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REVIEW 4 major objections 4 minor 98 references

Pricing Multi-strike Quanto Call Options on Multiple Assets with Stochastic Volatility, Correlation, and Exchange Rates

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A systematic Monte Carlo comparison of 60 stochastic model combinations for multi-strike quanto options finds that GARCH-Jump volatility, Weibull correlation, and Ornstein-Uhlenbeck exchange rates price best in the tested cases.

desk verdict A large MC model-comparison grid that is new but whose 'best model' ranking is circular because the true price is the average of the tested pack; worth reviewing, but the headline claim needs an external benchmark. read the letter →

arxiv 2411.16617 v1 pith:NPQ4PGDG submitted 2024-11-25 q-fin.PR q-fin.CPq-fin.MF

Boris Ter-Avanesov , Gunter A. Meissner This is my paper · ORCID
classification q-fin.PRq-fin.CPq-fin.MF MSC 91G2091G6065C05
keywords quantooptionsmulti-strikestochasticvolatilitycorrelationexchangeratesMonteCarlosimulationCoraGora
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 asks which combination of stochastic volatility, stochastic correlation, and stochastic exchange-rate models gives the most accurate Monte Carlo prices for quanto call options on multiple foreign-currency assets, where the payoff is the maximum of several asset-minus-strike terms converted at a fixed exchange rate. It tests 60 combinations of five volatility models, four correlation models, and three exchange-rate models, under three discretization schemes, for two option structures and two one-year periods. The authors' central finding is that GARCH-Jump volatility, Weibull correlation, and Ornstein-Uhlenbeck exchange rates perform best across all four test settings, with the Milstein scheme giving the best accuracy-to-runtime balance. The paper also finds mean reversion in correlation and exchange-rate models helpful, uses antithetic variates for variance reduction, and derives Cora and Gora as correlation risk sensitivities for hedging.

What carries the argument

The load-bearing machinery is the model grid: five stochastic volatility SDEs (a mean-reverting square-root process, a GARCH-type linear process, that process with jumps, a square-root process with jumps, and a 3/2 process) times four stochastic correlation SDEs (a bounded diffusion on [-1,1], its bounded variant, a mean-reverting extension, and a Weibull-based diffusion) times three exchange-rate SDEs (geometric Brownian motion, Ornstein-Uhlenbeck mean reversion, and exponential Levy with jumps). Each of the 60 combinations is run under Euler-Maruyama, Milstein, and Runge-Kutta discretizations, giving 180 variants whose price estimates are ranked by standard deviation and then by percentage error from the average of the 40 lowest-standard-deviation variants. The asset-price dynamics couple the USD underlying to each foreign underlying through one stochastic correlation process, while the FX processes are driven by independent Brownian motions; antithetic variates are applied to all Brownian increments. The Cora and Gora hedging parameters are obtained by applying the chain rule to the max-type payoff with respect to the stochastic correlation.

What would settle it

Take one of the four test settings, for example Case 1 with the 2021 start date, compute the option price with an independent high-accuracy method such as a nested Monte Carlo with $10^7$ outer paths or a PDE solver, and re-rank all 180 model variants by percentage error from that independent price; the paper's best-model conclusion fails if (GARCH-Jump, Weibull, OU) is not among the closest variants to that price.

Watch

Extended reading notes

Core claim

The paper claims that, under its simulation benchmark, the combination of GARCH-Jump stochastic volatility, Weibull stochastic correlation, and Ornstein-Uhlenbeck stochastic exchange rates gives the most accurate Monte Carlo price estimates for multi-strike quanto call options, for both a two-asset single-FX case and a three-asset two-FX case, and for both the 2021 and 2022 one-year test periods. The GARCH-Jump model adds compound-Poisson jumps to a GARCH-type linear volatility SDE; the Weibull model is a correlation SDE built so that correlations follow a Weibull distribution with exponential autocorrelation; the OU model adds mean reversion to GBM-style exchange rates. The paper also claims that mean reversion in correlation and FX processes improves Monte Carlo pricing, that the Milstein scheme best balances accuracy and runtime, and that antithetic variates reduce estimator variance. It further derives Cora and Gora, the first and second derivatives of option price with respect to correlation, so the correlation risk of these options can be hedged.

Load-bearing premise

The load-bearing premise is that the average price estimate of the 40 simulation variants with the lowest standard deviation is close enough to the true arbitrage-free option value to serve as the benchmark; if that average is biased, the reported best-model ranking only shows which models agree with each other, not which is most accurate.

