistent answers on which type of models perform better. Consequently, to tackle the current challenges in estimating and evaluating density predictions for asset prices, our thesis considers three interrelated topics: 1. Our first topic concerns the analysis of the numerical difficulties that characterize the estimation and calibration of complex stochastic price processes. Since the seminal papers of Black–Scholes and Merton1, processes where asset prices diffuse continuously have been extensively used in risk management and option pricing. However, to appropriately capture the asset price dynamics in the real-world, traditional diffusion models have been extended to encompass a variety of features including stochastic volatility, mean-reversion, or discontinuous jumps. While these refinements have increased the realism of stochastic asset processes, they also give rise to increasing complexity in the mathematical schemes that define future asset paths. In particular, evolution models which include stochastic volatility and/or jumps do not typically exhibit a tractable density that can be used to obtain the probability distribution at a future date T. Alternatively, the characteristic functions of complex stochastic processes are generally simpler and more tractable than their corresponding densities. Therefore, the use of Fourier transforms has rapidly gained traction and the pricing models developed in the last decades have mostly relied on characteristic functions to obtain option prices. However, the use of Fourier transforms also bring specific challenges due to discontinuities in the integrand functions, singularities at the lower or upper integration limits, or other mathematical nuisances. Furthermore, these challenges vary depending on the Fourier algorithm used to obtain option prices, the characteristic function employed to describe future asset prices, or even the parameter region considered in the calibration. As a result, specific integrals/summations routines can give rise to particular numerical problems, whereas instabilities can also arise under certain combinations of a Fourier-based method, characteristic function, and/or parameter region. Given these interdependencies, a better understanding of the different Fourier methods and their biases is paramount to avoid pricing errors and generate consistent price forecasts. 2. Our second topic concerns the statistical comparison of the main probabilistic models used to forecast future asset prices. Among forecast models, the most commonly used schemes to estimate future asset prices are: ▪ Historical-based predictions: Historical methods generate future predictions based on past prices. These models are easy to implement and extensively used in financial economics. However, it is well‐known that historical patterns do not repeat themselves, particularly in times of economic turmoil. Furthermore, historical models may yield different estimates depending on the length of the calibration window, introducing uncertainty and cherry‐picking concerns. 1 See Black and Scholes (1973) and Merton (1973). 8 ▪ Risk-neutral methods: Risk‐neutral estimates contain forward‐looking expectations and react immediately to changing market conditions, thus being conceptually better suited for forecasting purposes. However, risk‐neutral models do not consider investors’ risk preferences across different wealth states. ▪ Risk-adjusted models: By incorporating how investors value monetary outcomes in different states, these models provide a more realistic description of how investors operate in the real-world. However, risk-adjusted models are bounded by the assumption that investors are perfectly rational and always act without bias in their investment decision. ▪ Sentiment-adjusted forecasts: Since the pioneering work of Keynes (1936), increasing evidence shows that investors commit systematic behavioral mistakes that manifest in asset prices. If we accept that market prices can be affected by sentiment, it follows that market-implied forecasts should be appropriately adjusted to disentangle investor biases from fundamental expectations. Despite their differences, all these models are still extensively used for forecasting purposes. Historical models are mainly used by risk managers; risk-neutral models are used for asset pricing; risk-adjusted models are common in economics, and sentiment-based predictions are used in behavioral finance. The reason why all these models are still used is the disagreement across different stakeholders on the relative merits and drawbacks of each alternative approach. Conceptually, by including up-to-date expectations and higher realism in the processes used to describe market conditions and investors’ behavior, the forecasting ability should improve, leading to better predictions as we move from historical-based models to sentiment-adjusted predictions. However, comparisons across the different modelling approaches are scarce and when it comes to evaluating entire probability distributions there are no empirical analyses that comprehensively assess the information content of the competing schemes. 3. Our third topic concerns the framework to evaluate probabilistic forecasts. Currently, there are several statistical tests and scoring rules that are designed to measure specific angles of forecast performance (e.g.: statistical consistency, local accuracy, global errors, etc.). However, since different metrics can lead to diverging model choices, there is a need for a comprehensive evaluation framework that jointly considers the different partial aspects of probabilistic forecasts. The lack of a common framework further complicates the evaluation of competing forecasts, as partial evaluations can give rise to contradictory results and model scores that have been obtained under incomplete information.