A shape parameter that comes out at 0.89945 for eleven different stocks is measuring little about those stocks. The weakness lies inside the mechanism the authors want the reader to accept.
Sulistianingsih, Afrizal, Fikri and Pitriani address a familiar Black-Scholes bias. Equity log-returns have skew and fat tails, while the lognormal price model offers one volatility parameter and cannot accommodate either feature. Their answer is to replace the return density. They use the Skewed Laplace distribution, a two-sided exponential whose asymmetry parameter is bounded in (-1,1), re-parameterise it, then integrate the call payoff under a risk-neutral drift. The result is closed form. It divides into two cases according to the sign of d2 minus a location shift, with the usual Black-Scholes d2 inside the formula.
The derivation follows Theodossiou and Trigeorgis, as the paper says at the start of its results section. The paper's contribution is the estimator, expanded from the third-order approximation in the authors' earlier conference paper to fourth order. It also supplies an option-pricing demonstration and a fuller discussion of how the literature estimates the parameter.
Most of the Skewed Laplace studies cited in the paper use maximum likelihood. These authors instead combine the method of moments with a fourth-order Taylor expansion. The skew parameter is obtained from roots of two forms: plus or minus 0.5 times the square root of (β plus 1), and plus or minus 0.5 times the square root of (1 minus β). In those expressions, β* is the square root of (5 minus 4b). The term b is the fourth central moment scaled by 6n times sigma to the fourth. The selected root is real and falls inside (-1,1). This avoids a likelihood function and numerical maximisation. The stated attraction is practical ease.
The empirical exercise uses 11 US names (EL, K, V, XRX, ABT, CHTR, KO, PG, WMT, AAPL, AMZN), with four strikes per name and 44 call quotes. Volatility is calculated as the standard deviation of daily log-returns from 14 May 2018 to 14 May 2019. The authors assume maturity at ten days, a rate of 2.5% and no dividends. Against market prices, aggregate MSE is 65.6667 for the Skewed Laplace price and 87.1059 for Black-Scholes, roughly a quarter lower. Average pricing error is +0.0425 under the new model and -0.1929 under Black-Scholes. By the paper's sign convention, the new model prices below market while Black-Scholes prices above it.
This setup has limited relevance to a trade. The paper prices European calls, while listed US single-stock options are American-style. Any implementation must either choose contracts for which early exercise is negligible or acknowledge the approximation. We have not reproduced the paper's numbers and we report no backtest of our own here. End-of-day option data reveals nothing about intraday execution or bid-ask spreads. It also cannot establish whether a valuation gap remains after executable transaction costs.
Is the skew parameter doing any work?
Table 3's fourth column nearly repeats itself. PG receives 0.899448249 and AAPL receives 0.899453720, with every other name falling between them. Across eleven underlyings, the entire spread is about 5.5e-6.
Five millionths of a unit, across eleven different return distributions.
The underlying return samples differ sharply. XRX records a daily minimum of -0.1384, compared with PG's worst sample day of -0.0409. CHTR reaches +0.1327 at the top of its range; PG reaches only +0.0084. Mean daily returns extend from -0.0003 for K to +0.0015 for PG. Yet the skew parameter agrees to nine decimal places despite those differences in tails.
The authors explain why in the results section. They write that the fourth-order approximation generates very similar values of β*, which then produce similar estimates of ϖ̂. This is a candid arithmetic explanation for an estimator that has ceased to distinguish among the stocks. The case for the Skewed Laplace approach rests on a shape parameter that "can capture excess skewness and kurtosis frequently found in stock return underlying the option price". Once the fitted value is effectively constant near 0.9, any pricing gain comes from applying the same fixed, fat-tailed non-normal kernel to every name. Such a kernel might still improve prices. It supports a different claim, one that could be tested easily by fixing the parameter and checking whether the results move.
A skew of 0.9 is also large for daily equity log-returns and lies near the upper end of the admissible range. The summary statistics table reports only minimum, median, mean and maximum. Sample skewness and kurtosis do not appear, leaving the motivating stylised fact drawn from the literature rather than measured on this data.
