A Proposal for Multi-asset Generalised Variance Swaps

Diganta Mukherjee; Subhojit Biswas

REVIEW 4 major objections 5 minor 13 references

A Proposal for Multi-asset Generalised Variance Swaps

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

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper prices swaps on the trace and the largest eigenvalue of a multi-asset return covariance matrix under Markov-modulated volatility.

desk verdict A good-faith but unsound proposal: the 'maximum eigenvalue swap' is actually a constrained portfolio variance, and the pricing uses the historical measure. read the letter →

arxiv 1908.03899 v1 pith:AFJXCIKY submitted 2019-08-11 q-fin.MF

Subhojit Biswas , Diganta Mukherjee This is my paper · ORCID

classification q-fin.MF MSC 91G1091G80

keywords generalizedvarianceswapstraceofcovariancematrixmaximumeigenvalueMarkov-modulatedvolatilityswapmulti-assetderivativesconstrainedproblempricing

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

This paper proposes two new derivatives for a basket of stocks: a swap whose payoff is the trace of the return covariance matrix, and a swap whose payoff is the largest eigenvalue of that matrix. It prices both in a market where volatility is Markov-modulated, meaning the state of the economy switches randomly among a finite set of volatility regimes. For the trace swap, the price is the discounted expectation of the sum of future variances, computed through the semigroup $e^{tQ}$ of the Markov generator. For the largest-eigenvalue swap, the price incorporates the covariance structure and a target-return constraint through a constrained optimization problem solved by QR decomposition. The paper claims this is the first extension of covariance swaps to more than two assets, and a three-stock numerical example yields explicit prices for both swaps.

What carries the argument

The machinery that carries the argument is the semigroup of the Markov generator $Q$, combined with a QR-based reduction of a constrained eigenvalue problem. The paper uses the identity $E\{\sigma_i(x_t)\sigma_j(x_t)\} = e^{tQ}\sigma_i(x)\sigma_j(x)$ to turn expectations of future variance and covariance terms into matrix exponentials, so every swap price is assembled from integrals $\int_0^T e^{tQ}(\cdot)\,dt$. For the largest-eigenvalue swap, the portfolio weight vector $w$ maximizes $w^T\Omega w$ under norm, budget, and target-return constraints; a QR decomposition of the constraint matrix $A=[E[R]\ \mathbf{1}]$ transforms the problem into a lower-dimensional standard eigenvalue equation $Cr = g + \lambda r$, and the optimized eigenvalue becomes the payoff whose expectation is discounted.

What would settle it

Re-price either swap under an equivalent martingale measure that tilts the Markov chain by a nonzero market price of volatility risk, or build a self-financing dynamic hedge that replicates the realized trace or maximum-eigenvalue payoff; if either price differs from the paper's discounted historical expectation, the stated swap price is not arbitrage-free.

Watch

Extended reading notes

Core claim

The central claim is that generalized variance swaps can be defined and priced for any number of assets when volatility follows a finite-state Markov chain. For a portfolio covariance matrix $\Omega(x_t)$ whose entries $\sigma_i^2(x_t)$ and $\rho_{ij}\sigma_i(x_t)\sigma_j(x_t)$ depend on the Markov state $x_t$, the trace swap price is $e^{-rT}\big(\frac{1}{T}\int_0^T \sum_i e^{tQ}\sigma_i^2(x)\,dt - K\big)$, with $Q$ the generator of the chain. The largest-eigenvalue swap is priced by first solving $\max_w w^T\Omega w$ subject to $w^Tw=1$, $\mathbf{1}^Tw=1$, and $E[R]^Tw=k$, using a QR decomposition to reduce the problem to a smaller standard eigenvalue problem, then taking the discounted expectation of the resulting eigenvalue. For the three stocks in the numerical example, the trace swap price is 38.733 and the eigenvalue swap price is 14.011 per million units for the chosen strikes and horizon. The paper treats the historical transition probabilities as the pricing measure and concludes that, for the same basket, the eigenvalue swap is the cheaper contract.

