In low-sample settings, minimum variance portfolios often rely on covariance matrix inversion, introducing noise in such settings. This study explores the usage of Gaussian graphical models, which directly estimate the precision matrix for portfolio allocation and compare them with shrinkage, thresholding and other approaches, which estimate the covariance matrix. Using the Nifty 500 dataset, where true covariance and precision matrices are unknown, we compare portfolio allocation methods with the model confidence set approach, using portfolio variance as the performance metric. A comparative analysis of precision matrix estimators across different concentration ratios highlights that graphical Lasso and GreedyPrune outperform traditional covariance-based methods. Synthetic and empirical results consistently show that directly estimating the precision matrix yields superior portfolios in low-sample regimes. Among covariance estimators, nonlinear shrinkage delivers lower out-of-sample portfolio variance than linear shrinkage and thresholding methods. JEL Code: G11

An extension of the Hawkes process, the Marked Hawkes process distinguishes itself by featuring variable jump size across each event, in contrast to the constant jump size observed in a Hawkes process without marks. While extensive literature has been dedicated to the non-parametric estimation of both the linear and non-linear Hawkes process, there remains a significant gap in the literature regarding the marked Hawkes process. In response to this, we propose a methodology for estimating the conditional intensity of the marked Hawkes process. We introduce two distinct models: Shallow Neural Hawkes with marks - for Hawkes processes with excitatory kernels and Neural Network for Non-Linear Hawkes with Marks - for non-linear Hawkes processes. Both these approaches take the past arrival times and their corresponding marks as the input to obtain the arrival intensity. The proposed approach leverages neural networks to model the process, while preserving the interpretability associated with the marked Hawkes process.

We consider an investor who wants to hedge a path-dependent option with maturity $T$ using a static hedging portfolio using cash, the underlying, and vanilla put/call options on the same underlying with maturity $ t_1$, where $0 < t_1 < T$. We propose a model-free approach to construct such a portfolio. The framework is inspired by the \textit{primal-dual} Martingale Optimal Transport (MOT) problem, which was pioneered by \cite{beiglbock2013model}. The optimization problem is to determine the portfolio composition that minimizes the expected worst-case hedging error at $t_1$ (that coincides with the maturity of the options that are used in the hedging portfolio). The worst-case scenario corresponds to the distribution that yields the worst possible hedging performance. This formulation leads to a \textit{min-max} problem. We provide a numerical scheme for solving this problem when a finite number of vanilla option prices are available. Numerical results on the hedging performance of this model-free approach when the option prices are generated using a \textit{Black-Scholes} and a \textit{Merton Jump diffusion} model are presented. We also provide theoretical bounds on the hedging error at $T$, the maturity of the target option.

Market information events are generated intermittently and disseminated at high speeds in real-time. Market participants consume this high-frequency data to build limit order books, representing the current bids and offers for a given asset. The arrival processes, or the order flow of bid and offer events, are asymmetric and possibly dependent on each other. The quantum and direction of this asymmetry are often associated with the direction of the traded price movement. The Order Flow Imbalance (OFI) is an indicator commonly used to estimate this asymmetry. This paper uses Hawkes processes to estimate the OFI while accounting for the lagged dependence in the order flow between bids and offers. Secondly, we develop a method to forecast the near-term distribution of the OFI, which can then be used to compare models for forecasting OFI. Thirdly, we propose a method to compare the forecasts of OFI for an arbitrarily large number of models. We apply the approach developed to tick data from the Indian National Stock Exchange (NSE) and observe that the Hawkes process modeled with a Sum of Exponential's kernel gives the best forecast among all competing models.

This paper presents a Monte-Carlo-based artificial neural network framework for pricing Bermudan options, offering several notable advantages. These advantages encompass the efficient static hedging of the target Bermudan option and the effective generation of exposure profiles for risk management. We also introduce a novel optimisation algorithm designed to expedite the convergence of the neural network framework proposed by Lokeshwar et al. (2022) supported by a comprehensive error convergence analysis. We conduct an extensive comparative analysis of the Present Value (PV) distribution under Markovian and no-arbitrage assumptions. We compare the proposed neural network model in conjunction with the approach initially introduced by Longstaff and Schwartz (2001) and benchmark it against the COS model, the pricing model pioneered by Fang and Oosterlee (2009), across all Bermudan exercise time points. Additionally, we evaluate exposure profiles, including Expected Exposure and Potential Future Exposure, generated by our proposed model and the Longstaff-Schwartz model, comparing them against the COS model. We also derive exposure profiles at finer non-standard grid points or risk horizons using the proposed approach, juxtaposed with the Longstaff Schwartz method with linear interpolation and benchmark against the COS method. In addition, we explore the effectiveness of various interpolation schemes within the context of the Longstaff-Schwartz method for generating exposures at finer grid horizons.

