Working papers

Nonlinear Pricing Kernel via Explainable Neural Networks (2025)

We develop a framework for the optimal specification of a pricing kernel, also known as stochastic discount factor (SDF), by formulating a constrained optimization problem over a space of neural network functions. Departing from traditional linear kernel models, we approximate the SDF as a nonlinear function of pricing factors implemented with feedforward neural networks. The objective is to minimize a quadratic form of pricing errors, subject to a non-negativity constraint on the kernel that guarantees arbitrage-free pricing. This yields a non-convex, high-dimensional optimization problem over network weights, with the constraint incorporated via architecture and activation design. We develop a procedure through hypothesis testing to select the optimal network architecture among candidate specifications, enhancing model comparability and robustness. Empirically, we show that the worst-performing neural network outperforms the linear benchmark in pricing accuracy. Furthermore, statistical significance tests identify key pricing factors in the high-dimensional setting, improving the interpretability of the learned pricing kernel. Overall, our results demonstrate the advantages of neural networks in both flexibility and scalability for solving constrained asset pricing problems.

Presentation : QMUL Economics and Finance Workshop (2025), Adam Smith Business School Brown Bag Seminar (Glasgow, 2024).

Risk aversion and portfolio optimization for robo-advising (2023)

We develop a novel framework for learning investors' risk aversion using low-resolution data, a common issue arising from short trajectories recording investors' portfolio choices, particularly during disaster events. Furthermore, the observed portfolio choice is often affected by behavioural biases. Our approach combines online inverse optimization with deep RL to simultaneously estimate risk aversion and determine optimal investment strategies under both normal and disaster states. Utilizing real mutual fund data, we demonstrate that our algorithm's risk aversion estimation converges asymptotically to the optimal risk aversion during the learning process. Critically, based on the learned risk aversion and trained deep RL model, we show that robo-advisors adopting our approach can effectively tailor investment strategies to suit investor risk aversion under varying market conditions, outperforming traditional funds. This highlights the potential for our framework to enhance investment decision-making and better represent investor interests in both stable and volatile market environments.

Presentations: Adam Smith Business School Brown Bag Seminar (Glasgow, 2023), Statistics of Machine Learning (Prague, 2022).

Robo-advising under rare disasters (2022)

Robo-advisors provide automated portfolio management services to investors, and their growth has been unprecedented in the past few years. However, empirical evidence shows that robo-advisors underperformed during the recent COVID-19 pandemic. This may be because rare disasters are highly unlikely to occur and yet have a huge impact on financial markets. Our study develops a novel computational framework to improve the performance and robustness of robo-advising in the presence of rare disasters. It integrates RL with importance sampling. Instead of sampling the transition probability from a ground-truth probability distribution, we sample it from a proposal distribution, where the event of interest occurs more frequently. The proposed algorithm is validated by data covering the 2008 financial crisis and the COVID-19 pandemic, showing superior performance over benchmarked methods. The estimated quarterly return of the robo-advising portfolio using the optimal policy of the proposed algorithm is 0.512%, significantly higher than both the benchmarked policy and the average quarterly return, which are -0.639% and -14.55%, respectively. This improvement is attributed to targeted learning about rare disasters, enabling robo-advisors to reduce exposure to risky assets. The proposed algorithm is model-free and reduces the variance of value estimates through importance sampling. 

Presentations: The 6th International Conference on Econometrics and Statistics (Tokyo, 2023), IRTG 1792 Summar Camp (Buckow, 2022).