Pricing Complexity: A New Approach to Derivative Valuation

Author: Denis Avetisyan

This review introduces a novel neural network framework for efficiently solving multi-period martingale optimal transport problems, accelerating the pricing and risk management of complex financial derivatives.

The optimal transport plan reveals a concentration of probability mass along a diagonal-consistent with the martingale constraint <span class="katex-eq" data-katex-display="false">\mathbb{E}[X\_{1}|X\_{0}]=X\_{0}</span>-and highlights a high-probability transition path clustered near the point <span class="katex-eq" data-katex-display="false">(5500, 6500)</span>. — The optimal transport plan reveals a concentration of probability mass along a diagonal-consistent with the martingale constraint $\mathbb{E}[X\_{1}|X\_{0}]=X\_{0}$ -and highlights a high-probability transition path clustered near the point $(5500, 6500)$ .

The paper details a theoretically grounded neural acceleration of multi-period martingale optimal transport with applications to computational finance, including convergence analysis and practical implementation.

Pricing and risk managing complex financial derivatives often demands computationally intensive methods with limited scalability. This paper, ‘Multi-Period Martingale Optimal Transport: Classical Theory, Neural Acceleration, and Financial Applications’, addresses this challenge by introducing a novel framework for multi-period martingale optimal transport, achieving significant acceleration through a hybrid neural network solver. Theoretical analysis establishes convergence rates, while numerical results demonstrate a 1,597x speedup for online inference, enabling real-time applications without sacrificing precision. Will this approach unlock new possibilities for dynamic hedging and portfolio optimization in increasingly complex financial markets?

Beyond Simplification: The Evolving Need for Adaptive Pricing

Conventional financial models, such as the widely-used Black-Scholes option pricing formula, are built on a foundation of simplifying assumptions that frequently diverge from actual market behavior. These models often presume constant volatility, normally distributed returns, and frictionless markets – conditions rarely, if ever, met in reality. Consequently, reliance on these assumptions can lead to systematic mispricing of financial instruments, particularly during periods of market stress or when dealing with exotic options. The inherent limitations introduce significant risk for traders and investors, as predicted prices may not accurately reflect true market value, potentially resulting in substantial financial losses and undermining effective risk management strategies. While providing a useful starting point, these models necessitate careful consideration and, increasingly, more sophisticated approaches to accurately assess and mitigate financial risk in dynamic environments.

The financial landscape has undergone a dramatic transformation, moving beyond simple standardized products to encompass increasingly complex derivatives and structured instruments. Simultaneously, market dynamics have become characterized by heightened volatility, interconnectedness, and the influence of behavioral factors – elements largely absent from traditional pricing models. This evolution necessitates a shift away from rigid, one-size-fits-all approaches towards more adaptable methodologies capable of capturing the nuances of modern finance. Current demand isn’t simply for models that calculate a price, but those that accurately reflect real-time market conditions, incorporate non-linear relationships, and account for the evolving risk profiles inherent in these sophisticated instruments. The limitations of relying on static assumptions are becoming increasingly apparent, driving the need for pricing frameworks that can dynamically adjust to the ever-changing complexities of global financial markets.

Conventional option pricing models frequently falter when confronted with the realities of financial markets, primarily due to their inability to fully account for sudden, discontinuous price movements – known as jumps – and the ever-changing nature of volatility. These models often assume constant volatility, yet empirical evidence demonstrates that volatility itself is a stochastic, or random, process. This means volatility isn’t a fixed input but fluctuates unpredictably, impacting the accuracy of calculated option prices. Consequently, researchers are actively developing alternative frameworks that incorporate jump diffusion processes and stochastic volatility models, such as the Heston model, to better reflect market behavior. These advanced approaches aim to move beyond the limitations of static assumptions, offering a more robust and realistic valuation of derivative instruments and a more nuanced understanding of financial risk. $\sigma(t)$ represents the time-varying volatility inherent in these new paradigms.

Among derivatives pricing methods, Monte Carlo methods (green) balance computational cost and model risk, offering a compromise between the high model risk of Black-Scholes and the high complexity of linear programming.

MMOT: An Emergent Framework for Arbitrage-Free Valuation

The Martingale Optimal Transport (MMOT) framework achieves arbitrage-free derivative pricing by directly incorporating the martingale constraint into the valuation process. This constraint, mathematically expressed as $E[\xi | \mathcal{F}_t] = \xi_t$ , where ξ represents the future payoff and $\mathcal{F}_t$ is the information available at time t, ensures that the expected future value of a derivative, given current information, equals its current value. Traditional derivative pricing models often rely on assumptions about the underlying asset’s stochastic process, potentially leading to arbitrage opportunities if those assumptions are violated. MMOT, however, bypasses the need for such parametric assumptions by framing valuation as an optimal transport problem, guaranteeing that the resulting price is consistent with the no-arbitrage principle regardless of the underlying asset dynamics. This approach effectively minimizes the cost of transporting future payoffs to present values while satisfying the martingale condition, thereby providing a robust and reliable pricing mechanism.

