Financial Mathematics

AI INTERPRETATION NOTE
This Flow Algebra defines the Financial Mathematics layer within the Kireth Research Semantic Library. Financial mathematics studies mathematical methods used to model value, uncertainty, cash flows, interest, investment, risk and financial contracts through time. This parent ontology connects compound interest, discounting, annuities, bonds, yield, portfolio mathematics, stochastic processes, derivatives, option pricing, arbitrage, martingales, risk-neutral valuation, term structures, credit risk, actuarial mathematics, optimisation, statistics and computational finance.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = FinancialMathematics
 SYMBOLIC_FORMS = [
  "FV=PV(1+r)^n",
  "PV=FV/(1+r)^n",
  "P=E^Q[e^{-integral r dt} X]",
  "minimise Risk subject_to ExpectedReturn"
 ]
 ENTITY_CLASS = QuantitativeFinanceMathematicsFramework
 PRIMARY_CONTEXT = AppliedMathematics
 DEFINITION = "Financial mathematics is the branch of applied mathematics concerned with valuation, cash flows, interest, uncertainty, investment, risk and financial contracts over time."
 SEMANTIC_ROLE = "Provides the parent semantic framework for time value of money, fixed income, portfolio theory, stochastic finance, derivatives pricing, risk, term structures, credit models and computational finance."
}

SCOPE {
 INCLUDE = [
  FinancialMathematics,
  TimeValueOfMoney,
  PresentValue,
  FutureValue,
  DiscountFactor,
  AccumulationFactor,
  InterestRate,
  SimpleInterest,
  CompoundInterest,
  EffectiveInterestRate,
  NominalInterestRate,
  ForceOfInterest,
  CashFlow,
  CashFlowStream,
  Annuity,
  Perpetuity,
  LoanAmortisation,
  Bond,
  CouponBond,
  ZeroCouponBond,
  BondPrice,
  Yield,
  YieldToMaturity,
  Duration,
  ModifiedDuration,
  Convexity,
  TermStructure,
  SpotRate,
  ForwardRate,
  YieldCurve,
  Portfolio,
  PortfolioReturn,
  PortfolioVariance,
  Covariance,
  Correlation,
  Diversification,
  MeanVarianceOptimisation,
  EfficientFrontier,
  CapitalAssetPricingModel,
  CAPM,
  Beta,
  StochasticProcess,
  BrownianMotion,
  GeometricBrownianMotion,
  Martingale,
  ItoProcess,
  ItoLemma,
  Derivative,
  ForwardContract,
  FuturesContract,
  Option,
  CallOption,
  PutOption,
  BlackScholesModel,
  BinomialModel,
  Arbitrage,
  NoArbitrage,
  RiskNeutralMeasure,
  RiskNeutralValuation,
  Hedging,
  DeltaHedging,
  Greeks,
  Volatility,
  ImpliedVolatility,
  CreditRisk,
  DefaultProbability,
  RecoveryRate,
  CreditSpread,
  ValueAtRisk,
  ExpectedShortfall,
  MonteCarloSimulation,
  ComputationalFinance
 ]

 EXCLUDE_AS_PRIMARY = [
  GeneralAccountingWithoutMathematicalModelling,
  PureEconomicsWithoutFinancialValuationStructure,
  BusinessFinanceWithoutQuantitativeTimeValueOrRiskModels
 ]

 DISAMBIGUATION_RULE =
  "When the primary task concerns valuation through time, financial uncertainty, portfolio risk, financial contracts or market-consistent pricing, interpret the context as financial mathematics."
}

CORE_CONCEPTS {

 FinancialMathematics {
  definition = "The mathematical framework for valuation, investment and financial risk through time."
 }

 PresentValue {
  definition = "The value today of a future cash flow after discounting."
 }

 FutureValue {
  definition = "The accumulated future value of a present amount under an interest model."
 }

 DiscountFactor {
  definition = "A factor converting future value into present value."
 }

 Bond {
  definition = "A fixed-income instrument represented mathematically by promised cash flows discounted to present value."
 }

