Implied variance at a strike reflects the paths ending there. Fukasawa makes that statement precise through a first-order representation along a sequence of models whose relative perturbation scale r_n vanishes.
The paper proves a theorem without an empirical test. There is no data, calibration or options market check of the approximation's accuracy. A desk implementation must replace E[A_T | S_T = K] with a fitted risk-neutral conditional-variance estimator, approximate quadratic variation using daily or minute bars, and trade at the close.
What exactly does Theorem 2.5 say?
Start with a strictly positive continuous local martingale S^n. Let A^n denote the quadratic variation of log S^n at T_n. The Black-Scholes total implied variance recovered from the put struck at K_n = S_0 e^{k_n} is ŵ_n(k_n,T_n). Under Assumptions 2.1 and 2.3, Theorem 2.5 states that, for bounded {z_n},
ŵ_n(k_n,T_n) = E[A^n | S^n_{T_n} = K_n] + o(v_n r_n),
where v_n = E[A^n], while r_n is a relative perturbation scale chosen by the modeller. Expressed in annualized units, the remainder becomes o(v_n r_n / T_n). The expectation is conditional on the terminal price under the pricing measure, assuming zero rates and dividends.
The conditioning gives the result its force. A variance swap rate averages realized variance without conditioning. At a strike whose standardized log-moneyness z_n = (k_n + v_n/2)/√v_n stays bounded, implied variance is, to leading order along the sequence, realized variance averaged over paths that terminate at that strike. Two strikes on one surface therefore represent two conditional averages of the same random variable. Skew can be read as a relationship between quadratic variation and terminal price, which can be simulated.
The proof also marks the limits of the formula. Conditional on an environment sigma-field, the log price is Gaussian, with conditional density h_n(k) written explicitly in (51). The theorem uses the continuous-density conditional expectation defined in (53): E[A^n h_n(k)] / E[h_n(k)]. A stopped Black-Scholes martingale argument and the Gamma-Vega identity produce the option-price expansion. Monotone inversion of the put price then yields implied variance. The entire construction remains inside a model; the paper provides no method for turning a screen of quotes directly into the conditional expectation.
Assumption 2.3 does the heavy lifting
Assumption 2.3 asks for a decomposition of the log-price martingale as M = J + N, with orthogonal brackets. Conditional on everything else, N must be Gaussian. Its clock also satisfies D_T ≥ c A_T for a constant c > 0 independent of n. The Brownian-factor example (2.4) sets c = 1 − ρ². For the multifactor case, c = 1 − |ρ|², where |ρ|² = Σ ρ_i² < 1.
A genuinely independent volatility shock must therefore contribute a fixed share of total variance, uniformly along the sequence. The proof pays for this through 1/√c. Its density bound is h_n(k) ≤ 1/√(2πcA^n), and the numerator is bounded by √(v_n/(2πc)). The stopping threshold is b_n = (1 − c/2)v_n, while the localization window contracts with c. These examples satisfy the assumption for any |ρ| < 1 as long as c remains independent of n. The paper does not quantify how quickly the remainder worsens when c shrinks.
Concentration supplies the other condition. With Z_n = A^n/E[A^n], the requirement is Q(|Z_n − 1| > η) = o(r_n √v_n). The weighted variables Y_n = (Z_n − 1)/(r_n √Z_n) must also be uniformly integrable. Fukasawa explicitly says this weighted requirement is stronger than uniform integrability of the unweighted perturbation.
Proposition 2.2 gives a practical sufficient condition. Assume r_n(1+√v_n) → 0, impose a p-th moment bound on Y_n for some p > 1, and require r_n^{p−1}/√v_n → 0. The corresponding Markov tail has order O(r_n^p).
One rate across three regimes
The common treatment of three asymptotic regimes is the paper's strongest contribution. For small vol-of-vol, r_n = a_n and the remainder is o(a_n), provided ‖sup_t |V^n_t − V̄t|‖{L^p} ≤ C_{p,T} a_n for p ≥ 2 and a_n/√v_n → 0. Under fast mean reversion, r_n = n^{−1/2}, while v_n = T v̄ + O(n^{−1}) and ‖A^n − v_n‖_{L⁴} = O(n^{−1/2}); the remainder is then o(n^{−1/2}). Short maturity uses r_n = T^H and produces a remainder of o(T_n^H).
All three regimes follow from the same two assumptions. Maturity may vary, and the argument requires no joint weak limit or differentiable limiting regression. Fukasawa's SIAM note uses a martingale expansion with those additional features, and this paper extends that treatment.
The short-maturity corollary assumes k_n = O(√T_n) together with Q(|Z_n − 1| > η) = o(a_n √T_n). Taking a(T) = T^H changes the tail requirement to o(T_n^{H+1/2}). Proposition 2.9 derives it from spot-variance moment conditions when p > 1 + 1/(2H).
The Gaussian multifactor example then generates the roughness split directly. Its finite-difference skew is Σ_12 T_n^{H−1/2}/(√v_0 (2H+3)). Ornstein-Uhlenbeck kernels have H = 1/2 and a finite limiting skew. Power kernels ν_i(t−s)^{H−1/2} with H < 1/2 produce the T^{H−1/2} divergence whenever Σ_12 ≠ 0. This skew is a finite difference evaluated between two fixed standardized strikes x and y.
The wings remain outside the theorem
The abstract states the restriction plainly: the representation applies "for bounded standardized log-strikes", meaning bounded z_n = (k_n + v_n/2)/√v_n. Its coverage across several regimes is a fair defence of that limitation. Even so, bounded z describes roughly the area between the wings. In the short-maturity setting, the restriction becomes k_n = O(√T_n). Deep index puts, where crash premium and the convexity trade reside, fall beyond the theorem.
First order, no constant.
The stated remainder is o(v_n r_n), without an explicit constant. It therefore supplies no error magnitude for a fixed maturity or vol-of-vol level.
Fukasawa draws the boundary of the proof carefully. The text describes the links to both the martingale expansion and the earlier short-time expansion as formal, because weak convergence alone cannot establish convergence of prelimit regressions at a specified point. That candour carries more value than another corollary. The paper also discloses that generative AI was used for language editing and draft organization, with the author retaining responsibility for the mathematics.
Turning the formula into a trade
The formula can serve as a benchmark for a fitted surface. Using it requires E[A_T | S_T = K] under the pricing measure. In practice, a desk would fit a stochastic-volatility model, simulate terminal-conditioned integrated variance for each strike, and trade its gap against market implied variance. The resulting exercise tests the estimator and execution.
We are running a strategy of that general form on US listed equity and ETF options as this is written. Three gaps separate our implementation from the theorem:
- The conditional expectation is risk-neutral and conditions on a future terminal price. Neither quantity is historically observable, so our backtest replaces the theorem's object with a fitted estimator.
- Continuous quadratic variation must be estimated from daily or resampled minute bars. Variance between observations and microstructure noise remain measured imperfectly.
- Our options data is end of day. Entry and hedging are therefore simulated at the close rather than continuously.
We discussed a related result last year in our note. Fukasawa's result is more useful for a simulation desk because a Monte Carlo already generates the quantity on the right-hand side. A finite-sample error bound with an explicit constant could change my view of the wings by showing that bounded z is merely a technical convenience rather than the boundary of validity. Until such a result appears, this is a leading-order approximation for the belly of the surface. The wings need another price.