SfDx

Sparse Functional Derivatives Exchange

Trade futures, options, and text markets via a common framework. Futures (1D) and options (2D) are special cases. SfDx generalizes to nD by listing a basis instead of contracts.

The Core Idea

Traditional exchanges list specific contracts. SfDx lists a basis {φᵢ}. Any contract payoff f can be written as f = Σ αᵢφᵢ. Traders pick the weights α.

This solves two problems: (1) liquidity pools across all contracts sharing the basis, (2) infinite-dimensional payoffs become tractable via sparsity.

Dimensional Spectrum

0D — Binary
f = 1{event}
Single outcome
1D — Futures
f = S(T) - K
Linear in price at expiry
2D — Options
f = max(S(T) - K, 0)
Nonlinear payoff surface
nD — Path-Dependent
f = F[S(·)]
Functional of entire path

Pattern: each step adds structure. 0D = point. 1D = line. 2D = surface. nD = function space.

How It Works

1. Basis Listing
Exchange lists basis functions {φᵢ(z)} where z ∈ ℝⁿ is the state space
2. Functional Trading
Any payoff f = Σ αᵢφᵢ is tradeable. Quote f → implied quotes on {φᵢ} via arbitrage
3. Settlement
Outcome z_actual occurs. Payoff = f(z_actual) = Σ αᵢφᵢ(z_actual). Standard functional evaluation.

No-Arbitrage Constraint

If f = Σ αᵢφᵢ and g = Σ βᵢφᵢ, then price(f) + price(g) = price(Σ (αᵢ+βᵢ)φᵢ) or arbitrage exists. This propagates liquidity across all functionals.

Regularization

Fees ∝ ||α||₀ (L0 norm = number of non-zero coefficients). Complex bets cost more. This solves curse of dimensionality: infinite dimensions, finite complexity per contract.

Text Markets

Basis: LLM compresses world state W → latent vector z ∈ ℝⁿ. This compressed representation becomes the basis. Functionals: LLM outputs (continuations) from z become tradeable contracts: text generated from the compressed state.

Example: Compress "disease outbreak state" → z. Trade on LLM outputs: "peak estimate", "policy recommendation", "CDC guidance". When actual state z_actual occurs, settlement = LLM(z_actual, prompt). Bet on what the LLM will say about the compressed world.

Why This Works

Examples

Disease Outbreaks
Known case: "Peak cases" futures (1D)
SfDx: Basis {infection(x,y,t), healthcare_capacity, interventions} → functionals "Peak NYC < 50k" · "Lockdown duration" · "Hospital capacity alerts" all share liquidity
1D futures → 3D spatial + time + text
Healthcare Solutions
Known case: Binary "treatment works" bet
SfDx: Basis {patient_outcomes(disease, treatment), costs, adoption} → functionals "Cost per cure < $X" · "Adoption > Y%" · "Side effects rate" all priced consistently
Binary → nD treatment+outcome space
Product Launches
Known case: Stock option at launch (2D)
SfDx: Basis {features, pricing, reception, technical_performance} → functionals "Sales > $1M" · "Review score" · "Stock Δ" all share implied quotes
2D option → nD product state
LLM Continuations
Known case: Binary "LLM says X" bet
SfDx: Basis {z} = LLM-compressed outbreak state → functionals = LLM(z, prompt): "policy text" · "peak estimate" · "guidance summary" — bet on what LLM outputs from compressed reality
Trade LLM outputs from compressed states

Mathematical Foundation

Basis Expansion
Any L² payoff: f = Σ αᵢφᵢ where {φᵢ} is complete basis. Fourier, wavelets, Hermite functions are special cases.
Pricing Functional
Price operator Π: L²(Ω) → ℝ maps payoffs to prices. No-arbitrage ⟺ Π is linear, positive, normalized.
Neural Operators
DeepONets, FNOs learn Π directly without discretization. Map f ↦ price(f) as operator approximation.
Sparsity Theorem
L0 regularization makes nD tractable. Most practical payoffs are k-sparse with k ≪ n.
Key Insight

Traditional markets trade outcomes. SfDx trades functionals over state space. All markets sharing a basis get liquidity from arbitrage-enforced consistency. LLMs make strategy loading intelligible — describe bets in natural language, settle on math.