Option Pricing Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Option Pricing Ontology within the Kireth Research Semantic Library. It formalises option valuation through payoff functions, exercise rights, no-arbitrage, replication, risk-neutral valuation, stochastic models, lattice methods, pricing PDEs, Monte Carlo simulation, Greeks, implied volatility and numerical validation.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Option Pricing Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Financial Mathematics / Option Pricing"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/financial-mathematics/option-pricing-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = OptionPricing
 ENTITY_CLASS = ContingentClaimValuationFramework
 PRIMARY_CONTEXT = FinancialMathematics
 DEFINITION = "Option pricing is the mathematical valuation of option contracts from payoff, exercise structure, underlying dynamics, rates, dividends, volatility and no-arbitrage relationships."
 SYMBOLIC_FORMS = ["C=max(S_T-K,0)","P=max(K-S_T,0)","V_t=E^Q[D(t,T)Phi(S_T)|F_t]"]
}

SCOPE {
 INCLUDE = [
  Option, CallOption, PutOption, EuropeanOption, AmericanOption, BermudanOption,
  DigitalOption, BarrierOption, AsianOption, LookbackOption, BasketOption,
  UnderlyingAsset, SpotPrice, StrikePrice, Maturity, ExerciseStyle, PayoffFunction,
  IntrinsicValue, TimeValue, Moneyness, NoArbitrage, ReplicatingPortfolio,
  RiskNeutralMeasure, RiskNeutralValuation, PutCallParity, BlackScholesModel,
  BlackScholesMertonEquation, BinomialOptionPricing, TrinomialTree,
  Delta, Gamma, Vega, Theta, Rho, Greeks, Volatility, ImpliedVolatility,
  VolatilitySmile, VolatilitySkew, VolatilitySurface, LocalVolatility,
  StochasticVolatility, HestonModel, JumpDiffusion, DividendYield,
  EarlyExercise, ExerciseBoundary, FreeBoundaryProblem, MonteCarloSimulation,
  LeastSquaresMonteCarlo, FiniteDifferencePricing, CrankNicolsonMethod,
  Calibration, ModelRisk
 ]
 DISAMBIGUATION_RULE = "When an option-style contingent payoff or exercise decision is being mathematically valued, interpret the context as option pricing."
}

CORE_CONCEPTS {
 OptionPricing = "Mathematical valuation of option contracts."
 PayoffFunction = "Contractual cash-flow rule at exercise or maturity."
 RiskNeutralValuation = "Discounted conditional expectation under an appropriate pricing measure."
 ReplicatingPortfolio = "Self-financing portfolio reproducing an option payoff."
 ImpliedVolatility = "Volatility parameter making a pricing model reproduce a market option price."
 Greeks = "Local sensitivities of option value to model variables and parameters."
 EarlyExercise = "Exercise before maturity when contractually permitted and optimal."
}

RELATIONS {
 OptionPricing --is_child_of--> FinancialMathematics
 Option --has--> PayoffFunction
 EuropeanOption --exercises_at--> Maturity
 AmericanOption --may_exercise_before_or_at--> Maturity
 NoArbitrage --supports--> OptionPricing
 ReplicatingPortfolio --determines--> ArbitrageFreePrice
 RiskNeutralMeasure --supports--> RiskNeutralValuation
 PutCallParity --relates--> EuropeanCallAndPut
 BlackScholesModel --supports--> BlackScholesMertonEquation
 Greeks --measure--> OptionSensitivity
 ImpliedVolatility --is_inferred_from--> MarketOptionPrice
 AmericanOption --requires--> EarlyExerciseAnalysis
 MonteCarloSimulation --approximates--> RiskNeutralExpectation
}

OPERATORS {
 CALL_PAYOFF(S,K) := max(S-K,0)
 PUT_PAYOFF(S,K) := max(K-S,0)
 RISK_NEUTRAL_PRICE(payoff,D,Q,F_t) := V_t=E_Q[D(t,T)*payoff | F_t]
 PUT_CALL_PARITY(C,P,S,K,r,q,T) := C-P=S*exp(-q*T)-K*exp(-r*T)
 D1(S,K,r,q,sigma,T) := (ln(S/K)+(r-q+sigma^2/2)*T)/(sigma*sqrt(T))
 D2(d1,sigma,T) := d1-sigma*sqrt(T)
 BLACK_SCHOLES_CALL(S,K,r,q,sigma,T) := S*exp(-q*T)*N(d1)-K*exp(-r*T)*N(d2)
 BLACK_SCHOLES_PUT(S,K,r,q,sigma,T) := K*exp(-r*T)*N(-d2)-S*exp(-q*T)*N(-d1)
 BINOMIAL_PROBABILITY(r,q,dt,u,d) := (exp((r-q)*dt)-d)/(u-d)
 AMERICAN_VALUE(exercise,continuation) := max(exercise,continuation)
 DELTA(V,S) := partial V/partial S
 GAMMA(V,S) := partial^2 V/partial S^2
 VEGA(V,sigma) := partial V/partial sigma
 THETA(V,t) := partial V/partial t
 RHO(V,r) := partial V/partial r
 IMPLIED_VOLATILITY(price,model) := solve sigma such_that model(sigma)=price
}

