Black-Scholes pricing with 10 Greeks (delta through color).

Use when you need to price a European option or compute Greeks (delta, gamma, theta, vega, rho, etc.) using the Black-Scholes model. Provide spot price, strike, time to expiry, risk-free rate, and volatility. Returns: option price, 10 Greeks, and intrinsic/time value breakdown.

Newton-Raphson implied volatility solver. Converges in 5-8 iterations.

Use when you know the market price of an option and need to back out the implied volatility. Uses Newton-Raphson iteration. Provide spot, strike, time to expiry, risk-free rate, market price, and option type. Returns: implied volatility, convergence info, and Greeks at that IV.

Multi-leg options strategy P&L, breakevens, max profit/loss, risk/reward.

Use when analyzing a multi-leg options strategy (spreads, straddles, iron condors, etc.). Provide an array of legs with strike, premium, quantity, and type. Returns: net premium, max profit/loss, breakeven points, P&L at various prices, and payoff data.

22 risk metrics: Sharpe, Sortino, Calmar, Omega, VaR, CVaR, drawdown, skew, kurtosis.

Use when you have a series of portfolio returns and need comprehensive risk analytics. Provide an array of periodic returns (e.g. daily). Returns: 22 metrics including Sharpe, Sortino, Calmar, Omega, VaR (95/99), CVaR, max drawdown, skewness, kurtosis, win rate, profit factor. Optionally provide benchmark returns for alpha, beta, tracking error, and information ratio.

Kelly Criterion: discrete (win/loss) or continuous (returns series) mode.

Use when determining optimal bet/position sizing using the Kelly Criterion. Provide win probability and win/loss ratio. Returns: full Kelly fraction, half-Kelly, quarter-Kelly, and expected growth rate.

GBM Monte Carlo with contributions/withdrawals. Up to 5000 paths.

Use when running a Monte Carlo simulation for asset price paths. Provide starting price, drift, volatility, time horizon, and number of simulations. Returns: simulated terminal prices, percentile distribution (5th/25th/50th/75th/95th), expected value, probability of profit, and path statistics. NOTE: Via MCP, keep simulations ≤ 1000 and years ≤ 30 for fastest response. For larger simulations (up to 5000 paths, 100 years), call the REST API directly at https://api.quantoracle.dev/v1/simulate/montecarlo.

13 technical indicators + composite signals.

Use when you need multiple technical indicators computed from a price series. Provide an array of prices and optional volumes. Returns: SMA, EMA, RSI, MACD, Bollinger Bands, Stochastic %K, ATR, ROC, composite signals (overbought/oversold), and trend classification.

N x N correlation and covariance matrices from return series.

Use when computing an N×N correlation matrix for multiple assets. Provide a 2D array of return series. Returns: Pearson correlation matrix, covariance matrix, and eigenvalues for PCA analysis.

Fixed fractional position sizing with risk/reward targets.

Use when calculating how many shares/contracts to buy given account size and risk tolerance. Provide account value, risk percentage, entry price, and stop-loss price. Returns: position size, dollar risk, and shares to trade.

Drawdown decomposition with underwater curve.

Use when analyzing drawdown characteristics of a return series. Provide an array of returns. Returns: max drawdown, drawdown duration, recovery time, current drawdown, and all drawdown periods with start/end indices.

Trend + volatility regime + composite risk classification.

Use when classifying market regime (trending vs ranging, high vs low volatility). Provide a price series. Returns: trend regime (bullish/bearish/neutral), volatility regime (high/low/normal), and regime change signals.

Golden/death cross detection with signal history.

Use when detecting moving average crossovers (golden cross, death cross). Provide prices and two MA periods. Returns: current MA values, crossover signals, crossover history, and signal strength.

Bond price, Macaulay/modified duration, convexity, DV01.

Use when pricing a bond or computing yield, duration, and convexity. Provide face value, coupon rate, maturity, and yield or price. Returns: bond price (or yield), Macaulay duration, modified duration, convexity, and accrued interest.

Full amortization schedule with extra payment savings analysis.

Use when generating a loan amortization schedule. Provide principal, annual rate, and term in months. Returns: monthly payment, total interest, and a period-by-period schedule of principal, interest, and remaining balance.