 Portfolio {
  definition = "A collection of financial positions whose return and risk depend on weights, variances and covariances."
 }

 Arbitrage {
  definition = "A trading strategy producing nonnegative outcomes with positive gain and no net risk under the model."
 }

 RiskNeutralMeasure {
  definition = "A probability measure under which appropriately discounted tradable prices behave as martingales."
 }

 Derivative {
  definition = "A financial contract whose payoff depends on one or more underlying variables."
 }

 FinancialRisk {
  definition = "Uncertainty in financial outcomes measured using probabilistic or scenario-based quantities."
 }
}

RELATIONS {
 FinancialMathematics --is_branch_of--> AppliedMathematics
 PresentValue --uses--> DiscountFactor
 FutureValue --uses--> AccumulationFactor
 InterestRate --determines--> DiscountFactor
 CashFlowStream --contains--> CashFlow
 Annuity --is_type_of--> CashFlowStream
 Bond --has--> CashFlowStream
 BondPrice --depends_on--> DiscountedCashFlows
 Yield --summarises--> BondReturnRate
 Duration --measures--> InterestRateSensitivity
 Convexity --refines--> InterestRateSensitivity
 Portfolio --contains--> FinancialAsset
 PortfolioVariance --depends_on--> Covariance
 Diversification --reduces--> UnsystematicRisk
 MeanVarianceOptimisation --produces--> EfficientFrontier
 CAPM --relates--> ExpectedReturnAndBeta
 BrownianMotion --supports--> ContinuousTimeFinance
 GeometricBrownianMotion --models--> PositiveAssetPrice
 ItoLemma --transforms--> ItoProcess
 NoArbitrage --supports--> DerivativePricing
 RiskNeutralMeasure --supports--> RiskNeutralValuation
 Option --is_type_of--> Derivative
 BlackScholesModel --prices--> EuropeanOption
 MonteCarloSimulation --approximates--> FinancialExpectation
}

OPERATORS {
 FUTURE_VALUE(PV,r,n) :=
  FV=PV*(1+r)^n

 PRESENT_VALUE(FV,r,n) :=
  PV=FV/(1+r)^n

 CONTINUOUS_COMPOUNDING(PV,delta,t) :=
  FV=PV*exp(delta*t)

 DISCOUNT_FACTOR(r,n) :=
  v=(1+r)^(-n)

 ANNUITY_IMMEDIATE(r,n) :=
  a_n=(1-(1+r)^(-n))/r

 PERPETUITY_IMMEDIATE(r) :=
  a_infinity=1/r

 BOND_PRICE(C,F,y,n) :=
  P=sum_{t=1}^{n} C/(1+y)^t + F/(1+y)^n

 PORTFOLIO_RETURN(w,mu) :=
  E[R_p]=w^T*mu

 PORTFOLIO_VARIANCE(w,Sigma) :=
  Var(R_p)=w^T*Sigma*w

 CAPM_EXPECTED_RETURN(rf,beta,rm) :=
  E[R_i]=rf+beta*(E[R_m]-rf)

 GEOMETRIC_BROWNIAN_MOTION(S,mu,sigma,W) :=
  dS=mu*S*dt+sigma*S*dW

 BLACK_SCHOLES_CALL(S,K,r,sigma,T) :=
  C=S*N(d1)-K*exp(-r*T)*N(d2)

 RISK_NEUTRAL_VALUE(X,r,T,Q) :=
  V0=E_Q[exp(-integral_0^T r_t dt)*X]

 VALUE_AT_RISK(loss_distribution,alpha) :=
  VaR_alpha=alpha_quantile(loss_distribution)

