Portfolio Optimization Managing Value at Risk under Heavy Tail Return, using Stochastic Maximum Principle

Diganta Mukherjee; Mrinal K.Ghosh; Subhojit Biswas

REVIEW 5 major objections 4 minor 23 references

Portfolio Optimization Managing Value at Risk under Heavy Tail Return, using Stochastic Maximum Principle

T0 review · 5 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims to provide a closed-form approximate optimal portfolio strategy for heavy-tailed stock returns by maximizing median wealth subject to a Value-at-Risk constraint, and shows empirically that higher risk-free rates reduce…

desk verdict Good idea for continuous-time quantile-based portfolio choice, but the central SDE is misspecified and the median/VaR labels are swapped, so the derived optimal strategy solves a different problem. read the letter →

arxiv 1908.03905 v2 pith:DRYYFVO2 submitted 2019-08-11 q-fin.PM q-fin.MF

Subhojit Biswas , Mrinal K.Ghosh , Diganta Mukherjee This is my paper · ORCID

classification q-fin.PMq-fin.MF MSC 90C3991G1091G80

keywords continuous-timeportfoliooptimizationValueatRiskheavy-tailedreturnsquantileasymptoticsstochasticmaximumprincipleHamiltoniansystemnonparametricdensityestimationmedianreturn

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that a continuous-time portfolio problem can be solved even when the risky asset's return distribution is so heavy-tailed that variance does not exist. The proposed solution replaces the return process by the pair of its 5% and 50% sample quantiles, which are asymptotically jointly normal, and maximizes the discounted expected power utility of median wealth subject to the lower quantile wealth staying above a floor with at least 95% probability. Using the stochastic maximum principle with a smooth sigmoid approximation of the Value-at-Risk constraint, the paper derives a closed-form approximate optimal strategy, formula (7), and calibrates the needed densities nonparametrically from real daily prices. This matters because, if the derivation holds, it supplies an implementable allocation rule for heavy-tailed assets without knowing the distribution's functional form.

What carries the argument

The load-bearing object is the asymptotic covariance matrix $\Lambda$ of two sample quantiles, with entries $\Lambda_{ij}=p_i(1-p_j)/(f_if_j)$ for $i,j\in\{1,2\}$, where $p_1=0.05$, $p_2=0.5$, and $f_1,f_2$ are the densities at those quantiles. The paper uses $\Lambda$ as the diffusion coefficient in the SDE governing the quantile processes, turning a statistical fact about sample estimates into a stochastic control model. The second piece of machinery is the sigmoid approximation of the indicator function in the VaR constraint, which makes the objective smooth; the stochastic maximum principle then supplies adjoint equations whose reduced solution leads to the first-order condition and, after a further approximation, to formula (7).

What would settle it

Simulate a heavy-tailed return process with known parameters, form rolling 5% and 50% sample quantiles over time, and compare the empirical quadratic covariation of those quantile paths with the value implied by $\Lambda$; a mismatch means the quantile-SDE assumption is false, so formula (7) would not be the optimum for the simulated process.

Watch

Extended reading notes

Core claim

On its own terms, the paper's result is a closed-form approximate solution to a constrained stochastic control problem, whose state is the pair of wealth processes associated with the two quantiles: the median process $\bar L_1$ carries the utility, and the 5% quantile process $\bar L_2$ carries the Value-at-Risk constraint. After forming the Hamiltonian, solving the two adjoint equations with zero matrix-valued terms, and setting the derivative with respect to the portfolio weight to zero, the paper obtains the optimal portfolio weight $$\bar\pi_t=\frac{$e^{{-\beta t}}$\psi_0 \bar L_1(t)^\gamma $r^{2}$-\psi k_1\bar L_2(t)(b_2-r)(r-\$\beta$)}{\left($e^{{-\beta t}}$\psi_0 \bar L_1(t)^\gamma-\psi k_1\bar L_2(t)\right)(b_2-r)r}.$$ The paper presents this as the optimal strategy for the approximated problem and, via Ekeland's theorem, as an approximate optimum of the original non-smooth problem; it then supports the formula with simulations showing that higher risk-free rates lower the optimal stock fraction and that changes in $\gamma$ and $\beta$ leave the strategy largely unchanged.

Load-bearing premise

The load-bearing assumption is that the asymptotic covariance matrix $\Lambda$ of quantile estimates, a statement about how estimates vary across samples, can be used directly as the instantaneous random-shock matrix of the quantile processes' own time-series SDE; if the quantile paths do not follow that SDE, formula (7) has not been derived for the actual heavy-tailed return process.

