mathfin-data-analysis

Numerical Experiments (mathfin-data-analysis)

Note on framing

This is a theory-first journal. Mathematical Finance explicitly states that **numerical

experiments are welcome only when accompanied by a rigorous analysis** supporting the

theoretical developments, and that **routine application of computational methods to financial

data will not be considered**. So "data analysis" here is not empirical estimation — it is

numerical work that illustrates or stress-tests a theorem. This skill is deliberately lighter

than its empirical-journal counterpart.

When to trigger

• You want to add simulations or a numerical scheme to a proof-based paper
• A referee may ask whether your theorem "does anything" beyond existence
• You need to show convergence, accuracy, or qualitative behavior predicted by the theory

How to keep numerics journal-appropriate

1. Tie every experiment to a result. Each figure/table should illustrate a specific

theorem, proposition, or rate (e.g., "Monte Carlo error decays at the proven $O(n^{-1/2})$

rate", "the free boundary matches the smooth-fit characterization").

1. State the method precisely. Discretization scheme (Euler–Maruyama, Milstein, PDE

finite-difference/finite-element), step sizes, number of paths, variance reduction,

truncation of the domain — enough that the experiment is reproducible.

1. Report error, not just output. Where the theory gives a rate or bound, show the

empirical rate against it; show convergence as the grid refines.

1. Choose parameters with financial meaning (volatilities, maturities, strikes) so the

illustration speaks to the modelling problem.

1. Keep numerics subordinate. They support the theory; they are never the contribution.

Do not let a numerical section grow into a stand-alone empirical study.

Reproducibility (light but real)

• Pin software/library versions; set and report random seeds for any Monte Carlo.
• Make illustrative code reproducible; consider archiving it (Zenodo/GitHub) and citing it.
• Include a Data Availability Statement even if no external data are used (see

mathfin-replication-and-data-policy).

Matching scheme to result type

| Result being illustrated | Natural scheme | What the exhibit must report |

| --- | --- | --- |

| Strong/weak SDE convergence rate | Euler–Maruyama or Milstein with halving steps | log–log error slope against the proven order |

| BSDE well-posedness or rate | Backward Euler / least-squares Monte Carlo / deep BSDE solver | terminal error and driver residual across grids |

| Optimal stopping / free boundary | Binomial tree or PDE variational-inequality solver | boundary location against the smooth-fit characterization |

| Rough-volatility approximation | Hybrid scheme for fractional kernels; Markovian lift | implied-vol skew slope against the proven power law |

| Duality gap = 0 | Primal candidate and dual bound computed independently | gap shrinking as the discretization refines |

| Mean-field limit | N-player simulation vs. McKean–Vlasov solver | distance to the limit decaying in N at the stated rate |

Worked micro-example: convergence exhibit for a rough-volatility paper

Suppose Theorem 3.2 proves that a Markovian multi-factor approximation of a rough volatility

model converges at a rate governed by the Hurst parameter H. The journal-appropriate exhibit:

simulate both models with the same Brownian increments, plot the implied-volatility error

against the number of factors on log axes, draw the theoretical slope as a reference line, and

caption with the scheme, step size, path count, seed, and the theorem number. What would NOT

fit: calibrating the approximation to index-option data and reporting fit quality — that turns

an illustration into the empirical study the journal screens out.

Pre-submission numerics audit

• Every exhibit names the theorem, proposition, or rate it illustrates — no orphan plots.
• The observed rate is computed (regression slope), not eyeballed, and stated next to the

proven one.

• Degenerate sanity cases (zero volatility, Black–Scholes limit, H → 1/2) reproduce known

closed forms before the general runs are trusted.

• The numerical section would survive deletion: the theorems stand alone without it.

Anti-patterns

• A numerical study with no theorem behind it (out of scope for this journal).
• Plots with no error/convergence analysis where the theory promises a rate.
• Unstated scheme, step size, or path count — irreproducible.
• Calibrating to real market data and presenting it as the paper's result.

Output format

【Experiment】what it illustrates (which theorem/rate)
【Method】scheme + step/paths + variance reduction
【Error reported】empirical vs. theoretical rate/bound
【Parameters】financial values used
【Reproducibility】seeds + versions + code location
【Next step】mathfin-tables-figures