Load-bearing premise

The load-bearing premise is that the historical transition probabilities of the Markov chain, estimated from the same returns that define the volatility states, can be used directly as the pricing measure, with no separate market price of volatility risk.

Editorial extensions

If this is right

For any $n$-asset basket, the trace swap price is the discounted expected sum of the individual variance terms, so its calibration needs only each asset's variance semigroup term.
The largest-eigenvalue swap absorbs covariances and expected returns through the constrained weights, so its price can lie below the trace swap for the same basket, as in the numerical example.
The formulas give commodity producers a way to hedge basket-wide volatility driven by common factors, since only the Markov generator and the covariance entries need to be estimated.
The framework works for any finite-state Markov modulation; changing the number of states only changes the dimension of $Q$ and the constraint matrix.
The paper identifies jumps as the natural next step, noting that an Itô formulation is then unsatisfactory and a Lévy-based version would be required.

Reading between the lines

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

Nothing in the derivation forces the historical transition matrix to be the risk-neutral one; read strictly, the prices are actuarial expectations under the physical Markov chain, and any market price of volatility risk would distort $Q$.
The constrained weight vector in the eigenvalue swap is a mean-variance frontier object: it picks the direction of maximum portfolio variance subject to a target return, so the swap can be interpreted as trading the variance of that specific portfolio direction.
A testable extension would calibrate $Q$ from option-implied data instead of historical returns and compare the two prices; the gap would quantify the multi-asset volatility risk premium embedded in basket variance.
The numerical ordering of the two swap prices is sample-specific; with different correlations or target returns, the trace swap could be the cheaper contract.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

You should know two things upfront. First, the paper is not what it claims: the 'maximum eigenvalue' of the covariance matrix that the authors price in Section 2.2 is not an eigenvalue. They solve max w'Ωw subject to w'w=1, sum w=1, and E(R)'w=k, which is a constrained portfolio variance, not the spectral maximum. The numerical example makes this concrete: the reported swap value 43.264 is far below the actual largest eigenvalue of their covariance matrix (about 124). So the central novelty—a swap on the maximum eigenvalue—does not exist in the paper. Second, the pricing is done under the historical measure: the transition matrix, state variances, and covariances are all estimated from the same return sample, and the expectation is discounted at the risk-free rate with no market price of volatility risk. That means the 'prices' are not arbitrage-free derivatives prices.

Credit where it is due: the paper is clearly written and engages seriously with the existing literature on variance and covariance swaps. The trace swap is correctly identified as a sum of individual variance swaps (though then they price it inconsistently, using a spot expression in the formula and an average realized variance in the numerical section). The idea of extending single-asset variance products to a multi-asset setting is real and worth thinking about. The QR-decomposition derivation is algebraically careful, even if it is solving the wrong problem.

The soft spots are load-bearing. The mislabeled eigenvalue object is not a minor terminological issue; it changes the product and makes the price depend on expected returns and an arbitrary target k rather than on the covariance structure alone. The trace payoff inconsistency between spot and realized variance is a genuine error. And the numerical example fits all parameters to one sample and then prices the swap as an expectation under that fit, so it is a deterministic output of the fitted inputs, not a validation. On top of that, several of the algebraic expressions in Section 2.2 are unwieldy and contain what look like typos (e.g., 'Y 2 x' and 'σ 3 3'), which further reduces confidence.

Who is this for? A reader interested in multi-asset variance derivatives might find the failure modes instructive, but the paper in its current form does not provide a correct pricing formula for a well-defined product. It needs major reframing: either rename the second product as a constrained portfolio variance swap and justify the risk-neutral measure, or actually compute the spectral maximum eigenvalue and price that.

Recommendation: this does not deserve a serious referee in its current form. It would be a desk reject, though with encouragement to resubmit after the product is defined honestly and the pricing measure is fixed.

