A variance swap offers no timing advantage under the pricing measure. Maeda proves that result in half a page, and the rest of the paper follows from it.
Accrued variance plus the expected remainder gives the undiscounted forward value of a variance swap. In any arbitrage-free model, this value is a Q-martingale. Discounting the unwind payoff cancels exactly with the discounting already embedded in the mark-to-market. Corollary 2.3 therefore puts the supremum over stopping times at every stopping time.
The timing problem must live under the physical measure, where the premium drives the result. Maeda writes it as lambda = kappa_Q - kappa_P, with kappa_P theta_P = kappa_Q theta_Q fixing the long-run product. The usual configuration has lambda < 0, so strikes exceed forecast realised variance.
Three reductions follow. Accrued variance separates exactly as V = eps A + w(t,v). The absence of a discount factor makes the separation possible; Remark 3.2 shows that adding one immediately breaks the additive ansatz. Maturity, strike and costs then enter through one forcing term, A_Phi(t,v) = eps lambda g_Q(t) v + (s - c_m). The third reduction replaces the dated swap with a perpetual contract that terminates at an independent exponential time with rate delta. The running reward disappears, while the entry strike cancels. The remaining reward is affine in variance, with slope beta = -eps lambda/((delta+kappa_Q)(delta+kappa_P)).
A CIR process with an affine reward, entered and exited once subject to costs, is already solved. Maeda credits Dayanik and Karatzas for the general one-dimensional theory, and Leung, Li and Wang for this exact pairing of starting and stopping problems. His contribution is the reduction. The two thresholds are unique roots of smooth-pasting equations involving the confluent hypergeometric functions M and U. At c_m = 200bp, the analytic entry threshold is 0.12161 against 0.12172 from the finite-difference solve. At 300bp, the figures are 0.16745 and 0.16769. Both matches are within grid resolution.
The abstract advertises the entry/exit asymmetry as the paper's main conclusion. An entry rule exists only across an interval of carrying charges, "and even there triggers only deep in the upper tail of the physical law: a trader who is out of the market pays nothing to stay out." This review adds the sensitivity accounting around that conclusion. The paper reports those numbers without foregrounding them.
There is no data in the paper. Every figure comes from a Heston calibration chosen by the author, together with Monte Carlo under P. Lambda is never estimated.
One solve covers every strike
Sigma appears in the payoff solely through the additive constant -eps T Sigma^2. Constants vanish under both the time derivative and the generator, leaving A_Phi independent of the strike. A single numerical solution of the variational inequality therefore covers every strike and every accrued-variance level.
The free boundary changes sharply through time. During three quarters of a one-year swap's life, the dated-contract boundary travels about 1200 basis points of volatility for a long position and 650 for a short. Maeda interprets the rule by fixing accrued variance and residual maturity, then applying one threshold. The resulting error dwarfs the transaction costs the rule is meant to save.
Why the short waits for the 95th percentile
With the premium carrying its empirical sign, only the short is worth opening. For a long, sup_v eps D = +infinity. Its critical carrying charge is c*_m = -infinity, so no holding cost makes entry worthwhile. The position pays the Q-rate, accrues P-variance and steadily loses the premium.
One calibration within the short-entry window deserves attention. With kappa_Q = 1.0, a premium of 3.07 volatility points and c_m = 120bp, the perpetual position is unwound when the rate falls to 18.20%. That level is the 12.4th percentile of the stationary law. Entry occurs at 22.62%, or the 94.8th percentile. Table 2 uses a separate calibration, equation (41), and recalibrates each row so kappa_Q produces a three-point premium at the relevant delta. Expected contract life runs from one year to thirty-two. Entry never falls below the 94.7th percentile and rises monotonically to 99.4.
Maeda supplies the paper's best explanation: a trader who is out of the market pays nothing to stay out. Delay is penalised only by the survival factor e^(-delta zeta). At delta = 0.5, that penalty cannot overcome the option to wait for a richer level.
An idle-capital charge, c_0, turns the setup into a usable rule. Table 3 applies the accounting to the Section 8.2 calibration with c_m = 120bp. At c_0 = 0, entry occurs at the 94.8th percentile. A charge of 25bp lowers it to the 83.1st, while 50bp takes it to the 58.0th. At 100bp or above, entry is immediate at every level. Fifty basis points a year is small beside any hurdle rate a desk would actually be set, in Maeda's own phrase, yet it shifts the threshold by 37 percentile points. I read this as a structural result about the problem, rather than a calibration anyone should trade. The paper reaches the same conclusion. Premium and spread determine when to close. Opening requires a number describing the trader rather than the market.
Exit uses observables. Entry is a preference parameter wearing a threshold.
A window narrower than the defaults
The paper does its best work when it maps where an entry rule can exist. The figures are unflattering. Fix the stationary median of v at 19% in volatility, with s-bar = s_e-bar = 20bp and delta = 0.5. Below a premium of 1.09 volatility points, no holding cost permits a non-degenerate entry-exit pair. The base calibration uses a premium of 0.60 points and therefore lies outside the window. An unwind rule remains; an entry rule does not.