Editorial extensions

If this is right

  • For Monte Carlo pricing of multi-asset quanto calls, the paper's recommendation is to start from GARCH-Jump stochastic volatility, Weibull stochastic correlation, and Ornstein-Uhlenbeck exchange-rate dynamics.
  • The Milstein discretization scheme offers the best trade-off between execution time and standard deviation; Runge-Kutta is the fallback when the diffusion derivative is hard to compute.
  • Adding mean reversion to stochastic correlation and to exchange-rate models improves the precision of Monte Carlo price estimates in the settings tested.
  • The derived Cora and Gora expressions give first- and second-order sensitivity of the quanto option price to the stochastic correlation, so correlation hedging can be implemented for these payoffs.
  • Stochastic correlation generally beats the constant-correlation benchmark, though the margin varies by case and start date.

Reading between the lines

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

  • Editorial extension: the ranking is measured against an internal benchmark, so a natural next step is to re-run the comparison against an independent price from a PDE or long-run nested Monte Carlo; the paper itself notes it has no market prices or closed-form benchmark.
  • Editorial extension: the Cora/Gora chain-rule structure should transfer to other foreign-currency-linked max or spread payoffs, because it only uses the independence of the FX processes from the correlation process; each new payoff only changes the partial derivatives of the option value with respect to asset prices.
  • Editorial extension: the authors conjecture that stochastic volatility matters more than stochastic correlation for pricing accuracy; the logged outputs from the 180 variants could be decomposed by factor to quantify this, but the paper does not report such a decomposition.
  • Editorial extension: since jumps are not added to the correlation process, a natural test of the ranking is whether jumpy correlation, which the authors flag as plausible in stressed markets, would displace Weibull correlation as the best stochastic correlation choice.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper develops a Monte Carlo framework for pricing basket call options on two or three foreign equity indices under stochastic volatility (SV), stochastic correlation (SC), and stochastic exchange rate (SER) models. For two option structures (one FX rate, two FX rates) and two historical start dates (2021, 2022), the authors simulate 180 model variants formed from five SV models, four SC models, three SER models, and three discretization schemes, using antithetic variates. They rank the variants by percentage error relative to a 'true' price defined as the average of the 40 variants with the lowest Monte Carlo standard deviation. They conclude that GARCH-Jump SV, Weibull SC, and OU SER is the best combination, that the Milstein scheme offers the best trade-off, and they propose formulas for the correlation risk measures Cora and Gora.

Significance. The paper has notable strengths: a large and systematic grid of model combinations, a transparent simulation protocol, published parameter choices, avoidance of look-ahead bias in calibration, and implementation of antithetic variates. If the ranking were anchored to an independent price, such a comparison could offer practical guidance. However, as it stands, the central model-selection claim is not supported. The 'true' price is constructed from the same model outputs being ranked, so the winning model is merely the one closest to the average of a low-standard-deviation subset of the tested models; the paper explicitly acknowledges the absence of observed prices, alternative pricing methods, or closed-form solutions. In addition, the Cora/Gora derivations in Section 6 are formal chain-rule identities that do not compute the required derivatives, and the payoff definition in Section 3 is inconsistent with the stated quanto structure and with a consistent risk-neutral measure. These issues affect the validity of the headline conclusions.