Editorial extensions

If this is right

If formula (7) is the optimal strategy, a higher risk-free rate $r$ reduces the fraction held in the risky asset, matching the paper's empirical finding.
The strategy is implementable without specifying the return distribution, because calibration needs only sample quantiles and kernel-density estimates of the quantile densities.
Putting more weight on the utility objective and less on the VaR constraint increases both the stock allocation and the final wealth, consistent with accepting more tail risk for higher median gains.
The framework extends to logarithmic and exponential utility, and other constant-risk-aversion forms, at the cost of algebraic complexity, so the median/VaR formulation is not tied to power utility.
Because the constraint is smoothed, the resulting strategy is an approximate optimum rather than an exact one; Ekeland's theorem is invoked to justify the approximation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension is to replace the 5% quantile process by an expected-shortfall or conditional-VaR functional in the same Hamiltonian machinery, since the calibration only requires quantile asymptotics and a smooth approximation of the constraint.
A cautious reading treats formula (7) as a calibrated heuristic rather than an exact optimum for the original heavy-tailed process, because the derivation replaces the return process with a quantile diffusion and then smooths the constraint; simulated fat-tailed data would show how large the gap is.
The empirical insensitivity to $\gamma$ and $\beta$ suggests the allocation is governed mainly by the ratio of median to tail quantile levels and by the risk-free rate; if confirmed, the rule could be condensed into a simple tail-thickness-dependent trading formula.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The paper attacks a real problem: continuous-time portfolio choice when returns are heavy-tailed and moments don't exist, using quantiles as state variables and a VaR constraint. That is a sensible idea, and the paper correctly notes that sample quantiles are asymptotically joint normal with a known covariance. Formulating the optimization via the stochastic maximum principle with a smoothed indicator constraint is also a reasonable route. So the setup is not without merit.

Unfortunately, the central derivation is not sound. The authors take the asymptotic covariance matrix Lambda of sample quantiles and write dX = mu dt + Lambda dW for independent Brownians. That gives increments with covariance Lambda Lambda^T, not Lambda. To match the stated asymptotic distribution you need a matrix square root A with A A^T = Lambda. This is not a minor typo; every later equation, including the Hamiltonian, the adjoint BSDEs, and the optimal strategy (7), is derived from the wrong dynamics. The stress-test note is correct.

There is also a label swap that compounds the problem. X1 is the 5% quantile and X2 is the median by the definitions, but L(1), called the median wealth, is driven by dX1, and L(2), called the VaR process, is driven by dX2. So the problem actually maximizes the 5% quantile subject to the median, not the intended median maximization with a VaR constraint. That is a load-bearing inconsistency, not a notation choice.

The adjoint BSDEs are "solved" by simply setting q=0 and Q=0. That is not a derivation; it is an assumption that the martingale integrands vanish, which has to be justified or at least shown to be consistent. As it stands, the proof does not go through.

The empirical section is in-sample only: parameters are calibrated on the same Entergy series used to illustrate the strategy, with no out-of-sample check, no baselines, no code, and the figures are placeholders. The qualitative claim that higher risk-free rates reduce stock investment is plausible but not validated by the analysis.

What the paper does well: the idea of using quantile processes for heavy-tailed returns is worth pursuing, and the paper cites relevant literature, including the authors' own discrete-time work. The nonparametric calibration approach is sensible.

Bottom line: as it stands this is not a reliable result. The topic deserves serious work, but this version needs a major rethinking of the state dynamics. If the authors fix the diffusion matrix and the label swap, and then actually solve or convincingly approximate the adjoint equations, the paper could have value. I would not cite it in its current form, but I think a serious referee should see it—not for acceptance, but to give detailed feedback. I would send it to review rather than desk-reject, mainly because the errors are instructive and the topic is significant.

Referee Report

5 major / 4 minor

Summary. The paper considers a continuous-time portfolio problem with one risky asset and one risk-free asset when the risky return is heavy-tailed. Since moments may not exist, the authors propose to work with two return quantiles (the 5% and the 50% quantiles), use a multivariate normal approximation for their joint distribution, and model the quantile dynamics by an Itô SDE. They then formulate a constrained stochastic control problem that maximizes discounted power utility of a median-wealth process subject to a Value-at-Risk constraint, solve it by the stochastic maximum principle, and derive a closed-form approximate optimal strategy in equation (7). The model is calibrated nonparametrically with a kernel density estimator on daily Entergy Corporation returns, and the resulting strategy and wealth paths are simulated for various parameter choices. The paper claims that the results accord with financial intuition.