Portfolio optimization: max Sharpe, min vol, or risk parity weights.

Use when optimizing portfolio weights for max Sharpe, min volatility, or risk parity. Provide expected returns and a covariance matrix. Returns: optimal weights, expected return, volatility, Sharpe ratio, and efficient frontier points.

CRR binomial tree pricing for American and European options.

Use when pricing American or European options via the CRR binomial lattice. Provide spot, strike, time, rate, volatility, steps, and exercise style. Returns: option price, early exercise boundary, and tree node values.

Barrier option pricing using analytical formulas.

Use when pricing knock-in or knock-out barrier options. Provide spot, strike, barrier level, barrier type, and standard option parameters. Returns: barrier option price, vanilla equivalent, and barrier adjustment factors.

Asian option pricing: geometric closed-form or arithmetic approximation.

Use when pricing Asian (average-price) options. Provide spot, strike, time, rate, volatility, and averaging type. Returns: option price via geometric closed-form or Turnbull-Wakeman approximation.

Lookback option pricing (floating/fixed strike).

Use when pricing lookback options (floating or fixed strike). Provide spot, strike, min/max price, time, rate, and volatility. Returns: lookback option price via Goldman-Sosin-Gatto formulas.

Option chain analytics: skew, max pain, put-call ratios.

Use when analyzing an options chain for skew, max pain, and put-call ratios. Provide arrays of strikes, calls, puts, and open interest. Returns: max pain strike, put-call ratio, skew metrics, and implied volatility smile data.

Put-call parity check and arbitrage detection.

Use when checking put-call parity or detecting arbitrage opportunities. Provide call price, put price, spot, strike, rate, and time. Returns: parity check, theoretical values, and any arbitrage amount.

Build implied volatility surface from market data.

Use when constructing an implied volatility surface from market data. Provide arrays of strikes, expiries, and IV values. Returns: interpolated IV surface, skew metrics, term structure, and smile parameters.

OLS linear regression with R-squared, t-stats, and standard errors.

Use when fitting a linear regression (OLS). Provide x and y arrays. Returns: slope, intercept, R², adjusted R², t-statistics, p-values, standard errors, confidence intervals, and residuals.

Polynomial regression of degree n with goodness-of-fit metrics.

Use when fitting a polynomial of degree n to data. Provide x, y arrays, and degree. Returns: coefficients, R², fitted values, and residuals.

Engle-Granger cointegration test with hedge ratio and half-life.

Use when testing if two time series are cointegrated (mean-reverting pair). Provide two price series. Returns: Engle-Granger test statistic, p-value, critical values, hedge ratio, and spread series.

Hurst exponent via rescaled range (R/S) analysis.

Use when determining if a time series is mean-reverting (H<0.5), random walk (H=0.5), or trending (H>0.5). Provide a price or return series. Returns: Hurst exponent via R/S analysis, classification, and confidence.

GARCH(1,1) volatility forecast using maximum likelihood estimation.

Use when forecasting future volatility using a GARCH(1,1) model. Provide a return series. Returns: GARCH parameters (omega, alpha, beta), current conditional volatility, and multi-step ahead volatility forecasts.

Rolling and static z-scores with extreme value detection.

Use when computing z-scores for statistical analysis or detecting extremes. Provide a value or array and reference statistics. Returns: z-scores, mean, standard deviation, and flags for values beyond 2σ or 3σ.

Fit data to common distributions and rank by goodness of fit.

Use when fitting data to standard distributions (normal, lognormal, uniform). Provide a data array. Returns: best-fit distribution, parameters (mean, std, etc.), goodness-of-fit statistics (KS test, chi-squared), and Q-Q plot data.

Correlation and covariance matrices with optional eigenvalue decomposition.

Use when computing a correlation matrix with eigenvalue decomposition for multiple assets. Provide a 2D array of return series. Returns: Pearson and Spearman correlation matrices, eigenvalues, eigenvectors, and explained variance ratios.

Impermanent loss calculator for Uniswap v2/v3 AMM positions.

Use when calculating impermanent loss for a liquidity provider position. Provide initial prices and current prices for two tokens. Returns: impermanent loss percentage, hold value vs LP value, and breakeven price ratios.

Convert between APY and APR with configurable compounding frequency.

