Martin's equity-premium floor is too low whenever a Q-bubble exists, though Jarrow and Kwok leave the size of the error unmeasured. Their upward correction follows cleanly from the paper's definitions. The quantity responsible remains latent.
The authors acknowledge the problem immediately. Their opening paragraph refers to "the latent nature" of stock price bubbles, then cites two option-based procedures for inferring Q-bubbles, Jarrow and Kwok (2021) and Fusari, Jarrow and Lamichhaine (2024). Section 4 says the risk premium characterisation "forms the basis for our subsequent empirical estimation". Those citations and that promise leave the correction unquantified. This version contains no estimation.
Pure theory.
There are Nine propositions, six lemmas, six corollaries, with every proof housed in a twenty-one-section appendix. The version first circulated February 2025 and revised April 2026 has no data, universe, sample period or test statistic. Since the paper reports no estimates, our figures stand alone and cannot be read as a comparison with theirs.
One price, three pieces
Jarrow and Kwok combine two bubble literatures that have largely proceeded in parallel. One begins with Blanchard (1979). The other comes from Loewenstein and Willard (2000).
The first concerns the rational bubble. In an infinite-horizon model with expected return mu, the price equation (1+mu)S_t = E^P_t(S_{t+1} + x_{t+1}) admits a discounted-dividend solution F^P and a residual B^P satisfying E^P_t(B^P_{t+1}) = (1+mu)B^P_t. This residual is the P-bubble. It is consistent with no-arbitrage, is in principle detectable from historical prices under the statistical measure, and underlies the Phillips-Wu-Yu family of explosive-root tests cited by the paper.
The second literature studies local-martingale bubbles. No-free-lunch-with-vanishing-risk delivers an equivalent local martingale measure Q. Under Q, discounted price plus cumulative dividends is a local martingale, while true-martingale status can fail. Section 3.1 defines the Q-bubble as observed price plus cumulative dividends, less the discounted Q-expectation of terminal price plus cumulative dividends: B^Q_t(T):= S_t + D_{0:t} - F^Q_t(T). A strict local martingale makes this quantity positive. Construction sets it to zero at the horizon.
Proposition 8 connects the two strands: S_t = F^P_t(T) + B^P_t(T) + B^Q_t(T). One observed price now has three components. Taking T to infinity recovers the familiar rational decomposition once lim B^Q_t(T) = 0 is assumed. The reduction therefore rests on that assumption.
Two diagnostic quantities surround the decomposition. The P-martingale deviation Pi^P_t(tau) measures how far expected price plus reinvested dividends exceeds today's price. The Q-martingale deviation Pi^Q_t(tau) measures how far today's price exceeds the discounted Q-expectation of price plus dividends. Supermartingality makes Pi^Q nonnegative.
Martin's missing term
Proposition 9 writes the equity risk premium, E^P_t(r) minus E^Q_t(r), as Pi^P_t(tau)/S_t plus R^f Pi^Q_t(tau)/S_t minus r^f. Under the standard equivalent martingale measure, Pi^Q is identically zero and the second term collapses. E^Q_t(r) then equals the risk-free rate.
A Q-bubble pushes the risk-neutral expected return below r^f. Investors require compensation for holding an asset expected to fall under Q, and Corollary 6 sets that compensation exactly at R^f Pi^Q_t(tau)/S_t.
This changes Martin's bound. In the paper's statement, ERP^0 >= R^f times tau times SVIX-squared applies to ERP^0, the standard no-arbitrage premium. It requires the standard no-arbitrage condition and a negative correlation condition. The bound is a second-moment result assembled from the option surface, spot and futures.
Jarrow and Kwok append a first-moment term. The terms add together and arise under separate conditions, which gives the NFLVR correction its own economic role. Whenever Q-bubbles exist, an SVIX-based estimate of the expected market return is biased low by the Q-component representing the expected decline in the discounted Q-bubble over the horizon. The expected decline can peak away from the bubble's maximum size.
Given the premises, the argument is watertight.
Can Pi^Q be observed?
Pi^Q_t(tau) is a risk-neutral conditional expectation for a selected horizon. Estimating it requires the Q-distribution of price plus dividends at t+tau, inferred from the option surface, futures curve and rate curve, and then differenced against spot. The paper identifies two possible routes: its authors' own 2021 option-based work and the 2024 options-based bubble test it cites. Model and interpolation error from those procedures remain unquantified.
An estimated addition to a variance-based lower bound brings its estimation uncertainty with it. Before the correction can affect an allocation, someone needs to publish R^f Pi^Q/S with a standard error and demonstrate that the estimate is large beside that error.
One more assumption matters. Dividends are reinvested at the risk-free rate, a convention the authors say is required for no-arbitrage consistency between the P and Q expectations. They also observe that reinvestment in the stock would complicate the expressions. The three-way decomposition therefore depends on how dividends are reinvested.
Explosive prices identify the wrong bubble
Footnote 11 contains the paper's most useful practical statement. Every local martingale in discrete time is a martingale, so Q-bubbles cannot exist there. They belong to continuous time.