Referee Report

4 major / 5 minor

Summary. The paper proposes two new multi-asset variance swap products: one written on the trace of the return covariance matrix and one written on the 'maximum eigenvalue' of the same matrix, in a market with Markov-modulated volatilities. It derives pricing formulas using Markov-generator results for conditional expectations, and it demonstrates the approach on a numerical example with three US utility stocks using daily data over one year. The stated central claim is that this is the first extension of covariance/variance swap pricing to a multidimensional, Markov-modulated setting.

Significance. If the formulas were correct and the products genuinely new, the trace swap would be a modest but useful extension of single-asset variance swap pricing, and the eigenvalue swap would be a novel derivative. However, the 'maximum eigenvalue' product is not a spectral eigenvalue at all, and the numerical price shown is less than half of the actual largest eigenvalue of the fitted covariance matrix. In addition, the pricing is done under the historical measure with no risk adjustment, so the reported prices are not arbitrage-free. The paper also provides no code or machine-checked derivations, and the long algebraic formulas contain apparent typographical errors. The central claims are therefore not supported.

major comments (4)

[Section 2.2, Eq. (8); Section 3.2] The quantity labeled 'maximum eigenvalue' is not an eigenvalue of the covariance matrix Ω. The spectral maximum eigenvalue is the maximum of wᵀΩw subject only to wᵀw=1; the additional constraints 1ᵀw=1 and E(R)ᵀw=k shrink the feasible set. In the numerical example of Section 3.2, the reported vector w=(0.9569597, 0.2239916, -0.1822877) gives wᵀΩw=43.264×10⁻⁶, while the largest eigenvalue of the covariance matrix in Tables 1–2 is approximately 124×10⁻⁶. The paper therefore prices a constrained portfolio variance, not a swap on the maximum eigenvalue of the covariance matrix. This is a load-bearing error because the claimed novelty is precisely the eigenvalue swap.
[Section 2.1] The trace swap is introduced as e^{-rT}E[(tr Ω(x_T) - K], a payoff depending on the covariance matrix at maturity. The subsequent pricing formula, however, replaces tr Ω(x_T) with (1/T)∫₀ᵀ E[σ₁²(x_t)+σ₂²(x_t)+σ₃²(x_t)]dt, which is a time-averaged realized variance over the swap's life rather than the trace at the terminal date. These two payoffs are not equivalent, and the paper does not indicate which one is being priced. The derivation must be aligned to a single consistent payoff definition.
[Section 3] The pricing measure is the historical measure. The state thresholds, transition probabilities Π, state-dependent variances and covariances, and expected returns are all estimated from the same daily sample, and the expectation in the pricing formulas is computed under this fitted physical measure, then discounted at the risk-free rate. In a stochastic-volatility model, the physical measure is not the risk-neutral measure unless volatility risk is unpriced, which is neither assumed nor justified. Without a change of measure—for instance, a modified generator Q under the pricing measure—the numbers in Section 3 are not valid swap prices.
[Section 2.2, derivation after Eq. (2)] The constrained maximization is a quadratically constrained program, but the paper reduces it to the linear system (C−λI)r=g and then to an eigenvalue decomposition D u = d + λ u. For general n, this is not the correct optimality system: with d≠0, the equation (C−λI)r=g combined with rᵀr=s² is a nonlinear eigenvalue-type problem, not a standard eigen-decomposition, and the paper provides no proof that the resulting w is even a local maximizer, let alone the global one. This affects the general-n claim, not only the three-asset example.

minor comments (5)