Significance. If the derivation were correct, the paper would offer a genuinely useful construction: a quantile-based, nonparametric dynamic allocation rule that bypasses nonexistent moments under heavy tails. The choice of quantiles and the use of asymptotic normality of sample quantiles are appropriate building blocks, and the empirical setting is relevant. However, the central analytical chain contains load-bearing errors: the quantile SDE is misspecified, the objective and constraint labels are reversed, the adjoint BSDEs are solved by setting the martingale integrands to zero without justification, and the closed-form formula (7) rests on an unvalidated exponential approximation. These are not local presentation issues; they affect the claimed optimal strategy and therefore the entire contribution. The paper provides no machine-checked proofs, no code, and no out-of-sample validation, so the empirical conclusions are also not independently verifiable.

major comments (5)

[Section 2, quantile SDE after the Proposition] The diffusion matrix in the SDE for (dX1, dX2) is set equal to the asymptotic covariance matrix Lambda, but with independent Brownian motions the instantaneous covariance of the increments is Lambda Lambda^T, not Lambda. For example, Var(dX1) would be (p1(1-p1)/f1^2)^2 + (p1(1-p2)/(f1 f2))^2, not p1(1-p1)/f1^2. A correct diffusion model requires a matrix square root A with A A^T = Lambda. Moreover, the displayed Lambda also omits the 1/N factor from the standard asymptotic covariance of sample quantiles. Since equations (1), (2), the Hamiltonian, the adjoint equations, and formula (7) are all derived from this misspecified SDE, the closed-form strategy is not the optimal strategy for the process the authors intend to model.
[Section 2, wealth processes L(1) and L(2)] The paper says L(1) is the median wealth, but its dynamics are driven by dX1 = dX(p1) with p1 = 0.05, the lower 5% quantile; it then says L(2) is the lower-quantile (VaR) process, but its dynamics are driven by dX2 = dX(p2) with p2 = 0.5, the median. Consequently the objective in equation (2) maximizes a process driven by the 5% return quantile, while the constraint is applied to a process driven by the median. This reverses the roles of the objective and the risk constraint, so the optimization problem actually solved is not the stated one.
[Section 2, adjoint equations (3) and (4)] The adjoint BSDEs are not solved; the martingale integrands q(t) and Q(t) are simply set to zero. In the stochastic maximum principle, q and Q are part of the solution and are determined by the BSDE, and q appears in the Hamiltonian through a trace term involving sigma. Because sigma depends on the control pi, dropping q changes the first-order condition used to derive the strategy. No argument is given that q = 0 and Q = 0 are the actual integrands for this problem, so the resulting s(t), S(t), and ultimately equation (7) are not established as the SMP solution.
[Section 3, derivation of equation (7) from equation (6)] The step from equation (6) to equation (7) replaces the ratio of exponential terms by 1 without any error bound or numerical justification. This approximation is the step that produces the claimed closed-form optimal strategy. Since the resulting formula is the central output of the paper, the approximation must be validated with explicit error estimates or at least a careful numerical check over the relevant parameter range; as written, equation (7) is at best a heuristic.
[Section 3, calibration and empirical illustration] The parameters b2, Q0.05, f0.05, f0.5, and the KDE bandwidth are calibrated on the same Entergy Corporation series that is then used to generate the optimal-strategy and wealth curves. The reported regularities, such as higher risk-free rates reducing stock investment, are therefore in-sample properties of the fitted formula rather than out-of-sample evidence. A holdout period or a proper cross-validation scheme is needed before the empirical claims can support the analytical conclusions.

minor comments (4)

[Section 3 and List of Figures] The figures are not actually included; the text and the figure list contain placeholders such as 'Figure 1 and Figure 2 should be placed here.' This makes it impossible to verify the quantitative claims about confidence intervals and wealth paths.
[Section 2, utility discussion] The text refers to negative exponential utility as 'Constant Absolute Relative Risk Aversion (CARA)' and says RRA = eta x; this appears to conflate CARA and CRRA. Negative exponential utility has constant absolute risk aversion, not constant relative risk aversion.
[Section 2, Remark after equation (6)] The Remark invokes 'Ekeland's theorem' but cites reference [6], which is Clarke's book, and it does not state the metric or the sense in which the solution is approximate. As written, the remark cannot justify the combined effect of the indicator smoothing, the q = 0 and Q = 0 simplifications, and the exponential approximation.
[Section 3, simulated wealth magnitudes] For r = 0.0004 the text reports portfolio wealth reaching an order of 10^15 starting from L1 = 1 over a 795-day horizon; such magnitudes suggest possible numerical instability in the discretization of equation (7) and should be diagnosed rather than reported as a financial outcome.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained; suspected issues are modeling/calibration concerns, not equivalence by construction.