The entire discrete-time explosive-root testing apparatus can therefore address only P-bubbles, however carefully specified the tests may be. A positive explosiveness statistic bears on Pi^P. An option-implied martingale defect bears on B^Q. The paper establishes nesting, with the local martingale model containing the classical rational bubble as a special case. My inference from Footnote 11 is that the two test families remain non-interchangeable.
Even the P-side is weaker than its textbook form, and the paper supplies the weakening. In discrete time with constant mu > 0, Lemma 2 makes B^P a P-submartingale, becoming strict after a P-bubble occurs at some date. The bubble consequently grows in expectation. Remark 1 shows that, within the same discrete-time constant-mu model, dividend timing alone can make F^P either a submartingale or a supermartingale.
The authors describe constant mu as a strong restriction. Section 2.3 replaces it with a stochastic discount factor. Under this repaired specification, Lemma 5 expresses E^P_t(B^P_{t+tau}) minus B^P_t as risk-free growth on the current bubble, less R^f Cov^P_t(m_{t+tau}/m_t, B^P_{t+tau}). Remark 4 leaves the sign of that risk adjustment indeterminate. The submartingale property returns under the sufficient condition that the covariance is at most zero. The discussion below uses this repaired version.
Reading a positive Pi^P as a rational bubble requires several conditions together: F^P must be a P-martingale, dividends must be absent over the test window, and Cov^P_t(m, B^P) must equal zero for the iff in Corollary 4(ii). Remark 2 gives the continuous-time counterpart and leaves the sign of the expected change in F^P unrestricted. Price explosiveness after removing an I(1) dividend series is therefore an ambiguous signal. Example 1 explains why these tests still fire. For an explosive AR(1) with phi > 0, Pi^P_t(tau) = [(1+phi)^tau - 1] p_t, which is mechanically positive.
Proposition 4(iii) deserves more attention than it will probably receive. B^P_t = 0 if and only if B^P_u = 0 for all u > t. Once a P-bubble exists, it exists at every date. Yet the date-stamping procedures cited by the paper, the Phillips, Wu and Yu (2011) and Phillips, Shi and Yu (2015a,b) family, produce bubble episodes with starting and ending dates. Those episodes cannot represent this model's P-bubble.
Our proxy run
Our nine-year run produced a total return of 220.64%, a Sharpe of 0.47 and a maximum drawdown of -131.64%. Sortino was 0.51, Calmar 0.11 and volatility 52.64%, over 2016-01-01 to 2025-01-01. The combination of a -131.64% drawdown and 52.64% volatility looks unnatural for the long-only equal-weight book described below. We regard that figure as ours to explain; it says nothing about the paper.
The paper is theoretical and provides no estimates. These results belong solely to our proxy construction and do not measure the paper's decomposition. We neither tested nor replicated the paper.
Its central objects were beyond what we could estimate. F^P, B^P, B^Q and the stochastic discount factor are latent. End-of-day options also provide no continuous-time Q process for differencing, so Pi^Q was never estimated at all. The paper addresses a generic dividend-paying equity. Our run used US large-cap equities, daily prices, dividends and rates, along with proxies for the quantities left latent. It should be read as a construction inspired by the paper's idea.
We formed a long-only US equity portfolio and ranked names monthly at the close. The universe contained the top 500 non-ADR names by trailing dollar volume, subject to a $10mm median daily dollar volume screen, over 2016-01-01 to 2025-01-01 using daily bars. For each stock, we estimated E^P[S_{t+63} + D_{t:t+63}] from a 252-day trailing ex-dividend log drift. Its difference from spot became a Pi^P proxy. We combined that proxy with a price-to-dividend-fundamental gap and explosiveness terms in a single score.
The portfolio held the top decile, capped at 50 names. Positions were equal weighted with a 10% single-name cap. A short-term momentum gate controlled entry, while a daily trend-break rule triggered exits.
We charged $0.004 a share with a $1 minimum, and modelled zero slippage.
Those choices frame the reported figures. Zero slippage favors the daily exit rule. The portfolio has no short leg or hedge, leaving its volatility and drawdown exposed to the long equity book and any beta added by the top-decile explosive-trend tilt. Its dividend-only fundamental proxy systematically removes non-dividend payers. Many of the names most often described as bubbles during 2016 to 2025 were consequently ineligible.
Two point-in-time details still require confirmation: the release timing of fundamentals, and whether the annual universe used trailing or realised dollar volume. This was one automated pass. A weak result describes that pass alone and carries no information about Proposition 9 because the signal never contained the Q term. We have previously examined ranker-plus-gate designs in which the gate contributes less than advertised, including the 0DTE abstention layer that never bound. Our trend-break exit raises the same question.
A published estimate of R^f Pi^Q_t(tau)/S_t for a major index would change my view if it included a standard error and showed a correction large relative to its own uncertainty. Until such an estimate exists, the paper contributes a sign and a warning about which test measures which bubble. Both deserve attention.