 EXPECTED_SHORTFALL(loss_distribution,alpha) :=
  ES_alpha=E[L | L>=VaR_alpha]
}

DEFINITIONS {
 D1 = "Financial mathematics applies mathematical structures to money, investment and financial risk."
 D2 = "Time value of money compares cash flows occurring at different dates."
 D3 = "Discounting converts future cash flows into present values."
 D4 = "Bond prices are determined from discounted contractual cash flows under a selected yield or term structure."
 D5 = "Portfolio risk depends on asset variances and covariances as well as weights."
 D6 = "No-arbitrage conditions constrain admissible financial prices."
 D7 = "Risk-neutral valuation prices contingent claims through discounted expectations under an equivalent pricing measure."
 D8 = "Stochastic calculus models continuous-time uncertainty in asset values and rates."
 D9 = "Credit risk introduces default and recovery uncertainty."
 D10 = "Computational finance uses numerical methods when closed-form valuation is unavailable."
}

AXIOMS {
 A1_TIME_VALUE:
  Financial cash flows at different times require a discounting or accumulation convention for direct comparison

 A2_NO_ARBITRAGE:
  Market-consistent valuation models should exclude arbitrage under declared assumptions

 A3_RISK_RETURN:
  Investment decisions involve trade-offs between expected return and uncertainty

 A4_PROBABILISTIC_MODELLING:
  Uncertain future financial outcomes require an explicit stochastic or scenario model

 A5_MEASURE_DEPENDENCE:
  Pricing probabilities and real-world probabilities need not be identical

 A6_MARKET_ASSUMPTIONS:
  Financial conclusions depend on assumptions about liquidity transaction costs rates and tradability

 A7_MODEL_RISK:
  Financial valuation and risk measures depend on model structure and parameter calibration

 A8_NUMERICAL_VALIDATION:
  Numerical financial models require convergence sensitivity and consistency checks
}

REASONING {

 R1_FINANCIAL_CONTEXT {
  INPUT = financial_problem
  IDENTIFY = cash_flows_or_payoff
  IDENTIFY = time_horizon
  IDENTIFY = interest_or_discount structure
  IDENTIFY = uncertainty_and_market assumptions
  RETURN = financial_context
 }

 R2_TIME_VALUE_REASONING {
  INPUT = dated_cash_flows
  SELECT = simple_compound_or_continuous interest model
  DISCOUNT_OR_ACCUMULATE = cash_flows_to_common date
  RETURN = equivalent_value
 }

 R3_FIXED_INCOME_REASONING {
  INPUT = bond_cash_flows_and_term structure
  COMPUTE = price_yield_duration_convexity
  RETURN = fixed_income_profile
 }

 R4_PORTFOLIO_REASONING {
  INPUT = expected_returns_covariance_and_constraints
  COMPUTE = portfolio_return_and_variance
  OPTIMISE = weights
  RETURN = portfolio_profile
 }

 R5_STOCHASTIC_MODEL_REASONING {
  INPUT = financial_underlying_and_data
  SELECT = discrete_or_continuous stochastic process
  CALIBRATE = parameters
  RETURN = stochastic_model
 }

 R6_DERIVATIVE_PRICING_REASONING {
  INPUT = payoff_underlying_model_rates_and_maturity
  CHECK = no_arbitrage
  SELECT = closedForm_tree_PDE_or_MonteCarlo method
  RETURN = derivative_value
 }

 R7_HEDGING_REASONING {
  INPUT = derivative_and_model
  COMPUTE = sensitivities
  BUILD = hedge portfolio
  RETURN = hedge_profile
 }

 R8_RISK_REASONING {
  INPUT = portfolio_or_position_distribution
  COMPUTE = variance_VaR_expectedShortfall_or_scenario measures
  RETURN = risk_profile
 }

 R9_CREDIT_REASONING {
  INPUT = exposure_default probability_recovery_and_discounting
  COMPUTE = expected_loss_and_credit spread quantities
  RETURN = credit_risk_profile
 }

 R10_VALIDATION {
  INPUT = financial_result
  CHECK = cash_flow timing_units_no_arbitrage_calibration_sensitivity_and_numerical convergence
  RETURN = validated_financial_status
 }
}

