An arXiv preprint finds that a partial differential equation, or PDE, can be used to build a self-financing hedge for European claims in general one-dimensional diffusion markets. The paper establishes that the PDE's starting value equals the minimum hedging capital when an equivalent local martingale measure, or ELMM, exists. A worked example also shows that the fundamental PDE value can be higher than necessary, while a simulation without an ELMM still produces close tracking without settling the minimum-capital question.
A price with a condition
The framework covers general one-dimensional diffusions described through a scale function, a speed measure and a constant interest rate. It includes markets that do not have a classical stochastic differential equation, or SDE, representation. From this setup, the paper derives a hedging PDE and uses its solution to construct a self-financing strategy.
The key no-arbitrage question is whether an ELMM exists. In the paper's formulation, ELMM existence is equivalent to no free lunch with vanishing risk, or NFLVR, and is characterized by the condition J* = J together with a Borel beta function that meets the displayed model requirements. Under an ELMM, the risky asset follows the diffusion law of an auxiliary process.
The same state-space condition sets the theorem's uniqueness boundary. When J* = J, the hedging PDE has one bounded, continuous good solution for bounded, continuous terminal data. Within that stated class of solutions and payoffs, the equation therefore has a single answer.
When an ELMM exists and the payoff is bounded and continuous, the PDE's initial value equals minimal hedging capital. The associated strategy is self-financing and admissible. This pricing statement is conditional on the ELMM case.
The boundary is important because the paper gives a worked example in which the fundamental value function produces a price higher than necessary, illustrating a possible non-minimal PDE price. That example does not establish that every model without an ELMM has the same outcome. It shows instead that a PDE value need not automatically be the cheapest hedging price.
The test in simulated models
The numerical experiment used a general diffusion on the real line and a European bear spread with a lower strike of -2, an upper strike of 2 and a time horizon of 10. It evaluated four configurations: a Bachelier benchmark and three models labeled Skew-Sticky 1, Skew-Sticky 2 and Skew-Sticky 3. The first three satisfied the balance condition used in the analysis; the fourth did not and was analyzed without an ELMM.
The computational pipeline used an implicit finite-difference scheme on the interval [-50, 50], with 500 time-grid points and alternative spatial grids of 1000, 2000 and 4000 points. For each model evaluation, 2000 simulated paths were rebalanced at frequencies ranging from 32 to 4096 instances. The analysis reported mean tracking error, or MTE, standard deviation of tracking error, or StDTE, and RMSTE, another tracking-error measure.
It also used asymptotic 95% confidence intervals, with a 1.96 multiplier for MTE and a chi-squared approximation for StDTE. The paper does not present these intervals as rigorous statistical analysis.
Tracking was close, but prices varied
Across the configurations, MTE stayed near zero, typically below 0.07 in absolute value, while StDTE declined as rebalancing became more frequent. The Bachelier benchmark showed the expected square-root pattern. The skew-sticky configurations had different error levels, so the numerical details varied across models even though the broad tracking pattern was similar.
The no-ELMM skew-sticky configuration produced near-zero MTE and StDTE values comparable to the Bachelier benchmark. That result indicates that numerical hedging consistency can persist in the absence of an ELMM, but it does not establish minimal hedging capital. The initial capital may still exceed the minimum.
The reported premiums varied sharply by configuration: 2.0 for Bachelier, 0.271 for Skew-Sticky 1, 14.778 for Skew-Sticky 2 and 2.0 for Skew-Sticky 3. They are model-based numerical outputs rather than observed market prices. With the same payoff and horizon, the spread illustrates how the computed PDE value can differ across diffusion configurations.
What the test cannot settle
A rough convergence check added four descriptive slopes from the first two points on log-log plots: -0.48, -0.31, -0.51 and -0.42 across the four model panels. The paper treats these figures as qualitative trends, not as a rigorous asymptotic analysis.
The study draws a clear line between tracking and pricing. Its simulations show close tracking, including in the no-ELMM configuration, but the no-ELMM result does not establish that the PDE's initial value is minimal. The minimum-capital equality is established when an ELMM exists for bounded continuous payoffs, while the uniqueness result depends on J* = J. The document is labeled an arXiv preprint, and its numerical evidence comes from simulated diffusion models.
Paper data and sources
Original title: On the hedging problem in general 1D diffusion markets
Authors: Alexis Anagnostakis, David Criens, Mikhail Urusov
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
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