Hidden Structure: Dr. Anubha Goel on Reading What Financial Markets Do Not Say Directly

From Risk to Structure 

Most standard tools in quantitative finance are designed to measure specific, individual things, such as the risk of a single position or the price of a specific derivative. Goel’s earlier work during her PhD at IIT Delhi focused on these areas, using statistical tools to move beyond the common assumption that financial returns always follow a normal distribution. 

"My PhD research was in financial mathematics; I was drawn to the fact that it uses rigorous mathematics in a setting where uncertainty, risk, and decision-making really matter.” 

“I worked on problems such as portfolio optimisation, derivatives pricing, dependence modelling, and risk measurement. In the earlier phase I focused on tools like mixed CVaR, deviation measures, and copulas, because I wanted to model downside risk and joint behaviour more realistically, especially beyond normality assumptions." 

As her research progressed, she realized that even the most sophisticated traditional tools were missing something important about the larger organization of the market. This led her to ask whether financial data could be better understood by looking at its overall shape. 

"Even with these advanced methods, something was still missing. They capture dependence and risk well, but they do not always tell you enough about the broader structure of the market.”  

“The important turning point for me was realising that topology could uncover meaningful asset groupings and structural relationships that standard methods did not always make visible." 

Simplification Without Loss of Shape 

In mathematics, topology is the study of shape and how different points are connected. When applied to data, it can identify structural patterns that remain stable even when the data itself is noisy. This is particularly useful for financial time series, which are high-dimensional and constantly changing.  

“Correlation is local and pairwise; topology is more global. It can tell you how assets cluster, how those clusters connect, and whether there are persistent structural patterns in the market.” 

One of the most consistent challenges Goel addresses is sparse index tracking, which involves building a small portfolio that closely follows a broad market index without having to own every single stock in that index. This is a practical issue for fund managers because holding hundreds of different positions leads to high trading costs and operational complexity. 

"At a practical level, sparse index tracking is a very simple idea with a difficult implementation problem behind it. If you want to follow a broad market index, the most direct solution is to buy every stock in that index. But for many investors that is not ideal." 

While traditional methods focus on which stocks move together, a topological approach looks for stocks that are structurally representative of different parts of the market as a whole. 

"Markets are not really organised as a collection of isolated pairs. They are structured systems, with sectors, subgroups, overlaps, and changing relationships across time. My approach tries to capture that larger organisation by looking at the shape of stock price movements in a high-dimensional space." 

By identifying representatives from distinct regions of the market, this method maintains diversification even when the portfolio is small, resulting in a leaner selection of assets that still captures the behavior of the full index. 

“If ten stocks are all expressing almost the same part of the market, you may not need all ten. You may need one or two that carry that information well.” 

Tracking Behavior and Capital Flow 

At Tampere, Goel has extended this structural thinking into market surveillance and capital flow. One line of work uses a topological method called Mapper to identify investors who might be trading on private, non-public information. Instead of just asking who looks statistically unusual, this method looks for investors whose trading fits a specific, structurally meaningful behavioral pattern around important announcement windows. 

"Standard methods mostly ask who looks rare. Mapper asks which investors fall into a structurally meaningful pattern once you view the data through expert-designed lenses. Keeping false positives low was central, because in this setting a false positive is not just a modelling error. It could mean wrongly flagging an innocent investor." 

The method is designed to be conservative, only isolating cases where behavioral evidence is consistent across different views of the data, which makes it more useful for real-world market monitoring. Another project involves reconstructing how capital moves between investors and securities over time using transaction data from the Helsinki Stock Exchange. When millions of individual trades are organized into directed networks, recurring patterns of concentration and reallocation become visible. 

“Instead of asking only who bought or sold a stock, we ask what an investor likely sold in order to fund what they bought, and then aggregate that across many investors.” 

"At the level of single trades, markets can look extremely noisy. But once you reconstruct flows between securities, you start to see recurring patterns, concentration into particular assets, diversification away from others, and chain-like reallocations that repeat over time." 

These money-flow networks provide a unique view of how investors collectively rebalance their portfolios and how price pressure might move from one asset to another. 

The Interdisciplinary Future 

Goel’s career path through IIT Delhi, the Swiss Finance Institute at EPFL, and finally Tampere University has allowed her to blend mathematical rigor with practical financial application. At Tampere, she works at the intersection of finance, computation, and network science. 

"At Tampere, the perspective has become even broader, because here the work sits very naturally at the intersection of finance, computation, network science, and data analytics. It is an environment where financial questions can be studied using tools from data analytics, network science, and machine learning, without treating those as separate conversations." 

I have moved from asking, ‘Is this mathematically elegant?’ to asking, ‘Is it mathematically sound, computationally feasible, and relevant to the structure of real financial data?’

Anubha Goel

“I have moved from asking, ‘Is this mathematically elegant?’ to asking, ‘Is it mathematically sound, computationally feasible, and relevant to the structure of real financial data?’

Her Marie Curie fellowship provides the freedom to define an independent research agenda that addresses technically challenging problems with real-world impact. Looking ahead, she is focused on three main areas: creating explainable models for financial forecasts, modeling how information spreads through systems, and embedding topology directly into financial decision-making. Interestingly, her work on information contagion in markets has potential applications in healthcare, such as tracking the early spread of diseases when data is incomplete. 

"COVID made that very clear: one of the biggest challenges was not only understanding what had already happened, but identifying hidden chains of transmission early enough to respond effectively. I see that as a very promising direction, taking a framework developed in markets and extending it into healthcare." 

Ultimately, Goel aims to build models that can uncover hidden structures and remain reliable under uncertainty while still being easy for practitioners to interpret. 

"These are really models for inference in complex networks where the underlying propagation is only partially visible.” 

“The direction that seems most significant to me is toward models that can recover hidden structure, work well under uncertainty, and still remain interpretable."