Derivatives¶

The derivatives namespace contains pricing functions, object-oriented models, payoff strategies, numerical simulations, and calibration workflows.

Mathematical frame¶

Most derivative pricing routines implement the discounted expected payoff idea:

\[V_0 = e^{-rT}\mathbb{E}^{\mathbb{Q}}[\Pi(S_T)]\]

where \(\Pi(S_T)\) is the terminal payoff and \(\mathbb{Q}\) is a risk-neutral pricing measure.

Vanilla pricing¶

from abaquant.derivatives import black_scholes, black_76, implied_volatility_bsm

call = black_scholes(100.0, 105.0, 0.04, 0.22, 1.0, is_call=True)
future_call = black_76(102.0, 100.0, 0.04, 0.20, 1.0, is_call=True)
implied_vol = implied_volatility_bsm(call, 100.0, 105.0, 0.04, 1.0)

Use functional pricing when you need compact calculations inside vectorized or tabular workflows.

Greeks¶

from abaquant.derivatives import calculate_greeks, second_order_greeks

first_order = calculate_greeks(100.0, 105.0, 0.04, 0.22, 1.0, is_call=True)
second_order = second_order_greeks(100.0, 105.0, 0.04, 0.0, 0.22, 1.0, is_call=True)

Common first-order Greeks:

Greek	Interpretation
Delta	sensitivity to spot.
Gamma	sensitivity of delta to spot.
Vega	sensitivity to volatility.
Theta	sensitivity to time decay.
Rho	sensitivity to interest rate.

Model classes¶

from abaquant.derivatives import BlackScholesMertonModel

model = BlackScholesMertonModel(
    spot_price=100.0,
    strike_price=105.0,
    maturity_years=1.0,
    risk_free_rate=0.04,
    volatility=0.22,
    dividend_yield=0.01,
)

price = model.price("call")
greeks = model.greeks()
diagnostics = model.diagnostics(option_type="call")
scenario_grid = model.scenario_grid(
    spot_prices=[90.0, 100.0, 110.0],
    volatilities=[0.18, 0.22, 0.26],
    option_type="call",
)

Use model classes when you want reusable state, diagnostics, reports, visualizations, or scenario grids.

Advanced model families¶

Model	Use case	Main risk
Bachelier	Normal underlying dynamics, negative rates or negative underlyings.	Normal volatility convention differs from lognormal volatility.
Heston	Stochastic volatility and volatility clustering.	Calibration can be non-unique or unstable.
Merton jump diffusion	Discontinuous price jumps.	Jump intensity and jump-size estimates are difficult.
SABR	Implied-volatility smile and skew interpolation.	Hagan approximation can break in extreme regimes.
NIG and Variance-Gamma	Heavy tails and skewed return distributions.	Parameter interpretation and calibration risk.

from abaquant.derivatives.advanced import HestonModel, SABRModel, MertonModel

Trees¶

from abaquant.derivatives import binomial_tree, crr_binomial_tree

price, tree = binomial_tree(
    100.0,
    100.0,
    1.0,
    0.05,
    0.20,
    80,
    option_type="put",
    american=True,
)
crr_price, stock_tree, option_tree = crr_binomial_tree(
    100.0,
    100.0,
    0.05,
    0.20,
    1.0,
    80,
    is_call=True,
)

Tree methods are useful for American exercise and educational inspection of state lattices.

Exotics¶

Representative exotic helpers include:

from abaquant.derivatives import (
    cash_or_nothing_options,
    asset_or_nothing_options,
    geometric_asian_options,
    down_and_out_barrier_option,
    exchange_options,
)

Exotic helpers use compact closed-form or approximate formulas where available. Always check the formula convention before comparing against another system.

Option strategies¶

from abaquant.derivatives import OptionStrategy

strategy = OptionStrategy.bull_call_spread(
    lower_strike=100.0,
    upper_strike=115.0,
    lower_premium=6.0,
    upper_premium=2.0,
)

profile = strategy.profile(points=101)
break_evens = strategy.break_even_points()

Common predefined constructors include spreads, straddles, strangles, butterflies, condors, and protective puts.

Calibration¶

from abaquant.derivatives.calibration import BSMFlatVolCalibration, SABRSmileCalibration

# Use calibration classes when you have market observations in a structured table.
# The returned CalibrationResult stores fitted parameters and diagnostics.

Calibration minimizes model-versus-market errors. Treat fitted parameters as conditional estimates, not physical truths.

Visualization and reports¶

fig = model.visualize(chart="price_surface", option_type="call")
report = model.report(option_type="call")
report.save("reports", "bsm_call", formats=("markdown", "html"))

Available chart names vary by model and result type. See the visualization examples for concrete galleries.