Evidence one contract wide
The table contains a genuine MSE gap, and the authors describe it cautiously. Their conclusion says that "its lower aggregate MSE does not imply better performance for every individual option contract". The abstract makes the same concession, then states that the findings "indicate the potential of the Skewed Laplace distribution as an alternative approach for European call option pricing, although its performance may vary across individual option contracts". Potential is as far as 44 contracts can take the result. One row, AMZN K=1617.5 with squared error 2480.1600, dominates the 65.6667 average. The paper reports no standard errors, so a stronger conclusion is unavailable.
For AMZN at K=1617.5, squared error is 2480.1600 under the Skewed Laplace price and 2354.1130 under Black-Scholes. Across 44 contracts, this row supplies the largest term in both averages. It is also the one large observation won by Black-Scholes. The next largest source of error is AAPL at K=110: 136.2262 against 136.3990. That difference is 0.17 on a squared error of 136. Both models produce 78.67 against a market price of 90.35.
Removing those two makes the remaining pattern easier to see. The deep in-the-money contracts are effectively tied. For ABT at K=50, the prices are 26.0373 and 26.0350 against a market price of 26.22. The model values differ by 0.0023, under a quarter of a cent, while both fall about 18 cents below the quote.
A handful of names produce the separation. The paper emphasizes CHTR at K=372.5. The stock is at 373.15, the strike is 372.5 and the market price is 2.95. Skewed Laplace gives 0.7638, squared error 4.7794 and pricing error 0.7410. Black-Scholes gives 9.1422, squared error 38.3433 and pricing error -2.0990. The Skewed Laplace value therefore marks a near-at-the-money ten-day call 74% below its quote, yet wins because Black-Scholes misses by more. Every CHTR strike follows that pattern.
The new model also produces its own blowup. V at K=160, with the stock at 160.21, is valued at 0.0212 against a market price of 1.30. Black-Scholes gives 3.0477 for the same contract. Pricing a near-at-the-money ten-day call at essentially zero creates a 98% relative error. Even so, the Skewed Laplace squared error of 1.6353 beats Black-Scholes' 3.0544 and wins the row on the scorecard. Such behaviour determines whether the formula can be used.
The paper supplies no standard errors, no t-statistics and no test of the MSE difference. With 44 contracts from what appears to be one cross-section, plus a mean dominated by one row, the evidence leaves the models close and the ranking unestablished.
Historical vol sets the benchmark
Volatility is the standard deviation of one year of daily log-returns, making this a comparison with historical-vol Black-Scholes. Trading desks quote options from market-implied vol rather than one-year realised vol. Black-Scholes' average pricing error of -0.1929 indicates that realised vol from May 2018 to May 2019 exceeded ten-day implied vol on the quote date. The figure belongs to this sample period.
There are two further mechanical problems. The setup discloses one and omits the other. Maturity is assumed at ten days instead of being taken from the contract terms. No quote date is provided, preventing a check of the alignment among spot, strike and market price. The model also assumes away dividends, although KO, PG, WMT, ABT, K and EL all pay them. Over ten days, the effect is small and runs in the same direction as Black-Scholes' overpricing (APE -0.1929).
From price comparison to a trade
The paper neither attempts nor claims the test a trader would need: whether the gap between model value and market price predicts a return. Contracts could be sorted by that gap, held to expiry or delta-hedged, and charged the spread to determine whether the difference survives.
The models separate most clearly at the high CHTR and AMZN strikes. For AMZN K=1792.5, the values are 48.0170 and 77.9100. At AMZN K=1797.5, they are 43.0170 and 74.7770, about 30 price points apart. Among the low-priced contracts in Table 3, CHTR K=372.5 offers the clearest split: 0.7638 against 9.1422 on a $2.95 option, enough to clear any plausible spread.
Across the 44 contracts shown, average pricing error of +0.0425 leaves the model below market overall, which reads as a signal to sell. The average conceals changes in sign. Fourteen of the 44 rows in Table 4 have a negative Skewed Laplace pricing error, meaning that the model prints above market. All four EL strikes belong to that group.
We have raised a version of this complaint before (our note on Sepp and Lucic). The evidence here remains 44 contracts observed on a single date.
The evidence needed to change my mind is narrow and inexpensive to produce. Estimate the skew parameter for the same eleven stocks using maximum likelihood, or use a longer sample, then show dispersion greater than 5.5e-6. Separation would support the mechanism described by the paper. Without it, the closed form still has value, though its accurate description would be a fixed non-Gaussian kernel rather than a distribution adapting to each name.