Untangling Illiquidity: A New Approach to Option Pricing

Author: Denis Avetisyan

Researchers are leveraging deep learning and transfer learning to accurately estimate risk-neutral densities even in severely illiquid options markets where data is scarce.

The Bates model establishes implied volatility curves by leveraging a liquid proxy (orange) to train and subsequently approximate the target implied volatility, even in illiquid markets (blue).

A Deep Log-Sum-Exp Neural Network with transfer learning enables robust sieve estimation of implied volatility and convexity in challenging market conditions.

Estimating the risk-neutral density from option prices is notoriously difficult, particularly when markets suffer from significant illiquidity. This paper, ‘Transfer Learning (Il)liquidity’, addresses this challenge by introducing a Deep Log-Sum-Exp Neural Network that leverages transfer learning to accurately recover the risk-neutral density even with sparse data. Through a novel architecture and statistical proofs, we demonstrate improved estimation performance under severe illiquidity conditions, outperforming traditional methods in both simulations and empirical analysis of SPX options. Could this framework unlock more robust and reliable derivative pricing in increasingly fragmented and illiquid markets?

The Emergent Order of Risk

The foundation of derivative pricing and robust risk management lies in the accurate determination of the risk-neutral density (RND), a probability distribution reflecting expected future asset prices under the assumption of no arbitrage. This density isn’t simply about predicting a single future value; instead, it details the probabilities associated with all possible outcomes, allowing financial institutions to fairly price complex instruments and quantify potential losses. A precise RND is crucial because it’s directly used in the integral that calculates the expected payoff of a derivative, and even slight inaccuracies can lead to significant mispricing. Furthermore, risk managers rely on the RND to perform stress tests and calculate Value-at-Risk ($VaR$), demanding a distribution that faithfully represents the underlying asset’s price dynamics – a task that proves particularly challenging in volatile or incomplete markets.

Despite their prevalence in quantitative finance, stochastic volatility models – including the Bates, Kou, and Andersen-Benzoni-Lund frameworks – encounter significant hurdles when applied to markets with limited trading activity. Accurate parameter estimation, crucial for these models to function effectively, becomes exceptionally challenging in illiquid environments due to sparse data and increased sensitivity to outliers. The models rely on historical price data to calibrate parameters governing volatility and jump processes; however, a lack of sufficient data points in illiquid markets introduces substantial estimation error. This, in turn, compromises the models’ ability to accurately price derivatives, assess risk, and ultimately, protect against potential financial losses. Consequently, practitioners often face a trade-off between model complexity and reliability when dealing with less frequently traded assets, frequently necessitating the incorporation of adjustments or alternative methodologies.

The efficacy of financial risk models, such as those predicting derivative pricing, is fundamentally linked to the validity of their underlying assumptions. Often, these models presume stable volatility, normally distributed returns, or liquid markets for accurate parameter calibration. However, real-world financial landscapes frequently deviate from these ideals; market shocks, unforeseen events, and inherent complexities can introduce volatility clustering, skewness, or illiquidity. Consequently, models built on unrealistic premises can generate inaccurate risk assessments, underestimate potential losses, and ultimately expose institutions to significant financial vulnerability. The reliance on simplified representations, while computationally convenient, introduces model risk – the potential for substantial financial harm stemming from flawed model outputs, particularly during periods of market stress or rapid change.

The Deep-LSE model (blue solid line) accurately recovers the illiquid implied volatility curve (orange dotted line) using selected strikes (green crosses), outperforming quadratic splines (green solid line) in this SPX data analysis.

Learning the Shape of Risk

The DeepLogSumExpNeuralNetwork represents a departure from conventional methods of Risk-Neutral Density (RND) estimation by integrating deep learning techniques. Traditional RND estimation models, often relying on parametric assumptions or non-parametric methods like kernel density estimation, struggle with high-dimensional data and can be computationally expensive. This network addresses these limitations through a flexible, data-driven approach. By employing a neural network, the model learns the underlying RND directly from market data, allowing it to capture complex, non-linear relationships that are difficult to model with simpler techniques. This capability is particularly valuable in markets exhibiting complex dynamics and limited historical data, where accurate RND estimation is crucial for pricing and risk management.

The DeepLogSumExpNeuralNetwork incorporates the LogSumExp function, $log(∑exp(x_i))$ , to enforce convexity within the model’s optimization landscape. This convexity is critical for ensuring training stability by preventing oscillations and local minima that frequently plague non-convex optimization problems. Furthermore, a convex function facilitates improved interpretability of the model’s parameters and outputs, allowing for a clearer understanding of the relationships between inputs and estimated RND values. The resulting optimization process exhibits more reliable convergence characteristics, reducing the computational resources required for training and increasing the likelihood of achieving a globally optimal solution.