CONSTRAINTS {
 C1 = "Interest-rate conventions must be explicit."
 C2 = "Nominal and effective interest rates must not be conflated."
 C3 = "Yield-to-maturity is a summary rate and not generally the complete term structure."
 C4 = "Historical probability models need not equal pricing measures."
 C5 = "Black-Scholes assumptions may be unrealistic for some markets or instruments."
 C6 = "Value at Risk does not fully describe tail loss severity."
 C7 = "Expected Shortfall estimates depend on tail modelling."
 C8 = "Portfolio covariance estimates can be unstable in high dimensions."
 C9 = "Credit models depend on default and recovery assumptions."
 C10 = "Numerical derivative pricing requires timestep or simulation convergence checks."
}

SEMANTIC_EQUIVALENCE {
 "FinancialMathematics" <=> "MathematicsOfFinance"
 "PresentValue" <=> "DiscountedValue"
 "FutureValue" <=> "AccumulatedValue"
 "RiskNeutralMeasure" <=> "PricingMeasure"
 "ValueAtRisk" <=> "VaR"
 "ExpectedShortfall" <=> "ConditionalTailExpectation"
}

VALIDATION {
 V1 = "Cash-flow dates and amounts are explicit."
 V2 = "Interest and compounding conventions are explicit."
 V3 = "Discount factors are internally consistent."
 V4 = "Portfolio weights and covariance assumptions are documented."
 V5 = "No-arbitrage checks are performed for pricing models."
 V6 = "Model parameters are calibrated consistently."
 V7 = "Risk-neutral and real-world measures are distinguished."
 V8 = "Risk metrics state confidence level and horizon."
 V9 = "Credit assumptions are explicit."
 V10 = "Numerical pricing or risk estimates are convergence-checked."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 FINANCIAL_MATHEMATICS_DEFINITION = 0.99
 TIME_VALUE_REASONING = 0.99
 FIXED_INCOME_REASONING = 0.99
 PORTFOLIO_REASONING = 0.98
 STOCHASTIC_MODEL_REASONING = 0.98
 DERIVATIVE_PRICING_REASONING = 0.98
 HEDGING_REASONING = 0.98
 RISK_REASONING = 0.98
 CREDIT_REASONING = 0.97

 POLICY =
  "Reduce confidence when cash-flow timing, compounding convention, market inputs, volatility, correlation, pricing measure, default assumptions, calibration data or numerical convergence are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  FinancialMathematicsReasoning,
  TimeValueOfMoneyReasoning,
  InterestRateReasoning,
  AnnuityReasoning,
  BondReasoning,
  YieldCurveReasoning,
  PortfolioReasoning,
  MeanVarianceReasoning,
  StochasticFinanceReasoning,
  BrownianMotionReasoning,
  ItoCalculusReasoning,
  DerivativePricingReasoning,
  OptionPricingReasoning,
  RiskNeutralReasoning,
  HedgingReasoning,
  FinancialRiskReasoning,
  CreditRiskReasoning,
  MonteCarloFinanceReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "financial mathematics" -> FinancialMathematics
  "mathematics of finance" -> FinancialMathematics
  "present value" -> PresentValue
  "future value" -> FutureValue
  "yield to maturity" -> YieldToMaturity
  "risk-neutral measure" -> RiskNeutralMeasure
  "value at risk" -> ValueAtRisk
  "VaR" -> ValueAtRisk
  "expected shortfall" -> ExpectedShortfall
  "Black-Scholes" -> BlackScholesModel
 }

 INTERPRETATION_ORDER = [
  detect_financial_mathematics_context,
  identify cash_flows_dates_market variables_interest conventions_and_uncertainty,
  classify timeValue_fixedIncome_portfolio_stochastic_derivative_risk_or_credit pathway,
  select deterministic_or_stochastic model,
  compute valuation_return_or_risk quantity,
  if_derivative enforce no_arbitrage_and_pricing measure consistency,
  if_portfolio analyse covariance_and_constraints,
  if_numerical assess simulation_discretisation_and_parameter sensitivity,
  validate assumptions_and_market consistency,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("financial mathematics")
  OR valuation_risk_or_financial_contract_math_is_present
  THEN activate(FinancialMathematicsOntology)