Use when converting between APY and APR with different compounding frequencies. Provide rate and compounding periods. Returns: equivalent APY, APR, daily/weekly/monthly rates, and effective annual rate.

Liquidation price calculator for leveraged positions.

Use when computing the liquidation price for a leveraged position. Provide entry price, leverage, position side, and maintenance margin. Returns: liquidation price, distance to liquidation, and margin call price.

Funding rate analysis with annualization and regime detection.

Use when analyzing perpetual futures funding rates. Provide funding rate, position size, and holding period. Returns: annualized funding cost, projected payments, and carry trade opportunity estimate.

DEX slippage estimator for constant-product AMM (x*y=k).

Use when estimating slippage on a DEX trade using constant-product AMM math. Provide trade size and pool reserves. Returns: effective price, price impact percentage, and output amount after slippage.

Token vesting schedule with cliff, linear/graded unlock, and TGE.

Use when computing a token vesting schedule with cliff and linear vesting. Provide total tokens, cliff period, vesting duration, and TGE unlock percentage. Returns: period-by-period unlock schedule with cumulative totals.

Portfolio rebalance analyzer: drift detection and trade computation.

Use when checking if a crypto portfolio needs rebalancing. Provide target weights and current weights. Returns: whether rebalancing is needed, drift per asset, and trade list to restore targets.

Interest rate parity calculator with arbitrage detection.

Use when computing covered/uncovered interest rate parity for FX pairs. Provide domestic rate, foreign rate, and spot rate. Returns: theoretical forward rate, parity-implied rate, and arbitrage opportunity if any.

Purchasing power parity fair value estimation.

Use when estimating fair value of an FX rate using PPP. Provide domestic and foreign price indices and base-period exchange rate. Returns: PPP-implied fair value rate and over/undervaluation percentage.

Bootstrap forward rates from a spot yield curve.

Use when bootstrapping forward rates from a yield curve. Provide spot rates at various tenors. Returns: implied forward rates between each tenor pair.

Currency carry trade P&L decomposition.

Use when analyzing carry trade P&L decomposition. Provide high-yield and low-yield rates, entry spot rate, and exit spot rate. Returns: carry return, spot return, total return, and annualized P&L breakdown.

Convert nominal returns to real returns using Fisher equation.

Use when converting nominal returns to real returns using the Fisher equation. Provide nominal rate and inflation rate. Returns: real return, purchasing power change, and cumulative real growth over a period.

Taylor Rule interest rate prescription.

Use when estimating the appropriate policy interest rate via the Taylor Rule. Provide inflation, target inflation, output gap, and neutral rate. Returns: Taylor Rule implied rate and deviation from current policy rate.

Real yield and breakeven inflation from nominal yields.

Use when computing real yield and breakeven inflation. Provide nominal yield and inflation expectation (or TIPS yield). Returns: real yield, breakeven inflation rate, and inflation risk premium estimate.

Parametric Value-at-Risk and Conditional VaR.

Use when computing Value-at-Risk and Conditional VaR using parametric methods. Provide returns and confidence level. Returns: VaR, CVaR, and distribution parameters under normal or Student-t assumptions.

Portfolio stress test across multiple scenarios.

Use when stress-testing a portfolio against multiple scenarios. Provide portfolio weights, asset returns, and scenario definitions (e.g. market crash, rate hike). Returns: portfolio P&L under each scenario with component-level breakdown.

Multi-leg options payoff diagram data generation.

Use when you need payoff/P&L data points for plotting an options strategy. Provide legs with strike, premium, quantity, and type. Returns: array of price/payoff pairs for charting, plus key metrics (breakevens, max profit/loss).

Yield curve interpolation: linear, cubic spline, or Nelson-Siegel.

Use when interpolating a yield curve at arbitrary maturities. Provide observed maturities and yields, plus query maturities. Returns: interpolated yields via linear, cubic spline, or Nelson-Siegel models.

Credit spread and Z-spread from bond price vs risk-free curve.

Use when computing Z-spread and implied default probability from a corporate bond price. Provide bond price, coupon, maturity, and risk-free curve. Returns: Z-spread, option-adjusted spread, implied default probability, and loss-given-default.

Bollinger Bands with %B, bandwidth, and squeeze detection.