major comments (4)
  1. [Section 5] The benchmark used for all percentage-error rankings is self-referential: the paper states 'we compute the average price estimate of the 40 best variants to be the "true" value of the option' and also states that there are no observed prices, another pricing method, or a closed-form/series solution. Consequently, the ranking measures which model price is closest to the center of a low-standard-deviation subset of the tested pack, not which model price is closest to an arbitrage-free or market price. Since the different SV/SC/SER combinations imply different price distributions, averaging a subset cannot remove model bias and may even favor models with artificially low variance. The Abstract's claim that the GARCH-Jump/Weibull/OU combination 'performs best' is therefore not supported by the evidence.
  2. [Equations (24), (35) and (14)-(16)] The payoff in Eq. (24) and Eq. (35) contains the stochastic exchange rate FX(T) (e.g., S_GBP(T)*FX(T) - K2), whereas the Abstract and Section 1 define a quanto option as converting the foreign payoff at a fixed exchange rate. Moreover, Eqs. (14)-(16) give the FX rate drift as (r_f - r_d - 0.5σ²), which is the opposite of the standard domestic-risk-neutral drift for a USD-per-foreign-currency rate, and the foreign asset's drift is not adjusted to make S_GBP*FX a martingale under the domestic measure. As a result, the simulated option prices are not computed under a consistent risk-neutral measure, which undermines the numerical basis of the model ranking.
  3. [Section 6, Eqs. (63)-(66), (72)-(79)] The derivations of Cora and Gora stop at formal chain-rule identities. The paper never evaluates the load-bearing terms ∂S_GBP/∂ρ, ∂S_USD/∂ρ, ∂²S_GBP/∂ρ², ∂²C/∂S∂ρ, or the ∂C/∂S derivatives for the max payoff. Consequently, no actual hedging parameters are obtained, and the Abstract's claim that the paper 'derives the correlation risk parameters Cora and Gora' is an overstatement. To be useful, the formulas would need to be specialized to the chosen SDEs and either computed analytically or numerically.
  4. [Section 5, Figures 2-3] The ranking of the 180 variants is based on point estimates of percentage error, but the paper does not assess whether the differences between the top-ranked models are statistically significant. Given that the reported 95% confidence intervals of the price estimates are non-negligible, the top model may be within Monte Carlo error of several alternatives. Without a significance test or a separation analysis, the conclusion that a particular combination 'performs best' is fragile even under the authors' own benchmark.
minor comments (4)
  1. [Section 2.2.4, Eq. (11)] The Weibull SC model is defined for non-negative ρ only (the authors note that ρ_t must stay non-negative and k > 0), but the paper later applies it to equity correlations that can be negative; the paper should discuss how negative correlations are handled or whether the calibrated correlations are always positive.
  2. [Section 4, Eq. (36)] The phrase 'A Itô process' should be 'An Itô process'.
  3. [Section 4.8.2] The claim that antithetic variates reduce variance relies on the payoff being monotonic and the discretization linear in the Brownian increments; for the Milstein and Runge-Kutta schemes used for volatility and FX processes, the quadratic terms mean the negative-covariance property is not automatic and should be justified.
  4. [Section 3.2.3, Eq. (34)] In Case 2, the same ρ(t) is used for both USD-GBP and USD-EUR correlations, which implies a correlation of ρ² between USD and EUR and thus a very specific dependence structure; this restriction is not mentioned or justified.

Circularity Check

1 steps flagged · score 7.0 of 10

The 'true' option price is the average of the 40 lowest-standard-deviation model prices, so the headline 'best model' ranking measures proximity to the tested pack, not to an independent price.

  1. fitted input called prediction [Section 5, 'Results Comparison & Discussion' (benchmark construction and model selection; claim propagated to abstract and Section 7)]
    "all 180 model variants are ordered from lowest standard deviation to highest, and we compute the average price estimate of the 40 best variants to be the 'true' value of the option. Then, all the models are re-ordered by the lowest to highest percentage error from this value ... Comparing models by percentage error relative to this calculated value is an attempt to perform model selection even though we do not have real observed prices, another pricing method, or a closed-form/series solution to compare our MC simulation prices and performance to."

    The 'true' value used for ranking is not an independent price: it is the average of 40 price estimates drawn from the same 180 model variants being ranked. The percentage error therefore measures how close each variant is to an internal consensus subset, not how close it is to an arbitrage-free or market price. The winning combination (GARCH-Jump, Weibull, OU) is selected for having the smallest distance to an average that contains the outputs of that same family; the ranking is forced by the benchmark construction. The paper explicitly concedes there is no external anchor ('we do not have real observed prices, another pricing method, or a closed-form/series solution').

full rationale

The central model-selection claim is circular by construction: the paper defines the benchmark 'true' option value as the average price estimate of the 40 lowest-standard-deviation variants from the same 180-variant set, then ranks all variants by percentage error from that self-generated average. The headline conclusion that GARCH-Jump/Weibull/OU 'performs best' is therefore a statement about proximity to an internal consensus, not about accuracy relative to an independent price or market data. The paper's own admission that no real observed prices, alternative pricing method, or closed-form solution exists confirms the absence of an external anchor. The rest of the paper—Monte Carlo simulation design, discretization schemes, antithetic variates, and path plotting—is self-contained and not circular. The Cora/Gora section is incomplete rather than circular: the formulas are chain-rule identities with the key derivatives ∂S/∂ρ and ∂²S/∂ρ² left unevaluated, so they do not yet deliver computable hedging parameters, but this is a gap in derivation, not a circular reduction. No load-bearing self-citation chain was found. Because the paper's primary comparative result is forced by the self-referential benchmark, the circularity score is 7.