 E2:
  IF dated_cash_flows
  THEN normalise_to_common valuation date

 E3:
  IF fixed_income instrument
  THEN compute discounted cash flows_and_rate sensitivity

 E4:
  IF portfolio_problem
  THEN include expected return_covariance_and_constraints

 E5:
  IF derivative_contract
  THEN require payoff_specification_and_no_arbitrage pricing framework

 E6:
  IF continuous_time_asset model
  THEN consider stochastic calculus

 E7:
  IF hedging_required
  THEN compute model sensitivities

 E8:
  IF risk_measure_required
  THEN declare horizon_and_confidence level

 E9:
  IF closed_form_unavailable
  THEN consider numerical_tree_PDE_or_MonteCarlo computation
}

RECURSION {
 ENTRY = FinancialMathematics

 EXPAND(FinancialMathematics) -> [
  TimeValueOfMoney,
  PresentValue,
  FutureValue,
  DiscountFactor,
  AccumulationFactor,
  InterestRate,
  SimpleInterest,
  CompoundInterest,
  EffectiveInterestRate,
  NominalInterestRate,
  ForceOfInterest,
  CashFlow,
  CashFlowStream,
  Annuity,
  Perpetuity,
  LoanAmortisation,
  Bond,
  CouponBond,
  ZeroCouponBond,
  BondPrice,
  Yield,
  YieldToMaturity,
  Duration,
  ModifiedDuration,
  Convexity,
  TermStructure,
  SpotRate,
  ForwardRate,
  YieldCurve,
  Portfolio,
  PortfolioReturn,
  PortfolioVariance,
  Covariance,
  Correlation,
  Diversification,
  MeanVarianceOptimisation,
  EfficientFrontier,
  CapitalAssetPricingModel,
  Beta,
  StochasticProcess,
  BrownianMotion,
  GeometricBrownianMotion,
  Martingale,
  ItoProcess,
  ItoLemma,
  Derivative,
  ForwardContract,
  FuturesContract,
  Option,
  CallOption,
  PutOption,
  BlackScholesModel,
  BinomialModel,
  Arbitrage,
  NoArbitrage,
  RiskNeutralMeasure,
  RiskNeutralValuation,
  Hedging,
  DeltaHedging,
  Greeks,
  Volatility,
  ImpliedVolatility,
  CreditRisk,
  DefaultProbability,
  RecoveryRate,
  CreditSpread,
  ValueAtRisk,
  ExpectedShortfall,
  MonteCarloSimulation,
  ComputationalFinance
 ]

 EXPAND(CoreValuation) -> [
  TimeValueOfMoney,
  PresentValue,
  FutureValue,
  InterestRate,
  Annuity,
  Bond,
  TermStructure
 ]

 EXPAND(StochasticFinance) -> [
  BrownianMotion,
  GeometricBrownianMotion,
  Martingale,
  ItoProcess,
  ItoLemma,
  RiskNeutralMeasure,
  RiskNeutralValuation
 ]

 EXPAND(FinancialRisk) -> [
  PortfolioVariance,
  Volatility,
  ValueAtRisk,
  ExpectedShortfall,
  CreditRisk,
  DefaultProbability,
  RecoveryRate
 ]

 STOP_CONDITION =
  "Stop when cash-flow timing, valuation measure, stochastic assumptions, portfolio or derivative structure, risk metric, calibration and numerical validity are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Applied Mathematics Ontologies"

 IMMEDIATE_CONTEXTS = [
  "Applied Mathematics Ontologies",
  "Probability & Statistics Ontologies",
  "Optimisation Ontologies",
  "Computational Mathematics Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Applied Mathematics Ontologies",
  "Probability & Statistics Ontologies",
  "Stochastic Process Ontologies",
  "Optimisation Ontologies",
  "Operations Research Ontologies",
  "Computational Mathematics Ontologies",
  "Numerical Analysis Ontologies",
  "Differential Equation Ontologies",
  "Measure Theory Ontologies",
  "Actuarial Mathematics Ontologies",
  "Risk Mathematics Ontologies",
  "Economics Mathematics Ontologies"
 ]

