Vanilla prices can survive skew Brownian returns, even when path-dependent valuations cannot. One repair comes first: Appendix B identifies and corrects the sign error in Zhu and He (2018) for delta < 0.
Two processes under one name
The literature claims to use Itô and McKean's construction. Begin with reflected Brownian motion. Whenever it visits zero, a Bernoulli draw with parameter alpha determines the sign of the next excursion. The marginals form a mixture of half-normals, with P(X_t > 0) = alpha at every t.
Most option-pricing papers instead adopted the driver passed down from Corns and Satchell (2007): X_t = sqrt(1 - delta^2) W_1t + delta |W_2t|, where W_1 and W_2 are independent. Its marginals are skew-normal in Azzalini's sense. The paper distinguishes the processes through their marginal densities, a half-normal mixture for Itô-McKean and a skew-normal for Azzalini. Under the Azzalini process, the excursion probability is 1/2 + arctan(delta / sqrt(1 - delta^2)) / pi. With the paper's illustration delta = -0.6, it equals 0.295.
The confusion arose in the original proof when equality in law for the summands was carried over to equality in law for their sums, despite different joint laws. The Azzalini construction also requires two state variables. It therefore loses the Markov property in its own filtration, a distinction that matters later.
There is no market data. Every numerical illustration uses one parameter set: S_0 = 100, sigma = 0.5, delta = -0.6, r = 0, T = 1.
Arbitrage on the zero set
Rossello (2012) established that the structure condition fails under the Itô-McKean specification. The authors extend his argument to the Azzalini process. After applying Tanaka and then Itô, the finite-variation component of S contains a local-time term, sigma delta times the integral of S against dL^{W_2}. The local martingale bracket is sigma^2 times the integral of S^2 dt.
The measure dL^{W_2} lives on the zero set of W_2, which has Lebesgue measure zero. Consequently, dF assigns mass where d<M> assigns none. The relation dF << d<M> fails, leaving no equivalent martingale measure.
The explicit strategy is the new contribution. Hold sgn(delta)/S_t when W_2t = 0 and hold nothing otherwise. Both the Riemann integral and stochastic integral vanish on that null set, reducing gains to G_t = sigma |delta| L^{W_2}_t. Local time is non-decreasing and half-normally distributed. Hence G is monotone, and P(G_t > 0) > 0 for every t > 0. Fontana (2015) classifies this as an increasing profit, positioned at the weak end of the no-arbitrage hierarchy. Its violation is the severe case.
The strategy cannot be placed as a practical trade. Its support is a Lebesgue-null set. For the Azzalini process, trading also requires observing W_2, information unavailable from the asset price alone. The zero set can be read directly from S in the Itô-McKean case. Without an EMM, a conditional expectation generated by the model cannot serve as a price. The authors conclude that path-dependent and simulation pricing methods based on these processes most likely need to be discarded.
Can the formula survive?
It can. Buckner, Dowd and Hulley (2024) showed that the reflected geometric Brownian pricing formulae are themselves arbitrageable. The skew-normal call formula escapes that result.
The authors start from skew-normal marginals and normalise location until the mean equals the forward. They then verify Roper's conditions A1 to A5. Convexity in strike, the strike limit, the intrinsic-value limit at T = 0 and the bounds follow routinely. Maturity monotonicity requires the main work.
Appendix A supplies a stochastic dominance result: the difference between a normal CDF and an extended skew-normal CDF takes the sign of the skew parameter, with an explicit integral representation. This yields a lower bound for the call's maturity derivative proportional to (1 - delta^2). The bound remains positive whenever delta lies inside (-1, 1). Kellerer then provides a supporting Markov martingale.
A bounded surface with one singular point
The local half-variance stays bounded and uniformly elliptic: sigma^2 (1 - delta^2)/2 <= v_L <= sigma^2/2. For sigma = 0.5 and delta = -0.6, local volatility remains between 0.40 and 0.50. Its spatial derivative is bounded by sigma delta^2 / (2 sqrt(1 - delta^2)) times t^{-1/2}. That rate is slow enough for Le Gall's pathwise uniqueness result. Yamada-Watanabe then gives strong existence, while Figalli's superposition result confirms that the SDE has the marginals used to construct the call formula.
The only discontinuity occurs at (0,0). At that point, v_L switches between its two bounds according to the sign of kappa times delta. Pigato (2019) and Friz, Pigato and Seibel (2020) obtain a power-law skew of order -1/2 from local volatility that is step-discontinuous across the whole time domain.
Here, one point suffices.
The ATM skew behaves as T^{-1/2}, with prefactor sqrt(2 pi) [ -1/2 + Phi(delta sqrt(2/pi); delta / sqrt(1 - delta^2)) ]. It is negative for delta < 0 and flat at delta = 0. Concentrating the discontinuity at one point weakens the assumption behind the established result.
Then the wings flatten
The tails bear the cost. Because the skew-normal moment generating function is finite for every theta, Lee's moment formula has nothing to say. The authors instead apply Benaim-Friz. Minus the log density is regularly varying with index 2 at both ends. For every T, implied volatility approaches sigma sqrt(1 - delta^2) on one side and sigma on the other.
Under the illustration parameters, those limits are 0.40 and 0.50. They match the local-volatility bounds and support the paper's observation that the local and implied surfaces have the same geometry. There is no log-linearisation and no wing growth. The authors state the consequence plainly: smile convexity is very reduced. They suggest that applications should conceivably be confined to equity, where skew matters more than smile.
Two parameters, sigma and delta, determine the whole surface, including the ATM level and both wing asymptotes. We did not find a market calibration or any fit-error comparison with Heston or SVI in the paper, and the paper makes no such claim. Its asymptotics cover kappa = 0 and the |kappa| to infinity limit. Non-zero moneyness remains future work. From Section 5 onward, the analysis sets r = 0; positive-rate formulae are described as similar without being written out.
Zhu and He (2018), corrected
Appendix B deserves attention from anyone citing Zhu and He (2018). Two separate problems appear.
Their martingale argument silently shifts t from the process's initial time to the current valuation time. The authors prove that no process can obey the stated relation for every 0 <= t < s <= T. Such a process would require the correction function l to be additive through time, although l is not additive.
The call formula has a separate sign problem. Delta enters only as sgn(delta) times delta, which equals |delta|. The formula works for delta in [0, 1] and fails for delta < 0. The source is the step delta^2/delta = |delta|. Equity smiles with negative skew are precisely the delta < 0 application, placing the error where the model would most likely be used.
At delta = -0.6, S_0 = 100, sigma = 0.5, T = 1, r = 0, the pricing error is positive and rises monotonically with log-moneyness. It reaches its largest value for deep OTM calls, while the ATM price is incorrectly symmetric in delta. At least 40 published papers cite Zhu and He, and the vast majority treat it as a core reference.
A calibration that beats a two-parameter alternative on a real index surface would change my view of the model. For now, the durable result is that a discontinuity at a single point can generate the T^{-1/2} skew explosion.