Assumptions & free parameters 7 free parameters · 8 assumptions · 0 invented entities

The central claim rests on a large number of model parameters calibrated to historical summary statistics (Section 1), none of which are reported numerically in the text. The most important free choice is the 40-best threshold that defines the true option value: it is an ad hoc, internal benchmark. Standard financial assumptions (risk-neutral discounting, constant interest rates, zero FX-equity correlation, volatility independent of price level, identical correlation across asset pairs) are stated in Sections 2-3; the FX-independence assumption is explicit in equations (23) and (34). The paper introduces no new physical entities; Cora and Gora are named but are only partial derivatives of option price with respect to correlation, and they are not actually evaluated.

free parameters (7)
  • SV model parameters: kappa, theta, sigma, lambda, mu_J, sigma_J, zeta = Not reported in text; see Figures 42-43
    Calibrated to historical averages, rolling standard deviations, and rolling correlations of SP500, FTSE100, STOXX600 (Section 1). These drive all simulated price estimates.
  • SC model parameters: kappa, rho_bar, sigma, Jacobi h/f, Weibull lambda/k = Not reported in text; see Figures 42-43
    Calibrated to rolling correlations; the ranking across SC models depends on these values.
  • SER model parameters: mu, theta, sigma_FX, lambda_L, mu_L, sigma_L = Not reported in text; see Figures 42-43
    Set from historical average exchange rates, autocorrelation analysis, and rolling standard deviations (Section 1).
  • Constant correlation benchmark rho = Not stated
    Used in Section 5 benchmark runs; the comparison between constant and stochastic correlation depends on this value.
  • 40-best threshold for true price = 40
    The target price is the average of the 40 model variants with the lowest standard deviation; the number 40 is chosen ad hoc and is not derived or externally justified.
  • N=500,000 paths, dt=1/252 = N=500000, dt=1/252
    Simulation settings chosen after preliminary testing (Section 5); not justified by convergence analysis.
  • Strikes K1, K2, K3 = Not stated in text
    The payoff depends on these strikes, but their values and the resulting moneyness are not reported in the main text, so the price levels cannot be independently reproduced.
assumptions (8)
  • domain assumption Risk-neutral valuation: option price equals discounted expectation of payoff under a risk-neutral measure.
    Standard finance framework used throughout; necessary for the Monte Carlo estimator in Equation (50).
  • domain assumption FX rates are independent of underlying asset Brownian motions (dW_FX = dZ_FX independent).
    Equations (23) and (34) set FX Brownian motions independent from asset Brownian motions; this zero-correlation assumption makes the foreign asset drift r_f valid under the domestic measure but is a modeling restriction.
  • domain assumption Constant interest rates over the option lifetime.
    Section 3 states domestic and foreign interest rates are modeled as constants, despite Figure 24 showing they vary; this could bias prices over the 2022-2023 period.
  • domain assumption Volatility is independent of asset price levels.
    Section 3.2 states volatility is modeled as stochastic and independent from the level of the underlying; this excludes leverage effects.
  • domain assumption Same stochastic correlation process drives both asset pairs in Case 2.
    Equations (34) use the same rho(t) for USD-GBP and USD-EUR correlations, forcing the two correlations to be equal at all times.
  • ad hoc to paper The constructed true price is the average of the 40 lowest-standard-deviation model estimates.
    Section 5 defines the target this way; all percentage-error rankings inherit this choice. This is the load-bearing benchmark assumption.
  • ad hoc to paper Jump-augmented Euler/Milstein/Runge-Kutta discretizations are valid strong approximations of the jump SDEs.
    Section 4 adds the jump component directly to each scheme without convergence or order analysis; if this approximation is not adequate, the ranking across models with jumps could change.
  • domain assumption Antithetic variates preserve unbiasedness and reduce variance for the payoff.
    Section 4.8 argues monotonic payoffs and linear-in-Brownian-motion SDEs give negative covariance; no formal proof or numerical verification is provided for the max payoff with jumps.