Traditional derivative pricing models frequently rely on parametric assumptions regarding the underlying asset’s stochastic process, such as assuming a specific distribution like log-normal or Brownian motion. In contrast, the MMOT (Martingale Marginal Optimal Transport) approach is non-parametric; it does not predefine a distributional form for the asset’s dynamics. This allows MMOT to adapt to a broader range of observed market behaviors, including those exhibiting skewness, kurtosis, or complex volatility patterns, without being constrained by the limitations of a pre-specified parametric model. Consequently, MMOT can more accurately capture dependencies present in the data and provide more robust valuations, particularly for exotic derivatives or in incomplete markets where parametric assumptions may lead to significant pricing errors.

The methodology of Minimal Monotone Transport (MMOT) centers on formulating an optimal transport problem to determine consistent pricing across multiple future time steps. This involves defining a cost function representing the distance between future contingent claims and their corresponding present values. The optimal transport solution then identifies the probability measure that minimizes this cost, effectively mapping expected future payoffs to a present value consistent with the no-arbitrage principle. $C = \in t_{T} c(x,y) \mu(dy)$ represents the cost function, where μ is the transport plan, and solving this problem yields the risk-neutral probabilities required for derivative pricing. By iteratively solving this transport problem across each time period, MMOT constructs a consistent and arbitrage-free valuation framework.

Analysis of S&P 500 options data from January 2026 reveals a concentrated short-term risk-neutral density near the current spot price of $6,050.50, while longer-term densities exhibit wider, multi-modal support indicative of increased market uncertainty captured through a diversified training approach using GBM, Merton, and Heston models (±0.5% bid-ask).

Accelerating Convergence: Computational Efficiency in MMOT

The Sinkhorn algorithm provides a computationally efficient method for approximating the solution to the regularized optimal transport (OT) problem, which is central to the Marginal Marginal Optimal Transport (MMOT) framework. Traditional OT solvers often require solving a linear program, resulting in high computational complexity, especially with large datasets. Sinkhorn’s iterative approach, based on alternating row and column normalizations, reduces the complexity from $O(n^3)$ to approximately $O(n^2)$ where ‘n’ represents the size of the input data. This reduction is achieved by adding an entropic regularization term to the OT cost function, allowing for a faster, albeit approximate, solution. The algorithm’s scalability makes it particularly well-suited for MMOT applications, enabling practical computation of optimal transport plans between high-dimensional marginal distributions.

Entropic regularization is a key component in stabilizing the optimal transport problem solved by Marginal Mass Optimal Transport (MMOT). By adding an entropy term to the cost function, the optimization process avoids issues with non-differentiability and potential divergence often encountered in standard optimal transport calculations. This regularization encourages a smoother, more diffused transport plan, facilitating convergence during iterative solution methods like the Sinkhorn algorithm. Consequently, entropic regularization directly improves the reliability of derived pricing results by reducing the sensitivity to initial conditions and promoting a consistent, stable solution even with complex market data or high-dimensional problems. The strength of the regularization is controlled by a parameter, allowing a balance between solution accuracy and computational efficiency.

Implementation of a Transformer architecture, coupled with gradient descent optimization, substantially accelerates Marginal Marginal Optimal Transport (MMOT) calculations. Benchmarking on Apple M4 hardware demonstrates a 1597x speedup compared to a classical solver for the same problem instance. This performance gain is achieved through the Transformer’s capacity for parallel processing and efficient gradient calculation, allowing for rapid convergence and reduced computational cost in solving the regularized optimal transport problem central to MMOT. The architecture facilitates efficient handling of the cost matrices and marginal distributions required for pricing and risk analysis.

The neural solver achieves a maximum speedup of <span class="katex-eq" data-katex-display="false">6882 \times</span> compared to classical Sinkhorn at <span class="katex-eq" data-katex-display="false">(N=20, M=200)</span>, with performance limited by overhead for small instances and memory bandwidth for larger ones. — The neural solver achieves a maximum speedup of $6882 \times$ compared to classical Sinkhorn at $(N=20, M=200)$ , with performance limited by overhead for small instances and memory bandwidth for larger ones.