full rationale

The paper's claimed derivation does not reduce to its inputs. The asymptotic joint normality of sample quantiles is quoted from Beach and Davidson [4], an external source, and the quantile SDE is then written as a modeling step rather than as a consequence of the optimality conditions. The maximum-principle framework is imported from Yong and Zhou [23] and applied to the stated Hamiltonian; the optimality condition (6) and the closed-form approximation (7) are obtained by solving the resulting first-order conditions, not by assuming the conclusion. The only self-citation, Biswas and Mukherjee [5], appears in the literature review as discrete-time background and is not load-bearing for any theorem or formula in the paper. The numerical section calibrates Q0.05, f0.05, f0.5, b2, and bandwidth on the Entergy series, but it presents this as an empirical illustration rather than as an out-of-sample prediction; the reported comparative static that higher r lowers stock investment follows from evaluating formula (7) at different values of r, so it is not a fitted parameter renamed as a prediction. Concerns such as using Lambda directly as a diffusion matrix and the labeling of L(1) and L(2) relative to the 5% and 50% quantiles are correctness/modeling issues, not circularity: no equation in the chain is equivalent to its own input by construction.

Assumptions & free parameters 9 free parameters · 8 assumptions · 0 invented entities

The central derivation rests on the asymptotic normality of sample quantiles, on treating the quantile covariance matrix as a diffusion coefficient, on the validity of Yong-Zhou's maximum principle for the smoothed problem, and on several approximations (sigmoid, exponential ratio, q=0/Q=0). These are mostly ad hoc to this paper. Nine parameters are fitted or hand-chosen from the same data used for illustration. No invented entities are introduced.

free parameters (9)

b2 = 0.00599
Drift of the second quantile process (equation of motion for X2), estimated from the Entergy daily returns used in the numerical illustration; it enters the optimal strategy formula (7).
Q0.05 = 0.00077
VaR threshold in the constraint and in the sigmoid approximation; estimated from the 5% sample quantile of the same return series.
f0.05 = 47.63579
Estimated density at the 5% quantile from a kernel density estimator; appears in the diffusion coefficients of both quantile SDEs.
f0.5 = 68.43975
Estimated density at the median from the same KDE; appears in the diffusion coefficients.
alpha = 10
Sigmoid slope chosen by hand to approximate the indicator function in the VaR constraint; no fitting procedure or sensitivity analysis is provided.
epsilon = 0.00001
Width of the sigmoid transition band, chosen by hand; no sensitivity analysis is provided.
KDE bandwidth = 0.00271447
Bandwidth for the kernel density estimate reported as part of log-quadratic fitting; the selection criterion is not described.
psi, psi0 = (0.6,0.8), (0.8,0.6), (0.95,0.312), (0.312,0.95)
Lagrange multiplier pairs from the maximum principle, chosen to satisfy the normalization psi0^2+psi^2=1 and varied in the numerical experiments.
rho = 0.22941
Listed in the parameter table as estimated from the data, though it does not appear in the displayed optimal strategy formula; its role is unclear.

assumptions (8)

domain assumption The return distribution F is continuous, differentiable to second order, and strictly monotonic at the quantiles of interest.
Invoked in Section 2 before the Beach-Davidson proposition to ensure sample quantile consistency and density values; heavy-tailed distributions need not be smooth.
standard math Sample quantiles from the return series are asymptotically jointly normal with covariance matrix Lambda from Beach and Davidson.
The proposition is quoted from [4], but it is stated for i.i.d. samples; the paper applies it to a daily return time series without modeling dependence.
ad hoc to paper The asymptotic quantile covariance matrix Lambda can serve directly as the diffusion coefficient in a continuous-time SDE for the quantile processes.
The paper writes dX = mu dt + Lambda dW. For this SDE the instantaneous covariance is Lambda Lambda^T, so the covariance of the quantile process is not Lambda unless a square-root factor is intended. No justification is given for this step.
domain assumption The market is frictionless and the portfolio is self-financing.
Assumed at the start of Section 2 to derive wealth dynamics; standard in the literature.
standard math Yong and Zhou's stochastic maximum principle applies to the sigmoid-approximated constrained problem and the adjoint BSDEs have unique solutions.
The theorem is cited from [23]; the paper does not verify the technical conditions beyond stating the result.
ad hoc to paper The martingale integrand q(t) and Q(t) in the adjoint BSDEs can be set to zero.
In the derivation after equations (3) and (4) the paper chooses q=0 and Q=0 to reduce the BSDEs to ODEs. These processes are part of the unique solution, not free parameters, so this assumption is unjustified.
ad hoc to paper The ratio of exponential terms in equation (6) is approximately 1 for all t in the horizon.
Used in Section 3 to convert equation (6) into the closed form (7); the approximation is not checked against the parameter values, and the exponents are not small over the full 795-day horizon.
ad hoc to paper The indicator function in the VaR constraint can be replaced by a sigmoid with parameters alpha=10 and epsilon=0.00001 without materially changing the optimum.
Introduced in Section 2 to make J2 smooth; the paper cites Ekeland's theorem for approximate optimality but does not quantify the approximation error.