[Notation, Section 2.2] The symbol P is used both for the orthogonal matrix in the QR decomposition and for the probability measure; this is confusing and should be disambiguated.
[Tables 1–2 and Section 3.1–3.2] The table headers say 'All the figures are in 10−6', but the subsequent price calculations in Sections 3.1 and 3.2 omit the 10⁻⁶ scaling until the '1 million units' sentence. The units should be stated consistently throughout the numerical example.
[Section 3.1 and 3.2] The strike prices K=90 (trace) and K=30 (eigenvalue) are chosen ad hoc, and the paper does not discuss how the strike affects the swap's value or how a market participant would select it.
[Section 2, Proposition 4] The phrase 'we consider σ as a martingale' is not a definition; the martingale property is a modeling assumption and should be stated explicitly as an assumption, with its consequences for the drift of the stock price dynamics.
[Section 2.2, displayed formulas for k2 and k3] The expressions for k2 and k3 contain apparent typographical errors (for example, 'Y2 x' and 'µ2 z') that make the formulas unreadable and hinder verification.

Circularity Check

1 steps flagged · score 6.0 of 10

The 'maximum eigenvalue swap' is defined as a constrained portfolio variance, so the price reduces by construction to wᵀ E[e^{-rT}∫e^{tQ}Ωdt] w and is not a swap on the spectral maximum eigenvalue.

self definitional [Section 2.2, optimization setup and Eq. (8)]
"we assume that the investor considers the maximum eigenvalue of the covariance matrix, for a given expected mean return. ... maximize_w w(t)^T Ω w(t) subject to w(t)^T w(t) = 1, I^T w(t) = 1, E(R)^T w(t) = k ... We can write the maximum eigenvalue as, λ(x_t) = w(t)^T Ω w(t) ... P(x) = e^{−rT} E{(λ(x_t) − K)} (8)"

The largest eigenvalue of Ω is the maximum of the unconstrained Rayleigh quotient max_{||v||=1} vᵀΩv. By adding the constraints Iᵀw=1 and E(R)ᵀw=k, the feasible set is strictly smaller, and the maximizing w is not in general an eigenvector of Ω. Thus λ(x_t) is by construction the variance of a target-return-constrained portfolio, not a spectral eigenvalue. Equation (8) then prices the discounted expectation of this constructed quadratic form, so the numerical 'eigenvalue swap' price (wᵀ times the fitted expected discounted covariance matrix times w) is forced by the definition. The paper has therefore not derived a price for the maximum eigenvalue of the covariance matrix; it has defined a different quantity and labeled it as the maximum eigenvalue.

full rationale

The trace-swap pricing in Section 2.1 is a direct, self-contained expectation calculation under the Markov-modulated volatility model, using external citations (Elliott and Swishchuk 2007; Salvi and Swishchuk 2012) that are not self-citations and are used only for standard Markov-process martingale and quadratic-variation results. The numerical section is an in-sample illustration that plugs fitted transition probabilities and state variances into the derived formula; the paper does not frame these numbers as out-of-sample predictions, so I do not treat the fitting as a separate circular step. The clear circularity is in the central eigenvalue-swap claim: Section 2.2 defines the 'maximum eigenvalue' as the solution of a constrained maximization that is not the eigenvalue problem, and then prices exactly that defined object. The result is not equivalent to the largest eigenvalue of the covariance matrix by any independent computation, and the paper's novelty claim therefore reduces to a renamed constrained portfolio variance. This is a self-definitional step in the paper's main claimed contribution, warranting a score of 6.

Assumptions & free parameters 4 free parameters · 4 assumptions · 1 invented entities

The central pricing formulas rest on the imported Markov-modulated expectation results, plus a large set of parameters fitted to the same historical sample. The 'eigenvalue' swap additionally rests on a novel definition of eigenvalue that is not standard, and on the unstated equivalence of physical and risk-neutral measures.

free parameters (4)