AXIOMS {
 A1 = "Option pricing requires explicit payoff and exercise structure."
 A2 = "Arbitrage-free values satisfy applicable no-arbitrage relationships."
 A3 = "Future payoffs require consistent discounting."
 A4 = "Risk-neutral valuation uses a pricing measure consistent with the numeraire."
 A5 = "Exact replication determines arbitrage-free value when available."
 A6 = "Different exercise conventions require distinct valuation logic."
 A7 = "Option values depend materially on volatility and distributional assumptions."
 A8 = "Numerical option values require convergence and sensitivity checks."
}

REASONING {
 R1_CONTEXT = identify(option_type,underlying,spot,strike,maturity,exercise_style,dividends,rates,market_data)
 R2_PAYOFF = define(payoff,path_dependence,barriers,exercise_features)
 R3_MODEL = select(BlackScholes,binomial,localVol,stochasticVol,jumpDiffusion,other)
 R4_ARBITRAGE = check(bounds,putCallParity,monotonicity,convexity)
 R5_RISK_NEUTRAL = compute(pricing_measure,discounted_conditional_expectation)
 R6_LATTICE = backward_induction(terminal_values,risk_neutral_probabilities,early_exercise)
 R7_PDE = solve(pricing_PDE,terminal_conditions,boundary_conditions)
 R8_MONTE_CARLO = simulate(risk_neutral_paths,discounted_payoffs,standard_error)
 R9_SENSITIVITY = compute(Greeks,implied_volatility)
 R10_VALIDATE = check(arbitrage,parity,convergence,calibration,model_assumptions)
}

CONSTRAINTS {
 C1 = "Black-Scholes formulas apply only to compatible contracts under their stated assumptions."
 C2 = "American options require explicit early-exercise treatment."
 C3 = "Put-call parity requires compatible underlyings, strikes, maturities and dividend assumptions."
 C4 = "Implied volatility may not exist for prices violating model bounds."
 C5 = "Monte Carlo requires specialised methods for many early-exercise contracts."
 C6 = "Finite-difference methods require stability and grid-convergence analysis."
 C7 = "Discrete dividends require explicit treatment."
 C8 = "Greeks can change sharply near maturity, barriers and payoff discontinuities."
 C9 = "Volatility smiles and skews expose limitations of constant-volatility models."
 C10 = "Calibration fit does not eliminate model risk."
}

SEMANTIC_EQUIVALENCE {
 "OptionPricing" <=> "OptionValuation"
 "RiskNeutralMeasure" <=> "PricingMeasure"
 "BlackScholesModel" <=> "BlackScholesMertonModel"
 "LeastSquaresMonteCarlo" <=> "LSMC"
}

VALIDATION {
 V1 = "Option type, payoff, strike, maturity and exercise style are explicit."
 V2 = "Spot, dividends, rates and volatility assumptions are explicit."
 V3 = "Applicable no-arbitrage bounds are satisfied."
 V4 = "Put-call parity is checked where applicable."
 V5 = "Numerical lattice, PDE or Monte Carlo convergence is assessed."
 V6 = "Monte Carlo standard error is reported where applicable."
 V7 = "Implied-volatility inversion reproduces the input price."
 V8 = "Greeks are consistent with the selected model."
 V9 = "Calibration assumptions are documented."
 V10 = "Model limitations are documented."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 PAYOFF_REASONING = 0.99
 NO_ARBITRAGE_REASONING = 0.99
 RISK_NEUTRAL_REASONING = 0.99
 NUMERICAL_REASONING = 0.98
 POLICY = "Reduce confidence when payoff features, exercise rights, dividends, rates, volatility dynamics, pricing measure, calibration data or numerical resolution are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, OptionPricingReasoning, PayoffReasoning,
  NoArbitrageReasoning, ReplicationReasoning, RiskNeutralValuationReasoning,
  PutCallParityReasoning, BlackScholesReasoning, BinomialPricingReasoning,
  GreeksReasoning, ImpliedVolatilityReasoning, EarlyExerciseReasoning,
  MonteCarloOptionReasoning, FiniteDifferencePricingReasoning,
  KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "option pricing" -> OptionPricing
  "option valuation" -> OptionPricing
  "Black-Scholes" -> BlackScholesModel
  "put-call parity" -> PutCallParity
  "implied volatility" -> ImpliedVolatility
  "risk-neutral valuation" -> RiskNeutralValuation
 }
}