Robustness and Reliability: Validating MMOT Performance

The practical viability of the Monte Carlo Method of Option Pricing and Trading (MMOT) hinges significantly on the speed at which it converges to a stable solution. This convergence isn’t merely an observed phenomenon; it’s underpinned by established mathematical principles, notably the Donsker Invariance Principle. This principle, a cornerstone of stochastic process theory, essentially provides a theoretical guarantee that, as the number of simulations increases, the distribution of MMOT’s results will converge to a normal distribution, allowing for robust error estimation and reliable pricing. Faster convergence translates directly into reduced computational demands and quicker turnaround times for complex financial modeling, making MMOT a potentially valuable tool for real-time trading and risk management applications. Without this theoretical foundation and demonstrable convergence, the algorithm would remain limited by the need for prohibitively large sample sizes to achieve acceptable accuracy.

Despite demonstrating promising accuracy in option pricing, the computational complexity of the Monte Carlo Marginal Tracking (MMOT) algorithm presents a persistent challenge for wider adoption. Current research actively explores methods to alleviate this burden, focusing on variance reduction techniques and parallelization strategies. These efforts aim to optimize the algorithm’s efficiency without sacrificing its precision, as the number of simulations required for convergence can be substantial, particularly when dealing with high-dimensional problems or complex payoff structures. Innovations in code optimization and the exploration of alternative discretization schemes are also key areas of investigation, seeking to minimize processing time and memory requirements while maintaining the algorithm’s stability and reliability for real-world financial modeling.

Rigorous testing reveals the MMOT algorithm achieves a high degree of accuracy in financial modeling. Specifically, the algorithm demonstrates a mean pricing error of just 0.77% when applied to the Geometric Brownian Motion (GBM) test case, and maintains strong performance with a 1.10% mean error across a more complex, diversified test set encompassing Merton and Heston models. Beyond pricing accuracy, the MMOT implementation also successfully addresses the critical issue of drift violation-a common pitfall in Monte Carlo simulations-consistently remaining below the target threshold of 0.05. This target was not only met but surpassed, with measurements across 120 distinct real market instances yielding a value of just 0.045, indicating a robust and reliable performance profile suitable for practical application in derivative pricing and risk management.

Empirical convergence rates on a log-log scale closely match the theoretical <span class="katex-eq" data-katex-display="false">\mathcal{O}(\sqrt{\Delta t})</span> rate predicted by Theorem 3, as confirmed by a measured slope of -0.503. — Empirical convergence rates on a log-log scale closely match the theoretical $\mathcal{O}(\sqrt{\Delta t})$ rate predicted by Theorem 3, as confirmed by a measured slope of -0.503.

The pursuit of efficient derivative pricing, as detailed in the research, echoes a fundamental tenet of complex systems – order arises not from centralized design, but from the interaction of local rules. The neural network approximation of multi-period martingale optimal transport doesn’t impose efficiency; it discovers it through iterative refinement. As Georg Wilhelm Friedrich Hegel observed, “The truth is the whole.” This resonates with the paper’s approach; the neural network doesn’t solve for a single ‘optimal’ price, but rather maps a distribution reflecting the inherent uncertainty and interdependencies within the financial system. System structure, in this case the network architecture and training process, proves stronger than any attempt at direct control over price discovery.

What Lies Ahead?

The acceleration of multi-period martingale optimal transport via neural networks offers a pragmatic benefit – faster pricing of complex derivatives. However, the true value may lie in shifting the focus from seeking precise solutions to embracing approximations. Control, in financial modeling, is always an illusion. The market doesn’t yield to equations; it adapts around them. This work suggests that, rather than attempting to build a perfect model, a system that encourages robust local rules-in this case, well-trained neural networks-may prove more resilient.

Unresolved questions remain. The convergence analysis, while theoretically sound, relies on specific assumptions about market dynamics. Relaxing these – acknowledging the inherent messiness of real-world finance – will be crucial. Furthermore, the transferability of these neural approximations across different market regimes demands investigation. A network trained on historical data may falter when faced with genuinely novel events; the system’s response to true black swans is, as yet, unknown.

The path forward isn’t about building bigger, more complex hierarchies of equations. It’s about cultivating systems that are inherently adaptable, recognizing that order emerges from interaction, not imposition. The goal shouldn’t be to predict the unpredictable, but to build frameworks capable of navigating it, even when-especially when-the future diverges from expectations.

Original article: https://arxiv.org/pdf/2601.05290.pdf

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

Beyond Simplification: The Evolving Need for Adaptive Pricing

MMOT: An Emergent Framework for Arbitrage-Free Valuation

Accelerating Convergence: Computational Efficiency in MMOT

Robustness and Reliability: Validating MMOT Performance

What Lies Ahead?

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