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Cite this review

Pith. "Pith review of Pricing Multi-strike Quanto Call Options on Multiple Assets with Stochastic Volatility, Correlation, and Exchange Rates." pith.science (2026). https://pith.science/paper/NPQ4PGDG

@misc{pith2026241116617,
  author       = {Pith},
  title        = {Pith review of: Pricing Multi-strike Quanto Call Options on Multiple Assets with Stochastic Volatility, Correlation, and Exchange Rates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NPQ4PGDG}},
  note         = {Machine review of arXiv:2411.16617}
}
read the original abstract

Quanto options allow the buyer to exchange the foreign currency payoff into the domestic currency at a fixed exchange rate. We investigate quanto options with multiple underlying assets valued in different foreign currencies each with a different strike price in the payoff function. We carry out a comparative performance analysis of different stochastic volatility (SV), stochastic correlation (SC), and stochastic exchange rate (SER) models to determine the best combination of these models for Monte Carlo (MC) simulation pricing. In addition, we test the performance of all model variants with constant correlation as a benchmark. We find that a combination of GARCH-Jump SV, Weibull SC, and Ornstein Uhlenbeck (OU) SER performs best. In addition, we analyze different discretization schemes and their results. In our simulations, the Milstein scheme yields the best balance between execution times and lower standard deviations of price estimates. Furthermore, we find that incorporating mean reversion into stochastic correlation and stochastic FX rate modeling is beneficial for MC simulation pricing. We improve the accuracy of our simulations by implementing antithetic variates variance reduction. Finally, we derive the correlation risk parameters Cora and Gora in our framework so that correlation hedging of quanto options can be performed.

Figures

Figures reproduced from arXiv: 2411.16617 by the authors.

