Exotic Derivatives¶

qufin supports pricing and analysis of exotic derivatives beyond vanilla European options.

Bermudan Options (Longstaff-Schwartz)¶

Bermudan options can be exercised at specific dates before expiration. Priced via Least-Squares Monte Carlo (LSM).

from qufin.derivatives.bermudan_lsm import lsm_price

# lsm_price returns a dict with a "price" key.
result = lsm_price(
    s0=100,
    k=105,
    r=0.05,
    sigma=0.2,
    T=1.0,
    n_steps=12,          # monthly exercise opportunities
    n_paths=100_000,
    exercise_dates=[i / 12 for i in range(1, 13)],
    is_call=False,
)
print(f"Bermudan put price: {result['price']:.4f}")

Basket Options¶

Options on a weighted basket of multiple underlying assets.

import numpy as np
from qufin.derivatives.basket import basket_mc, BasketOptionSpec

corr = np.eye(3)  # 3x3 correlation matrix
spec = BasketOptionSpec(
    s0=np.array([100.0, 110.0, 95.0]),
    k=100.0,
    r=0.05,
    sigma=np.array([0.2, 0.25, 0.18]),
    corr=corr,
    T=1.0,
    weights=np.array([0.4, 0.35, 0.25]),
    is_call=True,
)
result = basket_mc(spec, n_paths=100_000, seed=42)
print(f"Basket price: {result.price:.4f}")

Autocallable Notes¶

Structured products that automatically redeem if the underlying exceeds a barrier on observation dates.

from qufin.derivatives.autocallable import autocallable_mc, AutocallableSpec

spec = AutocallableSpec(
    s0=100,
    k=100,
    barrier=105,              # autocall barrier (absolute level)
    coupon=0.08,              # 8% coupon
    observation_dates=[0.25, 0.5, 0.75, 1.0],  # quarterly
    r=0.05,
    sigma=0.25,
    T=1.0,
)
result = autocallable_mc(spec, seed=42)
print(f"Autocallable price: {result['price']:.4f}")

Path-Dependent Options¶

Lookback Options¶

The payoff depends on the maximum or minimum price during the option's life.

from qufin.derivatives.path_dependent import lookback_mc, LookbackOptionSpec

spec = LookbackOptionSpec(s0=100, r=0.05, sigma=0.2, T=1.0, is_call=True, n_steps=252)
price = lookback_mc(spec, n_paths=100_000, seed=42)
print(f"Lookback price: {price:.4f}")

Cliquet Options¶

Options with periodic resets that lock in gains at each observation date.

from qufin.derivatives.path_dependent import cliquet_mc

price = cliquet_mc(
    s0=100,
    r=0.05,
    sigma=0.2,
    T=1.0,
    n_periods=4,         # quarterly resets
    cap=0.05,            # 5% cap per period
    floor=0.0,           # 0% floor per period
    n_paths=100_000,
    seed=42,
)
print(f"Cliquet price: {price:.4f}")

Pricing Methods Comparison¶

Derivative	Method	Convergence	Notes
Bermudan	LSM (Monte Carlo)	O(1/sqrt(N))	Basis function choice matters
Basket	Monte Carlo	O(1/sqrt(N))	Correlation structure critical
Autocallable	Monte Carlo	O(1/sqrt(N))	Path-dependent, many barriers
Lookback	Monte Carlo	O(1/sqrt(N))	Needs fine time discretization
Cliquet	Monte Carlo	O(1/sqrt(N))	Cap/floor interactions complex

Quantum pricing

Multi-asset and path-dependent QAE pricing is planned for v0.2.0. Currently, quantum amplitude estimation is available for single-asset European options via qufin.options.amplitude_estimation.