The DeepLogSumExpNeuralNetwork utilizes a neural network architecture to model the Risk-Neutral Density (RND) which allows for adaptation to non-linear and time-varying market conditions. Traditional parametric RND estimation methods often struggle with the complexities inherent in modern financial markets; this network, through its layered structure and adjustable weights, can approximate highly complex functional forms of the RND without being constrained by pre-defined functional assumptions. This adaptability improves the precision with which the model captures subtle variations and dependencies within the underlying asset price dynamics, leading to more accurate estimations of implied volatility and other risk metrics. The network’s capacity to learn from data enables it to better represent the RND even in the presence of stochastic volatility, jumps, and other non-standard market behaviors.

Deep-LSE (orange) and quadratic splines (green) effectively recover the target RND distribution (blue) despite illiquidity, demonstrating successful reconstruction of the underlying data.

Borrowing Wisdom from Liquid Markets

Illiquidity in financial markets directly impedes the reliable estimation of Realized Niche Density (RND). RND estimation relies on statistical parameterization derived from observed trade data; however, limited transaction volume and infrequent price updates in illiquid markets result in sparse datasets. This data scarcity leads to high variance and bias in parameter estimates, diminishing the accuracy of RND calculations. Consequently, models trained on insufficient data may fail to generalize effectively or provide meaningful insights into market microstructure, necessitating alternative approaches to improve estimation robustness.

Transfer learning for illiquid markets utilizes data from highly traded, liquid proxy markets to enhance parameter estimation in target markets with sparse transaction data. This process involves pre-training a model – specifically the DeepLogSumExpNeuralNetwork – on the liquid market data, effectively capturing generalizable relationships between option prices and underlying asset characteristics. The learned parameters are then transferred and fine-tuned using the limited data available in the illiquid target market. This approach bypasses the need for extensive data in the target market, mitigating the instability and inaccuracy inherent in parameter estimation with limited observations, and ultimately improving the reliability of risk-neutral density (RND) recovery.

Transfer learning substantially diminishes estimation error and improves the resilience of the DeepLogSumExpNeuralNetwork when applied to illiquid markets. Empirical results demonstrate accurate recovery of the Risk Neutral Density (RND) with a minimal data requirement of only three option quotes, a significant improvement over traditional methods requiring substantially larger datasets for reliable parameter estimation. This enhanced performance is attributable to the knowledge transferred from liquid proxy markets, effectively regularizing the estimation process and mitigating the impact of sparse data in the target illiquid markets. The technique’s robustness is particularly valuable in situations where obtaining sufficient historical data for accurate RND estimation is impractical or impossible.

Deep-LSE (orange) and quadratic splines (green) effectively recover the illiquid target RND (blue) in this scenario, demonstrating successful illiquidity recovery.

Validating the Emergent Model

The DeepLogSumExpNeuralNetwork underwent rigorous testing utilizing SPXOptionData, a widely recognized benchmark dataset within the field of option pricing. This dataset, comprising a comprehensive collection of Standard & Poor’s 500 index option prices, allows for standardized and comparable evaluations of different pricing models. By assessing the network’s ability to accurately predict prices against this established data, researchers could objectively measure its performance relative to existing methodologies. The use of SPXOptionData ensures that the reported results are not simply a consequence of the specific data used, but reflect a genuine advancement in option pricing capability, facilitating broader acceptance and practical application within financial markets.

Evaluations using the SPXOptionData benchmark reveal a substantial advancement in accuracy with the DeepLogSumExpNeuralNetwork when contrasted with QuadraticSplines, a conventional Reduced-Dimensionality (RND) estimation technique. This deep learning approach minimizes pricing errors, as quantified by the MeanAbsoluteError, consistently outperforming established models such as Kernel, Maximum-Entropy, and Mixture methods. The observed reduction in error suggests a superior ability to capture the complex dynamics inherent in option pricing, offering a more precise valuation framework compared to traditional RND estimation.

To rigorously confirm the dependability of the DeepLogSumExpNeuralNetwork’s pricing estimator, researchers employed SieveMEstimation, a powerful statistical technique used to demonstrate theoretical consistency. This process essentially proves that, as the complexity of the model increases, its estimations converge towards the true underlying value – a crucial validation step for any financial model. By establishing this consistency, the study provides a strong mathematical foundation for the model’s reliability, assuring that observed performance improvements aren’t merely due to overfitting the training data but reflect a genuine ability to accurately price options. The successful application of SieveMEstimation therefore substantially enhances confidence in the DeepLogSumExpNeuralNetwork as a robust and trustworthy tool for option pricing tasks, particularly when compared to traditional methods.

