Brownian Bridge™

1. The `ImpliedVolatilitySurface` constructor calls `evaluate_cubic_spline_coefficients()` and stores `_alpha`, `_beta`, `_gamma` matrices. Why is this computation done at construction time rather than on each call to `implied_volatility()`?

Because the constructor runs on a separate threadThe coefficients depend only on the market data, which is fixed at construction. Pre-computing them avoids redundant O(N) Thomas solves on every query; queries are O(log N) binary search onlyThe Thomas algorithm is not numerically stable if called repeatedlyBecause `evaluate_cubic_spline_coefficients` is a non-const method and cannot be called from const methods

2. Why does the `ImpliedVolatilitySurface` interpolate in **total variance** `w(T,K) = σ*(T,K)² · T` along the maturity axis rather than in implied volatility directly?

Total variance is easier to compute numericallyLinear interpolation in total variance guarantees that the interpolated surface is free of calendar spread arbitrage, whereas linear interpolation in implied vol can produce decreasing total varianceImplied volatility is not defined for non-market maturitiesThe Thomas algorithm requires variance as input, not volatility

3. In the Thomas algorithm for a tridiagonal system of size N, the number of floating-point operations is O(N³), the same as full Gaussian elimination.

truefalse

4. The `smile_implied_volatility` method uses **linear extrapolation** for strikes outside the market range `[K₁, Kₙ]`. What is the alternative, and what is its practical risk?

Flat extrapolation (constant vol beyond the last market strike). Risk: no skew for deep out-of-the-money options, underpricing barriers and digitalsCubic extrapolation using the spline curvature at the boundary. Risk: the spline can produce negative implied vols far from the market rangeReject any query outside the market range. Risk: the Dupire model cannot be simulated for paths that exit the strike rangeAll of the above are valid alternatives with different trade-offs

5. In the Dupire local volatility computation, the partial derivative `∂σ*/∂K` is approximated by a central finite difference: ``` ∂σ*/∂K ≈ (σ*(T, K+εK) - σ*(T, K-εK)) / (2εK) ``` If `εK` is too small (e.g., `εK = 1e-15`), what numerical problem arises?

The Thomas algorithm divergesFloating-point **cancellation**: `σ*(K+εK)` and `σ*(K-εK)` differ by a tiny amount; subtracting nearly equal numbers loses significant digits, and the result is dominated by rounding errorThe cubic spline extrapolates beyond its valid rangeThe implied vol surface constructor throws an exception

6. The forward-moneyness adjustment `K(i) = K · exp(r · (T_i - T))` used during maturity interpolation converts a strike at time T to the equivalent strike at maturity T_i. If T < T_i (querying between the first and second market maturities), then K(i) < K.

truefalse

Quiz: Implied Volatility Surface in C++

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