Figure 1
Figure 1. shows the starting values for the underlying assets and the exchange rates. These values are used as the starting parameters for the MC simulations as discussed in section 4 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Case 1 top 30 models by percentage error for 2021 (left) and 2022 [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Case 2 top 30 models by percentage error for 2021 (left) and 2022 [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figures from the paper (40 more)
Figure 4
Figure 4. Figure 4: Performance of MC Simulation by Percentage Error, 2021 Start, [PITH_FULL_IMAGE:figures/full_fig_p043_4.png]
Figure 5
Figure 5. Figure 5: Performance of MC Simulation by Percentage Error, 2021 Start, [PITH_FULL_IMAGE:figures/full_fig_p044_5.png]
Figure 6
Figure 6. Figure 6: Performance of MC Simulation by Percentage Error, 2021 Start, [PITH_FULL_IMAGE:figures/full_fig_p045_6.png]
Figure 7
Figure 7. Figure 7: Performance of MC Simulation by Percentage Error, 2021 Start, [PITH_FULL_IMAGE:figures/full_fig_p046_7.png]
Figure 8
Figure 8. Figure 8: Performance of MC Simulation by Percentage Error, 2021 Start, [PITH_FULL_IMAGE:figures/full_fig_p047_8.png]
Figure 9
Figure 9. Figure 9: Performance of MC Simulation by Percentage Error, 2021 Start, [PITH_FULL_IMAGE:figures/full_fig_p048_9.png]
Figure 10
Figure 10. Figure 10: Performance of MC Simulation by Percentage Error, 2021 Start, [PITH_FULL_IMAGE:figures/full_fig_p049_10.png]
Figure 11
Figure 11. Figure 11: Performance of MC Simulation by Percentage Error, 2021 Start, [PITH_FULL_IMAGE:figures/full_fig_p050_11.png]
Figure 12
Figure 12. Figure 12: Performance of MC Simulation by Percentage Error, 2022 Start, [PITH_FULL_IMAGE:figures/full_fig_p051_12.png]
Figure 13
Figure 13. Figure 13: Performance of MC Simulation by Percentage Error, 2022 Start, [PITH_FULL_IMAGE:figures/full_fig_p052_13.png]
Figure 14
Figure 14. Figure 14: Performance of MC Simulation by Percentage Error, 2022 Start, [PITH_FULL_IMAGE:figures/full_fig_p053_14.png]
Figure 15
Figure 15. Figure 15: Performance of MC Simulation by Percentage Error, 2022 Start, [PITH_FULL_IMAGE:figures/full_fig_p054_15.png]
Figure 16
Figure 16. Figure 16: Performance of MC Simulation by Percentage Error, 2022 Start, [PITH_FULL_IMAGE:figures/full_fig_p055_16.png]
Figure 17
Figure 17. Figure 17: Performance of MC Simulation by Percentage Error, 2022 Start, [PITH_FULL_IMAGE:figures/full_fig_p056_17.png]
Figure 18
Figure 18. Figure 18: Performance of MC Simulation by Percentage Error, 2022 Start, 57 [PITH_FULL_IMAGE:figures/full_fig_p057_18.png]
Figure 19
Figure 19. Figure 19: Performance of MC Simulation by Percentage Error, 2022 Start, [PITH_FULL_IMAGE:figures/full_fig_p058_19.png]
Figure 20
Figure 20. Figure 20: Performance of MC Simulation with Constant Correlation by [PITH_FULL_IMAGE:figures/full_fig_p059_20.png]
Figure 21
Figure 21. Figure 21: Performance of MC Simulation with Constant Correlation by [PITH_FULL_IMAGE:figures/full_fig_p060_21.png]
Figure 22
Figure 22. Figure 22: Performance of MC Simulation with Constant Correlation by [PITH_FULL_IMAGE:figures/full_fig_p061_22.png]
Figure 23
Figure 23. Figure 23: Performance of MC Simulation with Constant Correlation by [PITH_FULL_IMAGE:figures/full_fig_p062_23.png]
Figure 24
Figure 24. Figure 24: Interest Rates 63 [PITH_FULL_IMAGE:figures/full_fig_p063_24.png]
Figure 25
Figure 25. Figure 25: Plots of Observed Underlying Asset Prices and Exchange Rates [PITH_FULL_IMAGE:figures/full_fig_p064_25.png]
Figure 26
Figure 26. Figure 26: Plots of Observed Underlying Asset Prices and Exchange Rates [PITH_FULL_IMAGE:figures/full_fig_p065_26.png]
Figure 27
Figure 27. Figure 27: Plot of Observed Underlying Asset Prices and Exchange Rates in [PITH_FULL_IMAGE:figures/full_fig_p066_27.png]
Figure 28
Figure 28. Figure 28: Plot 2 of Observed Underlying Asset Prices and Exchange Rates [PITH_FULL_IMAGE:figures/full_fig_p067_28.png]
Figure 29
Figure 29. Figure 29: Plot of Observed Underlying Asset Prices and Exchange Rates in [PITH_FULL_IMAGE:figures/full_fig_p068_29.png]
Figure 30
Figure 30. Figure 30: Plot 2 of Observed Underlying Asset Prices and Exchange Rates [PITH_FULL_IMAGE:figures/full_fig_p069_30.png]
Figure 31
Figure 31. Figure 31: Observed Volatilities and Correlations 1/2 [PITH_FULL_IMAGE:figures/full_fig_p070_31.png]
Figure 32
Figure 32. Figure 32: Observed Volatilities and Correlations 2/2 [PITH_FULL_IMAGE:figures/full_fig_p071_32.png]
Figure 33
Figure 33. Figure 33: MC Simulation Paths 1/9 [PITH_FULL_IMAGE:figures/full_fig_p072_33.png]
Figure 34
Figure 34. Figure 34: MC Simulation Paths 2/9 [PITH_FULL_IMAGE:figures/full_fig_p072_34.png]
Figure 35
Figure 35. Figure 35: MC Simulation Paths 3/9 72 [PITH_FULL_IMAGE:figures/full_fig_p072_35.png]
Figure 36
Figure 36. Figure 36: MC Simulation Paths 4/9 [PITH_FULL_IMAGE:figures/full_fig_p073_36.png]
Figure 37
Figure 37. Figure 37: MC Simulation Paths 5/9 [PITH_FULL_IMAGE:figures/full_fig_p073_37.png]
Figure 38
Figure 38. Figure 38: MC Simulation Paths 6/9 73 [PITH_FULL_IMAGE:figures/full_fig_p073_38.png]
Figure 39
Figure 39. Figure 39: MC Simulation Paths 7/9 [PITH_FULL_IMAGE:figures/full_fig_p074_39.png]
Figure 40
Figure 40. Figure 40: MC Simulation Paths 8/9 [PITH_FULL_IMAGE:figures/full_fig_p074_40.png]
Figure 41
Figure 41. Figure 41: MC Simulation Paths 9/9 74 [PITH_FULL_IMAGE:figures/full_fig_p074_41.png]
Figure 42
Figure 42. Figure 42: MC Simulation Parameters for 2021-2022 75 [PITH_FULL_IMAGE:figures/full_fig_p075_42.png]
Figure 43
Figure 43. Figure 43: MC Simulation Parameters for 2022-2023 76 [PITH_FULL_IMAGE:figures/full_fig_p076_43.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.