Use when computing Bollinger Bands with squeeze detection. Provide prices and optional period/multiplier. Returns: upper/mid/lower bands, %B, bandwidth, squeeze flag, and current position relative to bands.

Fibonacci retracement and extension levels.

Use when computing Fibonacci retracement and extension levels. Provide a high price and low price. Returns: retracement levels (23.6%, 38.2%, 50%, 61.8%, 78.6%) and extension levels (127.2%, 161.8%, 261.8%).

Average True Range with normalized ATR and volatility regime.

Use when measuring volatility via Average True Range. Provide an array of prices. Returns: current ATR, normalized ATR (as % of price), ATR history, and volatility regime classification.

Equal risk contribution portfolio weights.

Use when computing equal risk contribution (risk parity) portfolio weights. Provide a covariance matrix. Returns: risk parity weights and each asset's contribution to total portfolio risk.

Transaction cost model: commission + spread + market impact estimation.

Use when estimating total transaction costs including commissions, spread, and market impact. Provide trade size, price, spread, and commission structure. Returns: total cost, cost breakdown, and cost as percentage of trade value.

Probabilistic Sharpe Ratio — is the observed Sharpe statistically significant? Based on Bailey & Lopez de Prado (2012).

Use when testing whether a Sharpe ratio is statistically significant. Provide returns and a benchmark Sharpe. Returns: probabilistic Sharpe ratio (probability observed Sharpe exceeds benchmark), p-value, and required track record length.

Present value of a future lump sum and/or annuity stream.

Use when computing the present value of a future cash flow. Provide future value, discount rate, and number of periods. Returns: present value and discount factor.

Future value of a present lump sum and/or annuity stream.

Use when computing the future value of a present sum. Provide present value, interest rate, and number of periods. Returns: future value, total interest earned, and growth factor.

Internal rate of return via Newton-Raphson. First cash flow is typically negative (investment).

Use when computing the internal rate of return for a cash flow series. Provide an array of cash flows. Returns: IRR (decimal), annualized IRR, and NPV at the computed IRR (should be ~0).

Net present value of a cash flow series at a given discount rate.

Use when computing net present value of a series of cash flows. Provide discount rate and an array of cash flows (first is typically negative for initial investment). Returns: NPV, profitability index, and discounted cash flow breakdown.

Realized volatility: close-to-close, Parkinson, Garman-Klass, Yang-Zhang from OHLC.

Use when computing historical/realized volatility from a return series. Provide returns and optional annualization factor. Returns: realized volatility (close-to-close), annualized vol, and rolling vol series.

Normal distribution: CDF, PDF, quantile, and confidence intervals.

Use when computing normal distribution CDF, PDF, quantiles, or confidence intervals. Provide x (for CDF/PDF), p (for quantile), or confidence_level (for interval), with optional mean and std. Returns: CDF probability, PDF density, z-score, quantile value, and/or confidence interval bounds.

Standalone Sharpe ratio from a returns series.

Use when computing the Sharpe ratio from a return series. Provide returns and risk-free rate. Returns: annualized Sharpe ratio, annualized return, annualized volatility, and risk-adjusted metrics.

Compound Annual Growth Rate with optional forward projections.

Use when computing compound annual growth rate. Provide beginning value, ending value, and number of years. Returns: CAGR (decimal), total return, and equivalent annual return.

Scan and rank vertical spreads by risk/reward. Replaces 8-16 individual options/price calls.

Use when you need to evaluate and rank multiple vertical spread candidates at once. Instead of calling options/price 8-16 times, this scans all strike combinations in one call. Provide spot price, volatility, DTE, and strategy type (bull_call_spread, bear_put_spread, etc.). Returns: ranked candidates with risk/reward ratios, breakevens, and full Greeks for each leg. PAID ONLY — no free tier.

Combined regime classification: trend, vol, RSI, direction, strategy suggestion. Replaces technical + regime + realized-vol.

Use when you need a complete market regime assessment combining trend, volatility, RSI, and directional signals. Instead of calling technical + regime + realized-vol separately, this returns everything in one call. Provide closing prices (min 30). Returns: trend direction, volatility regime, RSI, SMA, strategy suggestion (momentum/mean-reversion/risk-off). PAID ONLY — no free tier.