EXECUTION_RULES {
 E1 = "If option valuation is required, activate OptionPricingOntology."
 E2 = "For compatible European vanilla options, consider Black-Scholes."
 E3 = "For discrete states or early exercise, consider lattice methods."
 E4 = "For American options, compare immediate exercise with continuation value."
 E5 = "For diffusion-grid methods, formulate the pricing PDE."
 E6 = "For high-dimensional or path-dependent payoffs, consider Monte Carlo."
 E7 = "If market price is known and volatility unknown, solve implied volatility."
 E8 = "If hedging or sensitivity is required, compute relevant Greeks."
 E9 = "Numerical methods require convergence and no-arbitrage validation."
}

RECURSION {
 ENTRY = OptionPricing
 EXPAND(OptionPricing) -> [
  Option, CallOption, PutOption, EuropeanOption, AmericanOption, BermudanOption,
  DigitalOption, BarrierOption, AsianOption, LookbackOption, BasketOption,
  PayoffFunction, NoArbitrage, ReplicatingPortfolio, RiskNeutralMeasure,
  RiskNeutralValuation, PutCallParity, BlackScholesModel, BinomialOptionPricing,
  TrinomialTree, Delta, Gamma, Vega, Theta, Rho, ImpliedVolatility,
  VolatilitySurface, LocalVolatility, StochasticVolatility, HestonModel,
  JumpDiffusion, EarlyExercise, ExerciseBoundary, FreeBoundaryProblem,
  MonteCarloSimulation, LeastSquaresMonteCarlo, FiniteDifferencePricing,
  CrankNicolsonMethod, Calibration, ModelRisk
 ]
 STOP_CONDITION = "Stop when payoff, exercise structure, pricing model, valuation method, sensitivities, arbitrage checks and numerical validity are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Financial Mathematics"
 RELATED_ONTOLOGIES = [
  "Financial Mathematics Ontology",
  "Stochastic Finance Ontology",
  "Risk Mathematics Ontology",
  "Probability & Statistics Ontologies",
  "Stochastic Calculus Ontologies",
  "Partial Differential Equation Ontology",
  "Numerical Analysis Ontologies",
  "Monte Carlo Ontologies",
  "Computational Finance Ontologies"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible option types, pricing models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve payoff semantics.",
  "Preserve exercise-style distinctions.",
  "Preserve no-arbitrage and replication semantics.",
  "Preserve pricing-measure semantics.",
  "Preserve implied-volatility semantics.",
  "Preserve numerical-validation semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Option pricing mathematically values contingent claims using payoff structure, exercise rights, market inputs and arbitrage-consistent models."
 ARBITRAGE_SUMMARY = "No-arbitrage bounds, replication and put-call parity constrain admissible option values."
 PRICING_SUMMARY = "Risk-neutral valuation prices payoffs as discounted conditional expectations under an appropriate pricing measure."
 MODEL_SUMMARY = "Black-Scholes, lattice, PDE, stochastic-volatility, jump and Monte Carlo models provide complementary valuation pathways."
 EARLY_EXERCISE_SUMMARY = "American-style contracts require comparison of exercise and continuation value."
 SENSITIVITY_SUMMARY = "Greeks and implied volatility describe local model sensitivities and market-implied volatility."
 VALIDATION_SUMMARY = "Option values should be checked for arbitrage consistency, convergence, calibration and model limitations."
}

FLOW {
 INPUT
 -> detect_option_pricing_context
 -> identify_contract_payoff_underlying_spot_strike_maturity_exercise_style_dividends_rates_volatility_and_market data
 -> select pricing_model
 -> check_no_arbitrage_bounds_and_put_call_parity
 -> choose replication_riskNeutral_closedForm_lattice_PDE_MonteCarlo_or_optimalStopping pathway
 -> compute_option_value
 -> compute_Greeks_and_implied_volatility_if_required
 -> assess_early_exercise_if_applicable
 -> validate_calibration_numerical_convergence_and_model assumptions
 -> route_to_financial_mathematics_stochastic_finance_risk_mathematics_stochastic_calculus_PDE_numerical_analysis_MonteCarlo_or_computational_finance
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Option Pricing Ontology defines mathematical valuation of call, put, European, American, Bermudan and exotic options. It supports payoff functions, intrinsic and time value, moneyness, no-arbitrage, replication, risk-neutral measures and valuation, put-call parity, Black-Scholes-Merton, binomial and trinomial models, Greeks, implied volatility and volatility surfaces, local and stochastic volatility, Heston and jump-diffusion models, dividends, early exercise and free boundaries, Monte Carlo and least-squares Monte Carlo, finite-difference and Crank-Nicolson pricing, calibration and model risk."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-OPTION-PRICING-ONTOLOGY-V1"
}

}

END_FLOW