Parametric and Nonparametric Approaches for Estimating Risk-Neutral Density

In this dissertation, we propose one parametric method and one nonparametric method for estimating the risk-neutral density, which is a fundamental concept in pricing financial derivatives, risk management, and assessing financial markets' perceptions over significant political or economic events. In Chapter 2, we propose an innovative parametric estimate of risk-neutral density using the Normal Inverse Gaussian distribution (NIG). It has been known that the estimation bias comes from two sources, the discontinuity of available option strikes and the asymmetry of available put and call option strikes. To reduce the bias, we propose two new methods as opposed to the existing ones in the literature for each of the two bias sources, respectively. We thus have four combinations of the bias-reduction approaches. We evaluate the performance of all the four combinations by running a comprehensive empirical study using 20 years' S&P 500 index option data. The two new methods proposed by us significantly outperform the classical ones in the literature regarding feasible domain coverage and option price estimation. In Chapter 3, we propose a new nonparametric method which estimates the risk-neutral density by natural cubic splines (NCS hereafter). Our method targets the logarithm of the underlying asset price so that the restriction to the positive domain is released. We also deliberatively put the knots on all the unique option strikes to make our NCS estimate flexible enough to capture the market information contained in option prices. We run a comprehensive empirical study on the proposed NCS method, as well as other relevant approaches in the literature on 20 years of S&P 500 index option data. The empirical study shows that our proposed NCS method is more robust than the historical piecewise constant method which can only produce a discontinuous density, especially for the cases where the options have longer than half a year to maturity. Moreover, our NCS method outperforms other historical continuous methods in terms of optimization feasibility and option price estimation. In Chapter 4, we theoretically prove the consistency property of our NCS method. We prove that under reasonable assumptions, the fair prices of options based on the estimated risk-neutral density converge to their market values on average. We then verify the theoretical property numerically

Loss of whether the user adds to the shopping cart.

Loss of whether the user adds to the shopping cart.</p

Dataset 2: The relationship between purchasing or not.

Dataset 2: The relationship between purchasing or not.</p

Mathematical proof.

For the mathematical proof, please refer to the document Appendix_Mathematical_Proof. (PDF)</p

Dataset 1: The relationship between whether a user collects it or not.

Dataset 1: The relationship between whether a user collects it or not.</p

ROC for predicting user purchases.

In recent years, the global e-commerce landscape has witnessed rapid growth, with sales reaching a new peak in the past year and expected to rise further in the coming years. Amid this e-commerce boom, accurately predicting user purchase behavior has become crucial for commercial success. We introduce a novel framework integrating three innovative approaches to enhance the prediction model’s effectiveness. First, we integrate an event-based timestamp encoding within a time-series attention model, effectively capturing the dynamic and temporal aspects of user behavior. This aspect is often neglected in traditional user purchase prediction methods, leading to suboptimal accuracy. Second, we incorporate Graph Neural Networks (GNNs) to analyze user behavior. By modeling users and their actions as nodes and edges within a graph structure, we capture complex relationships and patterns in user behavior more effectively than current models, offering a nuanced and comprehensive analysis. Lastly, our framework transcends traditional learning strategies by implementing advanced meta-learning techniques. This enables the model to autonomously adjust learning parameters, including the learning rate, in response to new and evolving data environments, thereby significantly enhancing its adaptability and learning efficiency. Through extensive experiments on diverse real-world e-commerce datasets, our model demonstrates superior performance, particularly in accuracy and adaptability in large-scale data scenarios. This study not only overcomes the existing challenges in analyzing e-commerce user behavior but also sets a foundation for future exploration in this dynamic field. We believe our contributions provide significant insights and tools for e-commerce platforms to better understand and cater to their users, ultimately driving sales and improving user experiences.</div

Accuracy of whether the user adds to the shopping cart.

Accuracy of whether the user adds to the shopping cart.</p

Dataset 2: The relationship between whether a product is added to the shopping cart or not.

Dataset 2: The relationship between whether a product is added to the shopping cart or not.</p

Accuracy of whether the user has collected it or not.

Accuracy of whether the user has collected it or not.</p

Loss of whether the user has collected it or not.

In recent years, the global e-commerce landscape has witnessed rapid growth, with sales reaching a new peak in the past year and expected to rise further in the coming years. Amid this e-commerce boom, accurately predicting user purchase behavior has become crucial for commercial success. We introduce a novel framework integrating three innovative approaches to enhance the prediction model’s effectiveness. First, we integrate an event-based timestamp encoding within a time-series attention model, effectively capturing the dynamic and temporal aspects of user behavior. This aspect is often neglected in traditional user purchase prediction methods, leading to suboptimal accuracy. Second, we incorporate Graph Neural Networks (GNNs) to analyze user behavior. By modeling users and their actions as nodes and edges within a graph structure, we capture complex relationships and patterns in user behavior more effectively than current models, offering a nuanced and comprehensive analysis. Lastly, our framework transcends traditional learning strategies by implementing advanced meta-learning techniques. This enables the model to autonomously adjust learning parameters, including the learning rate, in response to new and evolving data environments, thereby significantly enhancing its adaptability and learning efficiency. Through extensive experiments on diverse real-world e-commerce datasets, our model demonstrates superior performance, particularly in accuracy and adaptability in large-scale data scenarios. This study not only overcomes the existing challenges in analyzing e-commerce user behavior but also sets a foundation for future exploration in this dynamic field. We believe our contributions provide significant insights and tools for e-commerce platforms to better understand and cater to their users, ultimately driving sales and improving user experiences.</div