Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2505.24435

Path-dependent option pricing with two-dimensional PDE using MPDATA

Paweł Magnuszewski·Sylwester Arabas
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we discuss a simple yet robust PDE method for evaluating path-dependent Asian-style options using the non-oscillatory forward-in-time second-order MPDATA finite-difference scheme. The valuation methodology involves casting the Black-Merton-Scholes equation as a transport problem by first transforming it into a homogeneous advection-diffusion PDE via variable substitution, and then expressing the diffusion term as an advective flux using the pseudo-velocity technique. As a result, all terms of the Black-Merton-Sholes equation are consistently represented using a single high-order numerical scheme for the advection operator. We detail the additional steps required to solve the two-dimensional valuation problem compared to MPDATA valuations of vanilla instruments documented in a prior study. Using test cases employing fixed-strike instruments, we validate the solutions against Monte Carlo valuations, as well as against an approximate analytical solution in which geometric instead of arithmetic averaging is used. The analysis highlights the critical importance of the MPDATA corrective steps that improve the solution over the underlying first-order "upwind" step. The introduced valuation scheme is robust: conservative, non-oscillatory, and positive-definite; yet lucid: explicit in time, engendering intuitive stability-condition interpretation and inflow/outflow boundary-condition heuristics. MPDATA is particularly well suited for two-dimensional problems as it is not a dimensionally split scheme. The documented valuation workflow also constitutes a useful two-dimensional case for testing advection schemes featuring both Monte Carlo solutions and analytic bounds. An implementation of the introduced valuation workflow, based on the PyMPDATA package and the Numba Just-In-Time compiler for Python, is provided as free and open source software.

Explore related subjects

q-fin.CP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Paweł Magnuszewski, Sylwester Arabas. 2025-05-30. Path-dependent option pricing with two-dimensional PDE using MPDATA. https://arxiv.org/abs/2505.24435

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

CFOs Meet LLMs

Business sentiment is a closely watched economic signal, but measuring it is slow and costly: surveys typically reach only a few hundred firms, arrive periodically, and take time to compile. We show that large language models hold the potential to address these shortcomings. We prompt an LLM to role-play as the CFO of a specific company on a specific date, for every public-company CFO who responded to the Duke--Federal Reserve CFO Survey between 2002 and 2025, and answer a question about economy-wide optimism. The LLM-generated optimism score predicts the individual CFO's actual answer, even in specifications that include firm and year-quarter fixed effects as well as a control variable measuring the human CFO's lagged response. Accuracy increases with the information provided to the LLM, and the relation persists under quarterly aggregation. We find the same patterns hold for two other questions measuring CFO expectations: the respondent's optimism about their own firm and their expectation of own-firm revenues. With appropriate conditioning, LLMs may in the future be able to serve as digital twins of executives, offering scalable, high-frequency expectations data for financial research and policy.

q-fin.CP↗
arXiv2026

The Physical Crash Frontier: What Finite Option Quotes Can and Cannot Reveal

Physical crash probabilities recovered from option prices depend on a pricing kernel and on a risk-neutral distribution that finitely many bid and ask quotes do not identify. For a power utility investor, we characterize the pairs of physical crash probability and expected loss below the crash threshold that the quotes admit; the boundary of this set is the physical crash frontier. Both coordinates are ratios of moments, yet when the index is bounded above the set is convex, and second-order cone programs compute it exactly at the calibrated risk aversion of two. In a decade of weekly S&P 500 cross sections, the quotes beyond the two puts nearest a 10 percent decline shrink the range of admissible crash probabilities by about 80 percent, yet its upper end remains two to three times its lower end. That lower end exists only because the index is bounded. Otherwise, for any investor more risk averse than the log investor, a vanishing probability far in the right tail inflates the denominator and drives the crash probability to zero while every quote stays inside its spread. A positive floor is therefore a joint statement about prices and a tail restriction; anything tighter than the frontier is an assumption.

q-fin.CP↗
arXiv2026

OrderFusion+: Probabilistic Buy--Sell Price Trajectory Forecasting in Intraday Electricity Markets

Intraday electricity markets enable participants to adjust energy positions close to delivery, with price forecasts necessary to support trading and the scheduling of flexible electricity resources as well as increasingly responsive consumers. Forecasting model specifications have progressed from using macro-features, such as renewable generation and load, to the micro-features of continuous orderbooks. A recent advanced deep learning model, OrderFusion, explicitly models micro-level buy-sell orderbook interactions. However, despite its superior comparative forecasting performance, it considers only one delivery product at a time, ignoring neighboring-product information. Moreover, when forecasting aggregated price indices such as the liquid German ID3, ID2, and ID1 products, it omits the information contained in price trajectories. In contrast, pretrained time-series foundation models have shown success in financial, renewable-energy, and day-ahead electricity price forecasting. However, their performance on intraday orderbook data remains an open research question of considerable practical importance. In this paper, we propose OrderFusion+, an open-source deep learning model that combines historical orders from the target and neighboring delivery products to forecast probabilistic buy-sell price trajectories. We benchmark OrderFusion+ against forecasting baselines and pretrained foundation models, and investigate dynamic market conditions through the designed dynamic masking mechanism, revealing insights into market efficiency. The implementation and forecasts can be found at: https://runyao-yu.com/OrderFusion/

q-fin.CP↗