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Pith. "Pith review of Portfolio Optimization Managing Value at Risk under Heavy Tail Return, using Stochastic Maximum Principle." pith.science (2026). https://pith.science/paper/DRYYFVO2

@misc{pith2026190803905,
  author       = {Pith},
  title        = {Pith review of: Portfolio Optimization Managing Value at Risk under Heavy Tail Return, using Stochastic Maximum Principle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DRYYFVO2}},
  note         = {Machine review of arXiv:1908.03905}
}

read the original abstract

We consider an investor, whose portfolio consists of a single risky asset and a risk free asset, who wants to maximize his expected utility of the portfolio subject to managing the Value at Risk (VaR) assuming a heavy tailed distribution of the stock prices return. We use a stochastic maximum principle to formulate the dynamic optimisation problem. The equations which we obtain does not have any explicit analytical solution, so we look for accurate approximations to estimate the value function and optimal strategy. As our calibration strategy is non-parametric in nature, no prior knowledge on the form of the distribution function is needed. We also provide detailed empirical illustration using real life data. Our results show close concordance with financial intuition.We expect that our results will add to the arsenal of the high frequency traders.

Figures

Figures reproduced from arXiv: 1908.03905 by the authors.

**Figure 2.** Portfolio wealth and its corresponding 95% interval for different rates of interest (r). [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗

**Figure 3.** Average optimal strategy and its corresponding 95% interval r = 0.00014, [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Average portfolio wealth and its corresponding 95% interval r = 0.00014, [PITH_FULL_IMAGE:figures/full_fig_p028_4.png]

**Figure 5.** Figure 5: Average optimal strategy and its corresponding 95% interval r = 0.00014, [PITH_FULL_IMAGE:figures/full_fig_p029_5.png]

**Figure 6.** Figure 6: Average wealth and its corresponding 95% interval r = 0.00014, [PITH_FULL_IMAGE:figures/full_fig_p030_6.png]

**Figure 7.** Figure 7: Average optimal strategy and its corresponding 95% interval r = 0.00014, [PITH_FULL_IMAGE:figures/full_fig_p031_7.png]

**Figure 8.** Figure 8: Average wealth and its corresponding 95% interval r = 0.00014, [PITH_FULL_IMAGE:figures/full_fig_p032_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Agarwal & R

A. Agarwal & R. Sircar, Portfolio Benchmarking under Drawdown Constraint and Stochastic Sharpe Ratio, SIAM J. Financial Math, 9-2, 435–464 (2018)

work page 2018

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Y . Aït-Sahalia & A. W. Lo, Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices, The Journal of Finance, 53 (2), 499 - 547 (1998)

work page 1998

[3] [3]

T. W. Archibald & E. Possani, Investment and operational decisions for start-up companies: a game theory and Markov decision process approach, Annals of Operations Research, https://doi.org/10.1007/s10479-019- 03426-5 (2019)

work page doi:10.1007/s10479-019- 2019

[4] [4]

C. M. Beach & R. Davidson, Distribution -Free Statistical Inference with Lorenz Curves and Income Shares, Review of Economic Studies, L 723-735 (1983)

work page 1983

[5] [5]

Biswas & D

S. Biswas & D. Mukherjee, Discrete Time Portfolio Optimization managing Value at Risk under heavy tail return distribution, International Journal of Mathematical Modelling and Numerical Optimization, Accepted (2020)

work page 2020

[6] [6]

F. H. Clarke, Optimization and Nonsmooth Analysis , Classics in Applied Mathematics, SIAM, Philadelphia, 1990

work page 1990

[7] [7]

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J. Cvitanic & I. Karatzas, On portfolio optimization under "drawdown" constraints, IMA V ol. Math.its Appl. 65, 35–35 (1995)

work page 1995

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G. B. Folland, Real analysis, Pure and Applied Mathematics, John Wiley & Sons Inc., New york (1999)

work page 1999