In this paper, we present an artificial neural network framework for portfolio compression of a large portfolio of European options with varying maturities (target portfolio) by a significantly smaller portfolio of European options with shorter or same maturity (compressed portfolio), which also represents a self-replicating static hedge portfolio of the target portfolio. For the proposed machine learning architecture, which is consummately interpretable by choice of design, we also define the algorithm to learn model parameters by providing a parameter initialisation technique and leveraging the optimisation methodology proposed in Lokeshwar and Jain (2024), which was initially introduced to price Bermudan options. We demonstrate the convergence of errors and the iterative evolution of neural network parameters over the course of optimization process, using selected target portfolio samples for illustration. We demonstrate through numerical examples that the Exposure distributions and Exposure profiles (Expected Exposure and Potential Future Exposure) of the target portfolio and compressed portfolio align closely across future risk horizons under risk-neutral and real-world scenarios. Additionally, we benchmark the target portfolio's Financial Greeks (Delta, Gamma, and Vega) against the compressed portfolio at future time horizons across different market scenarios generated by Monte-Carlo simulations. Finally, we compare the regulatory capital requirement under the standardised approach for counterparty credit risk of the target portfolio against the compressed portfolio and highlight that the capital requirement for the compact portfolio substantially reduces.

An extension of the Hawkes process, the Marked Hawkes process distinguishes itself by featuring variable jump size across each event, in contrast to the constant jump size observed in a Hawkes process without marks. While extensive literature has been dedicated to the non-parametric estimation of both the linear and non-linear Hawkes process, there remains a significant gap in the literature regarding the marked Hawkes process. In response to this, we propose a methodology for estimating the conditional intensity of the marked Hawkes process. We introduce two distinct models: \textit{Shallow Neural Hawkes with marks}- for Hawkes processes with excitatory kernels and \textit{Neural Network for Non-Linear Hawkes with Marks}- for non-linear Hawkes processes. Both these approaches take the past arrival times and their corresponding marks as the input to obtain the arrival intensity. This approach is entirely non-parametric, preserving the interpretability associated with the marked Hawkes process. To validate the efficacy of our method, we subject the method to synthetic datasets with known ground truth. Additionally, we apply our method to model cryptocurrency order book data, demonstrating its applicability to real-world scenarios.

This paper presents the development of a Digital Twin (DTwin) to detect and localize the leaks in water distribution networks (WDNs), using single-stage and two-stage data-driven models. In the single-stage model, we test the anomalies in the dataset using Logistic Regression and Random Forest. In the two-stage model, a linear regression model predicts pressure differences between sensor pairs in the first stage. Based on this, we compute the distribution of residuals. In the second stage, changes in the residual distribution are classified using Multinomial Logistic Regression and Random Forest models to compute possible leak locations’ posterior probabilities. We have tested these models on a real-time dataset.

In this paper, we estimate Expected Utility Portfolios (EUPs) in high-dimensional, low-sample settings using various covariance matrix estimation methods, including shrinkage and thresholding-based methods. We perform synthetic experiments comparing these methods, using Average Out-of-Sample Variance (AOV) for Global Minimum Variance (GMV) portfolios and Average Out-of-Sample Utility (AOU) for EUPs. Additionally, we propose a practical method for fund managers to select optimal models based on empirical data, relying on AOV and AOU performance measures. The results indicate that shrinkage-based methods outperform thresholding-based ones in high-dimensional settings, with non-linear shrinkage being particularly effective.

This paper introduces the Neural Network for Nonlinear Hawkes processes (NNNH), a non-parametric method based on neural networks to fit nonlinear Hawkes processes. Our method is suitable for analyzing large datasets in which events exhibit both mutually-exciting and inhibitive patterns. The NNNH approach models the individual kernels and the base intensity of the nonlinear Hawkes process using feed forward neural networks and jointly calibrates the parameters of the networks by maximizing the log-likelihood function. We utilize Stochastic Gradient Descent to search for the optimal parameters and propose an unbiased estimator for the gradient, as well as an efficient computation method. We demonstrate the flexibility and accuracy of our method through numerical experiments on both simulated and real-world data, and compare it with state-of-the-art methods. Our results highlight the effectiveness of the NNNH method in accurately capturing the complexities of nonlinear Hawkes processes.

