Model and assumptions

Black-Scholes-Merton, and where each assumption is violated.

Valuation is Black-Scholes-Merton with continuous dividend yield. The assumptions are stated here along with where each is violated, because a margin engine built on an unstated model cannot be audited.

Assumptions

European exercise. All listed contracts are European and cash settled. This is a design choice and removes early exercise entirely.
Frictionless continuous hedging. Violated in practice. The cost is borne by liquidity providers rather than by the margin engine.
Constant volatility to expiry. Violated, and the surface exists precisely because it is. Each strike carries its own implied volatility, so the model is used as a quoting convention and an interpolation device rather than as a claim about the law of the underlying.
Lognormal terminal distribution. Violated in the tails. Scenario margin does not rely on it.

On using a model known to be wrong

Black-Scholes is retained as the mapping between price and implied volatility, which is all that is asked of it. Because every strike carries its own volatility, the model imposes no distributional view of its own, and the risk calculation inherits no tail assumption from it.

The pricing functional

C = S e^{- q T} Φ (d_{1}) - K e^{- r T} Φ (d_{2})

d_{1} = \frac{\ln (S / K) + (r - q + \frac{σ^{2}}{2}) T}{σ \sqrt{T}}, d_{2} = d_{1} - σ \sqrt{T}

The normal distribution function

Φ is evaluated by the Hart rational approximation as refined by West, accurate to approximately 1e−15 across the real line. The precision is not decorative: the implied volatility solver differentiates through Φ, and a lower accuracy error function appears as noise in vega, which propagates into both quoting and margin.

Degenerate inputs

For T ≤ 0 or σ ≤ 0 the functional returns discounted forward intrinsic value, which is the correct limit. A pricer that returns NaN at expiry is a margin engine that fails at expiry.