That window depends on the author's chosen tolerance. Reachability is screened at eta = 0.5%, measuring how often the entry level is touched. Appendix B raises eta = 5%, which still means a rule firing on one observation in twenty. The window then disappears entirely below about three and a half volatility points of premium. With a three-point premium, the admissible 93 basis points of carrying charge fall to none. At four points, the interval contracts from 159bp to 14bp. Maeda acknowledges that the screen is carrying more weight than a tolerance should.
Changing the spread convention causes similar damage. Treat the perpetual tear-up concession as a rate over expected remaining life, s-bar = s/delta, instead of a flat charge. The opening of the window moves from 1.09 to 2.15 volatility points. Waiting stretches to 7.9 years and holding to 3.5 years, roughly doubling the round trip. Remark 6.2 says the qualitative conclusions survive. The opportunity's tradable size does not.
Even under the favourable calibration, stationarity to entry takes 4.0 years and the holding period lasts 2.3 years. A round trip consumes about six years, leaving few repetitions in any history.
No jumps, no stop-loss
For the short, the continuation region has no upper bound. The strategy never exits a losing short-variance position. D increases with variance, and the objective rewards continued holding through an arbitrarily large spike at c_m per unit time, regardless of the mark. Maeda identifies this directly as the familiar short-volatility payoff without a stop on the losing side.
His next observation is less obvious. A stop-loss abandons paths on which the premium accumulates fastest. Continuation consequently loses value everywhere below the barrier, pushing the entry threshold b* higher. The stop-loss alters the lower edge of the rule along with the upper one. Maeda does not solve that version.
Heston variance is continuous and affine. Those properties produce the affine reward and hypergeometric solution, while realised index variance jumps. The paper gives one sentence to a power-tail alternative that widens the window without making the entry level more likely to be reached. No solve accompanies it.
The perpetual assumes a distant-horizon premium
The conclusion names this limitation directly. Dew-Becker, Giglio, Le and Rodriguez find the price of variance risk at one and two months, yet cannot distinguish it from zero beyond a quarter. A perpetual contract places weight on precisely those horizons. Equation (59) reveals a deeper issue than calibration. Under any two-measure affine specification, the average per-horizon premium equals (theta_Q - theta_P)(1 - e^(-kappa_Q u)). The expression starts at zero when u = 0 and increases monotonically. Parameter choice cannot make it decay, even after freeing the affine link.
Appendix B offers a stronger answer than I expected. Set lambda = 0, making D identically zero and removing the premium at every horizon. Then allow the carrying charge to vary with the state: c(v) = c_m + c_1 v. The derivation still works. Its slope becomes beta = c_1/(delta+kappa_P), and c_1 alone preserves the two-threshold structure. Without the premium, the two sides become symmetric and position direction ceases to matter.
At c_1 = +0.30 and c_m = -120bp, entry occurs below 18.82% and unwind above 22.31%. Mean time to entry is 1.5 years, followed by 4.2 years of holding. Across the six rows in Table 4, the absolute charge at the median state never exceeds 60 basis points a year. In five rows it stays within 12. Equation (63) explains the pattern: beta depends only on c_1, while c_m affects only the intercept.
The escape has two problems. Maeda states that c_1 belongs to the same preference-parameter class as c_0, describing a book rather than a surface. Restoring the premium at three volatility points also makes entry worse when the financing charge rises with variance. Entry is at the 93rd percentile for c_1 = 0, the 97th at c_1 = 0.15 and the 99th at 0.20. From c_1 = 0.25 through 0.40, no holding cost allows a non-degenerate pair. Reaching the entry region from the other side requires c_1 around 0.50, together with a negative holding cost of 84 to 190 basis points a year.
We cannot hold the instrument
The closed form describes a perpetual, continuously settled swap that closes through tear-up. Remark 6.3 concedes that no exchange lists such an instrument. Maeda's answer is that the contract remains the right idealisation: a randomised-maturity dated contract carrying the same exposure. A power perpetual on ETH squared already trades in decentralised markets.
The economics survive that answer; tradability does not. Cboe's S&P 500 variance futures, listed 23 September 2024, contain neither the perpetual termination clock nor the tear-up concession, and we cannot hold it. Replacing the contract with VIX futures or an option strip would test another payoff. The reduction depends on accrued realised variance. A claim on a constant-maturity swap rate has no accrual.
The degeneracy result should last. Any variance-timing rule must arise through one of the four channels listed in the paper: a pricing-to-physical drift wedge, transaction costs, a non-linear objective, or model uncertainty. Discounting cannot supply it. Maeda's exit threshold is determined by premium and spread. His entry threshold instead reflects the trader's hurdle rate, as he says.
An estimate of lambda, or of c_1, drawn from something beyond a chosen calibration would change my view of the entry result. Lambda is assigned here and never estimated. The calibration that reaches the window slows kappa_Q to 1.0 and sets physical long-run volatility at 16.8%. Maeda deserves credit for reporting the sweep himself. Hold kappa_Q = 2 and vary lambda and gamma over their admissible ranges: entry never drops below the 97th percentile. He treats that as close to the model's best result, rather than a fortunate corner.