State classification thresholds (sample mean returns) = mu1=0.000664, mu2=0.000873, mu3=0.000725
The Up/Down states for each stock are defined by whether the daily return exceeds the sample mean, so the mean is a fitted threshold that determines the state path.
Transition probability matrix Pi = Rows: Down [0.325, 0.2875, 0.3875]; Middle [0.3636, 0.3377, 0.2987]; Up [0.2796, 0.3011, 0.4194]
Estimated by counting transitions in the same sample used to define states and variances.
State-dependent variances and covariances = Tables 1 and 2 values (e.g., P1var=42.978, PCov(12)=40.911, all x 10^-6)
Estimated separately for each of the three states from the same sample.
Target return k = 0.0007
Chosen by the authors for the numerical eigenvalue swap without justification.

assumptions (4)

domain assumption Stock prices follow dS_t = S_t (mu dt + sigma(x_t) dW_t) with Markov chain x_t independent of W_t
Section 2; this is the model under which swaps are priced.
standard math The Markov chain generator Q and the semigroup e^{tQ} give the conditional expectations of sigma^2(x_t) and sigma_i sigma_j products exactly as in Propositions 1-4 of Salvi and Swishchuk (2012) and Elliott and Swishchuk (2007)
The paper imports these results without proof.
ad hoc to paper The historical transition probabilities estimated from price data can be used as the pricing measure, so no risk-neutral adjustment is made
Section 3 uses the sample transition matrix and then discounts at the risk-free rate; this equates physical and pricing measures without justification.
ad hoc to paper The constrained optimization with constraints w^T w = 1, 1^T w = 1, E(R)^T w = k defines the 'maximum eigenvalue' swap
Section 2.2; this is the paper's invented definition of eigenvalue, not the spectral maximum.

invented entities (1)

Swaps on the 'maximum eigenvalue' of the covariance matrix as defined by the constrained quadratic program
purpose: To provide a multi-asset variance hedging instrument
This quantity is not the spectral maximum eigenvalue unless the constraints are vacuous; no market data or replication argument shows it is actually traded or hedgeable.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A Proposal for Multi-asset Generalised Variance Swaps." pith.science (2026). https://pith.science/paper/AFJXCIKY

@misc{pith2026190803899,
  author       = {Pith},
  title        = {Pith review of: A Proposal for Multi-asset Generalised Variance Swaps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AFJXCIKY}},
  note         = {Machine review of arXiv:1908.03899}
}

read the original abstract

This paper proposes swaps on two important new measures of generalized variance, namely the maximum eigen-value and trace of the covariance matrix of the assets involved. We price these generalized variance swaps for financial markets with Markov-modulated volatilities. We consider multiple assets in the portfolio for theoretical purpose and demonstrate our approach with numerical examples taking three stocks in the portfolio. The resultsobtained in this paper have important implications for the commodity sector where such swaps would be useful for hedging risk

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

F. E. Benth, M. Groth & R. Kufakunesu (September 2007) Valuing Volatility and Variance Swaps for a Non-Gaussian Ornstein-Uhlenbeck Stochastic Volatility Model . Applied Mathematical Finance, Vol. 14, No. 4, 347-363

work page 2007
[2]

Bossu (2005) Arbitrage Pricing of Equity Correlation Swaps

S. Bossu (2005) Arbitrage Pricing of Equity Correlation Swaps . JPMorgan Equity Derivatives, Working paper

work page 2005
[3]

Bossu (2007) A New Approach For Modelling and Pricing Correlation Swaps

S. Bossu (2007) A New Approach For Modelling and Pricing Correlation Swaps. Equity Structuring - ECD London , Working paper

work page 2007
[4]

Broadie & A

M. Broadie & A. Jain (2008a) Pricing and Hedging Volatility Derivatives . The Journal of Derivatives, Vol. 15, No. 3, pp. 7-24

work page 2008
[5]

Broadie & A

M. Broadie & A. Jain (2008b) The Effect of Jumps and Discrete Sampling on Volatility and Variance Swaps . International Journal of Theoretical and Applied Finance, Vol.11, No.8. pp. 761-797

work page 2008
[6]

