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Examples

This page summarizes the built-in gallery and highlights the principal numerical patterns across the current experiment set.

The built-in gallery currently contains twelve examples. These include directional payoffs, symmetric payoffs, widened marginals, a quadratic case whose interval nearly collapses in the present discretization, and causal multi-period studies.

Gallery overview

Cross-example view of robust lower and upper values together with interval widths.

Related files:

Summary Table

Example Lower Upper Width Smallest eps value
Uniform absolute spread 0.6087 0.9975 0.3888 0.9538
Call on spread 0.1894 0.3841 0.1947 0.3469
Put on spread 0.4394 0.6341 0.1947 0.5969
Quadratic spread 0.9969 0.9969 0.0000 0.9969
Centered spread straddle 0.6739 1.0894 0.4155 1.0289
Centered call on spread 0.0869 0.2947 0.2078 0.2462
Wide absolute spread 0.8754 1.3101 0.4347 1.2487
Wide put on spread 0.7133 0.9382 0.2249 0.8868
Broad spread straddle 0.6217 1.0194 0.3977 0.9587
Causal absolute spread T3 see generated gallery see generated gallery see generated gallery see generated gallery
Causal call T4 see generated gallery see generated gallery see generated gallery see generated gallery
Causal convergence study see generated gallery see generated gallery see generated gallery n/a

Reading Conventions

The examples are interpreted according to the following conventions:

  • the exact LP bounds are the benchmark
  • interval width is used as a measure of residual model uncertainty under the martingale restriction
  • the smallest shipped eps value is treated as an approximation diagnostic rather than as a substitute for the LP benchmark
  • structural diagnostics and convergence diagnostics are used together

Selected Comparisons

Reference Absolute Spread

  • S1 ~ Uniform[1, 3]
  • S2 ~ Uniform[0, 4]
  • payoff |S2 - S1|
  • exact interval [0.6087, 0.9975]

This remains the baseline experiment for the repository. The interval is substantial, and the unrestricted countermonotone benchmark is materially larger than the exact martingale upper value.

Uniform abs spread exact summary

Exact upper plan together with the benchmark comparison panel.

Uniform abs spread structural diagnostics

Marginal profiles, conditional dispersion, and convex-order call gap.

Directional Spread Comparison

The call-on-spread and put-on-spread examples provide a useful directional comparison on the same support pair.

  • call on spread: interval [0.1894, 0.3841]
  • put on spread: interval [0.4394, 0.6341]

The widths coincide in the current discretization, but the interval levels differ. This separates directional asymmetry from overall interval scale.

Call spread exact summary

Directional payoff emphasizing upward spread moves.

Put spread exact summary

Directional payoff emphasizing downward spread moves.

Centered Symmetric System

The centered straddle and centered call examples move to S1 ~ Uniform[-1, 1] and S2 ~ Uniform[-2, 2].

  • centered straddle: interval [0.6739, 1.0894]
  • centered call: interval [0.0869, 0.2947]

These examples are useful when geometric symmetry is desirable without eliminating nontrivial pricing intervals.

Centered straddle exact summary

Symmetric setup with one of the widest intervals in the current gallery.

Centered call exact summary

Directional payoff on the centered support pair.

Wider-Marginal Regime

The wide absolute-spread and wide put-on-spread examples increase the variance of the second marginal.

  • wide absolute spread: interval [0.8754, 1.3101]
  • wide put on spread: interval [0.7133, 0.9382]

These examples show how increased dispersion in the second marginal broadens or shifts the robust pricing range.

Wide abs exact summary

Absolute-spread behavior under a noticeably wider second marginal.

Wide put exact summary

Downside-oriented payoff in the wider-marginal regime.

Broad Straddle

The broad spread straddle retains the original support pair but changes the payoff to |(S2 - S1) - 0.25|.

  • lower bound 0.6217
  • upper bound 1.0194
  • interval width 0.3977

This example complements the original absolute-spread benchmark by introducing a symmetric payoff around a nonzero strike.

Broad straddle exact summary

Symmetric spread sensitivity around a shifted center.

Broad straddle structural diagnostics

Structural diagnostics for the shifted straddle configuration.

Causal Multi-Period Examples

The causal gallery entries use chains of marginals rather than a single two-period pair.

  • causal absolute spread T3: three uniform marginals widening from [1, 3] to [0, 4]
  • causal call T4: four marginals with an adjacent call-on-spread payoff
  • causal convergence study: selected T values for an additive absolute-spread time-step study

These entries write causal artifact names such as:

  • causal_transport_chain.png
  • causal_summary.json
  • causal_experiment_report.md
  • continuous_limit.png for convergence studies

Diagnostic Figures

Useful diagnostics include:

The structural diagnostics combine marginal profiles, conditional dispersion, and convex-order call gaps in a single figure.

Per-Example Reports

Each example directory includes:

  • experiment_report.md and summary.json for two-period entries
  • causal_experiment_report.md and causal_summary.json for causal entries
  • continuous_summary.json for convergence entries
  • matching exact, regularization, causal, or continuous-limit plots

These files are linked from the generated Gallery Casebook.