We consider the hedging of European options when the price of the underlying asset follows a single-factor Markovian framework. By working in such a setting, Carr and Wu \cite{carr2014static} derived a spanning relation between a given option and a continuum of shorter-term options written on the same asset. In this paper, we have extended their approach to simultaneously include options over multiple short maturities. We then show a practical implementation of this with a finite set of shorter-term options to determine the hedging error using a Gaussian Quadrature method. We perform a wide range of experiments for both the \textit{Black-Scholes} and \textit{Merton Jump Diffusion} models, illustrating the comparative performance of the two methods.

Localizing leakages in large water distribution systems is an important and ever-present problem. Due to the complexity originating from water pipeline networks, too few sensors, and noisy measurements, this is a highly challenging problem to solve. In this work, we present a methodology based on generative deep learning and Bayesian inference for leak localization with uncertainty quantification. A generative model, utilizing deep neural networks, serves as a probabilistic surrogate model that replaces the full equations, while at the same time also incorporating the uncertainty inherent in such models. By embedding this surrogate model into a Bayesian inference scheme, leaks are located by combining sensor observations with a model output approximating the true posterior distribution for possible leak locations. We show that our methodology enables producing fast, accurate, and trustworthy results. It showed a convincing performance on three problems with increasing complexity. For a simple test case, the Hanoi network, the average topological distance (ATD) between the predicted and true leak location ranged from 0.3 to 3 with a varying number of sensors and level of measurement noise. For two more complex test cases, the ATD ranged from 0.75 to 4 and from 1.5 to 10, respectively. Furthermore, accuracies upwards of 83%, 72%, and 42% were achieved for the three test cases, respectively. The computation times ranged from 0.1 to 13 s, depending on the size of the neural network employed. This work serves as an example of a digital twin for a sophisticated application of advanced mathematical and deep learning techniques in the area of leak detection.

Water utilities face the challenge of reducing water losses by promptly detecting, localizing, and repairing leaks during their operational stage. To address this challenge, utilities are exploring alternative approaches to detect leaks with high accuracy in a timely manner, while minimizing environmental and economic consequences. This research proposes a two-stage model that relies on data analysis to predict leak incidents and their specific locations in water distribution networks (WDNs). By leveraging pressure and flow rate data collected from multiple points in the network, the model first calculates prediction errors in pressure heads. Subsequently, statistical measures applied to these error distributions are used to classify the occurrence and location of leaks. The suggested approach is both cost-effective and easily deployable. Through simulation-based case studies conducted on various benchmark networks, the efficacy of the proposed model is demonstrated. The results show that the model effectively predicts leak occurrences and their respective locations. However, it should be noted that as the network size increases, the model's performance diminishes, resulting in reduced accuracy. Later, the accuracy of leak prediction has been evaluated by examining its sensitivity to varying numbers of sensors and different levels of noise.

We present a semi-static replication algorithm for Bermudan swaptions under an affine, multi-factor term structure model. In contrast to dynamic replication, which needs to be continuously updated as the market moves, a semi-static replication needs to be rebalanced on just a finite number of instances. We show that the exotic derivative can be decomposed into a portfolio of vanilla discount bond options, which mirrors its value as the market moves and can be priced in closed form. This paves the way toward the efficient numerical simulation of xVA, market, and credit risk metrics for which forward valuation is the key ingredient. The static portfolio composition is obtained by regressing the target option’s value using an interpretable, artificial neural network. Leveraging the universal approximation power of neural networks, we prove that the replication error can be arbitrarily small for a sufficiently large portfolio. A direct, a lower bound, and an upper bound estimator for the Bermudan swaption price are inferred from the replication algorithm. Additionally, closed-form error margins to the price statistics are determined. We practically study the accuracy and convergence of the method through several numerical experiments. The results indicate that the semi-static replication approaches the LSM benchmark with basis point accuracy and provides tight, efficient error bounds. For in-model simulations, the semi-static replication outperforms a traditional dynamic hedge.

This paper presents a data-driven interpretable machine learning algorithm for semi-static hedging of Exchange Traded options, considering transaction costs with efficient run-time. Further, we provide empirical evidence on the performance of hedging longer-term National Stock Exchange (NSE) Index options using a self-replicating portfolio of shorter-term options and cash position, achieved by the automated algorithm, under different modeling assumptions and market conditions, including Covid period. We also systematically assess the model's performance using the Superior Predictive Ability (SPA) test by benchmarking against the static hedge proposed by Peter Carr and Liuren Wu and industry-standard dynamic hedging. We finally perform a thorough Profit and Loss (PnL) attribution analysis on the target option and hedge portfolios (dynamic and static) to discern the factors explaining the superior performance of static hedging.

