Machine Learning With Kernels for Portfolio Valuation and Risk Management

Abstract

We introduce a statistical simulation method for dynamic portfolio valuation and risk management building on machine learning with kernels. We learn the dynamic value process of a portfolio from a finite sample of its terminal cumulative cash flow. The learned value process is given in closed form thanks to a suitable choice of the kernel. We develop an asymptotic theory and prove convergence and a central limit theorem. We derive dimension-free sample error bounds and concentration inequalities. Numerical examples show good results for a relatively small training sample size. Keywords: dynamic portfolio valuation, kernel ridge regression, learning theory, reproducing kernel Hilbert space, portfolio risk management MSC (2010) Classification: 68T05, 91G60 JEL Classification: C15, G32 1 Introduction Valuation, risk measurement, and hedging form an integral task in portfolio risk management for banks, insurance companies, and other financial institutions. Portfolio risk arises because the values of constituent assets and liabilities of the portfolio change over time in response to changes in the underlying risk factors, e.g., interest rates, equity prices, real-estate prices, foreign exchange rates, credit spreads, etc. The quantification and management of this risk requires a stochastic model of the dynamic portfolio value process.In the absence of a liquid market for price discovery of such a portfolio, its value has to be derived from the cash flow that it generates. Examples of such portfolios include path-dependent options, structured products, such as barrier reverse convertibles, mortgage-backed securities, and life insurance products featuring mortality and longevity risks. More formally, we henceforth assume that all financial values and cash flows are discounted by some numeraire, e.g., the money market account that earns the short-term interest rates.

Kernel Minimum Divergence Portfolios

“…Our contributions are two-fold: 1. on the theoretical side: (i) we compute analytically the mean embedding for various target distribution -kernel pairs (Section 3.1). (ii) We show (Theorem 3.4) that such analytical knowledge leads to better concentration properties of MMD estimators, (iii) extend the result to the case of unbounded kernels (Theorem 3.5; recently motivated in finance for instance by [5]), and (iv) present minimax lower bounds (Theorem 3.6). 2. on the practical front: we demonstrate that the flexibility of considered measures result in more efficient portfolios in terms of skewness and kurtosis (Section 4-5), which provides a novel application of these divergence measures.…”

“…We showed that prior knowledge available on target distribution leads to analytical forms for the mean embedding (Lemma 3.1 and Lemma 3.2) and improved MMD concentration properties for bounded kernels (Theorem 3.4). Motivated by recent financial studies relying on unbounded kernels [5], using the Burkholder inequality we proved that the 1/ √ N rate (with N denoting the sample size) of MMD estimators can be extended to unbounded kernels (Theorem 3.5); we illustrated the idea for the exponential kernel. We showed matching minimax lower bounds (Theorem 3.6) under slightly more restrictive conditions.…”

“…• They are flexible in terms of the target distribution: Stein discrepancy requires the knowledge of the pdfs up to normalizing constant only, the Wasserstein distance can be written explicitly as a function of the inverse cdf (see (2.7)), the mean embedding (which is the underlying representation of probability measures used in MMD) can be computed analytically for different target distribution and kernel pairs. • These discrepancy measures show excellent performance in various (complementary) applications such as model criticism [37,31], two-sample [24,20], independence [21,46] and goodness-of-fit testing [36,8,28,17,26,19,15,1], portfolio valuation [5], statistical inference of generative models [6] and post selection inference [59], causal discovery [41,46], generative adversarial networks [11,35,3], assessing and tuning MCMC samplers [16,17,26,18], or designing Monte-Carlo control functionals for variance reduction [44,43,52], among many others.…”

“…skewness and lower kurtosis. 5 Targeting lighter-tailed distribution (via for example generalized normal distribution) or more positively skewed distributions (for instance with skew Gaussian target) should have a favorable impact regarding the out-of-sample moments of the minimum-divergence portfolio compared to traditional mean-variance portfolios. In our numerical studies, we are going to focus on three families Divergence (D)…”

Ensemble learning for portfolio valuation and risk management

“…Our method is different as it learns the entire value process V in one go, as opposed to any method relying on nested Monte Carlo simulation, which estimates V t for one fixed t at a time. Our method shares some similarities with the kernel-based method in our previous paper [Boudabsa and Filipović, 2022]. There we applied kernel ridge regression to derive a closed-form estimator of the value process V .…”