Option prices contain the market's whole probability distribution for where a price ends up, not just its volatility. This package extracts it.
The result is due to Breeden and Litzenberger (1978): the risk-neutral density is the second derivative of the call price in the strike. The intuition is a butterfly spread — buy a call either side of a strike, sell two in the middle, and it pays off only if the price lands there. Its price, per unit width, is the market's probability of that outcome.
Applied literally to quoted prices that recipe does not work. Two numerical derivatives amplify the bid-ask noise until the density oscillates and goes negative, which is the failure everyone hits first. This package differentiates the smile instead, in closed form, with non-negativity enforced during the fit rather than repaired afterwards.
pip install -e ".[dev]"
rnd demoSAMPLE method=svi T=0.2500y quotes=89
forward 4,999.9024 discount 0.990735 implied rate 3.7231%
atm vol 17.4193% smile fit rmse 0.028 vol points
mass 1.000005 martingale error +0.00 bp min density 3.641e-137
quotes: 0 bound, 7 monotonicity and 22 convexity violations across 89 strikes
mean 4,999.9024 sd 453.9989 skew -0.3638 excess kurtosis +1.7407
p(-20%) 0.0200 p(-20%) lognormal 0.0059 p(-10%) 0.1124 p(-10%) lognormal 0.1220
The headline is the last line. At the same forward and the same at-the-money volatility, the market prices a 20% drop at 2.0% where the lognormal says 0.6% — about three and a half times as likely. That gap is the crash risk the Black-Scholes bell curve does not contain, read straight off live prices.
On a CSV of quotes (strike, option_type, bid, ask):
rnd estimate quotes.csv --expiry-date 2026-12-18 --out results/On a live chain from Yahoo, with pip install "implied-rnd[yahoo]":
rnd fetch SPY --expiry-date 2026-12-18 --out results/Each run writes density.csv, fitted_quotes.csv, diagnostics.json and a
four-panel report.png.
From Python:
from rnd import OptionChain, estimate_density
chain = OptionChain.from_csv("quotes.csv", expiry=0.25, underlying="SPX")
result = estimate_density(chain, method="svi")
result.density.prob_below(4500) # risk-neutral P(S_T < 4500)
result.density.quantile(0.05) # 5% downside level
result.density.moments().skewness # implied skew
result.diagnostics() # every check, as a dictFull derivation in docs/METHOD.md. The pipeline in order:
- Clean the quotes. Zero bids, crossed markets and spreads wider than their own information content are removed, not down-weighted.
- Recover the forward from put-call parity.
C(K) - P(K) = D(F - K), so a regression of the call-put spread on the strike gives the discount factor as minus the slope and the forward as the intercept over it. No interest rate or dividend assumption enters anywhere — the forward comes from the same quotes as everything else, borrow and expected dividends included. - Keep the out-of-the-money wing on each side, where the information is.
- Invert to implied volatility. Quotes outside the no-arbitrage bounds return
nanand drop out on their own. - Fit total variance against log-moneyness, weighted by how tightly each
quote pins its own volatility —
half_spread / vega, times the Jacobian from volatility to total variance. - Differentiate the smile analytically. Given
w(k)and its first two derivatives the density is closed-form; nothing differentiates a price curve. - Check the result. Mass, martingale error, non-negativity and the butterfly condition, all reported rather than assumed.
Two smile models share step 6 through one interface:
- SVI (default) — Gatheral's raw parameterisation. Analytic derivatives, box bounds for admissibility, an escalating penalty on butterfly violations, and an exact repair step for whatever the penalty leaves behind. A penalty method approaches the feasible boundary from the wrong side and never quite arrives, so a converged-looking fit can still carry a negative probability a fraction of a basis point deep. The repair removes it.
- Weighted smoothing spline — more flexible, less protected. Smoothing defaults to the chi-square expectation, so the fit may miss each quote by about its own standard error and no more.
Where the two disagree, the chain is telling you something about its own quality.
Beyond the last quoted strike there is no data. Total variance continues linearly with the slope capped at 2 by Lee's moment formula — steeper implies a distribution with no finite moments — and scaled back further if the butterfly condition still dips, which it can right at the join where the spline's curvature falls away. A flat wing always satisfies the condition, so the search cannot fail.
pytest --cov=rnd --cov-report=term-missing213 tests, 97% coverage. The ones that matter score the estimator against ground truth: chains are generated from a mixture of lognormals, which has closed-form option prices and a closed-form density, so the estimate is compared with the exact answer rather than eyeballed on a chart. Recovery is within about 1% in L1 on clean quotes, and the density stays non-negative on noisy ones.
Two tests are worth reading on their own:
test_density_from_smile_when_the_smile_is_flat_matches_the_lognormal— with a flat smile the machinery must reproduce Black-Scholes exactly. It does, to 1e-12.test_density_from_prices_when_the_curve_is_noisy_goes_negative— the naive second difference, failing on purpose, so the problem being solved is visible in the suite rather than only claimed in the documentation.
This is the risk-neutral measure, not the real-world one. It is a probability weighted by risk aversion. The density says what the market charges for a payoff, which is not what it expects to happen; the two differ by a pricing kernel this package does not attempt to estimate. Read a risk-neutral tail probability as a forecast and you will overstate crash odds.
The bundled synthetic_index_chain.csv is generated, not recorded. It comes
from a two-component lognormal mixture so the demo has a known answer. It is not
market data and should not be used as any.
src/rnd/
black.py Black-76 pricing and the implied-volatility solver
chain.py Quote container and the cleaning rules
forward.py Implied forward and discount from put-call parity
smile.py Smile interface, flat control case, spline fit, wing rules
svi.py Raw SVI and its calibration
density.py Breeden-Litzenberger, closed form and finite difference
arbitrage.py Static checks on the raw call curve
pipeline.py End to end, with diagnostics
plotting.py The four-panel report
cli.py rnd demo / estimate / fetch
data/ Synthetic markets with known answers, and the Yahoo loader
Breeden and Litzenberger (1978); Figlewski (2008); Gatheral, The Volatility Surface (2006); Gatheral and Jacquier, Arbitrage-free SVI volatility surfaces (2014); Lee, The moment formula for implied volatility at extreme strikes (2004).
MIT licensed.