Asset pricing revolves around determining the theoretical value of an asset (stock) by evaluating a wide range of influence factors. High-dimensional data poses extraordinary challenges to asset pricing research as it results in a low signal-to-noise ratio environment that affects the computation of risk premia. Over the years data-driven solutions to the problems of empirical asset pricing have been found by developing a close bond with machine learning techniques. In practical applications, the use of dimensionality reduction techniques such as Statistical Factor Models exploit the concept of low dimensional spaces accounting for most of the variation in high dimensional data. This work explores the use of Statistical Factor Models based on conventional Principal Components Analysis and a novel Maximum Likelihood Estimation - computed numerically with the Expectation-Maximization algorithm using a centroid initialization method for smoother convergence. On comparing the implementations for 3 and 5 factors across various sample sizes, it was noted that that the proposed Maximum Likelihood Estimation based model outperforms the conventional Principal Components based factor model with a lower average mean deviation in L2 norm value for factor loadings.

In this paper, we perform a comprehensive study of different covariance and precision matrix estimation methods in the context of minimum variance portfolio allocation. The set of models studied by us can be broadly categorized as: Gaussian Graphical Model (GGM) based methods, Shrinkage Methods, Thresholding and Random Matrix Theory (RMT) based methods. Among these, GGM methods estimate the precision matrix directly while the other approaches estimate the covariance matrix. We perform a synthetic experiment to study the network learning and sample complexity performance of GGM methods. Thereafter, we compare all the covariance and precision matrix estimation methods in terms of their predictive ability for daily, weekly and monthly horizons. We consider portfolio risk as an indicator of estimation error and employ it as a loss function for comparison of the methods under consideration. We find that GGM methods outperform shrinkage and other approaches. Our observations for the performance of GGM methods are consistent with the synthetic experiment. We also propose a new criterion for the hyperparameter tuning of GGM methods. Our tuning approach outperforms the existing methodology in the synthetic setup. We further perform an empirical experiment where we study the properties of the estimated precision matrix. The properties of the estimated precision matrices calculated using our tuning approach are in agreement with the algorithm performances observed in the synthetic experiment and the empirical experiment for predictive ability performance comparison. Apart from this, we perform another synthetic experiment which demonstrates the direct relation between estimation error of the precision matrix and portfolio risk.

The Multi-dimensional Hawkes Process (MHP) is a class of self and mutually exciting point processes that find many applications–from predicting earthquakes to modelling order books in high-frequency trading. This paper makes two significant contributions; we first find an unbiased estimator for the gradient of the Hawkes process’s log-likelihood estimator. The estimator enables the efficient implementation of the stochastic gradient descent method for the maximum likelihood estimation. The second contribution is that we propose a specific neural network for the non-parametric estimation of the underlying kernels of the MHP. We evaluate the proposed model on synthetic and natural datasets and find the method has comparable or better performance than existing estimation methods. The use of neural networks for modelling the excitation kernel ensures that we do not compromise on the Hawkes model’s interpretability. At the same time, the proposed algorithm has the flexibility to estimate any non-standard Hawkes excitation kernel.

Text similarity algorithms are the core building blocks for several natural language processing (NLP) applications. Question-Answering, Search Engines, Plagiarism check, and Summarization are a few of the several applications with text similarity as the building block. Over the last two decades, immense research has been conducted by the NLP and computer science community to develop more accurate algorithms to detect semantic and syntactic similarities. But, more than often than not these works have overlooked the importance of chronological order of text, which is very essential in cases like a manufacturing-process text where we try to find the order in which ingredients are added, processed, etc. When we additionally capture word order similarity, the final text similarity tends to be more accurate. In this chapter, we present a simple ensemble method to capture semantic, syntactic, and chronological similarities between two sentences. The ensemble consists of a proportioned modified edit distance score and cosine similarity score from pretrained state-of-the-art embedding. The embedding captures the semantic similarity and the modified edit distance helps capture the word order, thus making the model more robust. Further, we have studied the effects of various ensemble ratios to help achieve maximum accuracy. The proposed model was demonstrated on the standard STS-B dataset and custom dataset. The results show that the ensemble achieves far better accuracy than the individual models to measure similarity in sentences.