Three ways to price an option, and the surface to price it on

Three pricers, one answer

The engine prices a European option three independent ways and treats any disagreement as a bug. Black-Scholes-Merton with a continuous dividend yield $q$ is the closed form:

The Cox-Ross-Rubinstein tree moves the stock up by $u = e^{\sigma\sqrt{\Delta t}}$ or down by $d = 1/u$ with risk-neutral probability $p = (e^{(r-q)\Delta t} - d)/(u - d)$, then rolls the payoff back to today. An American option takes the larger of holding and exercising at every node. The tree oscillates as the strike falls between nodes at a position that shifts with the number of steps; averaging N and N+1 steps removes most of it (Exhibit 7). With the same inputs, the tree values the American put at 10.390 against 10.056 for the European, an early-exercise premium of 0.334.

Monte Carlo draws the terminal price exactly in one step and reports every estimate with its standard error, so agreement is judged in standard errors rather than digits. On a million paths, antithetic pairs cut the variance by only 1.4x, because an out-of-the-money call’s payoff is far from linear in the normal draw. A control variate on the discounted terminal price, whose expectation is known, cuts it by 4.2x (Exhibit 8).

Greeks four ways

Each Greek is computed four ways: the closed form, central differences on the closed form, central differences on Monte Carlo, and pathwise Monte Carlo, which differentiates the payoff along each simulated path:

That works because a call or put payoff is Lipschitz. It fails for gamma: $f'$ is a step at the strike, so its derivative is a spike the simulation never samples. Gamma uses the likelihood-ratio estimator instead, which differentiates the density rather than the payoff:

The bumped Monte Carlo reuses the same draws for the up and down bumps, so most of the noise cancels path by path. Every estimate in Exhibit 9 sits within 0.63 standard errors of the closed form.

What discrete hedging leaves behind

The engine sells a six-month at-the-money call for $6.37, hedges it at Black-Scholes delta, and lets the stock drift at 8% a year in the real world. The P&L is centered on zero whatever the drift, which is the replication argument at work. What is left is the error from hedging at intervals: with 26 rebalances, about weekly, its standard deviation is $0.93, 15% of the premium; with 126, about daily, it falls to $0.44 (7%). The fitted slope of −0.49 in Exhibit 10 is close to the theoretical one half.

From option chains to a surface

The second half of the engine builds the SPY and QQQ surfaces in the lab above from a snapshot of their option chains at the October 9, 2026 close. Every filter counts what it drops (Exhibit 11): expiries under a week or over 800 days, zero bids, crossed markets, prices under five cents, spreads wider than half the mid, no open interest, and the in-the-money side of each strike, which keeps the early-exercise bias small.

Discount rates from the right options

Put-call parity gives each expiry’s discount factor and forward without assuming a rate: regressing $C - P$ on the strike gives slope $-DF$ and intercept $DF \cdot F$. On SPY and QQQ that regression is biased, because the early-exercise premium in in-the-money puts lowers $C - P$ at high strikes; it implies rates as low as −35%. European SPX options carry no such premium, so the curve comes from SPX (Exhibit 12). Fitted as $r(T) = c_0 + c_1 \ln T$, it gives 4.42% at three months and 4.88% at a year.

De-Americanizing the quotes

Each quote is then inverted on the American tree rather than the closed form. The bias from treating it as European is small in the middle of the chain (a median of 0.04 vol points for SPY) but reaches 4.3 vol points for long-dated puts near the forward. With rates well above the dividend yield the forward sits far above spot, so a long-dated put can be out of the money against the forward and in the money against spot, where early exercise is worth something.

A first attempt took the dividend yield from the parity forward and over-corrected, because that forward came from American prices and already carried the put’s premium. The fix calibrates the yield per expiry so that American-model put and call vols agree at the six strikes nearest the money. Checked out of sample at strikes further out, the median put-call gap falls from 0.65 to 0.04 vol points for SPY, and from 0.56 to 0.02 for QQQ. The calibrated yields also recover SPY’s quarterly dividend without being told about it (Exhibit 13): within 0.15% of zero for the November expiries, then 0.6% for December 18 and 1.1% for December 31, after the mid-December ex-dividend date. The yield also absorbs any cost of borrowing the ETF.

An arbitrage-free fit

Each expiry is fitted with raw SVI, in total variance $w = \sigma^2 T$ against log-moneyness $k = \ln(K/F)$:

The fit minimizes implied-vol errors scaled by the bid-ask half-spread, with three no-arbitrage conditions as penalties: the Gatheral-Jacquier butterfly function $g(k) \ge 0$ over the quoted strikes plus a margin (a non-negative density), total variance that never falls with maturity where adjacent expiries are both quoted, and Roger Lee’s wing bound $b(1 + |\rho|) \le 2$. Fitted freely, the SPY surface had butterfly arbitrage in 4 slices (QQQ 0) and 1 calendar violation (QQQ 1). With the constraints there are none, and the median fit error for SPY is unchanged at 0.13 vol points.

The constraints hold the fitted slices right at their boundary, which is why the live check in the surface section flags arbitrage after only a small change: a skew 5% steeper than fitted is enough for SPY. At the money, vol rises with maturity, from 9.5% at a week to 17.0% at 2.2 years for SPY (14.6% to 22.1% for QQQ), and QQQ carries 5.1 to 6.1 vol points more than SPY at every expiry both list (Exhibit 5).

How it is checked

• Known values. Hull’s textbook examples, including the five-step American put, and the standard at-the-money reference call.
• Invariants. Put-call parity on thousands of random inputs, prices inside the model-free bounds, the call and put deltas a dividend discount apart, an American call without dividends worth exactly the European, and an American put never below intrinsic value.
• Inversion. Implied vol reprices every option it inverts, and returns nothing rather than a wrong answer outside the no-arbitrage bounds.
• Simulation. Monte Carlo within four standard errors of the closed form for every variance-reduction choice, a standard error that falls with the square root of the paths, and results that repeat from a seed.
• The surface pipeline, on synthetic chains with known answers. The parity regression recovers the forward and discount factor and ignores a stale quote, each cleaning rule drops exactly its target, the American inversion recovers a known vol, and the dividend calibration recovers a known yield.
• Two implementations. The TypeScript engine on this page reproduces the closed-form Greeks committed in the Python repository to twelve significant digits (the bumped ones to nine), and recomputes every fitted slice’s at-the-money vol, butterfly minimum, wing slope and calendar gap from its parameters.

What it leaves out

• The pricers assume constant volatility and rates; the tree is plain Cox-Ross-Rubinstein.
• The hedging experiment ignores transaction costs, which would penalize frequent rebalancing.
• The surface is one snapshot of end-of-day quotes, with no check that calls and puts were quoted at the same moment. Your numbers would differ on another day.
• Dividends are a continuous yield per expiry rather than cash amounts on known ex-dates.
• Raw SVI has five parameters per expiry and cannot follow the steepest short-dated put wings: the worst slice is QQQ at 42 days, 0.74 vol points of fit error. SSVI or eSSVI would give a jointly arbitrage-free surface with fewer slice-by-slice compromises.
• Next steps: price an exotic off the fitted surface, a Dupire local-volatility model, and a cross-check against QuantLib.

Model, data pipeline and fits from Mizan Nurani’s options-engine (commit f4376f5, October 11, 2026). Option chains from Yahoo Finance at the October 9, 2026 close. This is a study, not a recommendation.