brycewang-stanford Tables & Figures (mathfin-tables-figures)
When to trigger
- A numerical section needs figures or tables
- A proof-heavy manuscript needs diagrams, assumption maps, or appendix organization
- Exhibits risk looking like empirical finance rather than theorem support
Exhibit rules
Every exhibit should answer: which theorem, proposition, algorithm, or modelling insight does this support?
- Tie figures to formal results. A convergence plot should cite the theorem or rate it illustrates.
- Report numerical settings. Grid, time step, paths, tolerance, truncation, seed, and software version belong in the caption or appendix.
- Keep tables small. Use them for parameter values, convergence rates, boundary comparisons, or sensitivity to assumptions.
- Move detail to appendices. The author guidelines encourage detailed mathematical analysis in appendices; use them for long derivations and extended numerical diagnostics.
- Avoid empirical-style claims. A calibration illustration does not establish an empirical result unless the paper is designed for that, which this venue generally is not.
Good exhibits
- Error decay against the proven rate
- Free-boundary shape implied by an optimal stopping theorem
- Sensitivity of a pricing formula to parameters with financial meaning
- Algorithm pseudocode paired with convergence conditions
- Assumption dependency diagram for a complex theorem
Theorem-dependency map as an exhibit
For a paper with layered results, a compact dependency table orients referees faster than prose:
Assumptions: (A1) filtered space, usual conditions (A2) NFLVR (A3) bounded volatility
Lemma 3.1 <- (A1),(A2) uniform integrability of the deflated wealth process
Lemma 3.2 <- (A1),(A3) gradient bound for the value function
Theorem 4.1 <- L3.1, L3.2 duality and existence of the optimizer
Theorem 4.2 <- Thm 4.1,(A3) regularity of the free boundary
Place it at the end of the introduction or the head of the proofs section; it doubles as your own audit that no result silently uses an unlisted assumption.
Figure axes speak the model's notation
- Label axes with the manuscript's symbols (t, S_t, the boundary b(t), the Hurst index H), not plotting-library defaults — a referee matching the plot to Theorem 4.2 should never have to translate notation.
- State units only where financially meaningful (years to maturity, volatility in annual terms); pure mathematical objects need no units.
- If a curve depends on a parameter the theorem restricts, show only values inside the theorem's hypotheses, or mark excursions outside them explicitly as such.
Algorithm displays
Pseudocode earns its place only when the paper proves something about the algorithm (convergence, complexity, stability). Pair each algorithm float with the statement number of that guarantee, keep inputs and outputs in the manuscript's notation, and state the stopping criterion explicitly — referees check that the displayed loop matches the object analyzed in the proof, not a simplified cousin of it.
Caption rewrite micro-example
A draft caption for an American-put exhibit reads "Exercise boundary for different volatilities." Journal-ready version: "Free boundary b(t) of the American put under Assumptions (A1)–(A3), computed with the penalized PDE scheme of Section 5 (grid 400×400, tolerance 1e-8), illustrating the monotonicity proved in Theorem 4.2; sigma in {0.2, 0.3, 0.4}, r = 0.02, K = 100. The figure illustrates the theorem's qualitative content and makes no calibration claim." Every clause does one caption-contract job: result, financial object, settings, interpretation limit.
Caption contract
Each caption should state four things: the formal result illustrated, the financial object, the numerical settings, and the interpretation limit. For example, a pricing-sensitivity figure should not claim broad empirical validation; it should say which theorem or model implication the curve helps readers understand.
If the exhibit cannot name a theorem, proposition, or modelling implication, cut it or move it to an exploratory appendix. Mathematical Finance exhibits are evidence for theory comprehension, not substitutes for proof.
Output format
[Exhibit] figure / table / algorithm / appendix
[Formal result supported] theorem/proposition/lemma
[Numerical details needed] ...
[Caption claim] ...
[Next step] mathfin-writing-style