Complete risk tearsheet: Sharpe, Sortino, VaR, Kelly, drawdown, Hurst, CAGR. Replaces 7 individual calls.

Use when you need a complete risk tearsheet for a return series. Instead of calling 7 individual risk/stats endpoints, this returns Sharpe, Sortino, Calmar, VaR, CVaR, Kelly, max drawdown, Hurst exponent, CAGR, and win rate in one call. Provide daily returns. Returns: comprehensive risk profile with portfolio values. PAID ONLY — no free tier.

Complete trade evaluation: sizing, risk/reward, Kelly, costs, regime, signals. Replaces 5 individual calls.

Use when evaluating whether to take a specific trade. Combines position sizing, risk/reward analysis, transaction cost estimation, regime detection, and technical signals into a single go/no-go verdict. Provide entry/stop/target prices, account size, and recent price history. Returns: position size, costs, signals, regime, Kelly sizing, and FAVORABLE/CAUTION/UNFAVORABLE verdict. PAID ONLY — no free tier.

Full portfolio health check: risk metrics, correlation, drawdown, rebalance, stress test. Replaces 6 individual calls.

Use when you need a complete portfolio health check. Combines risk metrics, correlation matrix, drawdown analysis, rebalance detection, and stress testing (2008 Crisis, Rate Hike, Flash Crash) in one call. Provide holdings with values, target weights, and return series. Returns: Sharpe, VaR, drawdown, correlation matrix, rebalance trades, and stress test P&L. PAID ONLY — no free tier.

Complete pairs trading signal: cointegration, Hurst, z-score, half-life, hedge ratio. Replaces 4 individual calls.

Use when analyzing a pairs trading opportunity. Combines cointegration testing, Hurst exponent, z-score analysis, half-life estimation, and trade signal generation in one call. Provide two price series. Returns: cointegration result, hedge ratio, spread z-score, Hurst, and actionable signal (LONG/SHORT/WAIT/CLOSE/NO_TRADE). PAID ONLY — no free tier.

Execute multiple computations in a single request. Max 100 per batch.

Use when you need to execute multiple computations efficiently. Bundle up to 100 individual endpoint calls into a single request for ~6x throughput improvement. Provide an array of {endpoint, params} objects. Price equals the sum of individual endpoint prices. Ideal for backtests, parameter sweeps, and portfolio-wide calculations.

Deterministic backtest of SMA crossover, RSI mean reversion, momentum, or Bollinger breakout. Replaces 10+ individual calls.

Use when backtesting a trading strategy on price history. Strategies: sma_crossover (params: fast, slow), rsi_mean_reversion (params: period, oversold, overbought), momentum (params: lookback), bollinger_breakout (params: period, std). Provide prices, strategy name, params, initial_capital, commission_bps. Returns: total return, Sharpe, Calmar, max drawdown, number of trades, win rate, equity curve, vs buy-and-hold. PAID ONLY — no free tier.

Generate trade list to rebalance from current holdings to target weights with transaction cost estimate.

Use when generating the exact trades needed to rebalance a portfolio from current holdings to target weights. Provide current_holdings (asset -> USD value), target_weights (asset -> weight, sum to 1), transaction_cost_bps. Returns: list of trades (buy/sell with amounts), total cost, drift before/after, and post-rebalance weights. PAID ONLY — no free tier.

Rank top options strategies given market outlook + volatility view. Returns P&L, breakevens, max profit/loss for each.

Use when agents need to pick the best options strategy given a market outlook. Provide spot price, outlook (bullish/bearish/neutral), vol_view (rising/falling/stable), T, sigma, r. Returns: ranked list of strategies (Long Call, Bull Call Spread, Iron Condor, Long Straddle, etc.) each with legs, max profit/loss, breakevens, net debit/credit, and score. PAID ONLY — no free tier.

Rank cheapest effective hedges for a given position. Compares protective puts, collars, inverse hedges.

Use when agents need to hedge an existing position. Provide position_type (long_stock/short_stock/long_crypto/long_options), position_value, asset_price, volatility, time_horizon_days, max_hedge_cost_pct. Returns: ranked hedges (protective put, collar, futures short, partial hedge) each with cost, protection level, affordability flag, and max loss after hedge. PAID ONLY — no free tier.