P. Carr, H. Geman, D. B. Madan & M. Yor (2005) Pricing options on realized variance . Finance Stochast. 9, 453-475

work page 2005
[7]

Carr & R

P. Carr & R. Lee (2007) Realized volatility and variance: Options via swaps . Bloomberg LP and University of Chicago. Available at: http://math.uchicago.edu/˜rl/OVSwithAppendices.pdf

work page 2007
[8]

Elliott & A

R. Elliott & A. V. Swishchuk (2007) Pricing Options and Variance Swaps in Markov Modulated Brownian Markets. In: ‘Hidden Markov Model in Finance’ , Eds. R. Mamon and R. Elliott, Springer

work page 2007

Show all 13 references

[9]

Da Fonseca, F

J. Da Fonseca, F. Ielpo & M. Grasselli (2009) Hedging (Co)Variance Risk with Variance Swaps . Available at SSRN: http://ssrn.com/abstract=1341811

2009
[10]

Golub & Urs von Matt (1991) A Constrained Eigenvalue Problem, Numerical Linear Algebra , Digital Signal Processing and Parallel Algorithms, Vol 70, 677-686

Walter Gander, Gene H. Golub & Urs von Matt (1991) A Constrained Eigenvalue Problem, Numerical Linear Algebra , Digital Signal Processing and Parallel Algorithms, Vol 70, 677-686

1991
[11]

Semere Habtemicael & Indranil SenGupta (2016) Pricing covariance swaps for Barndorff-Nielsen and Shephard process driven financial markets , Annals of Financial Economics, Vol 11, pp. 1650012

2016
[12]

Salvi & A

G. Salvi & A. V. Swishchuk (2012) Pricing of Variance, Volatility, Covariance and Correlation Swaps in a Markov-modulated Volatility Model . Preprint

2012
[13]

https://www.quantopian.com , Accessed: 2018-10-10

2018

Pith tools

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

[1] [1]

F. E. Benth, M. Groth & R. Kufakunesu (September 2007) Valuing Volatility and Variance Swaps for a Non-Gaussian Ornstein-Uhlenbeck Stochastic Volatility Model . Applied Mathematical Finance, Vol. 14, No. 4, 347-363

work page 2007

[2] [2]

Bossu (2005) Arbitrage Pricing of Equity Correlation Swaps

S. Bossu (2005) Arbitrage Pricing of Equity Correlation Swaps . JPMorgan Equity Derivatives, Working paper

work page 2005

[3] [3]

Bossu (2007) A New Approach For Modelling and Pricing Correlation Swaps

S. Bossu (2007) A New Approach For Modelling and Pricing Correlation Swaps. Equity Structuring - ECD London , Working paper

work page 2007

[4] [4]

Broadie & A

M. Broadie & A. Jain (2008a) Pricing and Hedging Volatility Derivatives . The Journal of Derivatives, Vol. 15, No. 3, pp. 7-24

work page 2008

[5] [5]

Broadie & A

M. Broadie & A. Jain (2008b) The Effect of Jumps and Discrete Sampling on Volatility and Variance Swaps . International Journal of Theoretical and Applied Finance, Vol.11, No.8. pp. 761-797

work page 2008

[6] [6]

P. Carr, H. Geman, D. B. Madan & M. Yor (2005) Pricing options on realized variance . Finance Stochast. 9, 453-475

work page 2005

[7] [7]

Carr & R

P. Carr & R. Lee (2007) Realized volatility and variance: Options via swaps . Bloomberg LP and University of Chicago. Available at: http://math.uchicago.edu/˜rl/OVSwithAppendices.pdf

work page 2007

[8] [8]

Elliott & A

R. Elliott & A. V. Swishchuk (2007) Pricing Options and Variance Swaps in Markov Modulated Brownian Markets. In: ‘Hidden Markov Model in Finance’ , Eds. R. Mamon and R. Elliott, Springer

work page 2007