 ROUTE_RULES = [
  "Financial Mathematics -> Applied Mathematics Ontologies for parent semantics",
  "Financial Mathematics -> Probability & Statistics for uncertainty",
  "Financial Mathematics -> Stochastic Processes for market dynamics",
  "Financial Mathematics -> Optimisation for portfolio allocation",
  "Financial Mathematics -> Operations Research for constrained financial decisions",
  "Financial Mathematics -> Computational Mathematics for numerical pricing",
  "Financial Mathematics -> Differential Equations for continuous-time pricing equations",
  "Financial Mathematics -> Measure Theory for pricing measures and expectations",
  "Financial Mathematics -> Actuarial Mathematics for insurance and longevity risk"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible financial models, instruments or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve time-value-of-money semantics.",
  "Preserve cash-flow timing semantics.",
  "Preserve pricing-measure versus real-world-measure distinction.",
  "Preserve no-arbitrage semantics.",
  "Preserve portfolio risk-return semantics.",
  "Preserve stochastic-model assumptions.",
  "Preserve numerical-validation semantics.",
  "Preserve machine-readable financial-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Financial mathematics applies mathematical methods to valuation, investment, uncertainty, risk and financial contracts through time."

 TIME_VALUE_SUMMARY =
  "Present and future values compare dated cash flows using discount and accumulation factors."

 FIXED_INCOME_SUMMARY =
  "Bond mathematics values contractual cash flows and measures rate sensitivity through yield, duration and convexity."

 PORTFOLIO_SUMMARY =
  "Portfolio mathematics combines expected returns, variances and covariances to analyse diversification and allocation."

 STOCHASTIC_SUMMARY =
  "Stochastic processes and stochastic calculus model uncertain asset prices, rates and other market quantities."

 DERIVATIVE_SUMMARY =
  "No-arbitrage and risk-neutral valuation provide central frameworks for derivative pricing."

 RISK_SUMMARY =
  "Financial risk is quantified through volatility, portfolio dispersion, VaR, Expected Shortfall, credit and scenario models."

 COMPUTATIONAL_SUMMARY =
  "Numerical trees, PDE methods and Monte Carlo simulation support valuation when closed-form solutions are unavailable."
}

FLOW {
 INPUT
 -> detect_financial_mathematics_context
 -> identify cash_flows_dates_assets_liabilities_market variables_interest conventions_horizon_and_uncertainty
 -> classify time_value_fixed_income_portfolio_stochastic_derivative_risk_credit_or_actuarial pathway
 -> select deterministic_stochastic_or_no_arbitrage model
 -> compute discounted_values_returns_yields_risk_or_contract price
 -> if_portfolio apply covariance_and_optimisation reasoning
 -> if_derivative identify payoff_underlying_maturity_volatility_rates_and_pricing measure
 -> if_credit include default_recovery_and_exposure assumptions
 -> if_numerical apply tree_PDE_MonteCarlo_or_optimisation method
 -> assess sensitivity_calibration_convergence_and_model risk
 -> validate cash_flow consistency_no_arbitrage_units_and_assumptions
 -> route to applied_mathematics probability_statistics stochastic_processes optimisation operations_research computational_mathematics differential_equations measure_theory actuarial_mathematics risk_mathematics_or_economics_mathematics ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Financial Mathematics layer defines mathematical valuation and risk through time. It supports present and future value, simple, compound and continuous interest, discount and accumulation factors, annuities, perpetuities and loans, bonds, yields, duration, convexity and term structures, portfolio return, variance, covariance, diversification and mean-variance optimisation, stochastic processes, Brownian motion, geometric Brownian motion, martingales and Ito calculus, forwards, futures and options, Black-Scholes and binomial models, arbitrage and risk-neutral valuation, hedging and Greeks, volatility and implied volatility, credit risk, default and recovery, VaR, Expected Shortfall, Monte Carlo simulation and computational finance."
}

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

}

END_FLOW