Deep-LSE (orange) and quadratic splines (green) effectively recover the illiquid target RND (blue) in Scenario 2, demonstrating successful reconstruction of the ground truth.

Towards a More Adaptive Financial Landscape

The DeepLogSumExpNeuralNetwork, while demonstrating promise in option pricing, possesses a scalable architecture primed for incorporating increasingly granular market data. Future iterations can move beyond traditional inputs like implied volatility and interest rates to integrate alternative datasets – sentiment analysis from news articles, high-frequency trading data, and macroeconomic indicators – to refine predictive accuracy. Moreover, the network’s design facilitates adaptation to shifting market regimes; continuous learning algorithms and dynamic weighting of input features allow it to recalibrate its parameters in real-time, responding to changes in volatility surfaces and correlation structures. This inherent flexibility is crucial, as financial markets are rarely stationary, and models must evolve to maintain their efficacy; the network’s ability to learn these evolving dynamics represents a significant step towards more resilient financial modeling.

The architecture underpinning the DeepLogSumExpNeuralNetwork extends beyond the valuation of options contracts, offering a versatile framework for tackling diverse financial challenges. Its ability to model complex, non-linear relationships within data proves particularly valuable in credit risk assessment, where predicting the probability of default requires analyzing intricate borrower profiles and macroeconomic indicators. Furthermore, the methodology’s capacity to efficiently explore a vast solution space makes it well-suited to portfolio optimization, enabling the identification of asset allocations that maximize returns while adhering to specified risk tolerances. By adapting the network’s input variables and objective functions, researchers and practitioners can leverage this approach to refine risk management strategies across multiple financial domains, potentially enhancing the stability and efficiency of financial systems as a whole.

The confluence of deep learning and established statistical methodologies promises a paradigm shift in financial risk management. Traditional models, often reliant on simplifying assumptions, struggle to capture the nuances of modern, interconnected markets. Integrating deep learning’s capacity for pattern recognition and non-linear modeling with the rigor of statistical inference allows for the creation of tools that are both adaptable and reliable. This synergistic approach moves beyond mere prediction, enabling a more comprehensive understanding of risk factors and their complex interdependencies. Consequently, financial institutions can anticipate and mitigate potential losses with greater accuracy, fostering stability and resilience in an increasingly volatile global landscape. Such advancements are poised to reshape the field, delivering more robust assessments and informed decision-making in areas ranging from derivative pricing to systemic risk analysis.

The Deep-LSE model accurately fits the implied volatility curve derived from liquid proxy asset option quotes, as demonstrated in this first-step recovery scenario.

The study demonstrates that even in the face of limited data – a hallmark of illiquid markets – a system can learn to approximate complex functions, mirroring emergent order from local interactions. This resonates with Thomas Hobbes’ assertion, “The only security of a man lies in the strength of his reason.” The Deep-LSE network, through transfer learning, effectively leverages existing knowledge to ‘reason’ its way to accurate risk-neutral density estimations, even where direct observation is sparse. The network doesn’t require centralized control or pre-defined rules; rather, it self-organizes through iterative learning, exhibiting governance without interference, much like complex systems evolving from simple interactions. This highlights how influence, not control, shapes outcomes in dynamic environments.

Where Do We Go From Here?

The demonstrated capacity of Deep-LSE to extrapolate risk-neutral density from sparse data suggests a broader truth: stability and order emerge from the bottom up. The illusion of control, so comforting to those who seek to model and predict markets, is revealed as just that-an illusion. This isn’t to say prediction is impossible, but rather that attempts to impose a pre-defined structure often miss the subtle self-organization inherent in complex systems. Future work might explore how transfer learning can be adapted to even more severely constrained data environments, perhaps by focusing on the underlying structure of illiquidity itself rather than specific asset characteristics.

A natural extension lies in moving beyond static risk-neutral densities. Real markets evolve, and the network’s capacity to adapt to changing liquidity landscapes remains an open question. Investigating the computational cost of continual learning, and developing methods to prevent catastrophic forgetting, will be crucial. Moreover, the current reliance on sieve estimation, while effective, introduces its own limitations. Exploring alternative regularization techniques, or even entirely different approaches to model calibration, could yield further improvements.

Ultimately, the pursuit of accurate pricing models is less about finding the ‘true’ density and more about understanding the emergent properties of markets. The Deep-LSE offers a promising tool, but its true value may lie in prompting a shift in perspective – away from centralized control and towards a recognition of the inherent resilience and self-correcting mechanisms present even in the most illiquid corners of the financial world.

Original article: https://arxiv.org/pdf/2512.11731.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/