Title: Optimal Non-Linear Shrinkage Estimation of Covariance Matrix of Asset Returns by Imaginary Direction Smoothing

Abstract: While Markowitz portfolio selection requires an estimator of the covariance matrix of returns, sample covariance performs poorly when a large number of assets are considered. Using random matrix theory, the limiting optimal shrinkage function is recently found but it depends on unknowns such as a real-direction limit of a Stieltjes transform of the limiting spectral distribution, defined on the upper complex half-plane. Empirical estimation of such quantities is quite unstable and several estimators are recently proposed via discretization, smoothing in the real direction or by resampling. We consider an alternative by defining a smoothed loss function based on smoothed empirical spectral distribution, and its optimum in terms of a smoothed Stieljes transform, where smoothing is along the imaginary direction via a Cauchy kernel. Compared to smoothing in the real direction, our smoothing in the imaginary direction offers a data-adaptive choice of the smoothing parameter, due to a relationship between the imaginary part of a complex Stieltjes transform and the sample eigenvalue distribution. Some theoretical properties and numerical illustration will be discussed.

This is a joint work with Sheung Chi Phillip Yam, Xiaolong Li and Yifan Shi.

Title: Gaussian Process Surrogates for Delta Hedging

Abstract: I will discuss a machine learning approach to option Greeks approximation based on Gaussian process (GP) surrogates. The method takes in noisily observed option prices, fits a nonparametric input-output map and then analytically differentiates the latter to obtain the various price sensitivities. Our motivation is to compute Greeks in cases where direct computation is expensive, such as in local volatility models, or can only ever be done approximately. We provide a detailed analysis of numerous aspects of GP surrogates, including choice of kernel family, simulation design, choice of trend function and impact of noise. We further discuss the application to Delta hedging, including a new Lemma that relates quality of the Delta approximation to discrete-time hedging loss. Results are illustrated with two extensive case studies that consider estimation of Delta, Theta and Gamma and benchmark approximation quality and uncertainty quantification using a variety of statistical metrics. Among our key take-aways are the recommendation to use Matern kernels, the benefit of including virtual training points to capture boundary conditions, and the significant loss of fidelity when training on stock-path-based datasets.

This is joint work with Yuri Saporito (FGV Rio de Janeiro, Brazil) and based on the preprint arxiv.org/abs/2010.08407

Title: Attractive Long-term Pension Payouts for Persisting Low Interest Rates

Abstract: The persistence of low interest rates is spurring research on the question how to increase yields, while limiting the variability of long-term investment payouts. Under the benchmark approach it is possible to achieve attractive, almost riskless, non-fluctuating long-term investment results. Payouts of savings account units that achieve an almost riskless outcome over a long time period can be hedged reliably as contingent claims by using a stock index and a savings account. This dynamic asset allocation can be performed in a less expensive manner than by traditional valuation methods. The benchmark approach is using real-world pricing, which provides the least expensive hedging strategy for replicable contingent claims.

Title: Insurance Risk Analysis of Financial Networks Vulnerable to a Shock

Abstract: We conduct a quantitative risk analysis of non-core insurance business of selling financial products to protect financial firms against investment losses due to a shock. To achieve the goal, we construct a static structural model composed of a network of firms who cross-hold each other, multiple primitive assets that are vulnerable to a shock, and an insurer who resides external to the network and speculates in selling protection to financial firms. Assume that each firm in the network is rational and follows the mean-variance principle to decide how much protection to purchase to optimize its portfolio. As a result, the shock may impact the insurer but indirectly through the network, and thus both the network integration and the shock are risk factors of this non-core insurance business. Our study focuses on their interactive role in the insurance risk. Depending on the shock size, an increase in the network integration may help reduce the insurance risk, or drive the insurer to a more profitable position, or hinder the insurer from making profit.

Title: An Axiomatic Foundation for the Expected Shortfall

Abstract: In the recent Basel Accords, the Expected Shortfall (ES, also known as CVaR and TVaR) has replaced the Value-at-Risk (VaR) as the standard risk measure for market risk in the banking sector, making it the most important risk measure in financial regulation. Although ES is, among many other nice properties, a coherent risk measure, it does not yet have an axiomatic foundation until now. We put forward four intuitive economic axioms for portfolio risk assessment - monotonicity, law invariance, prudence and no reward for concentration - that uniquely characterize the family of ES. The herein developed results, therefore, provide the first economic foundation for using ES as a globally dominating regulatory risk measure. Some other related results on ES will be discussed.