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").
2. 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.
3. 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.
4. Choose parameters with financial meaning (volatilities, maturities, strikes) so the illustration speaks to the modelling problem.
5. 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