The 15.8% call discount reflects a choice of pricing measure; the market has yet to endorse it. The paper leads its capacity discussion with this result. With a ceiling 25% above spot, its logistic stock model prices a one-year at-the-money call at 7.7689. Black-Scholes-Merton prices the same call at 9.2270.
How the brake enters the price
A stochastic Verhulst equation governs the stock, with drift μS(1−S/N) and volatility σS. The carrying capacity N is the price where expected growth stops. Well below N, the stock behaves like geometric Brownian motion. Growth stalls near N. Above that level, negative drift draws the price back.
Mulambula begins with useful housekeeping. Delta hedging cancels the stock's drift regardless of its form. Earlier logistic papers that retained the logistic term in their PDE after hedging are therefore, as the paper puts it, "internally inconsistent." The repair assumes an incomplete market, then chooses a state-dependent market price of risk. This replaces μ with the cost of carry a = r−q and preserves the (1−S/N) brake. The paper explicitly calls this choice "is a modelling choice, not a consequence of no-arbitrage."
The Feynman-Kac PDE supplies the prices. Setting X = 1/S makes the SDE linear. The resulting price is a geometric Brownian motion divided by 1 plus (as/N) times an exponential functional of Brownian motion.
There is no market data.
The base case uses s = K = 100, r = 5%, q = 2%, σ = 20%, T = 1. As N increases, calls rise and puts fall toward Black-Scholes prices. The call costs 7.5844 at N = 110, then 8.2938 at N = 200 and 9.0343 at N = 1000. With N = 125, the put costs 7.2489 versus 6.3301, a premium of 14.5%.
What explains the discount at N = 125?
Put-call parity exposes the forward shortfall in the N = 125 row. Call minus put under Black-Scholes is 9.2270 − 6.3301 = 2.8969. The logistic prices give 7.7689 − 7.2489 = 0.5200. Add the call discount (1.4581) to the put premium (0.9188) and the total, 2.3769, is the discounted shortfall of the model forward. That forward is 100.547, against the traded 103.045. Since the paper extracts its implied forward from those same call and put prices, the equality follows by construction. The model forward is 2.425% below the traded forward.
The paper reports and calculates this well. It measures forward bias of −2.425% at N = 125, −1.530% at N = 200 and −0.310% at N = 1000. The first-order 1/N approximation produces −2.486%, −1.554% and −0.311%. For N = 125, the tightest capacity in Table 6, the approximation misses by about 0.06 points (−2.486% against −2.425%). The author states the consequence directly: "If the underlying is genuinely traded and liquid, a violation of that size is an arbitrage and the model must not be used."
For untraded or constrained underlyings, the author argues, this gap measures capacity's effect on the effective forward. That defence needs A4, which says the underlying cannot be replicated using the bond and a liquid traded proxy. But A2 assumes frictionless trading without short-sale restrictions, and A3 gives the stock a continuous yield q. If that stock trades frictionlessly and pays q, the stock and bond replicate a forward at 103.045.
Incompleteness allows a choice of option-pricing measure. It still leaves the risk-neutral drift of a frictionlessly traded stock at r − q. In Section 3 the paper calls Q an equivalent martingale measure; in Section 8 it acknowledges that the discounted price is not a Q-martingale. Those claims conflict.
Order 2 convergence, directional Monte Carlo misses
At N → ∞, Crank-Nicolson, the paper's numerical PDE scheme, converges at order 2.00 with four implicit Rannacher half-steps at the start. Moving from a 100×100 grid to 1600×1600 reduces the call error from 3.88e-2 to 1.50e-4. The convergence table is clean.
The Monte Carlo comparison deserves more caution than its caption gives it. All three call discrepancies point upward; all three put discrepancies point downward. At N = 150, 200 and 400, the PDE call exceeds the simulation by 0.0332, 0.0339 and 0.0350, respectively, about 1.65 standard errors in each case. For puts, the PDE is lower by 0.0128, 0.0125 and 0.0121. The paper says 16,000 time steps removes the bias. We did not find that run reported.
Agreement between a PDE solver and a simulation of the same model would establish that the code solves equation (6). It cannot establish that equation (6) describes an options market.
A smaller issue concerns rigour in the Feynman-Kac step. The justification invokes coefficients "of linear growth," whereas aS(1−S/N) is quadratic in S. That justification needs replacing.
The dividend prediction meets US listed options
Mulambula identifies the dividend result as the clearest empirically testable prediction. Increasing q from 0 to 0.08 at N = 200 reduces the logistic call by 21.5%, compared with 41.2% under Black-Scholes. The paper attributes this to the presence of a = r−q inside the brake. The prediction therefore follows from the measure selected in equation (3).
Two rows have q > r. At q = 0.06 and 0.08, a turns negative and N becomes repelling rather than attracting. With N = 200 and q = 0.06, the logistic call costs 7.3468 against 7.0755, reversing the capacity discount. The paper limits its capacity-suppresses-growth interpretation to a > 0 and acknowledges the crossing curves in Figure 2. Its dividend discussion leaves these observations unconnected. At q = 0.06 and 0.08, a is below zero and N ceases to function as a ceiling.
We are building an end-of-day test using US listed options. We have no results to report. The test uses s/N as a relative-value feature for names with an identifiable ceiling, such as a licensed subscriber base or free float. The paper recommends estimating N from market-size data of this kind: prices far below capacity leave N on a flat estimation ridge. By design, our test does not treat the synthetic forward as an arbitrage signal.
The distance between the paper and a market test is specific. All tables use a single strike (K = 100) and maturity (T = 1). US single-name options allow American exercise; the paper assumes European payoffs in A5 and leaves American exercise for further work. It tabulates no Greek values, despite citing Greek oscillation as the reason for the Rannacher start. Its sole statement about the Greeks is that vega remains positive.
I would change my view if listed synthetic forwards for capacity-constrained names traded below the traded forward in the 1/N pattern of Table 6, around 1.5% with a ceiling twice spot. Until then, the 15.8% belongs to equation (3).